LMFDB - Embedded newform 800.4.c.i.449.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,4,Mod(449,800)] 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("800.449"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-52,0,-128] 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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.i.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.898979i q^{3} +28.4949i q^{7} +26.1918 q^{9} -2.60612 q^{11} -33.1918i q^{13} -119.576i q^{17} +143.192 q^{19} -25.6163 q^{21} -113.889i q^{23} +47.8184i q^{27} +67.6163 q^{29} -117.394 q^{31} -2.34285i q^{33} +256.384i q^{37} +29.8388 q^{39} +245.959 q^{41} +369.262i q^{43} -76.3133i q^{47} -468.959 q^{49} +107.496 q^{51} +40.4245i q^{53} +128.727i q^{57} +457.980 q^{59} -477.233 q^{61} +746.334i q^{63} -602.252i q^{67} +102.384 q^{69} +1091.98 q^{71} +117.878i q^{73} -74.2612i q^{77} +858.624 q^{79} +664.192 q^{81} +565.748i q^{83} +60.7857i q^{87} +625.151 q^{89} +945.798 q^{91} -105.535i q^{93} -805.959i q^{97} -68.2591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 52 q^{9} - 128 q^{11} + 416 q^{19} - 416 q^{21} + 584 q^{29} - 352 q^{31} + 864 q^{39} + 200 q^{41} - 1092 q^{49} + 2272 q^{51} + 1440 q^{59} - 2536 q^{61} + 96 q^{69} + 96 q^{71} + 64 q^{79} + 2500 q^{81}+ \cdots + 6272 q^{99}+O(q^{100}) $ 4 * q - 52 * q^9 - 128 * q^11 + 416 * q^19 - 416 * q^21 + 584 * q^29 - 352 * q^31 + 864 * q^39 + 200 * q^41 - 1092 * q^49 + 2272 * q^51 + 1440 * q^59 - 2536 * q^61 + 96 * q^69 + 96 * q^71 + 64 * q^79 + 2500 * q^81 + 1560 * q^89 + 3744 * q^91 + 6272 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0.898979i	0.173009i	0.996251	+	0.0865043i	$0.0275696\pi$
$3$	0.898979i	0.173009i	−0.996251	+	0.0865043i	$0.972430\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	28.4949i	1.53858i	0.638900	+	0.769290i	$0.279390\pi$
$7$	28.4949i	1.53858i	−0.638900	+	0.769290i	$0.720610\pi$
$8$	0	0
$9$	26.1918	0.970068
$10$	0	0
$11$	−2.60612	−0.0714342	−0.0357171	−	0.999362i	$-0.511372\pi$
$11$	−2.60612	−0.0714342	−0.0357171	+	0.999362i	$0.511372\pi$
$12$	0	0
$13$	− 33.1918i	− 0.708135i	−0.935220	−	0.354068i	$-0.884798\pi$
$13$	− 33.1918i	− 0.708135i	0.935220	−	0.354068i	$-0.115202\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 119.576i	− 1.70596i	−0.521944	−	0.852980i	$-0.674793\pi$
$17$	− 119.576i	− 1.70596i	0.521944	−	0.852980i	$-0.325207\pi$
$18$	0	0
$19$	143.192	1.72897	0.864486	−	0.502657i	$-0.167644\pi$
$19$	143.192	1.72897	0.864486	+	0.502657i	$0.167644\pi$
$20$	0	0
$21$	−25.6163	−0.266188
$22$	0	0
$23$	− 113.889i	− 1.03250i	−0.856439	−	0.516249i	$-0.827328\pi$
$23$	− 113.889i	− 1.03250i	0.856439	−	0.516249i	$-0.172672\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	47.8184i	0.340839i
$28$	0	0
$29$	67.6163	0.432967	0.216483	−	0.976286i	$-0.430541\pi$
$29$	67.6163	0.432967	0.216483	+	0.976286i	$0.430541\pi$
$30$	0	0
$31$	−117.394	−0.680147	−0.340074	−	0.940399i	$-0.610452\pi$
$31$	−117.394	−0.680147	−0.340074	+	0.940399i	$0.610452\pi$
$32$	0	0
$33$	− 2.34285i	− 0.0123587i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	256.384i	1.13917i	0.821933	+	0.569584i	$0.192896\pi$
$37$	256.384i	1.13917i	−0.821933	+	0.569584i	$0.807104\pi$
$38$	0	0
$39$	29.8388	0.122514
$40$	0	0
$41$	245.959	0.936887	0.468444	−	0.883493i	$-0.344815\pi$
$41$	245.959	0.936887	0.468444	+	0.883493i	$0.344815\pi$
$42$	0	0
$43$	369.262i	1.30958i	0.755811	+	0.654790i	$0.227243\pi$
$43$	369.262i	1.30958i	−0.755811	+	0.654790i	$0.772757\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 76.3133i	− 0.236839i	−0.992964	−	0.118420i	$-0.962217\pi$
$47$	− 76.3133i	− 0.236839i	0.992964	−	0.118420i	$-0.0377828\pi$
$48$	0	0
$49$	−468.959	−1.36723
$50$	0	0
$51$	107.496	0.295146
$52$	0	0
$53$	40.4245i	0.104769i	0.998627	+	0.0523843i	$0.0166821\pi$
$53$	40.4245i	0.104769i	−0.998627	+	0.0523843i	$0.983318\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	128.727i	0.299127i
$58$	0	0
$59$	457.980	1.01057	0.505287	−	0.862951i	$-0.331387\pi$
$59$	457.980	1.01057	0.505287	+	0.862951i	$0.331387\pi$
$60$	0	0
$61$	−477.233	−1.00169	−0.500847	−	0.865536i	$-0.666978\pi$
$61$	−477.233	−1.00169	−0.500847	+	0.865536i	$0.666978\pi$
$62$	0	0
$63$	746.334i	1.49253i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 602.252i	− 1.09816i	−0.835769	−	0.549081i	$-0.814978\pi$
$67$	− 602.252i	− 1.09816i	0.835769	−	0.549081i	$-0.185022\pi$
$68$	0	0
$69$	102.384	0.178631
$70$	0	0
$71$	1091.98	1.82527	0.912633	−	0.408780i	$-0.134046\pi$
$71$	1091.98	1.82527	0.912633	+	0.408780i	$0.134046\pi$
$72$	0	0
$73$	117.878i	0.188993i	0.995525	+	0.0944967i	$0.0301242\pi$
$73$	117.878i	0.188993i	−0.995525	+	0.0944967i	$0.969876\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 74.2612i	− 0.109907i
$78$	0	0
