LMFDB - Embedded newform 1323.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

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:	yes
Analytic conductor:	$78.0595269376$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.58258 q^{2} +13.0000 q^{4} +4.58258 q^{5} -22.9129 q^{8} +O(q^{10})$	$q-4.58258 q^{2} +13.0000 q^{4} +4.58258 q^{5} -22.9129 q^{8} -21.0000 q^{10} -22.9129 q^{11} -44.0000 q^{13} +1.00000 q^{16} +64.1561 q^{17} +49.0000 q^{19} +59.5735 q^{20} +105.000 q^{22} +77.9038 q^{23} -104.000 q^{25} +201.633 q^{26} +219.964 q^{29} -191.000 q^{31} +178.720 q^{32} -294.000 q^{34} +29.0000 q^{37} -224.546 q^{38} -105.000 q^{40} -371.189 q^{41} +386.000 q^{43} -297.867 q^{44} -357.000 q^{46} -64.1561 q^{47} +476.588 q^{50} -572.000 q^{52} +128.312 q^{53} -105.000 q^{55} -1008.00 q^{58} -705.717 q^{59} -128.000 q^{61} +875.272 q^{62} -827.000 q^{64} -201.633 q^{65} -988.000 q^{67} +834.029 q^{68} +811.116 q^{71} -266.000 q^{73} -132.895 q^{74} +637.000 q^{76} +146.000 q^{79} +4.58258 q^{80} +1701.00 q^{82} +1429.76 q^{83} +294.000 q^{85} -1768.87 q^{86} +525.000 q^{88} +801.951 q^{89} +1012.75 q^{92} +294.000 q^{94} +224.546 q^{95} -152.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 26 q^{4}+O(q^{10}) $ 2 * q + 26 * q^4	$ 2 q + 26 q^{4} - 42 q^{10} - 88 q^{13} + 2 q^{16} + 98 q^{19} + 210 q^{22} - 208 q^{25} - 382 q^{31} - 588 q^{34} + 58 q^{37} - 210 q^{40} + 772 q^{43} - 714 q^{46} - 1144 q^{52} - 210 q^{55} - 2016 q^{58} - 256 q^{61} - 1654 q^{64} - 1976 q^{67} - 532 q^{73} + 1274 q^{76} + 292 q^{79} + 3402 q^{82} + 588 q^{85} + 1050 q^{88} + 588 q^{94} - 304 q^{97}+O(q^{100}) $ 2 * q + 26 * q^4 - 42 * q^10 - 88 * q^13 + 2 * q^16 + 98 * q^19 + 210 * q^22 - 208 * q^25 - 382 * q^31 - 588 * q^34 + 58 * q^37 - 210 * q^40 + 772 * q^43 - 714 * q^46 - 1144 * q^52 - 210 * q^55 - 2016 * q^58 - 256 * q^61 - 1654 * q^64 - 1976 * q^67 - 532 * q^73 + 1274 * q^76 + 292 * q^79 + 3402 * q^82 + 588 * q^85 + 1050 * q^88 + 588 * q^94 - 304 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−4.58258	−1.62019	−0.810093	−	0.586302i	$-0.800583\pi$
$2$	−4.58258	−1.62019	−0.810093	+	0.586302i	$0.800583\pi$
$3$	0	0
$3$	0	0
$4$	13.0000	1.62500
$5$	4.58258	0.409878	0.204939	−	0.978775i	$-0.434300\pi$
$5$	4.58258	0.409878	0.204939	+	0.978775i	$0.434300\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−22.9129	−1.01262
$9$	0	0
$10$	−21.0000	−0.664078
$11$	−22.9129	−0.628045	−0.314022	−	0.949416i	$-0.601677\pi$
$11$	−22.9129	−0.628045	−0.314022	+	0.949416i	$0.601677\pi$
$12$	0	0
$13$	−44.0000	−0.938723	−0.469362	−	0.883006i	$-0.655516\pi$
$13$	−44.0000	−0.938723	−0.469362	+	0.883006i	$0.655516\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.0156250
$17$	64.1561	0.915302	0.457651	−	0.889132i	$-0.348691\pi$
$17$	64.1561	0.915302	0.457651	+	0.889132i	$0.348691\pi$
$18$	0	0
$19$	49.0000	0.591651	0.295826	−	0.955242i	$-0.404405\pi$
$19$	49.0000	0.591651	0.295826	+	0.955242i	$0.404405\pi$
$20$	59.5735	0.666052
$21$	0	0
$22$	105.000	1.01755
$23$	77.9038	0.706264	0.353132	−	0.935574i	$-0.385117\pi$
$23$	77.9038	0.706264	0.353132	+	0.935574i	$0.385117\pi$
$24$	0	0
$25$	−104.000	−0.832000
$26$	201.633	1.52091
$27$	0	0
$28$	0	0
$29$	219.964	1.40849	0.704245	−	0.709957i	$-0.251286\pi$
$29$	219.964	1.40849	0.704245	+	0.709957i	$0.251286\pi$
$30$	0	0
$31$	−191.000	−1.10660	−0.553300	−	0.832982i	$-0.686632\pi$
$31$	−191.000	−1.10660	−0.553300	+	0.832982i	$0.686632\pi$
$32$	178.720	0.987300
$33$	0	0
$34$	−294.000	−1.48296
$35$	0	0
$36$	0	0
$37$	29.0000	0.128853	0.0644266	−	0.997922i	$-0.479478\pi$
$37$	29.0000	0.128853	0.0644266	+	0.997922i	$0.479478\pi$
$38$	−224.546	−0.958584
$39$	0	0
$40$	−105.000	−0.415049
$41$	−371.189	−1.41390	−0.706950	−	0.707263i	$-0.749930\pi$
$41$	−371.189	−1.41390	−0.706950	+	0.707263i	$0.749930\pi$
$42$	0	0
$43$	386.000	1.36894	0.684470	−	0.729041i	$-0.260034\pi$
$43$	386.000	1.36894	0.684470	+	0.729041i	$0.260034\pi$
$44$	−297.867	−1.02057
$45$	0	0
$46$	−357.000	−1.14428
$47$	−64.1561	−0.199109	−0.0995545	−	0.995032i	$-0.531742\pi$
$47$	−64.1561	−0.199109	−0.0995545	+	0.995032i	$0.531742\pi$
$48$	0	0
$49$	0	0
$50$	476.588	1.34799
$51$	0	0
$52$	−572.000	−1.52543
$53$	128.312	0.332548	0.166274	−	0.986080i	$-0.446826\pi$
$53$	128.312	0.332548	0.166274	+	0.986080i	$0.446826\pi$
$54$	0	0
$55$	−105.000	−0.257422
$56$	0	0
$57$	0	0
$58$	−1008.00	−2.28202
$59$	−705.717	−1.55723	−0.778614	−	0.627503i	$-0.784077\pi$
$59$	−705.717	−1.55723	−0.778614	+	0.627503i	$0.784077\pi$
$60$	0	0
$61$	−128.000	−0.268668	−0.134334	−	0.990936i	$-0.542889\pi$
$61$	−128.000	−0.268668	−0.134334	+	0.990936i	$0.542889\pi$
$62$	875.272	1.79290
$63$	0	0
$64$	−827.000	−1.61523
$65$	−201.633	−0.384762
$66$	0	0
$67$	−988.000	−1.80154	−0.900772	−	0.434293i	$-0.856998\pi$
