LMFDB - Embedded newform 1323.4.a.bn.1.2

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9 $ x^8 - 54x^6 - 6x^5 + 555x^4 + 642x^3 - 218x^2 - 54x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.07765$ 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)

$f(q)$	$=$	$q-3.49236 q^{2} +4.19658 q^{4} -7.87531 q^{5} +13.2829 q^{8} +27.5034 q^{10} -54.0035 q^{11} +48.9137 q^{13} -79.9614 q^{16} +96.9775 q^{17} +142.327 q^{19} -33.0494 q^{20} +188.600 q^{22} -103.787 q^{23} -62.9795 q^{25} -170.824 q^{26} -24.5329 q^{29} -187.379 q^{31} +172.991 q^{32} -338.680 q^{34} +146.555 q^{37} -497.058 q^{38} -104.607 q^{40} +314.180 q^{41} +173.165 q^{43} -226.630 q^{44} +362.462 q^{46} -259.235 q^{47} +219.947 q^{50} +205.270 q^{52} -620.858 q^{53} +425.295 q^{55} +85.6778 q^{58} -443.110 q^{59} -113.296 q^{61} +654.396 q^{62} +35.5454 q^{64} -385.210 q^{65} +628.975 q^{67} +406.974 q^{68} -41.3042 q^{71} -447.111 q^{73} -511.822 q^{74} +597.287 q^{76} +434.706 q^{79} +629.720 q^{80} -1097.23 q^{82} -329.158 q^{83} -763.728 q^{85} -604.756 q^{86} -717.324 q^{88} -24.8709 q^{89} -435.551 q^{92} +905.340 q^{94} -1120.87 q^{95} +499.239 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 48 q^{4} - 44 q^{10} + 84 q^{13} + 156 q^{16} - 12 q^{19} + 224 q^{22} + 408 q^{25} - 800 q^{31} + 948 q^{34} + 692 q^{37} - 96 q^{40} + 1456 q^{43} + 1524 q^{46} - 1972 q^{52} + 1280 q^{55} + 2372 q^{58}+ \cdots - 584 q^{97}+O(q^{100}) $ 8 * q + 48 * q^4 - 44 * q^10 + 84 * q^13 + 156 * q^16 - 12 * q^19 + 224 * q^22 + 408 * q^25 - 800 * q^31 + 948 * q^34 + 692 * q^37 - 96 * q^40 + 1456 * q^43 + 1524 * q^46 - 1972 * q^52 + 1280 * q^55 + 2372 * q^58 - 216 * q^61 + 4964 * q^64 + 684 * q^67 + 4564 * q^73 + 380 * q^76 + 556 * q^79 - 3340 * q^82 + 1296 * q^85 + 6696 * q^88 + 492 * q^94 - 584 * q^97

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 $

$2$	−3.49236	−1.23474	−0.617368	−	0.786675i	$-0.711801\pi$
$2$	−3.49236	−1.23474	−0.617368	+	0.786675i	$0.711801\pi$
$3$	0	0
$3$	0	0
$4$	4.19658	0.524573
$5$	−7.87531	−0.704389	−0.352195	−	0.935927i	$-0.614564\pi$
$5$	−7.87531	−0.704389	−0.352195	+	0.935927i	$0.614564\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	13.2829	0.587027
$9$	0	0
$10$	27.5034	0.869735
$11$	−54.0035	−1.48024	−0.740122	−	0.672473i	$-0.765232\pi$
$11$	−54.0035	−1.48024	−0.740122	+	0.672473i	$0.765232\pi$
$12$	0	0
$13$	48.9137	1.04355	0.521777	−	0.853082i	$-0.325269\pi$
$13$	48.9137	1.04355	0.521777	+	0.853082i	$0.325269\pi$
$14$	0	0
$15$	0	0
$16$	−79.9614	−1.24940
$17$	96.9775	1.38356	0.691780	−	0.722109i	$-0.256827\pi$
$17$	96.9775	1.38356	0.691780	+	0.722109i	$0.256827\pi$
$18$	0	0
$19$	142.327	1.71853	0.859266	−	0.511530i	$-0.170921\pi$
$19$	142.327	1.71853	0.859266	+	0.511530i	$0.170921\pi$
$20$	−33.0494	−0.369503
$21$	0	0
$22$	188.600	1.82771
$23$	−103.787	−0.940918	−0.470459	−	0.882422i	$-0.655912\pi$
$23$	−103.787	−0.940918	−0.470459	+	0.882422i	$0.655912\pi$
$24$	0	0
$25$	−62.9795	−0.503836
$26$	−170.824	−1.28851
$27$	0	0
$28$	0	0
$29$	−24.5329	−0.157091	−0.0785456	−	0.996911i	$-0.525028\pi$
$29$	−24.5329	−0.157091	−0.0785456	+	0.996911i	$0.525028\pi$
$30$	0	0
$31$	−187.379	−1.08562	−0.542812	−	0.839855i	$-0.682640\pi$
$31$	−187.379	−1.08562	−0.542812	+	0.839855i	$0.682640\pi$
$32$	172.991	0.955647
$33$	0	0
$34$	−338.680	−1.70833
$35$	0	0
$36$	0	0
$37$	146.555	0.651174	0.325587	−	0.945512i	$-0.394438\pi$
$37$	146.555	0.651174	0.325587	+	0.945512i	$0.394438\pi$
$38$	−497.058	−2.12193
$39$	0	0
$40$	−104.607	−0.413496
$41$	314.180	1.19675	0.598374	−	0.801217i	$-0.295814\pi$
$41$	314.180	1.19675	0.598374	+	0.801217i	$0.295814\pi$
$42$	0	0
$43$	173.165	0.614127	0.307064	−	0.951689i	$-0.400654\pi$
$43$	173.165	0.614127	0.307064	+	0.951689i	$0.400654\pi$
$44$	−226.630	−0.776495
$45$	0	0
$46$	362.462	1.16179
$47$	−259.235	−0.804537	−0.402269	−	0.915522i	$-0.631778\pi$
$47$	−259.235	−0.804537	−0.402269	+	0.915522i	$0.631778\pi$
$48$	0	0
$49$	0	0
$50$	219.947	0.622104
$51$	0	0
$52$	205.270	0.547420
$53$	−620.858	−1.60908	−0.804541	−	0.593896i	$-0.797589\pi$
$53$	−620.858	−1.60908	−0.804541	+	0.593896i	$0.797589\pi$
$54$	0	0
$55$	425.295	1.04267
$56$	0	0
$57$	0	0
$58$	85.6778	0.193966
$59$	−443.110	−0.977764	−0.488882	−	0.872350i	$-0.662595\pi$
$59$	−443.110	−0.977764	−0.488882	+	0.872350i	$0.662595\pi$
$60$	0	0
$61$	−113.296	−0.237803	−0.118902	−	0.992906i	$-0.537937\pi$
$61$	−113.296	−0.237803	−0.118902	+	0.992906i	$0.537937\pi$
$62$	654.396	1.34046
$63$	0	0
$64$	35.5454	0.0694245
$65$	−385.210	−0.735069
$66$	0	0
$67$	628.975	1.14689	0.573444	−	0.819245i	$-0.305607\pi$
$67$	628.975	1.14689	0.573444	+	0.819245i	$0.305607\pi$
$68$	406.974	0.725777
$69$	0	0
$70$	0	0
$71$	−41.3042	−0.0690410	−0.0345205	−	0.999404i	$-0.510990\pi$