$79$	858.624	1.22282	0.611410	−	0.791314i	$-0.290603\pi$
$79$	858.624	1.22282	0.611410	+	0.791314i	$0.290603\pi$
$80$	0	0
$81$	664.192	0.911100
$82$	0	0
$83$	565.748i	0.748180i	0.927392	+	0.374090i	$0.122045\pi$
$83$	565.748i	0.748180i	−0.927392	+	0.374090i	$0.877955\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	60.7857i	0.0749070i
$88$	0	0
$89$	625.151	0.744560	0.372280	−	0.928120i	$-0.378576\pi$
$89$	625.151	0.744560	0.372280	+	0.928120i	$0.378576\pi$
$90$	0	0
$91$	945.798	1.08952
$92$	0	0
$93$	− 105.535i	− 0.117671i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 805.959i	− 0.843637i	−0.906680	−	0.421818i	$-0.861392\pi$
$97$	− 805.959i	− 0.843637i	0.906680	−	0.421818i	$-0.138608\pi$
$98$	0	0
$99$	−68.2591	−0.0692960
$100$	0	0
$101$	−260.988	−0.257121	−0.128561	−	0.991702i	$-0.541036\pi$
$101$	−260.988	−0.257121	−0.128561	+	0.991702i	$0.541036\pi$
$102$	0	0
$103$	1640.01i	1.56888i	0.620203	+	0.784441i	$0.287050\pi$
$103$	1640.01i	1.56888i	−0.620203	+	0.784441i	$0.712950\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	679.001i	0.613472i	0.951795	+	0.306736i	$0.0992369\pi$
$107$	679.001i	0.613472i	−0.951795	+	0.306736i	$0.900763\pi$
$108$	0	0
$109$	840.220	0.738335	0.369168	−	0.929363i	$-0.379643\pi$
$109$	840.220	0.738335	0.369168	+	0.929363i	$0.379643\pi$
$110$	0	0
$111$	−230.484	−0.197086
$112$	0	0
$113$	− 1609.15i	− 1.33961i	−0.742536	−	0.669806i	$-0.766377\pi$
$113$	− 1609.15i	− 1.33961i	0.742536	−	0.669806i	$-0.233623\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 869.355i	− 0.686939i
$118$	0	0
$119$	3407.29	2.62476
$120$	0	0
$121$	−1324.21	−0.994897
$122$	0	0
$123$	221.112i	0.162090i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1149.60i	− 0.803234i	−0.915808	−	0.401617i	$-0.868448\pi$
$127$	− 1149.60i	− 0.803234i	0.915808	−	0.401617i	$-0.131552\pi$
$128$	0	0
$129$	−331.959	−0.226569
$130$	0	0
$131$	−2436.99	−1.62535	−0.812673	−	0.582719i	$-0.801989\pi$
$131$	−2436.99	−1.62535	−0.812673	+	0.582719i	$0.801989\pi$
$132$	0	0
$133$	4080.24i	2.66016i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	764.465i	0.476735i	0.971175	+	0.238367i	$0.0766123\pi$
$137$	764.465i	0.476735i	−0.971175	+	0.238367i	$0.923388\pi$
$138$	0	0
$139$	2547.35	1.55441	0.777207	−	0.629245i	$-0.216636\pi$
$139$	2547.35	1.55441	0.777207	+	0.629245i	$0.216636\pi$
$140$	0	0
$141$	68.6041	0.0409752
$142$	0	0
$143$	86.5020i	0.0505850i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 421.585i	− 0.236542i
$148$	0	0
$149$	−2565.76	−1.41070	−0.705352	−	0.708857i	$-0.749211\pi$
$149$	−2565.76	−1.41070	−0.705352	+	0.708857i	$0.749211\pi$
$150$	0	0
$151$	524.949	0.282912	0.141456	−	0.989945i	$-0.454822\pi$
$151$	524.949	0.282912	0.141456	+	0.989945i	$0.454822\pi$
$152$	0	0
$153$	− 3131.90i	− 1.65490i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3569.59i	1.81455i	0.420537	+	0.907275i	$0.361842\pi$
$157$	3569.59i	1.81455i	−0.420537	+	0.907275i	$0.638158\pi$
$158$	0	0
$159$	−36.3408	−0.0181259
$160$	0	0
$161$	3245.25	1.58858
$162$	0	0
$163$	1450.96i	0.697227i	0.937267	+	0.348613i	$0.113347\pi$
$163$	1450.96i	0.697227i	−0.937267	+	0.348613i	$0.886653\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3281.38i	1.52048i	0.649640	+	0.760242i	$0.274920\pi$
$167$	3281.38i	1.52048i	−0.649640	+	0.760242i	$0.725080\pi$
$168$	0	0
$169$	1095.30	0.498544
$170$	0	0
$171$	3750.46	1.67722
$172$	0	0
$173$	1387.01i	0.609553i	0.952424	+	0.304776i	$0.0985817\pi$
$173$	1387.01i	0.609553i	−0.952424	+	0.304776i	$0.901418\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	411.714i	0.174838i
$178$	0	0
$179$	3009.90	1.25682	0.628409	−	0.777883i	$-0.283707\pi$
$179$	3009.90	1.25682	0.628409	+	0.777883i	$0.283707\pi$
$180$	0	0
$181$	−2304.14	−0.946217	−0.473109	−	0.881004i	$-0.656868\pi$
$181$	−2304.14	−0.946217	−0.473109	+	0.881004i	$0.656868\pi$
$182$	0	0
$183$	− 429.022i	− 0.173302i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	311.628i	0.121864i
$188$	0	0
$189$	−1362.58	−0.524408
$190$	0	0
$191$	−166.647	−0.0631317	−0.0315658	−	0.999502i	$-0.510049\pi$
$191$	−166.647	−0.0631317	−0.0315658	+	0.999502i	$0.510049\pi$
$192$	0	0
$193$	484.669i	0.180763i	0.995907	+	0.0903815i	$0.0288086\pi$
$193$	484.669i	0.180763i	−0.995907	+	0.0903815i	$0.971191\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2088.10i	− 0.755182i	−0.925973	−	0.377591i	$-0.876753\pi$
$197$	− 2088.10i	− 0.755182i	0.925973	−	0.377591i	$-0.123247\pi$
$198$	0	0
$199$	−3401.93	−1.21184	−0.605921	−	0.795524i	$-0.707195\pi$
$199$	−3401.93	−1.21184	−0.605921	+	0.795524i	$0.707195\pi$
$200$	0	0
$201$	541.412	0.189991
$202$	0	0
$203$	1926.72i	0.666154i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2982.96i	− 1.00159i
$208$	0	0
$209$	−373.176	−0.123508
$210$	0	0