$67$	−988.000	−1.80154	−0.900772	+	0.434293i	$0.856998\pi$
$68$	834.029	1.48737
$69$	0	0
$70$	0	0
$71$	811.116	1.35580	0.677900	−	0.735154i	$-0.262890\pi$
$71$	811.116	1.35580	0.677900	+	0.735154i	$0.262890\pi$
$72$	0	0
$73$	−266.000	−0.426479	−0.213239	−	0.977000i	$-0.568401\pi$
$73$	−266.000	−0.426479	−0.213239	+	0.977000i	$0.568401\pi$
$74$	−132.895	−0.208766
$75$	0	0
$76$	637.000	0.961433
$77$	0	0
$78$	0	0
$79$	146.000	0.207928	0.103964	−	0.994581i	$-0.466847\pi$
$79$	146.000	0.207928	0.103964	+	0.994581i	$0.466847\pi$
$80$	4.58258	0.00640434
$81$	0	0
$82$	1701.00	2.29078
$83$	1429.76	1.89081	0.945403	−	0.325903i	$-0.105668\pi$
$83$	1429.76	1.89081	0.945403	+	0.325903i	$0.105668\pi$
$84$	0	0
$85$	294.000	0.375162
$86$	−1768.87	−2.21794
$87$	0	0
$88$	525.000	0.635968
$89$	801.951	0.955130	0.477565	−	0.878596i	$-0.341519\pi$
$89$	801.951	0.955130	0.477565	+	0.878596i	$0.341519\pi$
$90$	0	0
$91$	0	0
$92$	1012.75	1.14768
$93$	0	0
$94$	294.000	0.322593
$95$	224.546	0.242505
$96$	0	0
$97$	−152.000	−0.159106	−0.0795529	−	0.996831i	$-0.525349\pi$
$97$	−152.000	−0.159106	−0.0795529	+	0.996831i	$0.525349\pi$
$98$	0	0
$99$	0	0
$100$	−1352.00	−1.35200
$101$	−1255.63	−1.23702	−0.618512	−	0.785775i	$-0.712264\pi$
$101$	−1255.63	−1.23702	−0.618512	+	0.785775i	$0.712264\pi$
$102$	0	0
$103$	−2063.00	−1.97353	−0.986764	−	0.162162i	$-0.948153\pi$
$103$	−2063.00	−1.97353	−0.986764	+	0.162162i	$0.948153\pi$
$104$	1008.17	0.950566
$105$	0	0
$106$	−588.000	−0.538789
$107$	−73.3212	−0.0662451	−0.0331226	−	0.999451i	$-0.510545\pi$
$107$	−73.3212	−0.0662451	−0.0331226	+	0.999451i	$0.510545\pi$
$108$	0	0
$109$	2171.00	1.90774	0.953872	−	0.300214i	$-0.0970580\pi$
$109$	2171.00	1.90774	0.953872	+	0.300214i	$0.0970580\pi$
$110$	481.170	0.417071
$111$	0	0
$112$	0	0
$113$	1182.30	0.984264	0.492132	−	0.870521i	$-0.336218\pi$
$113$	1182.30	0.984264	0.492132	+	0.870521i	$0.336218\pi$
$114$	0	0
$115$	357.000	0.289482
$116$	2859.53	2.28880
$117$	0	0
$118$	3234.00	2.52300
$119$	0	0
$120$	0	0
$121$	−806.000	−0.605560
$122$	586.570	0.435291
$123$	0	0
$124$	−2483.00	−1.79823
$125$	−1049.41	−0.750897
$126$	0	0
$127$	−1324.00	−0.925087	−0.462543	−	0.886597i	$-0.653063\pi$
$127$	−1324.00	−0.925087	−0.462543	+	0.886597i	$0.653063\pi$
$128$	2360.03	1.62968
$129$	0	0
$130$	924.000	0.623386
$131$	925.680	0.617382	0.308691	−	0.951162i	$-0.400109\pi$
$131$	925.680	0.617382	0.308691	+	0.951162i	$0.400109\pi$
$132$	0	0
$133$	0	0
$134$	4527.58	2.91883
$135$	0	0
$136$	−1470.00	−0.926849
$137$	−2511.25	−1.56606	−0.783032	−	0.621982i	$-0.786328\pi$
$137$	−2511.25	−1.56606	−0.783032	+	0.621982i	$0.786328\pi$
$138$	0	0
$139$	−404.000	−0.246524	−0.123262	−	0.992374i	$-0.539336\pi$
$139$	−404.000	−0.246524	−0.123262	+	0.992374i	$0.539336\pi$
$140$	0	0
$141$	0	0
$142$	−3717.00	−2.19665
$143$	1008.17	0.589560
$144$	0	0
$145$	1008.00	0.577309
$146$	1218.97	0.690974
$147$	0	0
$148$	377.000	0.209387
$149$	669.056	0.367860	0.183930	−	0.982939i	$-0.441118\pi$
$149$	669.056	0.367860	0.183930	+	0.982939i	$0.441118\pi$
$150$	0	0
$151$	2738.00	1.47560	0.737799	−	0.675021i	$-0.235865\pi$
$151$	2738.00	1.47560	0.737799	+	0.675021i	$0.235865\pi$
$152$	−1122.73	−0.599115
$153$	0	0
$154$	0	0
$155$	−875.272	−0.453571
$156$	0	0
$157$	1372.00	0.697436	0.348718	−	0.937228i	$-0.386617\pi$
$157$	1372.00	0.697436	0.348718	+	0.937228i	$0.386617\pi$
$158$	−669.056	−0.336881
$159$	0	0
$160$	819.000	0.404673
$161$	0	0
$162$	0	0
$163$	2792.00	1.34163	0.670817	−	0.741623i	$-0.265944\pi$
$163$	2792.00	1.34163	0.670817	+	0.741623i	$0.265944\pi$
$164$	−4825.45	−2.29759
$165$	0	0
$166$	−6552.00	−3.06346
$167$	412.432	0.191107	0.0955537	−	0.995424i	$-0.469538\pi$
$167$	412.432	0.191107	0.0955537	+	0.995424i	$0.469538\pi$
$168$	0	0
$169$	−261.000	−0.118798
$170$	−1347.28	−0.607832
$171$	0	0
$172$	5018.00	2.22453
$173$	2396.69	1.05328	0.526638	−	0.850090i	$-0.323452\pi$
$173$	2396.69	1.05328	0.526638	+	0.850090i	$0.323452\pi$
$174$	0	0
$175$	0	0
$176$	−22.9129	−0.00981320
$177$	0	0
$178$	−3675.00	−1.54749
$179$	−751.542	−0.313815	−0.156908	−	0.987613i	$-0.550152\pi$
$179$	−751.542	−0.313815	−0.156908	+	0.987613i	$0.550152\pi$
$180$	0	0
$181$	70.0000	0.0287462	0.0143731	−	0.999897i	$-0.495425\pi$
$181$	70.0000	0.0287462	0.0143731	+	0.999897i	$0.495425\pi$
$182$	0	0
$183$	0	0
$184$	−1785.00	−0.715174
$185$	132.895	0.0528141
$186$	0	0
$187$	−1470.00	−0.574851
$188$	−834.029	−0.323552
$189$	0	0
$190$	−1029.00	−0.392903
$191$	−2369.19	−0.897532	−0.448766	−	0.893649i	$-0.648136\pi$
$191$	−2369.19	−0.897532	−0.448766	+	0.893649i	$0.648136\pi$
$192$	0	0
$193$	−1042.00	−0.388626	−0.194313	−	0.980940i	$-0.562248\pi$
$193$	−1042.00	−0.388626	−0.194313	+	0.980940i	$0.562248\pi$
$194$	696.552	0.257781
$195$	0	0
$196$	0	0