$71$	−41.3042	−0.0690410	−0.0345205	+	0.999404i	$0.510990\pi$
$72$	0	0
$73$	−447.111	−0.716854	−0.358427	−	0.933558i	$-0.616687\pi$
$73$	−447.111	−0.716854	−0.358427	+	0.933558i	$0.616687\pi$
$74$	−511.822	−0.804028
$75$	0	0
$76$	597.287	0.901494
$77$	0	0
$78$	0	0
$79$	434.706	0.619092	0.309546	−	0.950884i	$-0.399823\pi$
$79$	434.706	0.619092	0.309546	+	0.950884i	$0.399823\pi$
$80$	629.720	0.880061
$81$	0	0
$82$	−1097.23	−1.47767
$83$	−329.158	−0.435299	−0.217649	−	0.976027i	$-0.569839\pi$
$83$	−329.158	−0.435299	−0.217649	+	0.976027i	$0.569839\pi$
$84$	0	0
$85$	−763.728	−0.974564
$86$	−604.756	−0.758285
$87$	0	0
$88$	−717.324	−0.868943
$89$	−24.8709	−0.0296214	−0.0148107	−	0.999890i	$-0.504715\pi$
$89$	−24.8709	−0.0296214	−0.0148107	+	0.999890i	$0.504715\pi$
$90$	0	0
$91$	0	0
$92$	−435.551	−0.493580
$93$	0	0
$94$	905.340	0.993391
$95$	−1120.87	−1.21051
$96$	0	0
$97$	499.239	0.522578	0.261289	−	0.965261i	$-0.415852\pi$
$97$	499.239	0.522578	0.261289	+	0.965261i	$0.415852\pi$
$98$	0	0
$99$	0	0
$100$	−264.299	−0.264299
$101$	−258.076	−0.254253	−0.127127	−	0.991887i	$-0.540575\pi$
$101$	−258.076	−0.254253	−0.127127	+	0.991887i	$0.540575\pi$
$102$	0	0
$103$	−960.317	−0.918669	−0.459334	−	0.888263i	$-0.651912\pi$
$103$	−960.317	−0.918669	−0.459334	+	0.888263i	$0.651912\pi$
$104$	649.716	0.612595
$105$	0	0
$106$	2168.26	1.98679
$107$	−2054.26	−1.85601	−0.928004	−	0.372570i	$-0.878477\pi$
$107$	−2054.26	−1.85601	−0.928004	+	0.372570i	$0.878477\pi$
$108$	0	0
$109$	1784.10	1.56776	0.783881	−	0.620911i	$-0.213237\pi$
$109$	1784.10	1.56776	0.783881	+	0.620911i	$0.213237\pi$
$110$	−1485.28	−1.28742
$111$	0	0
$112$	0	0
$113$	−1296.36	−1.07921	−0.539607	−	0.841917i	$-0.681427\pi$
$113$	−1296.36	−1.07921	−0.539607	+	0.841917i	$0.681427\pi$
$114$	0	0
$115$	817.356	0.662773
$116$	−102.954	−0.0824058
$117$	0	0
$118$	1547.50	1.20728
$119$	0	0
$120$	0	0
$121$	1585.38	1.19112
$122$	395.669	0.293624
$123$	0	0
$124$	−786.352	−0.569488
$125$	1480.40	1.05929
$126$	0	0
$127$	−312.058	−0.218037	−0.109018	−	0.994040i	$-0.534771\pi$
$127$	−312.058	−0.218037	−0.109018	+	0.994040i	$0.534771\pi$
$128$	−1508.06	−1.04137
$129$	0	0
$130$	1345.29	0.907616
$131$	−83.2704	−0.0555372	−0.0277686	−	0.999614i	$-0.508840\pi$
$131$	−83.2704	−0.0555372	−0.0277686	+	0.999614i	$0.508840\pi$
$132$	0	0
$133$	0	0
$134$	−2196.61	−1.41610
$135$	0	0
$136$	1288.14	0.812187
$137$	2909.58	1.81447	0.907233	−	0.420628i	$-0.138190\pi$
$137$	2909.58	1.81447	0.907233	+	0.420628i	$0.138190\pi$
$138$	0	0
$139$	−1414.91	−0.863392	−0.431696	−	0.902019i	$-0.642085\pi$
$139$	−1414.91	−0.863392	−0.431696	+	0.902019i	$0.642085\pi$
$140$	0	0
$141$	0	0
$142$	144.249	0.0852474
$143$	−2641.51	−1.54472
$144$	0	0
$145$	193.204	0.110653
$146$	1561.47	0.885125
$147$	0	0
$148$	615.029	0.341588
$149$	2186.63	1.20225	0.601125	−	0.799155i	$-0.294719\pi$
$149$	2186.63	1.20225	0.601125	+	0.799155i	$0.294719\pi$
$150$	0	0
$151$	1639.26	0.883450	0.441725	−	0.897151i	$-0.354367\pi$
$151$	1639.26	0.883450	0.441725	+	0.897151i	$0.354367\pi$
$152$	1890.52	1.00882
$153$	0	0
$154$	0	0
$155$	1475.67	0.764701
$156$	0	0
$157$	−1258.98	−0.639986	−0.319993	−	0.947420i	$-0.603680\pi$
$157$	−1258.98	−0.639986	−0.319993	+	0.947420i	$0.603680\pi$
$158$	−1518.15	−0.764415
$159$	0	0
$160$	−1362.35	−0.673147
$161$	0	0
$162$	0	0
$163$	−3159.01	−1.51799	−0.758995	−	0.651096i	$-0.774309\pi$
$163$	−3159.01	−1.51799	−0.758995	+	0.651096i	$0.774309\pi$
$164$	1318.48	0.627782
$165$	0	0
$166$	1149.54	0.537479
$167$	1848.90	0.856718	0.428359	−	0.903609i	$-0.359092\pi$
$167$	1848.90	0.856718	0.428359	+	0.903609i	$0.359092\pi$
$168$	0	0
$169$	195.548	0.0890068
$170$	2667.21	1.20333
$171$	0	0
$172$	726.703	0.322154
$173$	623.157	0.273860	0.136930	−	0.990581i	$-0.456277\pi$
$173$	623.157	0.273860	0.136930	+	0.990581i	$0.456277\pi$
$174$	0	0
$175$	0	0
$176$	4318.20	1.84941
$177$	0	0
$178$	86.8580	0.0365746
$179$	−2340.66	−0.977370	−0.488685	−	0.872460i	$-0.662523\pi$
$179$	−2340.66	−0.977370	−0.488685	+	0.872460i	$0.662523\pi$
$180$	0	0
$181$	461.896	0.189682	0.0948411	−	0.995492i	$-0.469766\pi$
$181$	461.896	0.189682	0.0948411	+	0.995492i	$0.469766\pi$
$182$	0	0
$183$	0	0
$184$	−1378.60	−0.552345
$185$	−1154.16	−0.458680
$186$	0	0
$187$	−5237.13	−2.04800
$188$	−1087.90	−0.422038
$189$	0	0
$190$	3914.48	1.49467
$191$	−2665.79	−1.00990	−0.504948	−	0.863150i	$-0.668488\pi$
$191$	−2665.79	−1.00990	−0.504948	+	0.863150i	$0.668488\pi$
$192$	0	0
$193$	130.569	0.0486972	0.0243486	−	0.999704i	$-0.492249\pi$
$193$	130.569	0.0486972	0.0243486	+	0.999704i	$0.492249\pi$
$194$	−1743.52	−0.645245
$195$	0	0
$196$	0	0
$197$	3729.51	1.34882	0.674408	−	0.738359i	$-0.264399\pi$
$197$	3729.51	1.34882	0.674408	+	0.738359i	$0.264399\pi$
$198$	0	0
$199$	3772.75	1.34394	0.671968	−	0.740580i	$-0.265449\pi$