$211$	2246.24	0.732879	0.366439	−	0.930442i	$-0.380577\pi$
$211$	2246.24	0.732879	0.366439	+	0.930442i	$0.380577\pi$
$212$	0	0
$213$	981.665i	0.315787i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3345.13i	− 1.04646i
$218$	0	0
$219$	−105.969	−0.0326975
$220$	0	0
$221$	−3968.93	−1.20805
$222$	0	0
$223$	1129.63i	0.339217i	0.985512	+	0.169608i	$0.0542503\pi$
$223$	1129.63i	0.339217i	−0.985512	+	0.169608i	$0.945750\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1693.88i	− 0.495273i	−0.968853	−	0.247637i	$-0.920346\pi$
$227$	− 1693.88i	− 0.495273i	0.968853	−	0.247637i	$-0.0796539\pi$
$228$	0	0
$229$	5250.72	1.51518	0.757592	−	0.652728i	$-0.226376\pi$
$229$	5250.72	1.51518	0.757592	+	0.652728i	$0.226376\pi$
$230$	0	0
$231$	66.7593	0.0190149
$232$	0	0
$233$	− 6042.60i	− 1.69899i	−0.527600	−	0.849493i	$-0.676908\pi$
$233$	− 6042.60i	− 1.69899i	0.527600	−	0.849493i	$-0.323092\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	771.886i	0.211559i
$238$	0	0
$239$	−385.416	−0.104312	−0.0521559	−	0.998639i	$-0.516609\pi$
$239$	−385.416	−0.104312	−0.0521559	+	0.998639i	$0.516609\pi$
$240$	0	0
$241$	4858.73	1.29866	0.649332	−	0.760505i	$-0.275049\pi$
$241$	4858.73	1.29866	0.649332	+	0.760505i	$0.275049\pi$
$242$	0	0
$243$	1888.19i	0.498467i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4752.80i	− 1.22435i
$248$	0	0
$249$	−508.596	−0.129442
$250$	0	0
$251$	−3784.74	−0.951757	−0.475878	−	0.879511i	$-0.657870\pi$
$251$	−3784.74	−0.951757	−0.475878	+	0.879511i	$0.657870\pi$
$252$	0	0
$253$	296.808i	0.0737556i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3702.36i	− 0.898626i	−0.893374	−	0.449313i	$-0.851669\pi$
$257$	− 3702.36i	− 0.898626i	0.893374	−	0.449313i	$-0.148331\pi$
$258$	0	0
$259$	−7305.63	−1.75270
$260$	0	0
$261$	1771.00	0.420007
$262$	0	0
$263$	− 3517.77i	− 0.824771i	−0.911009	−	0.412385i	$-0.864696\pi$
$263$	− 3517.77i	− 0.824771i	0.911009	−	0.412385i	$-0.135304\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	561.998i	0.128815i
$268$	0	0
$269$	5564.09	1.26115	0.630573	−	0.776130i	$-0.282820\pi$
$269$	5564.09	1.26115	0.630573	+	0.776130i	$0.282820\pi$
$270$	0	0
$271$	−2585.39	−0.579525	−0.289762	−	0.957099i	$-0.593576\pi$
$271$	−2585.39	−0.579525	−0.289762	+	0.957099i	$0.593576\pi$
$272$	0	0
$273$	850.253i	0.188497i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	884.433i	0.191843i	0.995389	+	0.0959213i	$0.0305797\pi$
$277$	884.433i	0.191843i	−0.995389	+	0.0959213i	$0.969420\pi$
$278$	0	0
$279$	−3074.76	−0.659789
$280$	0	0
$281$	3165.55	0.672032	0.336016	−	0.941856i	$-0.390920\pi$
$281$	3165.55	0.672032	0.336016	+	0.941856i	$0.390920\pi$
$282$	0	0
$283$	− 4425.40i	− 0.929550i	−0.885429	−	0.464775i	$-0.846135\pi$
$283$	− 4425.40i	− 0.929550i	0.885429	−	0.464775i	$-0.153865\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7008.58i	1.44148i
$288$	0	0
$289$	−9385.30	−1.91030
$290$	0	0
$291$	724.541	0.145956
$292$	0	0
$293$	2871.70i	0.572582i	0.958143	+	0.286291i	$0.0924223\pi$
$293$	2871.70i	0.572582i	−0.958143	+	0.286291i	$0.907578\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 124.621i	− 0.0243475i
$298$	0	0
$299$	−3780.18	−0.731148
$300$	0	0
$301$	−10522.1	−2.01489
$302$	0	0
$303$	− 234.623i	− 0.0444842i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3145.45i	0.584757i	0.956303	+	0.292379i	$0.0944468\pi$
$307$	3145.45i	0.584757i	−0.956303	+	0.292379i	$0.905553\pi$
$308$	0	0
$309$	−1474.33	−0.271430
$310$	0	0
$311$	−306.614	−0.0559052	−0.0279526	−	0.999609i	$-0.508899\pi$
$311$	−306.614	−0.0559052	−0.0279526	+	0.999609i	$0.508899\pi$
$312$	0	0
$313$	− 3120.66i	− 0.563547i	−0.959481	−	0.281773i	$-0.909077\pi$
$313$	− 3120.66i	− 0.563547i	0.959481	−	0.281773i	$-0.0909227\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8131.12i	1.44066i	0.693632	+	0.720330i	$0.256010\pi$
$317$	8131.12i	1.44066i	−0.693632	+	0.720330i	$0.743990\pi$
$318$	0	0
$319$	−176.216	−0.0309286
$320$	0	0
$321$	−610.408	−0.106136
$322$	0	0
$323$	− 17122.2i	− 2.94956i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	755.341i	0.127738i
$328$	0	0
$329$	2174.54	0.364396
$330$	0	0
$331$	−7877.80	−1.30817	−0.654083	−	0.756423i	$-0.726945\pi$
$331$	−7877.80	−1.30817	−0.654083	+	0.756423i	$0.726945\pi$
$332$	0	0
$333$	6715.16i	1.10507i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2169.15i	0.350627i	0.984513	+	0.175313i	$0.0560938\pi$
$337$	2169.15i	0.350627i	−0.984513	+	0.175313i	$0.943906\pi$
$338$	0	0
$339$	1446.59	0.231765
$340$	0	0
$341$	305.943	0.0485857
$342$	0	0
$343$	− 3589.19i	− 0.565009i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	325.475i	0.0503527i	0.999683	+	0.0251764i	$0.00801473\pi$
$347$	325.475i	0.0503527i	−0.999683	+	0.0251764i	$0.991985\pi$
$348$	0	0
$349$	−9311.11	−1.42812	−0.714058	−	0.700087i	$-0.753145\pi$
