LMFDB - Embedded newform 1323.4.a.bn.1.8

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			1.8
Root			$-2.12000$ 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)

$f(q)$	$=$	$q+5.46178 q^{2} +21.8311 q^{4} -0.199136 q^{5} +75.5424 q^{8} -1.08764 q^{10} +28.4233 q^{11} -32.5809 q^{13} +237.948 q^{16} +115.488 q^{17} -21.1500 q^{19} -4.34736 q^{20} +155.242 q^{22} +93.7656 q^{23} -124.960 q^{25} -177.950 q^{26} +231.571 q^{29} -281.742 q^{31} +695.280 q^{32} +630.770 q^{34} -146.554 q^{37} -115.517 q^{38} -15.0432 q^{40} +111.001 q^{41} +392.361 q^{43} +620.513 q^{44} +512.128 q^{46} -273.168 q^{47} -682.506 q^{50} -711.277 q^{52} -340.403 q^{53} -5.66011 q^{55} +1264.79 q^{58} +696.817 q^{59} +370.002 q^{61} -1538.81 q^{62} +1893.89 q^{64} +6.48804 q^{65} +87.1362 q^{67} +2521.23 q^{68} -88.3772 q^{71} +803.036 q^{73} -800.446 q^{74} -461.727 q^{76} +364.616 q^{79} -47.3840 q^{80} +606.264 q^{82} -921.684 q^{83} -22.9978 q^{85} +2142.99 q^{86} +2147.17 q^{88} +211.396 q^{89} +2047.01 q^{92} -1491.99 q^{94} +4.21172 q^{95} -845.718 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.46178	1.93103	0.965516	−	0.260343i	$-0.0838356\pi$
$2$	5.46178	1.93103	0.965516	+	0.260343i	$0.0838356\pi$
$3$	0	0
$3$	0	0
$4$	21.8311	2.72889
$5$	−0.199136	−0.0178113	−0.00890564	−	0.999960i	$-0.502835\pi$
$5$	−0.199136	−0.0178113	−0.00890564	+	0.999960i	$0.502835\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	75.5424	3.33854
$9$	0	0
$10$	−1.08764	−0.0343941
$11$	28.4233	0.779087	0.389544	−	0.921008i	$-0.372633\pi$
$11$	28.4233	0.779087	0.389544	+	0.921008i	$0.372633\pi$
$12$	0	0
$13$	−32.5809	−0.695101	−0.347551	−	0.937661i	$-0.612987\pi$
$13$	−32.5809	−0.695101	−0.347551	+	0.937661i	$0.612987\pi$
$14$	0	0
$15$	0	0
$16$	237.948	3.71793
$17$	115.488	1.64764	0.823821	−	0.566849i	$-0.191838\pi$
$17$	115.488	1.64764	0.823821	+	0.566849i	$0.191838\pi$
$18$	0	0
$19$	−21.1500	−0.255376	−0.127688	−	0.991814i	$-0.540756\pi$
$19$	−21.1500	−0.255376	−0.127688	+	0.991814i	$0.540756\pi$
$20$	−4.34736	−0.0486049
$21$	0	0
$22$	155.242	1.50444
$23$	93.7656	0.850065	0.425032	−	0.905178i	$-0.360263\pi$
$23$	93.7656	0.850065	0.425032	+	0.905178i	$0.360263\pi$
$24$	0	0
$25$	−124.960	−0.999683
$26$	−177.950	−1.34226
$27$	0	0
$28$	0	0
$29$	231.571	1.48282	0.741408	−	0.671055i	$-0.234159\pi$
$29$	231.571	1.48282	0.741408	+	0.671055i	$0.234159\pi$
$30$	0	0
$31$	−281.742	−1.63233	−0.816167	−	0.577817i	$-0.803905\pi$
$31$	−281.742	−1.63233	−0.816167	+	0.577817i	$0.803905\pi$
$32$	695.280	3.84092
$33$	0	0
$34$	630.770	3.18165
$35$	0	0
$36$	0	0
$37$	−146.554	−0.651171	−0.325585	−	0.945513i	$-0.605561\pi$
$37$	−146.554	−0.651171	−0.325585	+	0.945513i	$0.605561\pi$
$38$	−115.517	−0.493138
$39$	0	0
$40$	−15.0432	−0.0594636
$41$	111.001	0.422816	0.211408	−	0.977398i	$-0.432195\pi$
$41$	111.001	0.422816	0.211408	+	0.977398i	$0.432195\pi$
$42$	0	0
$43$	392.361	1.39150	0.695750	−	0.718284i	$-0.255072\pi$
$43$	392.361	1.39150	0.695750	+	0.718284i	$0.255072\pi$
$44$	620.513	2.12604
$45$	0	0
$46$	512.128	1.64150
$47$	−273.168	−0.847780	−0.423890	−	0.905714i	$-0.639336\pi$
$47$	−273.168	−0.847780	−0.423890	+	0.905714i	$0.639336\pi$
$48$	0	0
$49$	0	0
$50$	−682.506	−1.93042
$51$	0	0
$52$	−711.277	−1.89685
$53$	−340.403	−0.882225	−0.441113	−	0.897452i	$-0.645416\pi$
$53$	−340.403	−0.882225	−0.441113	+	0.897452i	$0.645416\pi$
$54$	0	0