$199$	3772.75	1.34394	0.671968	+	0.740580i	$0.265449\pi$
$200$	−836.551	−0.295765
$201$	0	0
$202$	901.296	0.313935
$203$	0	0
$204$	0	0
$205$	−2474.27	−0.842977
$206$	3353.77	1.13431
$207$	0	0
$208$	−3911.20	−1.30381
$209$	−7686.17	−2.54384
$210$	0	0
$211$	2398.28	0.782484	0.391242	−	0.920288i	$-0.372046\pi$
$211$	2398.28	0.782484	0.391242	+	0.920288i	$0.372046\pi$
$212$	−2605.48	−0.844081
$213$	0	0
$214$	7174.22	2.29168
$215$	−1363.73	−0.432585
$216$	0	0
$217$	0	0
$218$	−6230.73	−1.93577
$219$	0	0
$220$	1784.78	0.546955
$221$	4743.53	1.44382
$222$	0	0
$223$	3338.23	1.00244	0.501220	−	0.865320i	$-0.332885\pi$
$223$	3338.23	1.00244	0.501220	+	0.865320i	$0.332885\pi$
$224$	0	0
$225$	0	0
$226$	4527.35	1.33254
$227$	−1269.04	−0.371054	−0.185527	−	0.982639i	$-0.559399\pi$
$227$	−1269.04	−0.371054	−0.185527	+	0.982639i	$0.559399\pi$
$228$	0	0
$229$	2051.48	0.591990	0.295995	−	0.955190i	$-0.404349\pi$
$229$	2051.48	0.591990	0.295995	+	0.955190i	$0.404349\pi$
$230$	−2854.50	−0.818349
$231$	0	0
$232$	−325.868	−0.0922169
$233$	4950.44	1.39191	0.695953	−	0.718088i	$-0.254982\pi$
$233$	4950.44	1.39191	0.695953	+	0.718088i	$0.254982\pi$
$234$	0	0
$235$	2041.55	0.566707
$236$	−1859.55	−0.512908
$237$	0	0
$238$	0	0
$239$	262.671	0.0710910	0.0355455	−	0.999368i	$-0.488683\pi$
$239$	262.671	0.0710910	0.0355455	+	0.999368i	$0.488683\pi$
$240$	0	0
$241$	−2902.37	−0.775761	−0.387880	−	0.921710i	$-0.626793\pi$
$241$	−2902.37	−0.775761	−0.387880	+	0.921710i	$0.626793\pi$
$242$	−5536.73	−1.47072
$243$	0	0
$244$	−475.454	−0.124745
$245$	0	0
$246$	0	0
$247$	6961.75	1.79338
$248$	−2488.94	−0.637290
$249$	0	0
$250$	−5170.08	−1.30794
$251$	6265.68	1.57564	0.787821	−	0.615904i	$-0.211209\pi$
$251$	6265.68	1.57564	0.787821	+	0.615904i	$0.211209\pi$
$252$	0	0
$253$	5604.87	1.39279
$254$	1089.82	0.269218
$255$	0	0
$256$	4982.33	1.21639
$257$	3420.96	0.830325	0.415163	−	0.909747i	$-0.363725\pi$
$257$	3420.96	0.830325	0.415163	+	0.909747i	$0.363725\pi$
$258$	0	0
$259$	0	0
$260$	−1616.57	−0.385597
$261$	0	0
$262$	290.810	0.0685738
$263$	8222.52	1.92784	0.963921	−	0.266188i	$-0.0857642\pi$
$263$	8222.52	1.92784	0.963921	+	0.266188i	$0.0857642\pi$
$264$	0	0
$265$	4889.45	1.13342
$266$	0	0
$267$	0	0
$268$	2639.54	0.601626
$269$	5493.33	1.24511	0.622555	−	0.782576i	$-0.286095\pi$
$269$	5493.33	1.24511	0.622555	+	0.782576i	$0.286095\pi$
$270$	0	0
$271$	1096.48	0.245781	0.122891	−	0.992420i	$-0.460784\pi$
$271$	1096.48	0.245781	0.122891	+	0.992420i	$0.460784\pi$
$272$	−7754.45	−1.72861
$273$	0	0
$274$	−10161.3	−2.24039
$275$	3401.12	0.745800
$276$	0	0
$277$	−1555.47	−0.337398	−0.168699	−	0.985668i	$-0.553957\pi$
$277$	−1555.47	−0.337398	−0.168699	+	0.985668i	$0.553957\pi$
$278$	4941.39	1.06606
$279$	0	0
$280$	0	0
$281$	995.950	0.211435	0.105718	−	0.994396i	$-0.466286\pi$
$281$	995.950	0.211435	0.105718	+	0.994396i	$0.466286\pi$
$282$	0	0
$283$	8635.47	1.81387	0.906936	−	0.421269i	$-0.138415\pi$
$283$	8635.47	1.81387	0.906936	+	0.421269i	$0.138415\pi$
$284$	−173.337	−0.0362170
$285$	0	0
$286$	9225.11	1.90732
$287$	0	0
$288$	0	0
$289$	4491.64	0.914236
$290$	−674.739	−0.136628
$291$	0	0
$292$	−1876.34	−0.376042
$293$	−2180.01	−0.434667	−0.217333	−	0.976097i	$-0.569736\pi$
$293$	−2180.01	−0.434667	−0.217333	+	0.976097i	$0.569736\pi$
$294$	0	0
$295$	3489.63	0.688726
$296$	1946.67	0.382257
$297$	0	0
$298$	−7636.49	−1.48446
$299$	−5076.61	−0.981900
$300$	0	0
$301$	0	0
$302$	−5724.88	−1.09083
$303$	0	0
$304$	−11380.7	−2.14713
$305$	892.238	0.167506
$306$	0	0
$307$	−5281.18	−0.981800	−0.490900	−	0.871216i	$-0.663332\pi$
$307$	−5281.18	−0.981800	−0.490900	+	0.871216i	$0.663332\pi$
$308$	0	0
$309$	0	0
$310$	−5153.57	−0.944204
$311$	4297.47	0.783561	0.391780	−	0.920059i	$-0.371859\pi$
$311$	4297.47	0.783561	0.391780	+	0.920059i	$0.371859\pi$
$312$	0	0
$313$	2360.96	0.426356	0.213178	−	0.977013i	$-0.431619\pi$
$313$	2360.96	0.426356	0.213178	+	0.977013i	$0.431619\pi$
$314$	4396.82	0.790213
$315$	0	0
$316$	1824.28	0.324759
$317$	9092.66	1.61102	0.805512	−	0.592580i	$-0.201891\pi$
$317$	9092.66	1.61102	0.805512	+	0.592580i	$0.201891\pi$
$318$	0	0
$319$	1324.86	0.232533
$320$	−279.931	−0.0489019
$321$	0	0
$322$	0	0
$323$	13802.5	2.37769
$324$	0	0
$325$	−3080.56	−0.525780
$326$	11032.4	1.87432
$327$	0	0
$328$	4173.23	0.702524
$329$	0	0
$330$	0	0
$331$	4379.01	0.727167	0.363584	−	0.931562i	$-0.381553\pi$
$331$	4379.01	0.727167	0.363584	+	0.931562i	$0.381553\pi$
$332$	−1381.34	−0.228346
$333$	0	0
$334$	−6457.02	−1.05782
$335$	−4953.37	−0.807856
$336$	0	0
$337$	3314.59	0.535778	0.267889	−	0.963450i	$-0.413674\pi$
$337$	3314.59	0.535778	0.267889	+	0.963450i	$0.413674\pi$
$338$	−682.924	−0.109900
$339$	0	0
$340$	−3205.05	−0.511230
$341$	10119.1	1.60699
$342$	0	0
$343$	0	0
$344$	2300.14	0.360510