$349$	−9311.11	−1.42812	−0.714058	+	0.700087i	$0.753145\pi$
$350$	0	0
$351$	1587.18	0.241360
$352$	0	0
$353$	3144.94i	0.474188i	0.971487	+	0.237094i	$0.0761949\pi$
$353$	3144.94i	0.474188i	−0.971487	+	0.237094i	$0.923805\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3063.09i	0.454106i
$358$	0	0
$359$	570.424	0.0838603	0.0419302	−	0.999121i	$-0.486649\pi$
$359$	570.424	0.0838603	0.0419302	+	0.999121i	$0.486649\pi$
$360$	0	0
$361$	13644.9	1.98934
$362$	0	0
$363$	− 1190.44i	− 0.172126i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1629.17i	0.231722i	0.993265	+	0.115861i	$0.0369627\pi$
$367$	1629.17i	0.231722i	−0.993265	+	0.115861i	$0.963037\pi$
$368$	0	0
$369$	6442.12	0.908844
$370$	0	0
$371$	−1151.89	−0.161195
$372$	0	0
$373$	− 6985.76i	− 0.969728i	−0.874589	−	0.484864i	$-0.838869\pi$
$373$	− 6985.76i	− 0.969728i	0.874589	−	0.484864i	$-0.161131\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2244.31i	− 0.306599i
$378$	0	0
$379$	−11101.3	−1.50458	−0.752291	−	0.658831i	$-0.771051\pi$
$379$	−11101.3	−1.50458	−0.752291	+	0.658831i	$0.771051\pi$
$380$	0	0
$381$	1033.47	0.138967
$382$	0	0
$383$	− 6608.71i	− 0.881695i	−0.897582	−	0.440848i	$-0.854678\pi$
$383$	− 6608.71i	− 0.881695i	0.897582	−	0.440848i	$-0.145322\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9671.66i	1.27038i
$388$	0	0
$389$	−503.208	−0.0655878	−0.0327939	−	0.999462i	$-0.510440\pi$
$389$	−503.208	−0.0655878	−0.0327939	+	0.999462i	$0.510440\pi$
$390$	0	0
$391$	−13618.3	−1.76140
$392$	0	0
$393$	− 2190.80i	− 0.281199i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1885.18i	− 0.238323i	−0.992875	−	0.119162i	$-0.961979\pi$
$397$	− 1885.18i	− 0.238323i	0.992875	−	0.119162i	$-0.0380206\pi$
$398$	0	0
$399$	−3668.05	−0.460231
$400$	0	0
$401$	−13212.7	−1.64541	−0.822706	−	0.568467i	$-0.807537\pi$
$401$	−13212.7	−1.64541	−0.822706	+	0.568467i	$0.807537\pi$
$402$	0	0
$403$	3896.52i	0.481636i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 668.167i	− 0.0813755i
$408$	0	0
$409$	8441.37	1.02054	0.510268	−	0.860016i	$-0.329546\pi$
$409$	8441.37	1.02054	0.510268	+	0.860016i	$0.329546\pi$
$410$	0	0
$411$	−687.239	−0.0824793
$412$	0	0
$413$	13050.1i	1.55485i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2290.02i	0.268927i
$418$	0	0
$419$	−12823.2	−1.49512	−0.747559	−	0.664195i	$-0.768774\pi$
$419$	−12823.2	−1.49512	−0.747559	+	0.664195i	$0.768774\pi$
$420$	0	0
$421$	−2474.24	−0.286431	−0.143215	−	0.989692i	$-0.545744\pi$
$421$	−2474.24	−0.286431	−0.143215	+	0.989692i	$0.545744\pi$
$422$	0	0
$423$	− 1998.78i	− 0.229750i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 13598.7i	− 1.54119i
$428$	0	0
$429$	−77.7635	−0.00875165
$430$	0	0
$431$	12461.3	1.39267	0.696334	−	0.717718i	$-0.254813\pi$
$431$	12461.3	1.39267	0.696334	+	0.717718i	$0.254813\pi$
$432$	0	0
$433$	− 12063.0i	− 1.33883i	−0.742890	−	0.669414i	$-0.766545\pi$
$433$	− 12063.0i	− 1.33883i	0.742890	−	0.669414i	$-0.233455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 16307.9i	− 1.78516i
$438$	0	0
$439$	−16480.7	−1.79176	−0.895879	−	0.444299i	$-0.853453\pi$
$439$	−16480.7	−1.79176	−0.895879	+	0.444299i	$0.853453\pi$
$440$	0	0
$441$	−12282.9	−1.32630
$442$	0	0
$443$	2159.19i	0.231572i	0.993274	+	0.115786i	$0.0369387\pi$
$443$	2159.19i	0.231572i	−0.993274	+	0.115786i	$0.963061\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 2306.56i	− 0.244064i
$448$	0	0
$449$	−5552.38	−0.583592	−0.291796	−	0.956481i	$-0.594253\pi$
$449$	−5552.38	−0.583592	−0.291796	+	0.956481i	$0.594253\pi$
$450$	0	0
$451$	−641.000	−0.0669257
$452$	0	0
$453$	471.918i	0.0489463i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 16667.9i	− 1.70611i	−0.521820	−	0.853055i	$-0.674747\pi$
$457$	− 16667.9i	− 1.70611i	0.521820	−	0.853055i	$-0.325253\pi$
$458$	0	0
$459$	5717.91	0.581457
$460$	0	0
$461$	15255.5	1.54126	0.770629	−	0.637284i	$-0.219942\pi$
$461$	15255.5	1.54126	0.770629	+	0.637284i	$0.219942\pi$
$462$	0	0
$463$	− 6806.85i	− 0.683242i	−0.939838	−	0.341621i	$-0.889024\pi$
$463$	− 6806.85i	− 0.683242i	0.939838	−	0.341621i	$-0.110976\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6455.78i	− 0.639696i	−0.947469	−	0.319848i	$-0.896368\pi$
$467$	− 6455.78i	− 0.639696i	0.947469	−	0.319848i	$-0.103632\pi$
$468$	0	0
$469$	17161.1	1.68961
$470$	0	0
$471$	−3208.99	−0.313933
$472$	0	0
$473$	− 962.343i	− 0.0935488i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1058.79i	0.101633i
$478$	0	0
$479$	−8481.34	−0.809024	−0.404512	−	0.914533i	$-0.632559\pi$
$479$	−8481.34	−0.809024	−0.404512	+	0.914533i	$0.632559\pi$
$480$	0	0
$481$	8509.84	0.806685
$482$	0	0
$483$	2917.41i	0.274838i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10966.2i	− 1.02038i	−0.860061	−	0.510191i	$-0.829575\pi$
$487$	− 10966.2i	− 1.02038i	0.860061	−	0.510191i	$-0.170425\pi$