$197$	−2447.10	−0.885017	−0.442508	−	0.896764i	$-0.645911\pi$
$197$	−2447.10	−0.885017	−0.442508	+	0.896764i	$0.645911\pi$
$198$	0	0
$199$	−3575.00	−1.27349	−0.636746	−	0.771073i	$-0.719720\pi$
$199$	−3575.00	−1.27349	−0.636746	+	0.771073i	$0.719720\pi$
$200$	2382.94	0.842496
$201$	0	0
$202$	5754.00	2.00421
$203$	0	0
$204$	0	0
$205$	−1701.00	−0.579527
$206$	9453.85	3.19748
$207$	0	0
$208$	−44.0000	−0.0146676
$209$	−1122.73	−0.371583
$210$	0	0
$211$	−118.000	−0.0384998	−0.0192499	−	0.999815i	$-0.506128\pi$
$211$	−118.000	−0.0384998	−0.0192499	+	0.999815i	$0.506128\pi$
$212$	1668.06	0.540390
$213$	0	0
$214$	336.000	0.107329
$215$	1768.87	0.561099
$216$	0	0
$217$	0	0
$218$	−9948.77	−3.09090
$219$	0	0
$220$	−1365.00	−0.418310
$221$	−2822.87	−0.859215
$222$	0	0
$223$	511.000	0.153449	0.0767244	−	0.997052i	$-0.475554\pi$
$223$	511.000	0.153449	0.0767244	+	0.997052i	$0.475554\pi$
$224$	0	0
$225$	0	0
$226$	−5418.00	−1.59469
$227$	−5654.90	−1.65343	−0.826715	−	0.562620i	$-0.809793\pi$
$227$	−5654.90	−1.65343	−0.826715	+	0.562620i	$0.809793\pi$
$228$	0	0
$229$	2140.00	0.617534	0.308767	−	0.951138i	$-0.400084\pi$
$229$	2140.00	0.617534	0.308767	+	0.951138i	$0.400084\pi$
$230$	−1635.98	−0.469014
$231$	0	0
$232$	−5040.00	−1.42626
$233$	−5159.98	−1.45082	−0.725411	−	0.688316i	$-0.758350\pi$
$233$	−5159.98	−1.45082	−0.725411	+	0.688316i	$0.758350\pi$
$234$	0	0
$235$	−294.000	−0.0816104
$236$	−9174.32	−2.53050
$237$	0	0
$238$	0	0
$239$	4124.32	1.11623	0.558117	−	0.829762i	$-0.311524\pi$
$239$	4124.32	1.11623	0.558117	+	0.829762i	$0.311524\pi$
$240$	0	0
$241$	1510.00	0.403600	0.201800	−	0.979427i	$-0.435321\pi$
$241$	1510.00	0.403600	0.201800	+	0.979427i	$0.435321\pi$
$242$	3693.56	0.981119
$243$	0	0
$244$	−1664.00	−0.436585
$245$	0	0
$246$	0	0
$247$	−2156.00	−0.555397
$248$	4376.36	1.12056
$249$	0	0
$250$	4809.00	1.21659
$251$	−2749.55	−0.691433	−0.345717	−	0.938339i	$-0.612364\pi$
$251$	−2749.55	−0.691433	−0.345717	+	0.938339i	$0.612364\pi$
$252$	0	0
$253$	−1785.00	−0.443565
$254$	6067.33	1.49881
$255$	0	0
$256$	−4199.00	−1.02515
$257$	−7666.65	−1.86083	−0.930413	−	0.366512i	$-0.880552\pi$
$257$	−7666.65	−1.86083	−0.930413	+	0.366512i	$0.880552\pi$
$258$	0	0
$259$	0	0
$260$	−2621.23	−0.625238
$261$	0	0
$262$	−4242.00	−1.00027
$263$	−6319.37	−1.48163	−0.740816	−	0.671708i	$-0.765561\pi$
$263$	−6319.37	−1.48163	−0.740816	+	0.671708i	$0.765561\pi$
$264$	0	0
$265$	588.000	0.136304
$266$	0	0
$267$	0	0
$268$	−12844.0	−2.92751
$269$	−691.969	−0.156840	−0.0784202	−	0.996920i	$-0.524988\pi$
$269$	−691.969	−0.156840	−0.0784202	+	0.996920i	$0.524988\pi$
$270$	0	0
$271$	−2504.00	−0.561281	−0.280641	−	0.959813i	$-0.590547\pi$
$271$	−2504.00	−0.561281	−0.280641	+	0.959813i	$0.590547\pi$
$272$	64.1561	0.0143016
$273$	0	0
$274$	11508.0	2.53731
$275$	2382.94	0.522533
$276$	0	0
$277$	−6595.00	−1.43052	−0.715262	−	0.698856i	$-0.753693\pi$
$277$	−6595.00	−1.43052	−0.715262	+	0.698856i	$0.753693\pi$
$278$	1851.36	0.399414
$279$	0	0
$280$	0	0
$281$	−5911.52	−1.25499	−0.627494	−	0.778621i	$-0.715919\pi$
$281$	−5911.52	−1.25499	−0.627494	+	0.778621i	$0.715919\pi$
$282$	0	0
$283$	−1244.00	−0.261301	−0.130650	−	0.991429i	$-0.541707\pi$
$283$	−1244.00	−0.261301	−0.130650	+	0.991429i	$0.541707\pi$
$284$	10544.5	2.20317
$285$	0	0
$286$	−4620.00	−0.955197
$287$	0	0
$288$	0	0
$289$	−797.000	−0.162223
$290$	−4619.24	−0.935348
$291$	0	0
$292$	−3458.00	−0.693028
$293$	−5911.52	−1.17869	−0.589343	−	0.807883i	$-0.700613\pi$
$293$	−5911.52	−1.17869	−0.589343	+	0.807883i	$0.700613\pi$
$294$	0	0
$295$	−3234.00	−0.638274
$296$	−664.473	−0.130479
$297$	0	0
$298$	−3066.00	−0.596002
$299$	−3427.77	−0.662986
$300$	0	0
$301$	0	0
$302$	−12547.1	−2.39074
$303$	0	0
$304$	49.0000	0.00924455
$305$	−586.570	−0.110121
$306$	0	0
$307$	−6659.00	−1.23795	−0.618973	−	0.785413i	$-0.712451\pi$
$307$	−6659.00	−1.23795	−0.618973	+	0.785413i	$0.712451\pi$
$308$	0	0
$309$	0	0
$310$	4011.00	0.734869
$311$	971.506	0.177135	0.0885676	−	0.996070i	$-0.471771\pi$
$311$	971.506	0.177135	0.0885676	+	0.996070i	$0.471771\pi$
$312$	0	0
$313$	4900.00	0.884870	0.442435	−	0.896801i	$-0.354115\pi$
$313$	4900.00	0.884870	0.442435	+	0.896801i	$0.354115\pi$
$314$	−6287.29	−1.12998
$315$	0	0
$316$	1898.00	0.337882
$317$	9302.63	1.64823	0.824113	−	0.566425i	$-0.191674\pi$
$317$	9302.63	1.64823	0.824113	+	0.566425i	$0.191674\pi$
$318$	0	0
$319$	−5040.00	−0.884595
$320$	−3789.79	−0.662049
$321$	0	0
$322$	0	0
$323$	3143.65	0.541539
$324$	0	0
$325$	4576.00	0.781018
$326$	−12794.6	−2.17370
$327$	0	0
$328$	8505.00	1.43174
$329$	0	0
$330$	0	0
$331$	68.0000	0.0112919	0.00564595	−	0.999984i	$-0.498203\pi$
$331$	68.0000	0.0112919	0.00564595	+	0.999984i	$0.498203\pi$
$332$	18586.9	3.07256
$333$	0	0
$334$	−1890.00	−0.309629
$335$	−4527.58	−0.738413
$336$	0	0