$345$	0	0
$346$	−2176.29	−0.338144
$347$	−8553.87	−1.32333	−0.661666	−	0.749799i	$-0.730150\pi$
$347$	−8553.87	−1.32333	−0.661666	+	0.749799i	$0.730150\pi$
$348$	0	0
$349$	−9832.96	−1.50816	−0.754078	−	0.656785i	$-0.771916\pi$
$349$	−9832.96	−1.50816	−0.754078	+	0.656785i	$0.771916\pi$
$350$	0	0
$351$	0	0
$352$	−9342.10	−1.41459
$353$	−6171.41	−0.930513	−0.465257	−	0.885176i	$-0.654038\pi$
$353$	−6171.41	−0.930513	−0.465257	+	0.885176i	$0.654038\pi$
$354$	0	0
$355$	325.284	0.0486317
$356$	−104.373	−0.0155386
$357$	0	0
$358$	8174.43	1.20679
$359$	9547.02	1.40354	0.701772	−	0.712401i	$-0.252392\pi$
$359$	9547.02	1.40354	0.701772	+	0.712401i	$0.252392\pi$
$360$	0	0
$361$	13398.0	1.95335
$362$	−1613.11	−0.234207
$363$	0	0
$364$	0	0
$365$	3521.14	0.504944
$366$	0	0
$367$	4028.68	0.573012	0.286506	−	0.958078i	$-0.407506\pi$
$367$	4028.68	0.573012	0.286506	+	0.958078i	$0.407506\pi$
$368$	8298.96	1.17558
$369$	0	0
$370$	4030.76	0.566349
$371$	0	0
$372$	0	0
$373$	20.0509	0.00278336	0.00139168	−	0.999999i	$-0.499557\pi$
$373$	20.0509	0.00278336	0.00139168	+	0.999999i	$0.499557\pi$
$374$	18289.9	2.52874
$375$	0	0
$376$	−3443.39	−0.472285
$377$	−1199.99	−0.163933
$378$	0	0
$379$	−3787.79	−0.513366	−0.256683	−	0.966496i	$-0.582630\pi$
$379$	−3787.79	−0.513366	−0.256683	+	0.966496i	$0.582630\pi$
$380$	−4703.82	−0.635003
$381$	0	0
$382$	9309.92	1.24695
$383$	8001.72	1.06754	0.533771	−	0.845629i	$-0.320774\pi$
$383$	8001.72	1.06754	0.533771	+	0.845629i	$0.320774\pi$
$384$	0	0
$385$	0	0
$386$	−455.994	−0.0601282
$387$	0	0
$388$	2095.10	0.274130
$389$	−3484.64	−0.454186	−0.227093	−	0.973873i	$-0.572922\pi$
$389$	−3484.64	−0.454186	−0.227093	+	0.973873i	$0.572922\pi$
$390$	0	0
$391$	−10065.0	−1.30182
$392$	0	0
$393$	0	0
$394$	−13024.8	−1.66543
$395$	−3423.45	−0.436082
$396$	0	0
$397$	7037.76	0.889710	0.444855	−	0.895603i	$-0.353255\pi$
$397$	7037.76	0.889710	0.444855	+	0.895603i	$0.353255\pi$
$398$	−13175.8	−1.65941
$399$	0	0
$400$	5035.93	0.629491
$401$	8389.73	1.04480	0.522398	−	0.852702i	$-0.325038\pi$
$401$	8389.73	1.04480	0.522398	+	0.852702i	$0.325038\pi$
$402$	0	0
$403$	−9165.41	−1.13291
$404$	−1083.04	−0.133374
$405$	0	0
$406$	0	0
$407$	−7914.47	−0.963896
$408$	0	0
$409$	13545.5	1.63761	0.818804	−	0.574074i	$-0.194638\pi$
$409$	13545.5	1.63761	0.818804	+	0.574074i	$0.194638\pi$
$410$	8641.03	1.04085
$411$	0	0
$412$	−4030.05	−0.481908
$413$	0	0
$414$	0	0
$415$	2592.22	0.306620
$416$	8461.61	0.997270
$417$	0	0
$418$	26842.9	3.14098
$419$	1450.22	0.169089	0.0845443	−	0.996420i	$-0.473057\pi$
$419$	1450.22	0.169089	0.0845443	+	0.996420i	$0.473057\pi$
$420$	0	0
$421$	−923.118	−0.106865	−0.0534324	−	0.998571i	$-0.517016\pi$
$421$	−923.118	−0.106865	−0.0534324	+	0.998571i	$0.517016\pi$
$422$	−8375.64	−0.966161
$423$	0	0
$424$	−8246.80	−0.944575
$425$	−6107.60	−0.697087
$426$	0	0
$427$	0	0
$428$	−8620.87	−0.973611
$429$	0	0
$430$	4762.64	0.534128
$431$	1238.21	0.138382	0.0691908	−	0.997603i	$-0.477958\pi$
$431$	1238.21	0.138382	0.0691908	+	0.997603i	$0.477958\pi$
$432$	0	0
$433$	−10786.5	−1.19715	−0.598575	−	0.801067i	$-0.704266\pi$
$433$	−10786.5	−1.19715	−0.598575	+	0.801067i	$0.704266\pi$
$434$	0	0
$435$	0	0
$436$	7487.13	0.822405
$437$	−14771.7	−1.61700
$438$	0	0
$439$	−3353.49	−0.364586	−0.182293	−	0.983244i	$-0.558352\pi$
$439$	−3353.49	−0.364586	−0.182293	+	0.983244i	$0.558352\pi$
$440$	5649.15	0.612074
$441$	0	0
$442$	−16566.1	−1.78274
$443$	−8090.36	−0.867685	−0.433843	−	0.900989i	$-0.642843\pi$
$443$	−8090.36	−0.867685	−0.433843	+	0.900989i	$0.642843\pi$
$444$	0	0
$445$	195.866	0.0208650
$446$	−11658.3	−1.23775
$447$	0	0
$448$	0	0
$449$	−4211.19	−0.442625	−0.221312	−	0.975203i	$-0.571034\pi$
$449$	−4211.19	−0.442625	−0.221312	+	0.975203i	$0.571034\pi$
$450$	0	0
$451$	−16966.8	−1.77148
$452$	−5440.27	−0.566126
$453$	0	0
$454$	4431.95	0.458154
$455$	0	0
$456$	0	0
$457$	−9777.62	−1.00083	−0.500413	−	0.865787i	$-0.666819\pi$
$457$	−9777.62	−1.00083	−0.500413	+	0.865787i	$0.666819\pi$
$458$	−7164.51	−0.730951
$459$	0	0
$460$	3430.10	0.347672
$461$	12360.4	1.24876	0.624381	−	0.781120i	$-0.285351\pi$
$461$	12360.4	1.24876	0.624381	+	0.781120i	$0.285351\pi$
$462$	0	0
$463$	−16107.0	−1.61675	−0.808377	−	0.588666i	$-0.799653\pi$
$463$	−16107.0	−1.61675	−0.808377	+	0.588666i	$0.799653\pi$
$464$	1961.68	0.196269
$465$	0	0
$466$	−17288.7	−1.71864
$467$	13485.4	1.33625	0.668125	−	0.744049i	$-0.267097\pi$
$467$	13485.4	1.33625	0.668125	+	0.744049i	$0.267097\pi$
$468$	0	0
$469$	0	0
$470$	−7129.84	−0.699734
$471$	0	0
$472$	−5885.80	−0.573974
$473$	−9351.55	−0.909058
$474$	0	0
$475$	−8963.69	−0.865858
$476$	0	0
$477$	0	0
$478$	−917.340	−0.0877786
$479$	390.965	0.0372936	0.0186468	−	0.999826i	$-0.494064\pi$
$479$	390.965	0.0372936	0.0186468	+	0.999826i	$0.494064\pi$
$480$	0	0