$488$	0	0
$489$	−1304.38	−0.120626
$490$	0	0
$491$	6784.15	0.623553	0.311777	−	0.950155i	$-0.399076\pi$
$491$	6784.15	0.623553	0.311777	+	0.950155i	$0.399076\pi$
$492$	0	0
$493$	− 8085.26i	− 0.738624i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31115.8i	2.80832i
$498$	0	0
$499$	17785.8	1.59560	0.797799	−	0.602924i	$-0.205998\pi$
$499$	17785.8	1.59560	0.797799	+	0.602924i	$0.205998\pi$
$500$	0	0
$501$	−2949.89	−0.263057
$502$	0	0
$503$	8440.25i	0.748175i	0.927393	+	0.374087i	$0.122044\pi$
$503$	8440.25i	0.748175i	−0.927393	+	0.374087i	$0.877956\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	984.654i	0.0862525i
$508$	0	0
$509$	−10923.4	−0.951222	−0.475611	−	0.879656i	$-0.657773\pi$
$509$	−10923.4	−0.951222	−0.475611	+	0.879656i	$0.657773\pi$
$510$	0	0
$511$	−3358.91	−0.290782
$512$	0	0
$513$	6847.20i	0.589301i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	198.882i	0.0169184i
$518$	0	0
$519$	−1246.90	−0.105458
$520$	0	0
$521$	−9879.76	−0.830787	−0.415394	−	0.909642i	$-0.636356\pi$
$521$	−9879.76	−0.830787	−0.415394	+	0.909642i	$0.636356\pi$
$522$	0	0
$523$	105.197i	0.00879531i	0.999990	+	0.00439766i	$0.00139982\pi$
$523$	105.197i	0.00879531i	−0.999990	+	0.00439766i	$0.998600\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14037.4i	1.16030i
$528$	0	0
$529$	−803.653	−0.0660519
$530$	0	0
$531$	11995.3	0.980325
$532$	0	0
$533$	− 8163.84i	− 0.663443i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2705.84i	0.217440i
$538$	0	0
$539$	1222.17	0.0976668
$540$	0	0
$541$	19055.6	1.51435	0.757175	−	0.653212i	$-0.226579\pi$
$541$	19055.6	1.51435	0.757175	+	0.653212i	$0.226579\pi$
$542$	0	0
$543$	− 2071.37i	− 0.163704i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3910.22i	0.305647i	0.988253	+	0.152824i	$0.0488366\pi$
$547$	3910.22i	0.305647i	−0.988253	+	0.152824i	$0.951163\pi$
$548$	0	0
$549$	−12499.6	−0.971712
$550$	0	0
$551$	9682.11	0.748587
$552$	0	0
$553$	24466.4i	1.88141i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1727.52i	− 0.131413i	−0.997839	−	0.0657067i	$-0.979070\pi$
$557$	− 1727.52i	− 0.131413i	0.997839	−	0.0657067i	$-0.0209302\pi$
$558$	0	0
$559$	12256.5	0.927360
$560$	0	0
$561$	−280.148	−0.0210835
$562$	0	0
$563$	20905.0i	1.56491i	0.622709	+	0.782453i	$0.286032\pi$
$563$	20905.0i	1.56491i	−0.622709	+	0.782453i	$0.713968\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18926.1i	1.40180i
$568$	0	0
$569$	256.523	0.0188998	0.00944990	−	0.999955i	$-0.496992\pi$
$569$	256.523	0.0188998	0.00944990	+	0.999955i	$0.496992\pi$
$570$	0	0
$571$	−15793.4	−1.15750	−0.578749	−	0.815506i	$-0.696459\pi$
$571$	−15793.4	−1.15750	−0.578749	+	0.815506i	$0.696459\pi$
$572$	0	0
$573$	− 149.812i	− 0.0109223i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5923.65i	0.427391i	0.976900	+	0.213696i	$0.0685501\pi$
$577$	5923.65i	0.427391i	−0.976900	+	0.213696i	$0.931450\pi$
$578$	0	0
$579$	−435.708	−0.0312736
$580$	0	0
$581$	−16120.9	−1.15113
$582$	0	0
$583$	− 105.351i	− 0.00748405i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11730.7i	0.824832i	0.910996	+	0.412416i	$0.135315\pi$
$587$	11730.7i	0.824832i	−0.910996	+	0.412416i	$0.864685\pi$
$588$	0	0
$589$	−16809.8	−1.17596
$590$	0	0
$591$	1877.16	0.130653
$592$	0	0
$593$	2781.00i	0.192584i	0.995353	+	0.0962919i	$0.0306982\pi$
$593$	2781.00i	0.192584i	−0.995353	+	0.0962919i	$0.969302\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3058.27i	− 0.209659i
$598$	0	0
$599$	22999.7	1.56885	0.784425	−	0.620224i	$-0.212958\pi$
$599$	22999.7	1.56885	0.784425	+	0.620224i	$0.212958\pi$
$600$	0	0
$601$	−2680.42	−0.181925	−0.0909624	−	0.995854i	$-0.528994\pi$
$601$	−2680.42	−0.181925	−0.0909624	+	0.995854i	$0.528994\pi$
$602$	0	0
$603$	− 15774.1i	− 1.06529i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16977.3i	− 1.13523i	−0.823293	−	0.567617i	$-0.807865\pi$
$607$	− 16977.3i	− 1.13523i	0.823293	−	0.567617i	$-0.192135\pi$
$608$	0	0
$609$	−1732.08	−0.115250
$610$	0	0
$611$	−2532.98	−0.167714
$612$	0	0
$613$	− 24124.3i	− 1.58951i	−0.606930	−	0.794755i	$-0.707599\pi$
$613$	− 24124.3i	− 1.58951i	0.606930	−	0.794755i	$-0.292401\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12286.0i	− 0.801644i	−0.916156	−	0.400822i	$-0.868725\pi$
$617$	− 12286.0i	− 0.801644i	0.916156	−	0.400822i	$-0.131275\pi$
$618$	0	0
$619$	6240.17	0.405192	0.202596	−	0.979262i	$-0.435062\pi$
$619$	6240.17	0.405192	0.202596	+	0.979262i	$0.435062\pi$
$620$	0	0
$621$	5445.98	0.351915
$622$	0	0
$623$	17813.6i	1.14557i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 335.477i	− 0.0213679i
$628$	0	0
$629$	30657.2	1.94338
$630$	0	0
$631$	−8142.22	−0.513687	−0.256844	−	0.966453i	$-0.582683\pi$
$631$	−8142.22	−0.513687	−0.256844	+	0.966453i	$0.582683\pi$
$632$	0	0