$337$	−6757.00	−1.09222	−0.546109	−	0.837714i	$-0.683891\pi$
$337$	−6757.00	−1.09222	−0.546109	+	0.837714i	$0.683891\pi$
$338$	1196.05	0.192475
$339$	0	0
$340$	3822.00	0.609638
$341$	4376.36	0.694995
$342$	0	0
$343$	0	0
$344$	−8844.37	−1.38621
$345$	0	0
$346$	−10983.0	−1.70650
$347$	−9536.34	−1.47532	−0.737662	−	0.675170i	$-0.764070\pi$
$347$	−9536.34	−1.47532	−0.737662	+	0.675170i	$0.764070\pi$
$348$	0	0
$349$	1090.00	0.167182	0.0835908	−	0.996500i	$-0.473361\pi$
$349$	1090.00	0.167182	0.0835908	+	0.996500i	$0.473361\pi$
$350$	0	0
$351$	0	0
$352$	−4095.00	−0.620069
$353$	−1626.81	−0.245288	−0.122644	−	0.992451i	$-0.539137\pi$
$353$	−1626.81	−0.245288	−0.122644	+	0.992451i	$0.539137\pi$
$354$	0	0
$355$	3717.00	0.555712
$356$	10425.4	1.55209
$357$	0	0
$358$	3444.00	0.508439
$359$	3666.06	0.538962	0.269481	−	0.963006i	$-0.413148\pi$
$359$	3666.06	0.538962	0.269481	+	0.963006i	$0.413148\pi$
$360$	0	0
$361$	−4458.00	−0.649949
$362$	−320.780	−0.0465741
$363$	0	0
$364$	0	0
$365$	−1218.97	−0.174804
$366$	0	0
$367$	−4487.00	−0.638200	−0.319100	−	0.947721i	$-0.603381\pi$
$367$	−4487.00	−0.638200	−0.319100	+	0.947721i	$0.603381\pi$
$368$	77.9038	0.0110354
$369$	0	0
$370$	−609.000	−0.0855687
$371$	0	0
$372$	0	0
$373$	−229.000	−0.0317887	−0.0158943	−	0.999874i	$-0.505060\pi$
$373$	−229.000	−0.0317887	−0.0158943	+	0.999874i	$0.505060\pi$
$374$	6736.39	0.931364
$375$	0	0
$376$	1470.00	0.201621
$377$	−9678.40	−1.32218
$378$	0	0
$379$	4166.00	0.564625	0.282313	−	0.959322i	$-0.408898\pi$
$379$	4166.00	0.564625	0.282313	+	0.959322i	$0.408898\pi$
$380$	2919.10	0.394070
$381$	0	0
$382$	10857.0	1.45417
$383$	11648.9	1.55413	0.777064	−	0.629421i	$-0.216708\pi$
$383$	11648.9	1.55413	0.777064	+	0.629421i	$0.216708\pi$
$384$	0	0
$385$	0	0
$386$	4775.04	0.629646
$387$	0	0
$388$	−1976.00	−0.258547
$389$	−5370.78	−0.700024	−0.350012	−	0.936745i	$-0.613822\pi$
$389$	−5370.78	−0.700024	−0.350012	+	0.936745i	$0.613822\pi$
$390$	0	0
$391$	4998.00	0.646444
$392$	0	0
$393$	0	0
$394$	11214.0	1.43389
$395$	669.056	0.0852250
$396$	0	0
$397$	−12962.0	−1.63865	−0.819325	−	0.573329i	$-0.805652\pi$
$397$	−12962.0	−1.63865	−0.819325	+	0.573329i	$0.805652\pi$
$398$	16382.7	2.06329
$399$	0	0
$400$	−104.000	−0.0130000
$401$	−9971.68	−1.24180	−0.620900	−	0.783889i	$-0.713233\pi$
$401$	−9971.68	−1.24180	−0.620900	+	0.783889i	$0.713233\pi$
$402$	0	0
$403$	8404.00	1.03879
$404$	−16323.1	−2.01016
$405$	0	0
$406$	0	0
$407$	−664.473	−0.0809256
$408$	0	0
$409$	12238.0	1.47954	0.739768	−	0.672862i	$-0.234935\pi$
$409$	12238.0	1.47954	0.739768	+	0.672862i	$0.234935\pi$
$410$	7794.96	0.938941
$411$	0	0
$412$	−26819.0	−3.20698
$413$	0	0
$414$	0	0
$415$	6552.00	0.775000
$416$	−7863.70	−0.926802
$417$	0	0
$418$	5145.00	0.602034
$419$	−10860.7	−1.26630	−0.633151	−	0.774029i	$-0.718239\pi$
$419$	−10860.7	−1.26630	−0.633151	+	0.774029i	$0.718239\pi$
$420$	0	0
$421$	−6235.00	−0.721794	−0.360897	−	0.932606i	$-0.617529\pi$
$421$	−6235.00	−0.721794	−0.360897	+	0.932606i	$0.617529\pi$
$422$	540.744	0.0623768
$423$	0	0
$424$	−2940.00	−0.336743
$425$	−6672.23	−0.761531
$426$	0	0
$427$	0	0
$428$	−953.176	−0.107648
$429$	0	0
$430$	−8106.00	−0.909084
$431$	7400.86	0.827116	0.413558	−	0.910478i	$-0.364286\pi$
$431$	7400.86	0.827116	0.413558	+	0.910478i	$0.364286\pi$
$432$	0	0
$433$	4384.00	0.486563	0.243281	−	0.969956i	$-0.421776\pi$
$433$	4384.00	0.486563	0.243281	+	0.969956i	$0.421776\pi$
$434$	0	0
$435$	0	0
$436$	28223.0	3.10008
$437$	3817.29	0.417862
$438$	0	0
$439$	−5432.00	−0.590559	−0.295279	−	0.955411i	$-0.595413\pi$
$439$	−5432.00	−0.590559	−0.295279	+	0.955411i	$0.595413\pi$
$440$	2405.85	0.260669
$441$	0	0
$442$	12936.0	1.39209
$443$	−1846.78	−0.198066	−0.0990328	−	0.995084i	$-0.531575\pi$
$443$	−1846.78	−0.198066	−0.0990328	+	0.995084i	$0.531575\pi$
$444$	0	0
$445$	3675.00	0.391487
$446$	−2341.70	−0.248616
$447$	0	0
$448$	0	0
$449$	15315.0	1.60971	0.804853	−	0.593474i	$-0.202244\pi$
$449$	15315.0	1.60971	0.804853	+	0.593474i	$0.202244\pi$
$450$	0	0
$451$	8505.00	0.887993
$452$	15370.0	1.59943
$453$	0	0
$454$	25914.0	2.67886
$455$	0	0
$456$	0	0
$457$	−979.000	−0.100209	−0.0501047	−	0.998744i	$-0.515955\pi$
$457$	−979.000	−0.100209	−0.0501047	+	0.998744i	$0.515955\pi$
$458$	−9806.71	−1.00052
$459$	0	0
$460$	4641.00	0.470408
$461$	6832.62	0.690297	0.345148	−	0.938548i	$-0.387829\pi$
$461$	6832.62	0.690297	0.345148	+	0.938548i	$0.387829\pi$
$462$	0	0
$463$	−10078.0	−1.01159	−0.505793	−	0.862655i	$-0.668800\pi$
$463$	−10078.0	−1.01159	−0.505793	+	0.862655i	$0.668800\pi$
$464$	219.964	0.0220077
$465$	0	0
$466$	23646.0	2.35060
$467$	14233.5	1.41038	0.705189	−	0.709019i	$-0.250862\pi$
$467$	14233.5	1.41038	0.705189	+	0.709019i	$0.250862\pi$
$468$	0	0
$469$	0	0
$470$	1347.28	0.132224
$471$	0	0
$472$	16170.0	1.57687
$473$	−8844.37	−0.859756
$474$	0	0