$55$	−5.66011	−0.0138765
$56$	0	0
$57$	0	0
$58$	1264.79	2.86336
$59$	696.817	1.53759	0.768795	−	0.639495i	$-0.220857\pi$
$59$	696.817	1.53759	0.768795	+	0.639495i	$0.220857\pi$
$60$	0	0
$61$	370.002	0.776622	0.388311	−	0.921528i	$-0.373059\pi$
$61$	370.002	0.776622	0.388311	+	0.921528i	$0.373059\pi$
$62$	−1538.81	−3.15209
$63$	0	0
$64$	1893.89	3.69900
$65$	6.48804	0.0123806
$66$	0	0
$67$	87.1362	0.158886	0.0794431	−	0.996839i	$-0.474686\pi$
$67$	87.1362	0.158886	0.0794431	+	0.996839i	$0.474686\pi$
$68$	2521.23	4.49623
$69$	0	0
$70$	0	0
$71$	−88.3772	−0.147725	−0.0738623	−	0.997268i	$-0.523533\pi$
$71$	−88.3772	−0.147725	−0.0738623	+	0.997268i	$0.523533\pi$
$72$	0	0
$73$	803.036	1.28751	0.643755	−	0.765231i	$-0.277375\pi$
$73$	803.036	1.28751	0.643755	+	0.765231i	$0.277375\pi$
$74$	−800.446	−1.25743
$75$	0	0
$76$	−461.727	−0.696891
$77$	0	0
$78$	0	0
$79$	364.616	0.519272	0.259636	−	0.965707i	$-0.416397\pi$
$79$	364.616	0.519272	0.259636	+	0.965707i	$0.416397\pi$
$80$	−47.3840	−0.0662211
$81$	0	0
$82$	606.264	0.816472
$83$	−921.684	−1.21889	−0.609446	−	0.792828i	$-0.708608\pi$
$83$	−921.684	−1.21889	−0.609446	+	0.792828i	$0.708608\pi$
$84$	0	0
$85$	−22.9978	−0.0293466
$86$	2142.99	2.68703
$87$	0	0
$88$	2147.17	2.60101
$89$	211.396	0.251774	0.125887	−	0.992045i	$-0.459822\pi$
$89$	211.396	0.251774	0.125887	+	0.992045i	$0.459822\pi$
$90$	0	0
$91$	0	0
$92$	2047.01	2.31973
$93$	0	0
$94$	−1491.99	−1.63709
$95$	4.21172	0.00454856
$96$	0	0
$97$	−845.718	−0.885254	−0.442627	−	0.896706i	$-0.645953\pi$
$97$	−845.718	−0.885254	−0.442627	+	0.896706i	$0.645953\pi$
$98$	0	0
$99$	0	0
$100$	−2728.02	−2.72802
$101$	156.952	0.154626	0.0773132	−	0.997007i	$-0.475366\pi$
$101$	156.952	0.154626	0.0773132	+	0.997007i	$0.475366\pi$
$102$	0	0
$103$	139.325	0.133283	0.0666414	−	0.997777i	$-0.478772\pi$
$103$	139.325	0.133283	0.0666414	+	0.997777i	$0.478772\pi$
$104$	−2461.24	−2.32062
$105$	0	0
$106$	−1859.21	−1.70361
$107$	−415.476	−0.375379	−0.187690	−	0.982228i	$-0.560100\pi$
$107$	−415.476	−0.375379	−0.187690	+	0.982228i	$0.560100\pi$
$108$	0	0
$109$	−844.080	−0.741727	−0.370863	−	0.928687i	$-0.620938\pi$
$109$	−844.080	−0.741727	−0.370863	+	0.928687i	$0.620938\pi$
$110$	−30.9143	−0.0267961
$111$	0	0
$112$	0	0
$113$	−1342.21	−1.11738	−0.558692	−	0.829375i	$-0.688697\pi$
$113$	−1342.21	−1.11738	−0.558692	+	0.829375i	$0.688697\pi$
$114$	0	0
$115$	−18.6721	−0.0151407
$116$	5055.45	4.04643
$117$	0	0
$118$	3805.86	2.96914
$119$	0	0
$120$	0	0
$121$	−523.113	−0.393023
$122$	2020.87	1.49968
$123$	0	0
$124$	−6150.73	−4.45445
$125$	49.7761	0.0356169
$126$	0	0
$127$	978.750	0.683858	0.341929	−	0.939726i	$-0.388920\pi$
$127$	978.750	0.683858	0.341929	+	0.939726i	$0.388920\pi$
$128$	4781.76	3.30197
$129$	0	0
$130$	35.4362	0.0239074
$131$	−2372.81	−1.58254	−0.791271	−	0.611465i	$-0.790580\pi$
$131$	−2372.81	−1.58254	−0.791271	+	0.611465i	$0.790580\pi$
$132$	0	0
$133$	0	0
$134$	475.919	0.306815
$135$	0	0
$136$	8724.24	5.50071
$137$	1602.67	0.999458	0.499729	−	0.866182i	$-0.333433\pi$
$137$	1602.67	0.999458	0.499729	+	0.866182i	$0.333433\pi$
$138$	0	0
$139$	−2101.78	−1.28252	−0.641262	−	0.767322i	$-0.721589\pi$
$139$	−2101.78	−1.28252	−0.641262	+	0.767322i	$0.721589\pi$
$140$	0	0
$141$	0	0
$142$	−482.697	−0.285261
$143$	−926.059	−0.541545
$144$	0	0
$145$	−46.1141	−0.0264108
$146$	4386.01	2.48623
$147$	0	0
$148$	−3199.43	−1.77697