$481$	7168.53	0.679536
$482$	10136.1	0.957860
$483$	0	0
$484$	6653.18	0.624830
$485$	−3931.66	−0.368098
$486$	0	0
$487$	−15791.5	−1.46937	−0.734683	−	0.678410i	$-0.762669\pi$
$487$	−15791.5	−1.46937	−0.734683	+	0.678410i	$0.762669\pi$
$488$	−1504.89	−0.139597
$489$	0	0
$490$	0	0
$491$	191.606	0.0176111	0.00880557	−	0.999961i	$-0.497197\pi$
$491$	191.606	0.0176111	0.00880557	+	0.999961i	$0.497197\pi$
$492$	0	0
$493$	−2379.14	−0.217345
$494$	−24312.9	−2.21435
$495$	0	0
$496$	14983.1	1.35637
$497$	0	0
$498$	0	0
$499$	5621.00	0.504270	0.252135	−	0.967692i	$-0.418867\pi$
$499$	5621.00	0.504270	0.252135	+	0.967692i	$0.418867\pi$
$500$	6212.61	0.555672
$501$	0	0
$502$	−21882.0	−1.94550
$503$	16936.3	1.50129	0.750647	−	0.660704i	$-0.229742\pi$
$503$	16936.3	1.50129	0.750647	+	0.660704i	$0.229742\pi$
$504$	0	0
$505$	2032.43	0.179093
$506$	−19574.2	−1.71973
$507$	0	0
$508$	−1309.58	−0.114376
$509$	4225.77	0.367984	0.183992	−	0.982928i	$-0.441098\pi$
$509$	4225.77	0.367984	0.183992	+	0.982928i	$0.441098\pi$
$510$	0	0
$511$	0	0
$512$	−5335.61	−0.460552
$513$	0	0
$514$	−11947.2	−1.02523
$515$	7562.80	0.647100
$516$	0	0
$517$	13999.6	1.19091
$518$	0	0
$519$	0	0
$520$	−5116.71	−0.431505
$521$	−4084.11	−0.343432	−0.171716	−	0.985146i	$-0.554931\pi$
$521$	−4084.11	−0.343432	−0.171716	+	0.985146i	$0.554931\pi$
$522$	0	0
$523$	−16971.4	−1.41894	−0.709471	−	0.704734i	$-0.751066\pi$
$523$	−16971.4	−1.41894	−0.709471	+	0.704734i	$0.751066\pi$
$524$	−349.451	−0.0291333
$525$	0	0
$526$	−28716.0	−2.38038
$527$	−18171.6	−1.50202
$528$	0	0
$529$	−1395.22	−0.114673
$530$	−17075.7	−1.39947
$531$	0	0
$532$	0	0
$533$	15367.7	1.24887
$534$	0	0
$535$	16177.9	1.30735
$536$	8354.62	0.673255
$537$	0	0
$538$	−19184.7	−1.53738
$539$	0	0
$540$	0	0
$541$	630.987	0.0501446	0.0250723	−	0.999686i	$-0.492018\pi$
$541$	630.987	0.0501446	0.0250723	+	0.999686i	$0.492018\pi$
$542$	−3829.32	−0.303475
$543$	0	0
$544$	16776.2	1.32219
$545$	−14050.4	−1.10431
$546$	0	0
$547$	14377.3	1.12382	0.561910	−	0.827198i	$-0.310067\pi$
$547$	14377.3	1.12382	0.561910	+	0.827198i	$0.310067\pi$
$548$	12210.3	0.951819
$549$	0	0
$550$	−11877.9	−0.920866
$551$	−3491.70	−0.269966
$552$	0	0
$553$	0	0
$554$	5432.27	0.416598
$555$	0	0
$556$	−5937.80	−0.452912
$557$	−12749.6	−0.969867	−0.484934	−	0.874551i	$-0.661156\pi$
$557$	−12749.6	−0.969867	−0.484934	+	0.874551i	$0.661156\pi$
$558$	0	0
$559$	8470.16	0.640876
$560$	0	0
$561$	0	0
$562$	−3478.21	−0.261067
$563$	−14080.0	−1.05400	−0.527001	−	0.849865i	$-0.676684\pi$
$563$	−14080.0	−1.05400	−0.527001	+	0.849865i	$0.676684\pi$
$564$	0	0
$565$	10209.2	0.760187
$566$	−30158.2	−2.23965
$567$	0	0
$568$	−548.640	−0.0405289
$569$	−977.908	−0.0720493	−0.0360246	−	0.999351i	$-0.511469\pi$
$569$	−977.908	−0.0720493	−0.0360246	+	0.999351i	$0.511469\pi$
$570$	0	0
$571$	−4877.53	−0.357475	−0.178737	−	0.983897i	$-0.557201\pi$
$571$	−4877.53	−0.357475	−0.178737	+	0.983897i	$0.557201\pi$
$572$	−11085.3	−0.810315
$573$	0	0
$574$	0	0
$575$	6536.46	0.474068
$576$	0	0
$577$	−8873.67	−0.640235	−0.320117	−	0.947378i	$-0.603722\pi$
$577$	−8873.67	−0.640235	−0.320117	+	0.947378i	$0.603722\pi$
$578$	−15686.4	−1.12884
$579$	0	0
$580$	810.797	0.0580457
$581$	0	0
$582$	0	0
$583$	33528.5	2.38183
$584$	−5938.93	−0.420813
$585$	0	0
$586$	7613.37	0.536699
$587$	17470.2	1.22840	0.614201	−	0.789150i	$-0.289479\pi$
$587$	17470.2	1.22840	0.614201	+	0.789150i	$0.289479\pi$
$588$	0	0
$589$	−26669.2	−1.86568
$590$	−12187.1	−0.850395
$591$	0	0
$592$	−11718.7	−0.813575
$593$	−17570.5	−1.21675	−0.608376	−	0.793649i	$-0.708179\pi$
$593$	−17570.5	−1.21675	−0.608376	+	0.793649i	$0.708179\pi$
$594$	0	0
$595$	0	0
$596$	9176.35	0.630668
$597$	0	0
$598$	17729.4	1.21239
$599$	−2470.61	−0.168525	−0.0842623	−	0.996444i	$-0.526853\pi$
$599$	−2470.61	−0.168525	−0.0842623	+	0.996444i	$0.526853\pi$
$600$	0	0
$601$	7760.05	0.526688	0.263344	−	0.964702i	$-0.415175\pi$
$601$	7760.05	0.526688	0.263344	+	0.964702i	$0.415175\pi$
$602$	0	0
$603$	0	0
$604$	6879.28	0.463433
$605$	−12485.4	−0.839013
$606$	0	0
$607$	23384.1	1.56365	0.781823	−	0.623500i	$-0.214290\pi$
$607$	23384.1	1.56365	0.781823	+	0.623500i	$0.214290\pi$
$608$	24621.3	1.64231
$609$	0	0
$610$	−3116.01	−0.206826
$611$	−12680.1	−0.839579
$612$	0	0
$613$	19483.4	1.28373	0.641865	−	0.766817i	$-0.278161\pi$
$613$	19483.4	1.28373	0.641865	+	0.766817i	$0.278161\pi$
$614$	18443.8	1.21226
$615$	0	0
$616$	0	0
$617$	11177.6	0.729327	0.364664	−	0.931139i	$-0.381184\pi$
$617$	11177.6	0.729327	0.364664	+	0.931139i	$0.381184\pi$
$618$	0	0
$619$	−6253.28	−0.406043	−0.203021	−	0.979174i	$-0.565076\pi$
$619$	−6253.28	−0.406043	−0.203021	+	0.979174i	$0.565076\pi$
$620$	6192.77	0.401141
$621$	0	0
$622$	−15008.3	−0.967491
$623$	0	0
$624$	0	0
$625$	−3786.15	−0.242313
$626$	−8245.33	−0.526437