$475$	−5096.00	−0.492254
$476$	0	0
$477$	0	0
$478$	−18900.0	−1.80851
$479$	−17120.5	−1.63310	−0.816551	−	0.577273i	$-0.804117\pi$
$479$	−17120.5	−1.63310	−0.816551	+	0.577273i	$0.804117\pi$
$480$	0	0
$481$	−1276.00	−0.120958
$482$	−6919.69	−0.653907
$483$	0	0
$484$	−10478.0	−0.984035
$485$	−696.552	−0.0652140
$486$	0	0
$487$	446.000	0.0414994	0.0207497	−	0.999785i	$-0.493395\pi$
$487$	446.000	0.0414994	0.0207497	+	0.999785i	$0.493395\pi$
$488$	2932.85	0.272057
$489$	0	0
$490$	0	0
$491$	783.620	0.0720250	0.0360125	−	0.999351i	$-0.488534\pi$
$491$	783.620	0.0720250	0.0360125	+	0.999351i	$0.488534\pi$
$492$	0	0
$493$	14112.0	1.28919
$494$	9880.03	0.899846
$495$	0	0
$496$	−191.000	−0.0172906
$497$	0	0
$498$	0	0
$499$	16598.0	1.48904	0.744518	−	0.667603i	$-0.232680\pi$
$499$	16598.0	1.48904	0.744518	+	0.667603i	$0.232680\pi$
$500$	−13642.3	−1.22021
$501$	0	0
$502$	12600.0	1.12025
$503$	10888.2	0.965171	0.482585	−	0.875849i	$-0.339698\pi$
$503$	10888.2	0.965171	0.482585	+	0.875849i	$0.339698\pi$
$504$	0	0
$505$	−5754.00	−0.507029
$506$	8179.90	0.718658
$507$	0	0
$508$	−17212.0	−1.50327
$509$	3510.25	0.305676	0.152838	−	0.988251i	$-0.451159\pi$
$509$	3510.25	0.305676	0.152838	+	0.988251i	$0.451159\pi$
$510$	0	0
$511$	0	0
$512$	362.023	0.0312487
$513$	0	0
$514$	35133.0	3.01488
$515$	−9453.85	−0.808906
$516$	0	0
$517$	1470.00	0.125049
$518$	0	0
$519$	0	0
$520$	4620.00	0.389616
$521$	−1626.81	−0.136799	−0.0683993	−	0.997658i	$-0.521789\pi$
$521$	−1626.81	−0.136799	−0.0683993	+	0.997658i	$0.521789\pi$
$522$	0	0
$523$	−18335.0	−1.53295	−0.766476	−	0.642273i	$-0.777991\pi$
$523$	−18335.0	−1.53295	−0.766476	+	0.642273i	$0.777991\pi$
$524$	12033.8	1.00325
$525$	0	0
$526$	28959.0	2.40052
$527$	−12253.8	−1.01287
$528$	0	0
$529$	−6098.00	−0.501192
$530$	−2694.55	−0.220838
$531$	0	0
$532$	0	0
$533$	16332.3	1.32726
$534$	0	0
$535$	−336.000	−0.0271524
$536$	22637.9	1.82427
$537$	0	0
$538$	3171.00	0.254111
$539$	0	0
$540$	0	0
$541$	−11947.0	−0.949430	−0.474715	−	0.880140i	$-0.657449\pi$
$541$	−11947.0	−0.949430	−0.474715	+	0.880140i	$0.657449\pi$
$542$	11474.8	0.909379
$543$	0	0
$544$	11466.0	0.903678
$545$	9948.77	0.781942
$546$	0	0
$547$	6236.00	0.487444	0.243722	−	0.969845i	$-0.421631\pi$
$547$	6236.00	0.487444	0.243722	+	0.969845i	$0.421631\pi$
$548$	−32646.3	−2.54485
$549$	0	0
$550$	−10920.0	−0.846601
$551$	10778.2	0.833335
$552$	0	0
$553$	0	0
$554$	30222.1	2.31771
$555$	0	0
$556$	−5252.00	−0.400601
$557$	−16121.5	−1.22637	−0.613187	−	0.789938i	$-0.710113\pi$
$557$	−16121.5	−1.22637	−0.613187	+	0.789938i	$0.710113\pi$
$558$	0	0
$559$	−16984.0	−1.28506
$560$	0	0
$561$	0	0
$562$	27090.0	2.03331
$563$	2786.21	0.208569	0.104285	−	0.994547i	$-0.466745\pi$
$563$	2786.21	0.208569	0.104285	+	0.994547i	$0.466745\pi$
$564$	0	0
$565$	5418.00	0.403428
$566$	5700.72	0.423356
$567$	0	0
$568$	−18585.0	−1.37290
$569$	−20227.5	−1.49030	−0.745150	−	0.666897i	$-0.767622\pi$
$569$	−20227.5	−1.49030	−0.745150	+	0.666897i	$0.767622\pi$
$570$	0	0
$571$	12938.0	0.948228	0.474114	−	0.880463i	$-0.342768\pi$
$571$	12938.0	0.948228	0.474114	+	0.880463i	$0.342768\pi$
$572$	13106.2	0.958036
$573$	0	0
$574$	0	0
$575$	−8101.99	−0.587611
$576$	0	0
$577$	−19514.0	−1.40793	−0.703967	−	0.710232i	$-0.748590\pi$
$577$	−19514.0	−1.40793	−0.703967	+	0.710232i	$0.748590\pi$
$578$	3652.31	0.262831
$579$	0	0
$580$	13104.0	0.938128
$581$	0	0
$582$	0	0
$583$	−2940.00	−0.208855
$584$	6094.83	0.431859
$585$	0	0
$586$	27090.0	1.90969
$587$	−12070.5	−0.848727	−0.424364	−	0.905492i	$-0.639502\pi$
$587$	−12070.5	−0.848727	−0.424364	+	0.905492i	$0.639502\pi$
$588$	0	0
$589$	−9359.00	−0.654721
$590$	14820.0	1.03412
$591$	0	0
$592$	29.0000	0.00201333
$593$	24915.5	1.72539	0.862694	−	0.505726i	$-0.168775\pi$
$593$	24915.5	1.72539	0.862694	+	0.505726i	$0.168775\pi$
$594$	0	0
$595$	0	0
$596$	8697.73	0.597773
$597$	0	0
$598$	15708.0	1.07416
$599$	−4422.19	−0.301645	−0.150823	−	0.988561i	$-0.548192\pi$
$599$	−4422.19	−0.301645	−0.150823	+	0.988561i	$0.548192\pi$
$600$	0	0
$601$	−22724.0	−1.54231	−0.771157	−	0.636644i	$-0.780322\pi$
$601$	−22724.0	−1.54231	−0.771157	+	0.636644i	$0.780322\pi$
$602$	0	0
$603$	0	0
$604$	35594.0	2.39785
$605$	−3693.56	−0.248206
$606$	0	0
$607$	3388.00	0.226548	0.113274	−	0.993564i	$-0.463866\pi$
$607$	3388.00	0.226548	0.113274	+	0.993564i	$0.463866\pi$
$608$	8757.30	0.584137
$609$	0	0
$610$	2688.00	0.178416
$611$	2822.87	0.186908
$612$	0	0
$613$	−12643.0	−0.833028	−0.416514	−	0.909129i	$-0.636748\pi$
$613$	−12643.0	−0.833028	−0.416514	+	0.909129i	$0.636748\pi$
$614$	30515.4	2.00570
$615$	0	0
$616$	0	0
$617$	12125.5	0.791174	0.395587	−	0.918429i	$-0.370541\pi$
$617$	12125.5	0.791174	0.395587	+	0.918429i	$0.370541\pi$
$618$	0	0
$619$	2455.00	0.159410	0.0797050	−	0.996818i	$-0.474602\pi$