$149$	524.400	0.288326	0.144163	−	0.989554i	$-0.453951\pi$
$149$	524.400	0.288326	0.144163	+	0.989554i	$0.453951\pi$
$150$	0	0
$151$	−867.666	−0.467613	−0.233807	−	0.972283i	$-0.575118\pi$
$151$	−867.666	−0.467613	−0.233807	+	0.972283i	$0.575118\pi$
$152$	−1597.72	−0.852580
$153$	0	0
$154$	0	0
$155$	56.1050	0.0290739
$156$	0	0
$157$	5.91999	0.00300934	0.00150467	−	0.999999i	$-0.499521\pi$
$157$	5.91999	0.00300934	0.00150467	+	0.999999i	$0.499521\pi$
$158$	1991.45	1.00273
$159$	0	0
$160$	−138.455	−0.0684116
$161$	0	0
$162$	0	0
$163$	−832.959	−0.400260	−0.200130	−	0.979769i	$-0.564136\pi$
$163$	−832.959	−0.400260	−0.200130	+	0.979769i	$0.564136\pi$
$164$	2423.28	1.15382
$165$	0	0
$166$	−5034.04	−2.35372
$167$	566.798	0.262636	0.131318	−	0.991340i	$-0.458079\pi$
$167$	566.798	0.262636	0.131318	+	0.991340i	$0.458079\pi$
$168$	0	0
$169$	−1135.48	−0.516834
$170$	−125.609	−0.0566693
$171$	0	0
$172$	8565.66	3.79724
$173$	4125.77	1.81316	0.906579	−	0.422036i	$-0.138684\pi$
$173$	4125.77	1.81316	0.906579	+	0.422036i	$0.138684\pi$
$174$	0	0
$175$	0	0
$176$	6763.27	2.89660
$177$	0	0
$178$	1154.60	0.486184
$179$	927.646	0.387349	0.193675	−	0.981066i	$-0.437959\pi$
$179$	927.646	0.387349	0.193675	+	0.981066i	$0.437959\pi$
$180$	0	0
$181$	−1211.67	−0.497585	−0.248792	−	0.968557i	$-0.580034\pi$
$181$	−1211.67	−0.497585	−0.248792	+	0.968557i	$0.580034\pi$
$182$	0	0
$183$	0	0
$184$	7083.28	2.83797
$185$	29.1842	0.0115982
$186$	0	0
$187$	3282.55	1.28366
$188$	−5963.56	−2.31350
$189$	0	0
$190$	23.0035	0.00878342
$191$	−1827.49	−0.692318	−0.346159	−	0.938176i	$-0.612514\pi$
$191$	−1827.49	−0.692318	−0.346159	+	0.938176i	$0.612514\pi$
$192$	0	0
$193$	−822.230	−0.306660	−0.153330	−	0.988175i	$-0.549000\pi$
$193$	−822.230	−0.306660	−0.153330	+	0.988175i	$0.549000\pi$
$194$	−4619.13	−1.70945
$195$	0	0
$196$	0	0
$197$	−4873.12	−1.76241	−0.881207	−	0.472731i	$-0.843268\pi$
$197$	−4873.12	−1.76241	−0.881207	+	0.472731i	$0.843268\pi$
$198$	0	0
$199$	1482.23	0.528001	0.264001	−	0.964523i	$-0.414958\pi$
$199$	1482.23	0.528001	0.264001	+	0.964523i	$0.414958\pi$
$200$	−9439.81	−3.33748
$201$	0	0
$202$	857.236	0.298589
$203$	0	0
$204$	0	0
$205$	−22.1043	−0.00753090
$206$	760.965	0.257373
$207$	0	0
$208$	−7752.55	−2.58434
$209$	−601.153	−0.198960
$210$	0	0
$211$	−4359.21	−1.42228	−0.711138	−	0.703053i	$-0.751820\pi$
$211$	−4359.21	−1.42228	−0.711138	+	0.703053i	$0.751820\pi$
$212$	−7431.37	−2.40749
$213$	0	0
$214$	−2269.24	−0.724869
$215$	−78.1332	−0.0247844
$216$	0	0
$217$	0	0
$218$	−4610.18	−1.43230
$219$	0	0
$220$	−123.566	−0.0378675
$221$	−3762.70	−1.14528
$222$	0	0
$223$	3312.73	0.994784	0.497392	−	0.867526i	$-0.334291\pi$
$223$	3312.73	0.994784	0.497392	+	0.867526i	$0.334291\pi$
$224$	0	0
$225$	0	0
$226$	−7330.85	−2.15770
$227$	798.321	0.233420	0.116710	−	0.993166i	$-0.462765\pi$
$227$	798.321	0.233420	0.116710	+	0.993166i	$0.462765\pi$
$228$	0	0
$229$	354.914	0.102417	0.0512083	−	0.998688i	$-0.483693\pi$
$229$	354.914	0.102417	0.0512083	+	0.998688i	$0.483693\pi$
$230$	−101.983	−0.0292372
$231$	0	0
$232$	17493.4	4.95043
$233$	−4473.91	−1.25792	−0.628960	−	0.777438i	$-0.716519\pi$
$233$	−4473.91	−1.25792	−0.628960	+	0.777438i	$0.716519\pi$
$234$	0	0
$235$	54.3976	0.0151000
$236$	15212.3	4.19591
$237$	0	0
$238$	0	0
$239$	−1494.65	−0.404523	−0.202262	−	0.979332i	$-0.564829\pi$
$239$	−1494.65	−0.404523	−0.202262	+	0.979332i	$0.564829\pi$