$633$	2019.32i	0.126794i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15565.6i	0.968182i
$638$	0	0
$639$	28600.9	1.77063
$640$	0	0
$641$	6380.36	0.393150	0.196575	−	0.980489i	$-0.437018\pi$
$641$	6380.36	0.393150	0.196575	+	0.980489i	$0.437018\pi$
$642$	0	0
$643$	− 18308.0i	− 1.12286i	−0.827524	−	0.561430i	$-0.810251\pi$
$643$	− 18308.0i	− 1.12286i	0.827524	−	0.561430i	$-0.189749\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 21497.9i	− 1.30629i	−0.757233	−	0.653145i	$-0.773449\pi$
$647$	− 21497.9i	− 1.30629i	0.757233	−	0.653145i	$-0.226551\pi$
$648$	0	0
$649$	−1193.55	−0.0721895
$650$	0	0
$651$	3007.20	0.181047
$652$	0	0
$653$	− 16035.4i	− 0.960972i	−0.877002	−	0.480486i	$-0.840460\pi$
$653$	− 16035.4i	− 0.960972i	0.877002	−	0.480486i	$-0.159540\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3087.43i	0.183337i
$658$	0	0
$659$	12259.7	0.724688	0.362344	−	0.932044i	$-0.381977\pi$
$659$	12259.7	0.724688	0.362344	+	0.932044i	$0.381977\pi$
$660$	0	0
$661$	7515.22	0.442221	0.221111	−	0.975249i	$-0.429032\pi$
$661$	7515.22	0.442221	0.221111	+	0.975249i	$0.429032\pi$
$662$	0	0
$663$	− 3567.99i	− 0.209003i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7700.74i	− 0.447037i
$668$	0	0
$669$	−1015.51	−0.0586874
$670$	0	0
$671$	1243.73	0.0715552
$672$	0	0
$673$	17438.7i	0.998829i	0.866363	+	0.499415i	$0.166452\pi$
$673$	17438.7i	0.998829i	−0.866363	+	0.499415i	$0.833548\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14679.1i	− 0.833331i	−0.909060	−	0.416665i	$-0.863199\pi$
$677$	− 14679.1i	− 0.833331i	0.909060	−	0.416665i	$-0.136801\pi$
$678$	0	0
$679$	22965.7	1.29800
$680$	0	0
$681$	1522.77	0.0856866
$682$	0	0
$683$	30494.6i	1.70841i	0.519937	+	0.854204i	$0.325955\pi$
$683$	30494.6i	1.70841i	−0.519937	+	0.854204i	$0.674045\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4720.29i	0.262140i
$688$	0	0
$689$	1341.76	0.0741903
$690$	0	0
$691$	−13887.9	−0.764576	−0.382288	−	0.924043i	$-0.624864\pi$
$691$	−13887.9	−0.764576	−0.382288	+	0.924043i	$0.624864\pi$
$692$	0	0
$693$	− 1945.04i	− 0.106617i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 29410.7i	− 1.59829i
$698$	0	0
$699$	5432.17	0.293939
$700$	0	0
$701$	11296.0	0.608622	0.304311	−	0.952573i	$-0.401574\pi$
$701$	11296.0	0.608622	0.304311	+	0.952573i	$0.401574\pi$
$702$	0	0
$703$	36712.0i	1.96959i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7436.82i	− 0.395602i
$708$	0	0
$709$	24405.6	1.29277	0.646384	−	0.763013i	$-0.276281\pi$
$709$	24405.6	1.29277	0.646384	+	0.763013i	$0.276281\pi$
$710$	0	0
$711$	22489.0	1.18622
$712$	0	0
$713$	13369.8i	0.702251i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 346.481i	− 0.0180468i
$718$	0	0
$719$	−11883.4	−0.616379	−0.308189	−	0.951325i	$-0.599723\pi$
$719$	−11883.4	−0.616379	−0.308189	+	0.951325i	$0.599723\pi$
$720$	0	0
$721$	−46731.9	−2.41385
$722$	0	0
$723$	4367.90i	0.224680i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21154.5i	1.07920i	0.841922	+	0.539599i	$0.181424\pi$
$727$	21154.5i	1.07920i	−0.841922	+	0.539599i	$0.818576\pi$
$728$	0	0
$729$	16235.7	0.824861
$730$	0	0
$731$	44154.7	2.23409
$732$	0	0
$733$	− 5806.54i	− 0.292591i	−0.989241	−	0.146296i	$-0.953265\pi$
$733$	− 5806.54i	− 0.292591i	0.989241	−	0.146296i	$-0.0467351\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1569.54i	0.0784462i
$738$	0	0
$739$	−14389.0	−0.716248	−0.358124	−	0.933674i	$-0.616584\pi$
$739$	−14389.0	−0.716248	−0.358124	+	0.933674i	$0.616584\pi$
$740$	0	0
$741$	4272.67	0.211822
$742$	0	0
$743$	18366.0i	0.906839i	0.891297	+	0.453420i	$0.149796\pi$
$743$	18366.0i	0.906839i	−0.891297	+	0.453420i	$0.850204\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14818.0i	0.725785i
$748$	0	0
$749$	−19348.1	−0.943876
$750$	0	0
$751$	−19355.0	−0.940444	−0.470222	−	0.882548i	$-0.655826\pi$
$751$	−19355.0	−0.940444	−0.470222	+	0.882548i	$0.655826\pi$
$752$	0	0
$753$	− 3402.41i	− 0.164662i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	494.408i	0.0237379i	0.999930	+	0.0118689i	$0.00377809\pi$
$757$	494.408i	0.0237379i	−0.999930	+	0.0118689i	$0.996222\pi$
$758$	0	0
$759$	−266.824	−0.0127604
$760$	0	0
$761$	−2310.20	−0.110045	−0.0550227	−	0.998485i	$-0.517523\pi$
$761$	−2310.20	−0.110045	−0.0550227	+	0.998485i	$0.517523\pi$
$762$	0	0
$763$	23942.0i	1.13599i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 15201.2i	− 0.715623i
$768$	0	0
$769$	−24590.9	−1.15315	−0.576573	−	0.817045i	$-0.695611\pi$
$769$	−24590.9	−1.15315	−0.576573	+	0.817045i	$0.695611\pi$
$770$	0	0
$771$	3328.34	0.155470
$772$	0	0
$773$	− 22102.6i	− 1.02843i	−0.857662	−	0.514214i	$-0.828084\pi$
$773$	− 22102.6i	− 1.02843i	0.857662	−	0.514214i	$-0.171916\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6567.61i	− 0.303232i
$778$	0	0
$779$	35219.3	1.61985
$780$	0	0