$619$	2455.00	0.159410	0.0797050	+	0.996818i	$0.474602\pi$
$620$	−11378.5	−0.737053
$621$	0	0
$622$	−4452.00	−0.286992
$623$	0	0
$624$	0	0
$625$	8191.00	0.524224
$626$	−22454.6	−1.43365
$627$	0	0
$628$	17836.0	1.13333
$629$	1860.53	0.117940
$630$	0	0
$631$	−8758.00	−0.552536	−0.276268	−	0.961081i	$-0.589098\pi$
$631$	−8758.00	−0.552536	−0.276268	+	0.961081i	$0.589098\pi$
$632$	−3345.28	−0.210551
$633$	0	0
$634$	−42630.0	−2.67043
$635$	−6067.33	−0.379173
$636$	0	0
$637$	0	0
$638$	23096.2	1.43321
$639$	0	0
$640$	10815.0	0.667969
$641$	9889.20	0.609360	0.304680	−	0.952455i	$-0.401450\pi$
$641$	9889.20	0.609360	0.304680	+	0.952455i	$0.401450\pi$
$642$	0	0
$643$	5581.00	0.342291	0.171146	−	0.985246i	$-0.445253\pi$
$643$	5581.00	0.342291	0.171146	+	0.985246i	$0.445253\pi$
$644$	0	0
$645$	0	0
$646$	−14406.0	−0.877394
$647$	−6644.73	−0.403758	−0.201879	−	0.979410i	$-0.564705\pi$
$647$	−6644.73	−0.403758	−0.201879	+	0.979410i	$0.564705\pi$
$648$	0	0
$649$	16170.0	0.978009
$650$	−20969.9	−1.26539
$651$	0	0
$652$	36296.0	2.18016
$653$	11667.2	0.699195	0.349597	−	0.936900i	$-0.386318\pi$
$653$	11667.2	0.699195	0.349597	+	0.936900i	$0.386318\pi$
$654$	0	0
$655$	4242.00	0.253051
$656$	−371.189	−0.0220922
$657$	0	0
$658$	0	0
$659$	10847.0	0.641180	0.320590	−	0.947218i	$-0.396119\pi$
$659$	10847.0	0.641180	0.320590	+	0.947218i	$0.396119\pi$
$660$	0	0
$661$	−12488.0	−0.734836	−0.367418	−	0.930056i	$-0.619758\pi$
$661$	−12488.0	−0.734836	−0.367418	+	0.930056i	$0.619758\pi$
$662$	−311.615	−0.0182950
$663$	0	0
$664$	−32760.0	−1.91466
$665$	0	0
$666$	0	0
$667$	17136.0	0.994765
$668$	5361.61	0.310549
$669$	0	0
$670$	20748.0	1.19637
$671$	2932.85	0.168735
$672$	0	0
$673$	−21310.0	−1.22056	−0.610282	−	0.792184i	$-0.708944\pi$
$673$	−21310.0	−1.22056	−0.610282	+	0.792184i	$0.708944\pi$
$674$	30964.5	1.76959
$675$	0	0
$676$	−3393.00	−0.193047
$677$	−14870.5	−0.844192	−0.422096	−	0.906551i	$-0.638706\pi$
$677$	−14870.5	−0.844192	−0.422096	+	0.906551i	$0.638706\pi$
$678$	0	0
$679$	0	0
$680$	−6736.39	−0.379895
$681$	0	0
$682$	−20055.0	−1.12602
$683$	16501.9	0.924489	0.462244	−	0.886753i	$-0.347044\pi$
$683$	16501.9	0.924489	0.462244	+	0.886753i	$0.347044\pi$
$684$	0	0
$685$	−11508.0	−0.641895
$686$	0	0
$687$	0	0
$688$	386.000	0.0213897
$689$	−5645.73	−0.312170
$690$	0	0
$691$	−21404.0	−1.17836	−0.589180	−	0.808002i	$-0.700549\pi$
$691$	−21404.0	−1.17836	−0.589180	+	0.808002i	$0.700549\pi$
$692$	31156.9	1.71157
$693$	0	0
$694$	43701.0	2.39030
$695$	−1851.36	−0.101045
$696$	0	0
$697$	−23814.0	−1.29415
$698$	−4995.01	−0.270865
$699$	0	0
$700$	0	0
$701$	−12208.0	−0.657759	−0.328880	−	0.944372i	$-0.606671\pi$
$701$	−12208.0	−0.657759	−0.328880	+	0.944372i	$0.606671\pi$
$702$	0	0
$703$	1421.00	0.0762362
$704$	18949.0	1.01444
$705$	0	0
$706$	7455.00	0.397412
$707$	0	0
$708$	0	0
$709$	15779.0	0.835815	0.417907	−	0.908490i	$-0.362764\pi$
$709$	15779.0	0.835815	0.417907	+	0.908490i	$0.362764\pi$
$710$	−17033.4	−0.900357
$711$	0	0
$712$	−18375.0	−0.967180
$713$	−14879.6	−0.781552
$714$	0	0
$715$	4620.00	0.241648
$716$	−9770.05	−0.509950
$717$	0	0
$718$	−16800.0	−0.873218
$719$	−12767.1	−0.662213	−0.331106	−	0.943593i	$-0.607422\pi$
$719$	−12767.1	−0.662213	−0.331106	+	0.943593i	$0.607422\pi$
$720$	0	0
$721$	0	0
$722$	20429.1	1.05304
$723$	0	0
$724$	910.000	0.0467125
$725$	−22876.2	−1.17186
$726$	0	0
$727$	−24848.0	−1.26762	−0.633811	−	0.773488i	$-0.718510\pi$
$727$	−24848.0	−1.26762	−0.633811	+	0.773488i	$0.718510\pi$
$728$	0	0
$729$	0	0
$730$	5586.00	0.283215
$731$	24764.2	1.25299
$732$	0	0
$733$	−34634.0	−1.74521	−0.872603	−	0.488430i	$-0.837570\pi$
$733$	−34634.0	−1.74521	−0.872603	+	0.488430i	$0.837570\pi$
$734$	20562.0	1.03400
$735$	0	0
$736$	13923.0	0.697294
$737$	22637.9	1.13145
$738$	0	0
$739$	4178.00	0.207971	0.103985	−	0.994579i	$-0.466841\pi$
$739$	4178.00	0.207971	0.103985	+	0.994579i	$0.466841\pi$
$740$	1727.63	0.0858229
$741$	0	0
$742$	0	0
$743$	14118.9	0.697137	0.348568	−	0.937283i	$-0.386668\pi$
$743$	14118.9	0.697137	0.348568	+	0.937283i	$0.386668\pi$
$744$	0	0
$745$	3066.00	0.150778
$746$	1049.41	0.0515035
$747$	0	0
$748$	−19110.0	−0.934132
$749$	0	0
$750$	0	0
$751$	950.000	0.0461598	0.0230799	−	0.999734i	$-0.492653\pi$
$751$	950.000	0.0461598	0.0230799	+	0.999734i	$0.492653\pi$
$752$	−64.1561	−0.00311108
$753$	0	0
$754$	44352.0	2.14218
$755$	12547.1	0.604815
$756$	0	0
$757$	−5158.00	−0.247650	−0.123825	−	0.992304i	$-0.539516\pi$
$757$	−5158.00	−0.247650	−0.123825	+	0.992304i	$0.539516\pi$
$758$	−19091.0	−0.914798
$759$	0	0
$760$	−5145.00	−0.245564
$761$	−20135.8	−0.959164	−0.479582	−	0.877497i	$-0.659212\pi$
$761$	−20135.8	−0.959164	−0.479582	+	0.877497i	$0.659212\pi$
$762$	0	0
$763$	0	0
$764$	−30799.5	−1.45849
$765$	0	0
$766$	−53382.0	−2.51798
$767$	31051.5	1.46181
$768$	0	0