$240$	0	0
$241$	156.885	0.0419329	0.0209665	−	0.999780i	$-0.493326\pi$
$241$	156.885	0.0419329	0.0209665	+	0.999780i	$0.493326\pi$
$242$	−2857.13	−0.758940
$243$	0	0
$244$	8077.55	2.11931
$245$	0	0
$246$	0	0
$247$	689.085	0.177512
$248$	−21283.5	−5.44960
$249$	0	0
$250$	271.866	0.0687774
$251$	−3498.68	−0.879819	−0.439909	−	0.898042i	$-0.644989\pi$
$251$	−3498.68	−0.879819	−0.439909	+	0.898042i	$0.644989\pi$
$252$	0	0
$253$	2665.13	0.662275
$254$	5345.72	1.32055
$255$	0	0
$256$	10965.9	2.67721
$257$	−4739.10	−1.15026	−0.575130	−	0.818062i	$-0.695049\pi$
$257$	−4739.10	−1.15026	−0.575130	+	0.818062i	$0.695049\pi$
$258$	0	0
$259$	0	0
$260$	141.641	0.0337854
$261$	0	0
$262$	−12959.8	−3.05594
$263$	−3364.44	−0.788823	−0.394411	−	0.918934i	$-0.629052\pi$
$263$	−3364.44	−0.788823	−0.394411	+	0.918934i	$0.629052\pi$
$264$	0	0
$265$	67.7865	0.0157136
$266$	0	0
$267$	0	0
$268$	1902.28	0.433583
$269$	5207.27	1.18027	0.590136	−	0.807304i	$-0.299074\pi$
$269$	5207.27	1.18027	0.590136	+	0.807304i	$0.299074\pi$
$270$	0	0
$271$	2312.00	0.518244	0.259122	−	0.965845i	$-0.416567\pi$
$271$	2312.00	0.518244	0.259122	+	0.965845i	$0.416567\pi$
$272$	27480.1	6.12583
$273$	0	0
$274$	8753.46	1.92999
$275$	−3551.79	−0.778840
$276$	0	0
$277$	2606.72	0.565425	0.282713	−	0.959205i	$-0.408766\pi$
$277$	2606.72	0.565425	0.282713	+	0.959205i	$0.408766\pi$
$278$	−11479.5	−2.47659
$279$	0	0
$280$	0	0
$281$	−5271.62	−1.11914	−0.559570	−	0.828783i	$-0.689034\pi$
$281$	−5271.62	−1.11914	−0.559570	+	0.828783i	$0.689034\pi$
$282$	0	0
$283$	−5808.21	−1.22001	−0.610004	−	0.792398i	$-0.708832\pi$
$283$	−5808.21	−1.22001	−0.610004	+	0.792398i	$0.708832\pi$
$284$	−1929.37	−0.403124
$285$	0	0
$286$	−5057.93	−1.04574
$287$	0	0
$288$	0	0
$289$	8424.45	1.71473
$290$	−251.865	−0.0510002
$291$	0	0
$292$	17531.2	3.51347
$293$	1979.70	0.394729	0.197364	−	0.980330i	$-0.436762\pi$
$293$	1979.70	0.394729	0.197364	+	0.980330i	$0.436762\pi$
$294$	0	0
$295$	−138.761	−0.0273864
$296$	−11071.0	−2.17396
$297$	0	0
$298$	2864.16	0.556766
$299$	−3054.97	−0.590881
$300$	0	0
$301$	0	0
$302$	−4739.00	−0.902977
$303$	0	0
$304$	−5032.59	−0.949469
$305$	−73.6808	−0.0138326
$306$	0	0
$307$	−924.005	−0.171778	−0.0858888	−	0.996305i	$-0.527373\pi$
$307$	−924.005	−0.171778	−0.0858888	+	0.996305i	$0.527373\pi$
$308$	0	0
$309$	0	0
$310$	306.433	0.0561427
$311$	10108.6	1.84311	0.921554	−	0.388251i	$-0.126920\pi$
$311$	10108.6	1.84311	0.921554	+	0.388251i	$0.126920\pi$
$312$	0	0
$313$	−6830.52	−1.23349	−0.616747	−	0.787161i	$-0.711550\pi$
$313$	−6830.52	−1.23349	−0.616747	+	0.787161i	$0.711550\pi$
$314$	32.3337	0.00581114
$315$	0	0
$316$	7959.96	1.41703
$317$	−7623.62	−1.35074	−0.675371	−	0.737478i	$-0.736016\pi$
$317$	−7623.62	−1.35074	−0.675371	+	0.737478i	$0.736016\pi$
$318$	0	0
$319$	6582.02	1.15524
$320$	−377.141	−0.0658839
$321$	0	0
$322$	0	0
$323$	−2442.56	−0.420768
$324$	0	0
$325$	4071.32	0.694881
$326$	−4549.45	−0.772916
$327$	0	0
$328$	8385.30	1.41159
$329$	0	0
$330$	0	0
$331$	2483.21	0.412355	0.206177	−	0.978515i	$-0.433898\pi$
$331$	2483.21	0.412355	0.206177	+	0.978515i	$0.433898\pi$
$332$	−20121.4	−3.32622
$333$	0	0
$334$	3095.73	0.507158
$335$	−17.3520	−0.00282997
$336$	0	0
$337$	7895.47	1.27624	0.638121	−	0.769936i	$-0.279712\pi$
$337$	7895.47	1.27624	0.638121	+	0.769936i	$0.279712\pi$
$338$	−6201.77	−0.998023
$339$	0	0
$340$	−502.067	−0.0800836