$781$	−2845.83	−0.130386
$782$	0	0
$783$	3233.30i	0.147572i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 29715.0i	− 1.34590i	−0.739686	−	0.672952i	$-0.765026\pi$
$787$	− 29715.0i	− 1.34590i	0.739686	−	0.672952i	$-0.234974\pi$
$788$	0	0
$789$	3162.40	0.142693
$790$	0	0
$791$	45852.6	2.06110
$792$	0	0
$793$	15840.2i	0.709335i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1237.06i	0.0549799i	0.999622	+	0.0274899i	$0.00875142\pi$
$797$	1237.06i	0.0549799i	−0.999622	+	0.0274899i	$0.991249\pi$
$798$	0	0
$799$	−9125.20	−0.404038
$800$	0	0
$801$	16373.9	0.722274
$802$	0	0
$803$	− 307.203i	− 0.0135006i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5002.00i	0.218189i
$808$	0	0
$809$	4675.39	0.203186	0.101593	−	0.994826i	$-0.467606\pi$
$809$	4675.39	0.203186	0.101593	+	0.994826i	$0.467606\pi$
$810$	0	0
$811$	−3243.36	−0.140431	−0.0702156	−	0.997532i	$-0.522369\pi$
$811$	−3243.36	−0.140431	−0.0702156	+	0.997532i	$0.522369\pi$
$812$	0	0
$813$	− 2324.21i	− 0.100263i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	52875.3i	2.26423i
$818$	0	0
$819$	24772.2	1.05691
$820$	0	0
$821$	33551.4	1.42625	0.713126	−	0.701036i	$-0.247279\pi$
$821$	33551.4	1.42625	0.713126	+	0.701036i	$0.247279\pi$
$822$	0	0
$823$	6365.66i	0.269615i	0.990872	+	0.134807i	$0.0430416\pi$
$823$	6365.66i	0.269615i	−0.990872	+	0.134807i	$0.956958\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18980.7i	0.798094i	0.916930	+	0.399047i	$0.130659\pi$
$827$	18980.7i	0.798094i	−0.916930	+	0.399047i	$0.869341\pi$
$828$	0	0
$829$	−33674.2	−1.41080	−0.705400	−	0.708809i	$-0.749233\pi$
$829$	−33674.2	−1.41080	−0.705400	+	0.708809i	$0.749233\pi$
$830$	0	0
$831$	−795.087	−0.0331904
$832$	0	0
$833$	56076.0i	2.33244i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5613.58i	− 0.231821i
$838$	0	0
$839$	4639.47	0.190909	0.0954544	−	0.995434i	$-0.469570\pi$
$839$	4639.47	0.190909	0.0954544	+	0.995434i	$0.469570\pi$
$840$	0	0
$841$	−19817.0	−0.812540
$842$	0	0
$843$	2845.77i	0.116267i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 37733.2i	− 1.53073i
$848$	0	0
$849$	3978.34	0.160820
$850$	0	0
$851$	29199.2	1.17619
$852$	0	0
$853$	− 1646.37i	− 0.0660850i	−0.999454	−	0.0330425i	$-0.989480\pi$
$853$	− 1646.37i	− 0.0660850i	0.999454	−	0.0330425i	$-0.0105197\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46767.8i	1.86413i	0.362295	+	0.932064i	$0.381993\pi$
$857$	46767.8i	1.86413i	−0.362295	+	0.932064i	$0.618007\pi$
$858$	0	0
$859$	−11466.7	−0.455460	−0.227730	−	0.973724i	$-0.573130\pi$
$859$	−11466.7	−0.455460	−0.227730	+	0.973724i	$0.573130\pi$
$860$	0	0
$861$	−6300.57	−0.249388
$862$	0	0
$863$	− 42704.1i	− 1.68443i	−0.539140	−	0.842216i	$-0.681250\pi$
$863$	− 42704.1i	− 1.68443i	0.539140	−	0.842216i	$-0.318750\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8437.19i	− 0.330498i
$868$	0	0
$869$	−2237.68	−0.0873511
$870$	0	0
$871$	−19989.9	−0.777647
$872$	0	0
$873$	− 21109.6i	− 0.818385i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18376.9i	0.707574i	0.935326	+	0.353787i	$0.115106\pi$
$877$	18376.9i	0.707574i	−0.935326	+	0.353787i	$0.884894\pi$
$878$	0	0
$879$	−2581.60	−0.0990616
$880$	0	0
$881$	−7024.26	−0.268619	−0.134310	−	0.990939i	$-0.542882\pi$
$881$	−7024.26	−0.268619	−0.134310	+	0.990939i	$0.542882\pi$
$882$	0	0
$883$	41747.1i	1.59106i	0.605917	+	0.795528i	$0.292806\pi$
$883$	41747.1i	1.59106i	−0.605917	+	0.795528i	$0.707194\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 5863.18i	− 0.221946i	−0.993823	−	0.110973i	$-0.964603\pi$
$887$	− 5863.18i	− 0.221946i	0.993823	−	0.110973i	$-0.0353967\pi$
$888$	0	0
$889$	32757.8	1.23584
$890$	0	0
$891$	−1730.97	−0.0650836
$892$	0	0
$893$	− 10927.4i	− 0.409488i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3398.30i	− 0.126495i
$898$	0	0
$899$	−7937.74	−0.294481
$900$	0	0
$901$	4833.78	0.178731
$902$	0	0
$903$	− 9459.14i	− 0.348594i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 13945.6i	− 0.510536i	−0.966870	−	0.255268i	$-0.917836\pi$
$907$	− 13945.6i	− 0.510536i	0.966870	−	0.255268i	$-0.0821638\pi$
$908$	0	0
$909$	−6835.75	−0.249425
$910$	0	0
$911$	−38202.8	−1.38937	−0.694685	−	0.719314i	$-0.744456\pi$
$911$	−38202.8	−1.38937	−0.694685	+	0.719314i	$0.744456\pi$
$912$	0	0
$913$	− 1474.41i	− 0.0534456i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 69441.7i	− 2.50073i
$918$	0	0
$919$	−8586.98	−0.308225	−0.154112	−	0.988053i	$-0.549252\pi$
$919$	−8586.98	−0.308225	−0.154112	+	0.988053i	$0.549252\pi$
$920$	0	0
$921$	−2827.70	−0.101168
$922$	0	0
$923$	− 36244.7i	− 1.29254i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	42954.9i	1.52192i
$928$	0	0
$929$	−50645.2	−1.78861	−0.894303	−	0.447461i	$-0.852328\pi$
$929$	−50645.2	−1.78861	−0.894303	+	0.447461i	$0.852328\pi$
$930$	0	0
$931$	−67151.1	−2.36390