$627$	0	0
$628$	−5283.42	−0.335719
$629$	14212.5	0.900938
$630$	0	0
$631$	−19733.2	−1.24496	−0.622478	−	0.782637i	$-0.713874\pi$
$631$	−19733.2	−1.24496	−0.622478	+	0.782637i	$0.713874\pi$
$632$	5774.16	0.363424
$633$	0	0
$634$	−31754.8	−1.98919
$635$	2457.55	0.153583
$636$	0	0
$637$	0	0
$638$	−4626.90	−0.287117
$639$	0	0
$640$	11876.5	0.733528
$641$	143.768	0.00885882	0.00442941	−	0.999990i	$-0.498590\pi$
$641$	143.768	0.00885882	0.00442941	+	0.999990i	$0.498590\pi$
$642$	0	0
$643$	−29565.4	−1.81329	−0.906645	−	0.421894i	$-0.861366\pi$
$643$	−29565.4	−1.81329	−0.906645	+	0.421894i	$0.861366\pi$
$644$	0	0
$645$	0	0
$646$	−48203.4	−2.93582
$647$	23780.7	1.44500	0.722501	−	0.691370i	$-0.242992\pi$
$647$	23780.7	1.44500	0.722501	+	0.691370i	$0.242992\pi$
$648$	0	0
$649$	23929.5	1.44733
$650$	10758.4	0.649200
$651$	0	0
$652$	−13257.0	−0.796296
$653$	−1741.59	−0.104370	−0.0521851	−	0.998637i	$-0.516619\pi$
$653$	−1741.59	−0.104370	−0.0521851	+	0.998637i	$0.516619\pi$
$654$	0	0
$655$	655.780	0.0391198
$656$	−25122.3	−1.49521
$657$	0	0
$658$	0	0
$659$	8493.45	0.502061	0.251030	−	0.967979i	$-0.419231\pi$
$659$	8493.45	0.502061	0.251030	+	0.967979i	$0.419231\pi$
$660$	0	0
$661$	−25014.7	−1.47195	−0.735976	−	0.677008i	$-0.763276\pi$
$661$	−25014.7	−1.47195	−0.735976	+	0.677008i	$0.763276\pi$
$662$	−15293.1	−0.897860
$663$	0	0
$664$	−4372.18	−0.255532
$665$	0	0
$666$	0	0
$667$	2546.20	0.147810
$668$	7759.05	0.449411
$669$	0	0
$670$	17299.0	0.997489
$671$	6118.36	0.352007
$672$	0	0
$673$	29696.5	1.70091	0.850456	−	0.526046i	$-0.176326\pi$
$673$	29696.5	1.70091	0.850456	+	0.526046i	$0.176326\pi$
$674$	−11575.7	−0.661544
$675$	0	0
$676$	820.632	0.0466905
$677$	−3713.27	−0.210801	−0.105401	−	0.994430i	$-0.533612\pi$
$677$	−3713.27	−0.210801	−0.105401	+	0.994430i	$0.533612\pi$
$678$	0	0
$679$	0	0
$680$	−10144.5	−0.572096
$681$	0	0
$682$	−35339.7	−1.98420
$683$	24254.7	1.35883	0.679414	−	0.733755i	$-0.262234\pi$
$683$	24254.7	1.35883	0.679414	+	0.733755i	$0.262234\pi$
$684$	0	0
$685$	−22913.8	−1.27809
$686$	0	0
$687$	0	0
$688$	−13846.5	−0.767289
$689$	−30368.4	−1.67917
$690$	0	0
$691$	−14298.7	−0.787191	−0.393595	−	0.919284i	$-0.628769\pi$
$691$	−14298.7	−0.787191	−0.393595	+	0.919284i	$0.628769\pi$
$692$	2615.13	0.143659
$693$	0	0
$694$	29873.2	1.63396
$695$	11142.9	0.608164
$696$	0	0
$697$	30468.4	1.65577
$698$	34340.2	1.86217
$699$	0	0
$700$	0	0
$701$	−28913.3	−1.55783	−0.778916	−	0.627128i	$-0.784230\pi$
$701$	−28913.3	−1.55783	−0.778916	+	0.627128i	$0.784230\pi$
$702$	0	0
$703$	20858.7	1.11906
$704$	−1919.57	−0.102765
$705$	0	0
$706$	21552.8	1.14894
$707$	0	0
$708$	0	0
$709$	10575.2	0.560169	0.280085	−	0.959975i	$-0.409637\pi$
$709$	10575.2	0.560169	0.280085	+	0.959975i	$0.409637\pi$
$710$	−1136.01	−0.0600473
$711$	0	0
$712$	−330.357	−0.0173886
$713$	19447.6	1.02148
$714$	0	0
$715$	20802.7	1.08808
$716$	−9822.77	−0.512701
$717$	0	0
$718$	−33341.6	−1.73301
$719$	5347.27	0.277357	0.138678	−	0.990337i	$-0.455715\pi$
$719$	5347.27	0.277357	0.138678	+	0.990337i	$0.455715\pi$
$720$	0	0
$721$	0	0
$722$	−46790.7	−2.41187
$723$	0	0
$724$	1938.39	0.0995021
$725$	1545.07	0.0791482
$726$	0	0
$727$	19629.1	1.00138	0.500689	−	0.865627i	$-0.333080\pi$
$727$	19629.1	1.00138	0.500689	+	0.865627i	$0.333080\pi$
$728$	0	0
$729$	0	0
$730$	−12297.1	−0.623473
$731$	16793.2	0.849682
$732$	0	0
$733$	−2404.87	−0.121181	−0.0605906	−	0.998163i	$-0.519298\pi$
$733$	−2404.87	−0.121181	−0.0605906	+	0.998163i	$0.519298\pi$
$734$	−14069.6	−0.707519
$735$	0	0
$736$	−17954.2	−0.899186
$737$	−33966.9	−1.69767
$738$	0	0
$739$	−19309.0	−0.961153	−0.480577	−	0.876953i	$-0.659573\pi$
$739$	−19309.0	−0.961153	−0.480577	+	0.876953i	$0.659573\pi$
$740$	−4843.54	−0.240611
$741$	0	0
$742$	0	0
$743$	−801.314	−0.0395658	−0.0197829	−	0.999804i	$-0.506297\pi$
$743$	−801.314	−0.0395658	−0.0197829	+	0.999804i	$0.506297\pi$
$744$	0	0
$745$	−17220.4	−0.846852
$746$	−70.0249	−0.00343672
$747$	0	0
$748$	−21978.0	−1.07433
$749$	0	0
$750$	0	0
$751$	14872.7	0.722654	0.361327	−	0.932439i	$-0.382324\pi$
$751$	14872.7	0.722654	0.361327	+	0.932439i	$0.382324\pi$
$752$	20728.7	1.00519
$753$	0	0
$754$	4190.81	0.202414
$755$	−12909.7	−0.622292
$756$	0	0
$757$	9127.52	0.438237	0.219118	−	0.975698i	$-0.429682\pi$
$757$	9127.52	0.438237	0.219118	+	0.975698i	$0.429682\pi$
$758$	13228.3	0.633871
$759$	0	0
$760$	−14888.4	−0.710605
$761$	−30994.4	−1.47641	−0.738203	−	0.674578i	$-0.764325\pi$
$761$	−30994.4	−1.47641	−0.738203	+	0.674578i	$0.764325\pi$
$762$	0	0
$763$	0	0
$764$	−11187.2	−0.529764
$765$	0	0
$766$	−27944.9	−1.31813
$767$	−21674.2	−1.02035
$768$	0	0
$769$	−12979.5	−0.608653	−0.304326	−	0.952568i	$-0.598431\pi$
$769$	−12979.5	−0.608653	−0.304326	+	0.952568i	$0.598431\pi$
$770$	0	0
$771$	0	0
$772$	547.944	0.0255452