$341$	−8008.05	−1.27173
$342$	0	0
$343$	0	0
$344$	29639.9	4.64557
$345$	0	0
$346$	22534.1	3.50127
$347$	889.423	0.137599	0.0687994	−	0.997631i	$-0.478083\pi$
$347$	889.423	0.137599	0.0687994	+	0.997631i	$0.478083\pi$
$348$	0	0
$349$	−6962.20	−1.06785	−0.533923	−	0.845533i	$-0.679283\pi$
$349$	−6962.20	−1.06785	−0.533923	+	0.845533i	$0.679283\pi$
$350$	0	0
$351$	0	0
$352$	19762.2	2.99241
$353$	11560.6	1.74308	0.871542	−	0.490320i	$-0.163120\pi$
$353$	11560.6	1.74308	0.871542	+	0.490320i	$0.163120\pi$
$354$	0	0
$355$	17.5991	0.00263116
$356$	4615.00	0.687064
$357$	0	0
$358$	5066.60	0.747984
$359$	8457.02	1.24330	0.621649	−	0.783296i	$-0.286463\pi$
$359$	8457.02	1.24330	0.621649	+	0.783296i	$0.286463\pi$
$360$	0	0
$361$	−6411.68	−0.934783
$362$	−6617.89	−0.960853
$363$	0	0
$364$	0	0
$365$	−159.914	−0.0229322
$366$	0	0
$367$	−6912.92	−0.983247	−0.491623	−	0.870808i	$-0.663596\pi$
$367$	−6912.92	−0.983247	−0.491623	+	0.870808i	$0.663596\pi$
$368$	22311.3	3.16048
$369$	0	0
$370$	159.398	0.0223965
$371$	0	0
$372$	0	0
$373$	423.911	0.0588453	0.0294226	−	0.999567i	$-0.490633\pi$
$373$	423.911	0.0588453	0.0294226	+	0.999567i	$0.490633\pi$
$374$	17928.6	2.47879
$375$	0	0
$376$	−20635.8	−2.83034
$377$	−7544.79	−1.03071
$378$	0	0
$379$	−9714.33	−1.31660	−0.658300	−	0.752756i	$-0.728724\pi$
$379$	−9714.33	−1.31660	−0.658300	+	0.752756i	$0.728724\pi$
$380$	91.9465	0.0124125
$381$	0	0
$382$	−9981.37	−1.33689
$383$	2214.15	0.295399	0.147699	−	0.989032i	$-0.452813\pi$
$383$	2214.15	0.295399	0.147699	+	0.989032i	$0.452813\pi$
$384$	0	0
$385$	0	0
$386$	−4490.84	−0.592171
$387$	0	0
$388$	−18462.9	−2.41576
$389$	−7665.46	−0.999111	−0.499556	−	0.866282i	$-0.666503\pi$
$389$	−7665.46	−0.999111	−0.499556	+	0.866282i	$0.666503\pi$
$390$	0	0
$391$	10828.8	1.40060
$392$	0	0
$393$	0	0
$394$	−26615.9	−3.40328
$395$	−72.6081	−0.00924889
$396$	0	0
$397$	−4978.20	−0.629341	−0.314671	−	0.949201i	$-0.601894\pi$
$397$	−4978.20	−0.629341	−0.314671	+	0.949201i	$0.601894\pi$
$398$	8095.60	1.01959
$399$	0	0
$400$	−29734.0	−3.71675
$401$	−4770.21	−0.594047	−0.297023	−	0.954870i	$-0.595994\pi$
$401$	−4770.21	−0.594047	−0.297023	+	0.954870i	$0.595994\pi$
$402$	0	0
$403$	9179.41	1.13464
$404$	3426.43	0.421958
$405$	0	0
$406$	0	0
$407$	−4165.56	−0.507319
$408$	0	0
$409$	597.257	0.0722065	0.0361033	−	0.999348i	$-0.488505\pi$
$409$	597.257	0.0722065	0.0361033	+	0.999348i	$0.488505\pi$
$410$	−120.729	−0.0145424
$411$	0	0
$412$	3041.62	0.363714
$413$	0	0
$414$	0	0
$415$	183.541	0.0217100
$416$	−22652.9	−2.66983
$417$	0	0
$418$	−3283.37	−0.384198
$419$	−9571.90	−1.11603	−0.558016	−	0.829830i	$-0.688437\pi$
$419$	−9571.90	−1.11603	−0.558016	+	0.829830i	$0.688437\pi$
$420$	0	0
$421$	−5954.33	−0.689303	−0.344651	−	0.938731i	$-0.612003\pi$
$421$	−5954.33	−0.689303	−0.344651	+	0.938731i	$0.612003\pi$
$422$	−23809.0	−2.74646
$423$	0	0
$424$	−25714.9	−2.94534
$425$	−14431.4	−1.64712
$426$	0	0
$427$	0	0
$428$	−9070.29	−1.02437
$429$	0	0
$430$	−426.747	−0.0478594
$431$	−2543.44	−0.284254	−0.142127	−	0.989848i	$-0.545394\pi$
$431$	−2543.44	−0.284254	−0.142127	+	0.989848i	$0.545394\pi$
$432$	0	0
$433$	−9781.07	−1.08556	−0.542781	−	0.839874i	$-0.682629\pi$
$433$	−9781.07	−1.08556	−0.542781	+	0.839874i	$0.682629\pi$
$434$	0	0
$435$	0	0
$436$	−18427.2	−2.02409
$437$	−1983.14	−0.217086
$438$	0	0
$439$	16820.3	1.82867	0.914336	−	0.404957i	$-0.132713\pi$