$932$	0	0
$933$	− 275.640i	− 0.00967208i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26068.7i	− 0.908887i	−0.890775	−	0.454444i	$-0.849838\pi$
$937$	− 26068.7i	− 0.908887i	0.890775	−	0.454444i	$-0.150162\pi$
$938$	0	0
$939$	2805.41	0.0974985
$940$	0	0
$941$	−4036.78	−0.139846	−0.0699230	−	0.997552i	$-0.522275\pi$
$941$	−4036.78	−0.139846	−0.0699230	+	0.997552i	$0.522275\pi$
$942$	0	0
$943$	− 28012.0i	− 0.967334i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47790.1i	1.63988i	0.572446	+	0.819942i	$0.305995\pi$
$947$	47790.1i	1.63988i	−0.572446	+	0.819942i	$0.694005\pi$
$948$	0	0
$949$	3912.57	0.133833
$950$	0	0
$951$	−7309.71	−0.249247
$952$	0	0
$953$	− 49284.8i	− 1.67523i	−0.546264	−	0.837613i	$-0.683950\pi$
$953$	− 49284.8i	− 1.67523i	0.546264	−	0.837613i	$-0.316050\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 158.415i	− 0.00535092i
$958$	0	0
$959$	−21783.4	−0.733495
$960$	0	0
$961$	−16009.7	−0.537400
$962$	0	0
$963$	17784.3i	0.595110i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	19830.2i	0.659459i	0.944076	+	0.329729i	$0.106957\pi$
$967$	19830.2i	0.659459i	−0.944076	+	0.329729i	$0.893043\pi$
$968$	0	0
$969$	15392.5	0.510299
$970$	0	0
$971$	24538.6	0.811000	0.405500	−	0.914095i	$-0.367097\pi$
$971$	24538.6	0.811000	0.405500	+	0.914095i	$0.367097\pi$
$972$	0	0
$973$	72586.5i	2.39159i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15113.6i	0.494909i	0.968900	+	0.247454i	$0.0795940\pi$
$977$	15113.6i	0.494909i	−0.968900	+	0.247454i	$0.920406\pi$
$978$	0	0
$979$	−1629.22	−0.0531870
$980$	0	0
$981$	22006.9	0.716235
$982$	0	0
$983$	13262.7i	0.430330i	0.976578	+	0.215165i	$0.0690289\pi$
$983$	13262.7i	0.430330i	−0.976578	+	0.215165i	$0.930971\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1954.87i	0.0630436i
$988$	0	0
$989$	42054.8	1.35214
$990$	0	0
$991$	−7053.73	−0.226104	−0.113052	−	0.993589i	$-0.536063\pi$
$991$	−7053.73	−0.226104	−0.113052	+	0.993589i	$0.536063\pi$
$992$	0	0
$993$	− 7081.98i	− 0.226324i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38686.1i	− 1.22889i	−0.788961	−	0.614443i	$-0.789381\pi$
$997$	− 38686.1i	− 1.22889i	0.788961	−	0.614443i	$-0.210619\pi$
$998$	0	0
$999$	−12259.8	−0.388273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.i.449.3		4
4.3	odd	2		800.4.c.k.449.2		4
5.2	odd	4		160.4.a.c.1.2	✓	2
5.3	odd	4		800.4.a.s.1.1		2
5.4	even	2	inner	800.4.c.i.449.2		4
15.2	even	4		1440.4.a.t.1.1		2
20.3	even	4		800.4.a.m.1.2		2
20.7	even	4		160.4.a.g.1.1	yes	2
20.19	odd	2		800.4.c.k.449.3		4
40.3	even	4		1600.4.a.cn.1.1		2
40.13	odd	4		1600.4.a.cd.1.2		2
40.27	even	4		320.4.a.o.1.2		2
40.37	odd	4		320.4.a.s.1.1		2
60.47	odd	4		1440.4.a.x.1.2		2
80.27	even	4		1280.4.d.q.641.3		4
80.37	odd	4		1280.4.d.x.641.2		4
80.67	even	4		1280.4.d.q.641.2		4
80.77	odd	4		1280.4.d.x.641.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.c.1.2	✓	2	5.2	odd	4
160.4.a.g.1.1	yes	2	20.7	even	4
320.4.a.o.1.2		2	40.27	even	4
320.4.a.s.1.1		2	40.37	odd	4
800.4.a.m.1.2		2	20.3	even	4
800.4.a.s.1.1		2	5.3	odd	4
800.4.c.i.449.2		4	5.4	even	2	inner
800.4.c.i.449.3		4	1.1	even	1	trivial
800.4.c.k.449.2		4	4.3	odd	2
800.4.c.k.449.3		4	20.19	odd	2
1280.4.d.q.641.2		4	80.67	even	4
1280.4.d.q.641.3		4	80.27	even	4
1280.4.d.x.641.2		4	80.37	odd	4
1280.4.d.x.641.3		4	80.77	odd	4
1440.4.a.t.1.1		2	15.2	even	4
1440.4.a.x.1.2		2	60.47	odd	4
1600.4.a.cd.1.2		2	40.13	odd	4
1600.4.a.cn.1.1		2	40.3	even	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.898979i q^{3} +28.4949i q^{7} +26.1918 q^{9} -2.60612 q^{11} -33.1918i q^{13} -119.576i q^{17} +143.192 q^{19} -25.6163 q^{21} -113.889i q^{23} +47.8184i q^{27} +67.6163 q^{29} -117.394 q^{31} -2.34285i q^{33} +256.384i q^{37} +29.8388 q^{39} +245.959 q^{41} +369.262i q^{43} -76.3133i q^{47} -468.959 q^{49} +107.496 q^{51} +40.4245i q^{53} +128.727i q^{57} +457.980 q^{59} -477.233 q^{61} +746.334i q^{63} -602.252i q^{67} +102.384 q^{69} +1091.98 q^{71} +117.878i q^{73} -74.2612i q^{77} +858.624 q^{79} +664.192 q^{81} +565.748i q^{83} +60.7857i q^{87} +625.151 q^{89} +945.798 q^{91} -105.535i q^{93} -805.959i q^{97} -68.2591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 52 q^{9} - 128 q^{11} + 416 q^{19} - 416 q^{21} + 584 q^{29} - 352 q^{31} + 864 q^{39} + 200 q^{41} - 1092 q^{49} + 2272 q^{51} + 1440 q^{59} - 2536 q^{61} + 96 q^{69} + 96 q^{71} + 64 q^{79} + 2500 q^{81}+ \cdots + 6272 q^{99}+O(q^{100}) \) 4 * q - 52 * q^9 - 128 * q^11 + 416 * q^19 - 416 * q^21 + 584 * q^29 - 352 * q^31 + 864 * q^39 + 200 * q^41 - 1092 * q^49 + 2272 * q^51 + 1440 * q^59 - 2536 * q^61 + 96 * q^69 + 96 * q^71 + 64 * q^79 + 2500 * q^81 + 1560 * q^89 + 3744 * q^91 + 6272 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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