$769$	−21476.0	−1.00708	−0.503540	−	0.863972i	$-0.667969\pi$
$769$	−21476.0	−1.00708	−0.503540	+	0.863972i	$0.667969\pi$
$770$	0	0
$771$	0	0
$772$	−13546.0	−0.631517
$773$	−40202.9	−1.87063	−0.935316	−	0.353812i	$-0.884885\pi$
$773$	−40202.9	−1.87063	−0.935316	+	0.353812i	$0.884885\pi$
$774$	0	0
$775$	19864.0	0.920692
$776$	3482.76	0.161113
$777$	0	0
$778$	24612.0	1.13417
$779$	−18188.2	−0.836536
$780$	0	0
$781$	−18585.0	−0.851503
$782$	−22903.7	−1.04736
$783$	0	0
$784$	0	0
$785$	6287.29	0.285864
$786$	0	0
$787$	22192.0	1.00516	0.502579	−	0.864531i	$-0.332385\pi$
$787$	22192.0	1.00516	0.502579	+	0.864531i	$0.332385\pi$
$788$	−31812.2	−1.43815
$789$	0	0
$790$	−3066.00	−0.138080
$791$	0	0
$792$	0	0
$793$	5632.00	0.252205
$794$	59399.3	2.65492
$795$	0	0
$796$	−46475.0	−2.06943
$797$	−7080.08	−0.314667	−0.157333	−	0.987546i	$-0.550290\pi$
$797$	−7080.08	−0.314667	−0.157333	+	0.987546i	$0.550290\pi$
$798$	0	0
$799$	−4116.00	−0.182245
$800$	−18586.9	−0.821434
$801$	0	0
$802$	45696.0	2.01195
$803$	6094.83	0.267848
$804$	0	0
$805$	0	0
$806$	−38512.0	−1.68303
$807$	0	0
$808$	28770.0	1.25263
$809$	14041.0	0.610205	0.305102	−	0.952320i	$-0.401309\pi$
$809$	14041.0	0.610205	0.305102	+	0.952320i	$0.401309\pi$
$810$	0	0
$811$	12355.0	0.534948	0.267474	−	0.963565i	$-0.413811\pi$
$811$	12355.0	0.534948	0.267474	+	0.963565i	$0.413811\pi$
$812$	0	0
$813$	0	0
$814$	3045.00	0.131114
$815$	12794.6	0.549906
$816$	0	0
$817$	18914.0	0.809935
$818$	−56081.6	−2.39712
$819$	0	0
$820$	−22113.0	−0.941731
$821$	−21712.2	−0.922975	−0.461488	−	0.887147i	$-0.652684\pi$
$821$	−21712.2	−0.922975	−0.461488	+	0.887147i	$0.652684\pi$
$822$	0	0
$823$	−6910.00	−0.292670	−0.146335	−	0.989235i	$-0.546748\pi$
$823$	−6910.00	−0.292670	−0.146335	+	0.989235i	$0.546748\pi$
$824$	47269.3	1.99843
$825$	0	0
$826$	0	0
$827$	−40597.0	−1.70701	−0.853505	−	0.521085i	$-0.825528\pi$
$827$	−40597.0	−1.70701	−0.853505	+	0.521085i	$0.825528\pi$
$828$	0	0
$829$	5530.00	0.231683	0.115841	−	0.993268i	$-0.463044\pi$
$829$	5530.00	0.231683	0.115841	+	0.993268i	$0.463044\pi$
$830$	−30025.0	−1.25564
$831$	0	0
$832$	36388.0	1.51626
$833$	0	0
$834$	0	0
$835$	1890.00	0.0783307
$836$	−14595.5	−0.603823
$837$	0	0
$838$	49770.0	2.05164
$839$	7918.69	0.325845	0.162922	−	0.986639i	$-0.447908\pi$
$839$	7918.69	0.325845	0.162922	+	0.986639i	$0.447908\pi$
$840$	0	0
$841$	23995.0	0.983845
$842$	28572.4	1.16944
$843$	0	0
$844$	−1534.00	−0.0625622
$845$	−1196.05	−0.0486928
$846$	0	0
$847$	0	0
$848$	128.312	0.00519606
$849$	0	0
$850$	30576.0	1.23382
$851$	2259.21	0.0910044
$852$	0	0
$853$	21112.0	0.847434	0.423717	−	0.905795i	$-0.360725\pi$
$853$	21112.0	0.847434	0.423717	+	0.905795i	$0.360725\pi$
$854$	0	0
$855$	0	0
$856$	1680.00	0.0670809
$857$	39313.9	1.56702	0.783511	−	0.621378i	$-0.213427\pi$
$857$	39313.9	1.56702	0.783511	+	0.621378i	$0.213427\pi$
$858$	0	0
$859$	−31295.0	−1.24304	−0.621520	−	0.783398i	$-0.713485\pi$
$859$	−31295.0	−1.24304	−0.621520	+	0.783398i	$0.713485\pi$
$860$	22995.4	0.911785
$861$	0	0
$862$	−33915.0	−1.34008
$863$	31216.5	1.23131	0.615656	−	0.788015i	$-0.288891\pi$
$863$	31216.5	1.23131	0.615656	+	0.788015i	$0.288891\pi$
$864$	0	0
$865$	10983.0	0.431715
$866$	−20090.0	−0.788321
$867$	0	0
$868$	0	0
$869$	−3345.28	−0.130588
$870$	0	0
$871$	43472.0	1.69115
$872$	−49743.9	−1.93181
$873$	0	0
$874$	−17493.0	−0.677013
$875$	0	0
$876$	0	0
$877$	−106.000	−0.00408137	−0.00204069	−	0.999998i	$-0.500650\pi$
$877$	−106.000	−0.00408137	−0.00204069	+	0.999998i	$0.500650\pi$
$878$	24892.6	0.956814
$879$	0	0
$880$	−105.000	−0.00402222
$881$	20277.9	0.775459	0.387730	−	0.921773i	$-0.373259\pi$
$881$	20277.9	0.775459	0.387730	+	0.921773i	$0.373259\pi$
$882$	0	0
$883$	−37816.0	−1.44123	−0.720617	−	0.693333i	$-0.756141\pi$
$883$	−37816.0	−1.44123	−0.720617	+	0.693333i	$0.756141\pi$
$884$	−36697.3	−1.39622
$885$	0	0
$886$	8463.00	0.320903
$887$	−48758.6	−1.84572	−0.922860	−	0.385135i	$-0.874155\pi$
$887$	−48758.6	−1.84572	−0.922860	+	0.385135i	$0.874155\pi$
$888$	0	0
$889$	0	0
$890$	−16841.0	−0.634281
$891$	0	0
$892$	6643.00	0.249354
$893$	−3143.65	−0.117803
$894$	0	0
$895$	−3444.00	−0.128626
$896$	0	0
$897$	0	0
$898$	−70182.0	−2.60802
$899$	−42013.1	−1.55864
$900$	0	0
$901$	8232.00	0.304381
$902$	−38974.8	−1.43871
$903$	0	0
$904$	−27090.0	−0.996681
$905$	320.780	0.0117824
$906$	0	0
$907$	24158.0	0.884403	0.442201	−	0.896916i	$-0.354198\pi$
$907$	24158.0	0.884403	0.442201	+	0.896916i	$0.354198\pi$
$908$	−73513.7	−2.68683
$909$	0	0
$910$	0	0
$911$	−13142.8	−0.477982	−0.238991	−	0.971022i	$-0.576817\pi$
$911$	−13142.8	−0.477982	−0.238991	+	0.971022i	$0.576817\pi$
$912$	0	0
$913$	−32760.0	−1.18751
$914$	4486.34	0.162358
$915$	0	0
$916$	27820.0	1.00349
$917$	0	0
$918$	0	0
$919$	22772.0	0.817387	0.408694	−	0.912672i	$-0.365984\pi$