$439$	16820.3	1.82867	0.914336	+	0.404957i	$0.132713\pi$
$440$	−427.579	−0.0463273
$441$	0	0
$442$	−20551.1	−2.21157
$443$	3726.75	0.399691	0.199846	−	0.979827i	$-0.435956\pi$
$443$	3726.75	0.399691	0.199846	+	0.979827i	$0.435956\pi$
$444$	0	0
$445$	−42.0965	−0.00448442
$446$	18093.4	1.92096
$447$	0	0
$448$	0	0
$449$	−9287.05	−0.976131	−0.488065	−	0.872807i	$-0.662297\pi$
$449$	−9287.05	−0.976131	−0.488065	+	0.872807i	$0.662297\pi$
$450$	0	0
$451$	3155.02	0.329411
$452$	−29301.9	−3.04921
$453$	0	0
$454$	4360.26	0.450742
$455$	0	0
$456$	0	0
$457$	12019.4	1.23029	0.615147	−	0.788413i	$-0.289097\pi$
$457$	12019.4	1.23029	0.615147	+	0.788413i	$0.289097\pi$
$458$	1938.47	0.197770
$459$	0	0
$460$	−407.633	−0.0413173
$461$	9571.71	0.967026	0.483513	−	0.875337i	$-0.339361\pi$
$461$	9571.71	0.967026	0.483513	+	0.875337i	$0.339361\pi$
$462$	0	0
$463$	2265.15	0.227366	0.113683	−	0.993517i	$-0.463735\pi$
$463$	2265.15	0.227366	0.113683	+	0.993517i	$0.463735\pi$
$464$	55101.8	5.51301
$465$	0	0
$466$	−24435.5	−2.42908
$467$	5320.94	0.527246	0.263623	−	0.964626i	$-0.415082\pi$
$467$	5320.94	0.527246	0.263623	+	0.964626i	$0.415082\pi$
$468$	0	0
$469$	0	0
$470$	297.108	0.0291587
$471$	0	0
$472$	52639.2	5.13330
$473$	11152.2	1.08410
$474$	0	0
$475$	2642.91	0.255294
$476$	0	0
$477$	0	0
$478$	−8163.48	−0.781148
$479$	−3766.98	−0.359327	−0.179663	−	0.983728i	$-0.557501\pi$
$479$	−3766.98	−0.359327	−0.179663	+	0.983728i	$0.557501\pi$
$480$	0	0
$481$	4774.86	0.452630
$482$	856.871	0.0809739
$483$	0	0
$484$	−11420.1	−1.07251
$485$	168.413	0.0157675
$486$	0	0
$487$	8857.92	0.824211	0.412105	−	0.911136i	$-0.364794\pi$
$487$	8857.92	0.824211	0.412105	+	0.911136i	$0.364794\pi$
$488$	27950.9	2.59278
$489$	0	0
$490$	0	0
$491$	−17311.1	−1.59112	−0.795560	−	0.605874i	$-0.792823\pi$
$491$	−17311.1	−1.59112	−0.795560	+	0.605874i	$0.792823\pi$
$492$	0	0
$493$	26743.6	2.44315
$494$	3763.63	0.342781
$495$	0	0
$496$	−67039.8	−6.06891
$497$	0	0
$498$	0	0
$499$	−7030.45	−0.630714	−0.315357	−	0.948973i	$-0.602124\pi$
$499$	−7030.45	−0.630714	−0.315357	+	0.948973i	$0.602124\pi$
$500$	1086.67	0.0971945
$501$	0	0
$502$	−19109.0	−1.69896
$503$	−1519.74	−0.134716	−0.0673578	−	0.997729i	$-0.521457\pi$
$503$	−1519.74	−0.134716	−0.0673578	+	0.997729i	$0.521457\pi$
$504$	0	0
$505$	−31.2547	−0.00275409
$506$	14556.4	1.27887
$507$	0	0
$508$	21367.2	1.86617
$509$	19455.4	1.69419	0.847096	−	0.531440i	$-0.178349\pi$
$509$	19455.4	1.69419	0.847096	+	0.531440i	$0.178349\pi$
$510$	0	0
$511$	0	0
$512$	21639.1	1.86782
$513$	0	0
$514$	−25883.9	−2.22119
$515$	−27.7447	−0.00237394
$516$	0	0
$517$	−7764.35	−0.660495
$518$	0	0
$519$	0	0
$520$	490.122	0.0413332
$521$	−10224.0	−0.859731	−0.429866	−	0.902893i	$-0.641439\pi$
$521$	−10224.0	−0.859731	−0.429866	+	0.902893i	$0.641439\pi$
$522$	0	0
$523$	−16607.0	−1.38847	−0.694237	−	0.719746i	$-0.744258\pi$
$523$	−16607.0	−1.38847	−0.694237	+	0.719746i	$0.744258\pi$
$524$	−51801.0	−4.31858
$525$	0	0
$526$	−18375.9	−1.52324
$527$	−32537.8	−2.68950
$528$	0	0
$529$	−3375.01	−0.277390
$530$	370.235	0.0303434
$531$	0	0
$532$	0	0
$533$	−3616.52	−0.293900
$534$	0	0
$535$	82.7363	0.00668598
$536$	6582.48	0.530447
$537$	0	0
$538$	28441.0	2.27914
$539$	0	0
$540$	0	0
$541$	6582.54	0.523116	0.261558	−	0.965188i	$-0.415764\pi$
$541$	6582.54	0.523116	0.261558	+	0.965188i	$0.415764\pi$
$542$	12627.7	1.00075
$543$	0	0
$544$	80296.4	6.32846