$773$	28808.1	1.34043	0.670217	−	0.742165i	$-0.266201\pi$
$773$	28808.1	1.34043	0.670217	+	0.742165i	$0.266201\pi$
$774$	0	0
$775$	11801.1	0.546976
$776$	6631.35	0.306767
$777$	0	0
$778$	12169.6	0.560799
$779$	44716.4	2.05665
$780$	0	0
$781$	2230.57	0.102197
$782$	35150.7	1.60740
$783$	0	0
$784$	0	0
$785$	9914.88	0.450799
$786$	0	0
$787$	3658.15	0.165691	0.0828456	−	0.996562i	$-0.473599\pi$
$787$	3658.15	0.165691	0.0828456	+	0.996562i	$0.473599\pi$
$788$	15651.2	0.707552
$789$	0	0
$790$	11955.9	0.538446
$791$	0	0
$792$	0	0
$793$	−5541.70	−0.248161
$794$	−24578.4	−1.09856
$795$	0	0
$796$	15832.7	0.704992
$797$	−20171.4	−0.896497	−0.448249	−	0.893909i	$-0.647952\pi$
$797$	−20171.4	−0.896497	−0.448249	+	0.893909i	$0.647952\pi$
$798$	0	0
$799$	−25139.9	−1.11312
$800$	−10894.9	−0.481489
$801$	0	0
$802$	−29300.0	−1.29005
$803$	24145.6	1.06112
$804$	0	0
$805$	0	0
$806$	32008.9	1.39884
$807$	0	0
$808$	−3428.01	−0.149253
$809$	30508.2	1.32585	0.662924	−	0.748686i	$-0.269315\pi$
$809$	30508.2	1.32585	0.662924	+	0.748686i	$0.269315\pi$
$810$	0	0
$811$	−2550.91	−0.110449	−0.0552247	−	0.998474i	$-0.517588\pi$
$811$	−2550.91	−0.110449	−0.0552247	+	0.998474i	$0.517588\pi$
$812$	0	0
$813$	0	0
$814$	27640.2	1.19016
$815$	24878.1	1.06926
$816$	0	0
$817$	24646.1	1.05540
$818$	−47305.7	−2.02201
$819$	0	0
$820$	−10383.5	−0.442202
$821$	32657.1	1.38824	0.694118	−	0.719861i	$-0.255794\pi$
$821$	32657.1	1.38824	0.694118	+	0.719861i	$0.255794\pi$
$822$	0	0
$823$	30390.4	1.28717	0.643587	−	0.765373i	$-0.277446\pi$
$823$	30390.4	1.28717	0.643587	+	0.765373i	$0.277446\pi$
$824$	−12755.8	−0.539284
$825$	0	0
$826$	0	0
$827$	13100.8	0.550859	0.275430	−	0.961321i	$-0.411180\pi$
$827$	13100.8	0.550859	0.275430	+	0.961321i	$0.411180\pi$
$828$	0	0
$829$	24522.6	1.02739	0.513694	−	0.857974i	$-0.328277\pi$
$829$	24522.6	1.02739	0.513694	+	0.857974i	$0.328277\pi$
$830$	−9052.97	−0.378594
$831$	0	0
$832$	1738.65	0.0724483
$833$	0	0
$834$	0	0
$835$	−14560.6	−0.603463
$836$	−32255.6	−1.33443
$837$	0	0
$838$	−5064.71	−0.208780
$839$	−38095.2	−1.56757	−0.783785	−	0.621032i	$-0.786714\pi$
$839$	−38095.2	−1.56757	−0.783785	+	0.621032i	$0.786714\pi$
$840$	0	0
$841$	−23787.1	−0.975322
$842$	3223.86	0.131950
$843$	0	0
$844$	10064.6	0.410470
$845$	−1540.00	−0.0626954
$846$	0	0
$847$	0	0
$848$	49644.6	2.01038
$849$	0	0
$850$	21329.9	0.860718
$851$	−15210.5	−0.612702
$852$	0	0
$853$	−27500.3	−1.10386	−0.551929	−	0.833891i	$-0.686108\pi$
$853$	−27500.3	−1.10386	−0.551929	+	0.833891i	$0.686108\pi$
$854$	0	0
$855$	0	0
$856$	−27286.6	−1.08953
$857$	13903.0	0.554163	0.277082	−	0.960846i	$-0.410633\pi$
$857$	13903.0	0.554163	0.277082	+	0.960846i	$0.410633\pi$
$858$	0	0
$859$	41524.3	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$859$	41524.3	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$860$	−5723.01	−0.226922
$861$	0	0
$862$	−4324.28	−0.170865
$863$	1744.09	0.0687942	0.0343971	−	0.999408i	$-0.489049\pi$
$863$	1744.09	0.0687942	0.0343971	+	0.999408i	$0.489049\pi$
$864$	0	0
$865$	−4907.55	−0.192904
$866$	37670.3	1.47816
$867$	0	0
$868$	0	0
$869$	−23475.7	−0.916407
$870$	0	0
$871$	30765.5	1.19684
$872$	23698.1	0.920319
$873$	0	0
$874$	51588.2	1.99656
$875$	0	0
$876$	0	0
$877$	46717.4	1.79879	0.899393	−	0.437140i	$-0.144009\pi$
$877$	46717.4	1.79879	0.899393	+	0.437140i	$0.144009\pi$
$878$	11711.6	0.450168
$879$	0	0
$880$	−34007.1	−1.30270
$881$	−17214.0	−0.658291	−0.329145	−	0.944279i	$-0.606761\pi$
$881$	−17214.0	−0.658291	−0.329145	+	0.944279i	$0.606761\pi$
$882$	0	0
$883$	20487.2	0.780804	0.390402	−	0.920645i	$-0.372336\pi$
$883$	20487.2	0.780804	0.390402	+	0.920645i	$0.372336\pi$
$884$	19906.6	0.757388
$885$	0	0
$886$	28254.5	1.07136
$887$	−18307.5	−0.693016	−0.346508	−	0.938047i	$-0.612633\pi$
$887$	−18307.5	−0.693016	−0.346508	+	0.938047i	$0.612633\pi$
$888$	0	0
$889$	0	0
$890$	−684.034	−0.0257628
$891$	0	0
$892$	14009.1	0.525853
$893$	−36896.1	−1.38262
$894$	0	0
$895$	18433.4	0.688449
$896$	0	0
$897$	0	0
$898$	14707.0	0.546524
$899$	4596.96	0.170542
$900$	0	0
$901$	−60209.3	−2.22626
$902$	59254.3	2.18731
$903$	0	0
$904$	−17219.4	−0.633528
$905$	−3637.58	−0.133610
$906$	0	0
$907$	38336.3	1.40346	0.701730	−	0.712443i	$-0.252411\pi$
$907$	38336.3	1.40346	0.701730	+	0.712443i	$0.252411\pi$
$908$	−5325.63	−0.194645
$909$	0	0
$910$	0	0
$911$	−30148.4	−1.09644	−0.548222	−	0.836333i	$-0.684695\pi$
$911$	−30148.4	−1.09644	−0.548222	+	0.836333i	$0.684695\pi$
$912$	0	0
$913$	17775.7	0.644348
$914$	34147.0	1.23576
$915$	0	0
$916$	8609.21	0.310542
$917$	0	0
$918$	0	0
$919$	−13799.4	−0.495323	−0.247661	−	0.968847i	$-0.579662\pi$
$919$	−13799.4	−0.495323	−0.247661	+	0.968847i	$0.579662\pi$
$920$	10856.9	0.389066
$921$	0	0
$922$	−43166.8	−1.54189
$923$	−2020.34	−0.0720480
$924$	0	0