$919$	22772.0	0.817387	0.408694	+	0.912672i	$0.365984\pi$
$920$	−8179.90	−0.293134
$921$	0	0
$922$	−31311.0	−1.11841
$923$	−35689.1	−1.27272
$924$	0	0
$925$	−3016.00	−0.107206
$926$	46183.2	1.63896
$927$	0	0
$928$	39312.0	1.39060
$929$	18852.7	0.665810	0.332905	−	0.942960i	$-0.391971\pi$
$929$	18852.7	0.665810	0.332905	+	0.942960i	$0.391971\pi$
$930$	0	0
$931$	0	0
$932$	−67079.7	−2.35759
$933$	0	0
$934$	−65226.0	−2.28507
$935$	−6736.39	−0.235619
$936$	0	0
$937$	49492.0	1.72554	0.862771	−	0.505595i	$-0.168727\pi$
$937$	49492.0	1.72554	0.862771	+	0.505595i	$0.168727\pi$
$938$	0	0
$939$	0	0
$940$	−3822.00	−0.132617
$941$	11021.1	0.381804	0.190902	−	0.981609i	$-0.438859\pi$
$941$	11021.1	0.381804	0.190902	+	0.981609i	$0.438859\pi$
$942$	0	0
$943$	−28917.0	−0.998587
$944$	−705.717	−0.0243317
$945$	0	0
$946$	40530.0	1.39296
$947$	32687.5	1.12165	0.560824	−	0.827935i	$-0.310484\pi$
$947$	32687.5	1.12165	0.560824	+	0.827935i	$0.310484\pi$
$948$	0	0
$949$	11704.0	0.400346
$950$	23352.8	0.797542
$951$	0	0
$952$	0	0
$953$	−28925.2	−0.983190	−0.491595	−	0.870824i	$-0.663586\pi$
$953$	−28925.2	−0.983190	−0.491595	+	0.870824i	$0.663586\pi$
$954$	0	0
$955$	−10857.0	−0.367879
$956$	53616.1	1.81388
$957$	0	0
$958$	78456.0	2.64593
$959$	0	0
$960$	0	0
$961$	6690.00	0.224564
$962$	5847.37	0.195974
$963$	0	0
$964$	19630.0	0.655850
$965$	−4775.04	−0.159289
$966$	0	0
$967$	−39502.0	−1.31365	−0.656825	−	0.754043i	$-0.728101\pi$
$967$	−39502.0	−1.31365	−0.656825	+	0.754043i	$0.728101\pi$
$968$	18467.8	0.613199
$969$	0	0
$970$	3192.00	0.105659
$971$	33837.7	1.11834	0.559168	−	0.829054i	$-0.311121\pi$
$971$	33837.7	1.11834	0.559168	+	0.829054i	$0.311121\pi$
$972$	0	0
$973$	0	0
$974$	−2043.83	−0.0672367
$975$	0	0
$976$	−128.000	−0.00419793
$977$	31693.1	1.03782	0.518911	−	0.854828i	$-0.326338\pi$
$977$	31693.1	1.03782	0.518911	+	0.854828i	$0.326338\pi$
$978$	0	0
$979$	−18375.0	−0.599865
$980$	0	0
$981$	0	0
$982$	−3591.00	−0.116694
$983$	17661.2	0.573048	0.286524	−	0.958073i	$-0.407500\pi$
$983$	17661.2	0.573048	0.286524	+	0.958073i	$0.407500\pi$
$984$	0	0
$985$	−11214.0	−0.362749
$986$	−64669.3	−2.08873
$987$	0	0
$988$	−28028.0	−0.902520
$989$	30070.9	0.966833
$990$	0	0
$991$	14456.0	0.463380	0.231690	−	0.972790i	$-0.425574\pi$
$991$	14456.0	0.463380	0.231690	+	0.972790i	$0.425574\pi$
$992$	−34135.6	−1.09255
$993$	0	0
$994$	0	0
$995$	−16382.7	−0.521977
$996$	0	0
$997$	35170.0	1.11720	0.558598	−	0.829438i	$-0.311339\pi$
$997$	35170.0	1.11720	0.558598	+	0.829438i	$0.311339\pi$
$998$	−76061.6	−2.41251
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.u.1.1		2
3.2	odd	2	inner	1323.4.a.u.1.2		2
7.6	odd	2		189.4.a.h.1.1	✓	2
21.20	even	2		189.4.a.h.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.h.1.1	✓	2	7.6	odd	2
189.4.a.h.1.2	yes	2	21.20	even	2
1323.4.a.u.1.1		2	1.1	even	1	trivial
1323.4.a.u.1.2		2	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-4.58258 q^{2} +13.0000 q^{4} +4.58258 q^{5} -22.9129 q^{8} +O(q^{10})\)	\(q-4.58258 q^{2} +13.0000 q^{4} +4.58258 q^{5} -22.9129 q^{8} -21.0000 q^{10} -22.9129 q^{11} -44.0000 q^{13} +1.00000 q^{16} +64.1561 q^{17} +49.0000 q^{19} +59.5735 q^{20} +105.000 q^{22} +77.9038 q^{23} -104.000 q^{25} +201.633 q^{26} +219.964 q^{29} -191.000 q^{31} +178.720 q^{32} -294.000 q^{34} +29.0000 q^{37} -224.546 q^{38} -105.000 q^{40} -371.189 q^{41} +386.000 q^{43} -297.867 q^{44} -357.000 q^{46} -64.1561 q^{47} +476.588 q^{50} -572.000 q^{52} +128.312 q^{53} -105.000 q^{55} -1008.00 q^{58} -705.717 q^{59} -128.000 q^{61} +875.272 q^{62} -827.000 q^{64} -201.633 q^{65} -988.000 q^{67} +834.029 q^{68} +811.116 q^{71} -266.000 q^{73} -132.895 q^{74} +637.000 q^{76} +146.000 q^{79} +4.58258 q^{80} +1701.00 q^{82} +1429.76 q^{83} +294.000 q^{85} -1768.87 q^{86} +525.000 q^{88} +801.951 q^{89} +1012.75 q^{92} +294.000 q^{94} +224.546 q^{95} -152.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 26 q^{4}+O(q^{10}) \) 2 * q + 26 * q^4	\( 2 q + 26 q^{4} - 42 q^{10} - 88 q^{13} + 2 q^{16} + 98 q^{19} + 210 q^{22} - 208 q^{25} - 382 q^{31} - 588 q^{34} + 58 q^{37} - 210 q^{40} + 772 q^{43} - 714 q^{46} - 1144 q^{52} - 210 q^{55} - 2016 q^{58} - 256 q^{61} - 1654 q^{64} - 1976 q^{67} - 532 q^{73} + 1274 q^{76} + 292 q^{79} + 3402 q^{82} + 588 q^{85} + 1050 q^{88} + 588 q^{94} - 304 q^{97}+O(q^{100}) \) 2 * q + 26 * q^4 - 42 * q^10 - 88 * q^13 + 2 * q^16 + 98 * q^19 + 210 * q^22 - 208 * q^25 - 382 * q^31 - 588 * q^34 + 58 * q^37 - 210 * q^40 + 772 * q^43 - 714 * q^46 - 1144 * q^52 - 210 * q^55 - 2016 * q^58 - 256 * q^61 - 1654 * q^64 - 1976 * q^67 - 532 * q^73 + 1274 * q^76 + 292 * q^79 + 3402 * q^82 + 588 * q^85 + 1050 * q^88 + 588 * q^94 - 304 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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