$545$	168.087	0.0132111
$546$	0	0
$547$	3407.73	0.266369	0.133185	−	0.991091i	$-0.457480\pi$
$547$	3407.73	0.266369	0.133185	+	0.991091i	$0.457480\pi$
$548$	34988.1	2.72741
$549$	0	0
$550$	−19399.1	−1.50397
$551$	−4897.72	−0.378675
$552$	0	0
$553$	0	0
$554$	14237.4	1.09185
$555$	0	0
$556$	−45884.2	−3.49986
$557$	−9480.11	−0.721158	−0.360579	−	0.932729i	$-0.617421\pi$
$557$	−9480.11	−0.721158	−0.360579	+	0.932729i	$0.617421\pi$
$558$	0	0
$559$	−12783.5	−0.967233
$560$	0	0
$561$	0	0
$562$	−28792.5	−2.16110
$563$	21864.4	1.63673	0.818363	−	0.574702i	$-0.194882\pi$
$563$	21864.4	1.63673	0.818363	+	0.574702i	$0.194882\pi$
$564$	0	0
$565$	267.282	0.0199020
$566$	−31723.2	−2.35588
$567$	0	0
$568$	−6676.23	−0.493184
$569$	−1143.39	−0.0842414	−0.0421207	−	0.999113i	$-0.513411\pi$
$569$	−1143.39	−0.0842414	−0.0421207	+	0.999113i	$0.513411\pi$
$570$	0	0
$571$	10471.1	0.767429	0.383715	−	0.923452i	$-0.374645\pi$
$571$	10471.1	0.767429	0.383715	+	0.923452i	$0.374645\pi$
$572$	−20216.9	−1.47781
$573$	0	0
$574$	0	0
$575$	−11717.0	−0.849795
$576$	0	0
$577$	−4759.60	−0.343405	−0.171703	−	0.985149i	$-0.554927\pi$
$577$	−4759.60	−0.343405	−0.171703	+	0.985149i	$0.554927\pi$
$578$	46012.6	3.31119
$579$	0	0
$580$	−1006.72	−0.0720721
$581$	0	0
$582$	0	0
$583$	−9675.39	−0.687331
$584$	60663.3	4.29840
$585$	0	0
$586$	10812.7	0.762234
$587$	6307.45	0.443503	0.221751	−	0.975103i	$-0.428823\pi$
$587$	6307.45	0.443503	0.221751	+	0.975103i	$0.428823\pi$
$588$	0	0
$589$	5958.83	0.416858
$590$	−757.885	−0.0528841
$591$	0	0
$592$	−34872.2	−2.42101
$593$	−17731.9	−1.22793	−0.613963	−	0.789335i	$-0.710426\pi$
$593$	−17731.9	−1.22793	−0.613963	+	0.789335i	$0.710426\pi$
$594$	0	0
$595$	0	0
$596$	11448.2	0.786808
$597$	0	0
$598$	−16685.6	−1.14101
$599$	19110.1	1.30353	0.651766	−	0.758420i	$-0.274028\pi$
$599$	19110.1	1.30353	0.651766	+	0.758420i	$0.274028\pi$
$600$	0	0
$601$	13118.3	0.890357	0.445179	−	0.895442i	$-0.353140\pi$
$601$	13118.3	0.890357	0.445179	+	0.895442i	$0.353140\pi$
$602$	0	0
$603$	0	0
$604$	−18942.1	−1.27606
$605$	104.171	0.00700024
$606$	0	0
$607$	−17473.4	−1.16841	−0.584204	−	0.811607i	$-0.698593\pi$
$607$	−17473.4	−1.16841	−0.584204	+	0.811607i	$0.698593\pi$
$608$	−14705.1	−0.980876
$609$	0	0
$610$	−402.429	−0.0267112
$611$	8900.06	0.589293
$612$	0	0
$613$	3095.39	0.203951	0.101975	−	0.994787i	$-0.467484\pi$
$613$	3095.39	0.203951	0.101975	+	0.994787i	$0.467484\pi$
$614$	−5046.71	−0.331708
$615$	0	0
$616$	0	0
$617$	26334.8	1.71831	0.859157	−	0.511712i	$-0.170989\pi$
$617$	26334.8	1.71831	0.859157	+	0.511712i	$0.170989\pi$
$618$	0	0
$619$	2686.04	0.174412	0.0872061	−	0.996190i	$-0.472206\pi$
$619$	2686.04	0.174412	0.0872061	+	0.996190i	$0.472206\pi$
$620$	1224.83	0.0793395
$621$	0	0
$622$	55211.0	3.55910
$623$	0	0
$624$	0	0
$625$	15610.1	0.999048
$626$	−37306.8	−2.38192
$627$	0	0
$628$	129.240	0.00821216
$629$	−16925.2	−1.07290
$630$	0	0
$631$	22414.2	1.41409	0.707047	−	0.707167i	$-0.250027\pi$
$631$	22414.2	1.41409	0.707047	+	0.707167i	$0.250027\pi$
$632$	27543.9	1.73361
$633$	0	0
$634$	−41638.6	−2.60833
$635$	−194.904	−0.0121804
$636$	0	0
$637$	0	0
$638$	35949.6	2.23081
$639$	0	0
$640$	−952.222	−0.0588123
$641$	22139.1	1.36418	0.682092	−	0.731266i	$-0.261070\pi$
$641$	22139.1	1.36418	0.682092	+	0.731266i	$0.261070\pi$
$642$	0	0
$643$	−22523.7	−1.38141	−0.690707	−	0.723134i	$-0.742701\pi$