$925$	−9229.94	−0.328085
$926$	56251.5	1.99626
$927$	0	0
$928$	−4243.96	−0.150124
$929$	42953.7	1.51697	0.758485	−	0.651691i	$-0.225940\pi$
$929$	42953.7	1.51697	0.758485	+	0.651691i	$0.225940\pi$
$930$	0	0
$931$	0	0
$932$	20774.9	0.730155
$933$	0	0
$934$	−47095.8	−1.64991
$935$	41244.0	1.44259
$936$	0	0
$937$	6369.72	0.222081	0.111040	−	0.993816i	$-0.464582\pi$
$937$	6369.72	0.222081	0.111040	+	0.993816i	$0.464582\pi$
$938$	0	0
$939$	0	0
$940$	8567.54	0.297279
$941$	−6002.48	−0.207944	−0.103972	−	0.994580i	$-0.533155\pi$
$941$	−6002.48	−0.207944	−0.103972	+	0.994580i	$0.533155\pi$
$942$	0	0
$943$	−32607.9	−1.12604
$944$	35431.7	1.22161
$945$	0	0
$946$	32659.0	1.12245
$947$	12113.1	0.415652	0.207826	−	0.978166i	$-0.433361\pi$
$947$	12113.1	0.415652	0.207826	+	0.978166i	$0.433361\pi$
$948$	0	0
$949$	−21869.8	−0.748077
$950$	31304.4	1.06911
$951$	0	0
$952$	0	0
$953$	10130.6	0.344346	0.172173	−	0.985067i	$-0.444921\pi$
$953$	10130.6	0.344346	0.172173	+	0.985067i	$0.444921\pi$
$954$	0	0
$955$	20994.0	0.711360
$956$	1102.32	0.0372924
$957$	0	0
$958$	−1365.39	−0.0460478
$959$	0	0
$960$	0	0
$961$	5320.00	0.178578
$962$	−25035.1	−0.839047
$963$	0	0
$964$	−12180.0	−0.406943
$965$	−1028.27	−0.0343018
$966$	0	0
$967$	−17651.7	−0.587013	−0.293506	−	0.955957i	$-0.594822\pi$
$967$	−17651.7	−0.587013	−0.293506	+	0.955957i	$0.594822\pi$
$968$	21058.5	0.699221
$969$	0	0
$970$	13730.8	0.454504
$971$	−32208.7	−1.06450	−0.532248	−	0.846588i	$-0.678653\pi$
$971$	−32208.7	−1.06450	−0.532248	+	0.846588i	$0.678653\pi$
$972$	0	0
$973$	0	0
$974$	55149.6	1.81428
$975$	0	0
$976$	9059.27	0.297111
$977$	24098.8	0.789138	0.394569	−	0.918866i	$-0.370894\pi$
$977$	24098.8	0.789138	0.394569	+	0.918866i	$0.370894\pi$
$978$	0	0
$979$	1343.11	0.0438469
$980$	0	0
$981$	0	0
$982$	−669.158	−0.0217451
$983$	−15156.9	−0.491790	−0.245895	−	0.969296i	$-0.579082\pi$
$983$	−15156.9	−0.491790	−0.245895	+	0.969296i	$0.579082\pi$
$984$	0	0
$985$	−29371.1	−0.950092
$986$	8308.82	0.268364
$987$	0	0
$988$	29215.5	0.940759
$989$	−17972.4	−0.577844
$990$	0	0
$991$	18357.4	0.588437	0.294219	−	0.955738i	$-0.404941\pi$
$991$	18357.4	0.588437	0.294219	+	0.955738i	$0.404941\pi$
$992$	−32414.9	−1.03747
$993$	0	0
$994$	0	0
$995$	−29711.6	−0.946654
$996$	0	0
$997$	−18552.6	−0.589335	−0.294668	−	0.955600i	$-0.595209\pi$
$997$	−18552.6	−0.589335	−0.294668	+	0.955600i	$0.595209\pi$
$998$	−19630.6	−0.622640
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bn.1.2		8
3.2	odd	2	inner	1323.4.a.bn.1.7		8
7.3	odd	6		189.4.e.h.163.7	yes	16
7.5	odd	6		189.4.e.h.109.7	yes	16
7.6	odd	2		1323.4.a.bo.1.2		8
21.5	even	6		189.4.e.h.109.2	✓	16
21.17	even	6		189.4.e.h.163.2	yes	16
21.20	even	2		1323.4.a.bo.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.h.109.2	✓	16	21.5	even	6
189.4.e.h.109.7	yes	16	7.5	odd	6
189.4.e.h.163.2	yes	16	21.17	even	6
189.4.e.h.163.7	yes	16	7.3	odd	6
1323.4.a.bn.1.2		8	1.1	even	1	trivial
1323.4.a.bn.1.7		8	3.2	odd	2	inner
1323.4.a.bo.1.2		8	7.6	odd	2
1323.4.a.bo.1.7		8	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.49236 q^{2} +4.19658 q^{4} -7.87531 q^{5} +13.2829 q^{8} +27.5034 q^{10} -54.0035 q^{11} +48.9137 q^{13} -79.9614 q^{16} +96.9775 q^{17} +142.327 q^{19} -33.0494 q^{20} +188.600 q^{22} -103.787 q^{23} -62.9795 q^{25} -170.824 q^{26} -24.5329 q^{29} -187.379 q^{31} +172.991 q^{32} -338.680 q^{34} +146.555 q^{37} -497.058 q^{38} -104.607 q^{40} +314.180 q^{41} +173.165 q^{43} -226.630 q^{44} +362.462 q^{46} -259.235 q^{47} +219.947 q^{50} +205.270 q^{52} -620.858 q^{53} +425.295 q^{55} +85.6778 q^{58} -443.110 q^{59} -113.296 q^{61} +654.396 q^{62} +35.5454 q^{64} -385.210 q^{65} +628.975 q^{67} +406.974 q^{68} -41.3042 q^{71} -447.111 q^{73} -511.822 q^{74} +597.287 q^{76} +434.706 q^{79} +629.720 q^{80} -1097.23 q^{82} -329.158 q^{83} -763.728 q^{85} -604.756 q^{86} -717.324 q^{88} -24.8709 q^{89} -435.551 q^{92} +905.340 q^{94} -1120.87 q^{95} +499.239 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 48 q^{4} - 44 q^{10} + 84 q^{13} + 156 q^{16} - 12 q^{19} + 224 q^{22} + 408 q^{25} - 800 q^{31} + 948 q^{34} + 692 q^{37} - 96 q^{40} + 1456 q^{43} + 1524 q^{46} - 1972 q^{52} + 1280 q^{55} + 2372 q^{58}+ \cdots - 584 q^{97}+O(q^{100}) \) 8 * q + 48 * q^4 - 44 * q^10 + 84 * q^13 + 156 * q^16 - 12 * q^19 + 224 * q^22 + 408 * q^25 - 800 * q^31 + 948 * q^34 + 692 * q^37 - 96 * q^40 + 1456 * q^43 + 1524 * q^46 - 1972 * q^52 + 1280 * q^55 + 2372 * q^58 - 216 * q^61 + 4964 * q^64 + 684 * q^67 + 4564 * q^73 + 380 * q^76 + 556 * q^79 - 3340 * q^82 + 1296 * q^85 + 6696 * q^88 + 492 * q^94 - 584 * q^97

Introduction

L-functions

Modular forms

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Representations

Groups

Database

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