$643$	−22523.7	−1.38141	−0.690707	+	0.723134i	$0.742701\pi$
$644$	0	0
$645$	0	0
$646$	−13340.8	−0.812516
$647$	−11523.5	−0.700209	−0.350105	−	0.936711i	$-0.613854\pi$
$647$	−11523.5	−0.700209	−0.350105	+	0.936711i	$0.613854\pi$
$648$	0	0
$649$	19805.9	1.19792
$650$	22236.7	1.34184
$651$	0	0
$652$	−18184.4	−1.09226
$653$	461.937	0.0276830	0.0138415	−	0.999904i	$-0.495594\pi$
$653$	461.937	0.0276830	0.0138415	+	0.999904i	$0.495594\pi$
$654$	0	0
$655$	472.511	0.0281871
$656$	26412.5	1.57200
$657$	0	0
$658$	0	0
$659$	−14373.3	−0.849627	−0.424813	−	0.905281i	$-0.639660\pi$
$659$	−14373.3	−0.849627	−0.424813	+	0.905281i	$0.639660\pi$
$660$	0	0
$661$	−67.8490	−0.00399246	−0.00199623	−	0.999998i	$-0.500635\pi$
$661$	−67.8490	−0.00399246	−0.00199623	+	0.999998i	$0.500635\pi$
$662$	13562.8	0.796271
$663$	0	0
$664$	−69626.3	−4.06931
$665$	0	0
$666$	0	0
$667$	21713.4	1.26049
$668$	12373.8	0.716702
$669$	0	0
$670$	−94.7727	−0.00546476
$671$	10516.7	0.605056
$672$	0	0
$673$	21970.2	1.25838	0.629190	−	0.777252i	$-0.283387\pi$
$673$	21970.2	1.25838	0.629190	+	0.777252i	$0.283387\pi$
$674$	43123.4	2.46447
$675$	0	0
$676$	−24788.9	−1.41038
$677$	2993.29	0.169928	0.0849642	−	0.996384i	$-0.472922\pi$
$677$	2993.29	0.169928	0.0849642	+	0.996384i	$0.472922\pi$
$678$	0	0
$679$	0	0
$680$	−1737.31	−0.0979747
$681$	0	0
$682$	−43738.2	−2.45575
$683$	−5461.90	−0.305994	−0.152997	−	0.988227i	$-0.548892\pi$
$683$	−5461.90	−0.305994	−0.152997	+	0.988227i	$0.548892\pi$
$684$	0	0
$685$	−319.150	−0.0178016
$686$	0	0
$687$	0	0
$688$	93361.4	5.17350
$689$	11090.6	0.613236
$690$	0	0
$691$	15779.0	0.868685	0.434342	−	0.900748i	$-0.356981\pi$
$691$	15779.0	0.868685	0.434342	+	0.900748i	$0.356981\pi$
$692$	90070.0	4.94790
$693$	0	0
$694$	4857.84	0.265708
$695$	418.540	0.0228434
$696$	0	0
$697$	12819.3	0.696650
$698$	−38026.0	−2.06204
$699$	0	0
$700$	0	0
$701$	−8058.91	−0.434209	−0.217105	−	0.976148i	$-0.569661\pi$
$701$	−8058.91	−0.434209	−0.217105	+	0.976148i	$0.569661\pi$
$702$	0	0
$703$	3099.61	0.166293
$704$	53830.6	2.88184
$705$	0	0
$706$	63141.5	3.36595
$707$	0	0
$708$	0	0
$709$	26018.9	1.37823	0.689113	−	0.724654i	$-0.258001\pi$
$709$	26018.9	1.37823	0.689113	+	0.724654i	$0.258001\pi$
$710$	96.1225	0.00508086
$711$	0	0
$712$	15969.4	0.840558
$713$	−26417.7	−1.38759
$714$	0	0
$715$	184.412	0.00964560
$716$	20251.5	1.05703
$717$	0	0
$718$	46190.4	2.40085
$719$	4663.46	0.241889	0.120944	−	0.992659i	$-0.461408\pi$
$719$	4663.46	0.241889	0.120944	+	0.992659i	$0.461408\pi$
$720$	0	0
$721$	0	0
$722$	−35019.2	−1.80510
$723$	0	0
$724$	−26452.1	−1.35785
$725$	−28937.2	−1.48234
$726$	0	0
$727$	−35484.5	−1.81024	−0.905121	−	0.425154i	$-0.860220\pi$
$727$	−35484.5	−1.81024	−0.905121	+	0.425154i	$0.860220\pi$
$728$	0	0
$729$	0	0
$730$	−873.413	−0.0442828
$731$	45312.9	2.29269
$732$	0	0
$733$	15257.1	0.768805	0.384402	−	0.923166i	$-0.374408\pi$
$733$	15257.1	0.768805	0.384402	+	0.923166i	$0.374408\pi$
$734$	−37756.9	−1.89868
$735$	0	0
$736$	65193.4	3.26503
$737$	2476.70	0.123786
$738$	0	0
$739$	25753.7	1.28196	0.640978	−	0.767559i	$-0.278529\pi$
$739$	25753.7	1.28196	0.640978	+	0.767559i	$0.278529\pi$
$740$	637.123	0.0316501
$741$	0	0
$742$	0	0
$743$	5703.55	0.281619	0.140809	−	0.990037i	$-0.455029\pi$
$743$	5703.55	0.281619	0.140809	+	0.990037i	$0.455029\pi$
$744$	0	0
$745$	−104.427	−0.00513545
$746$	2315.31	0.113632
$747$	0	0
$748$	71661.7	3.50296
$749$	0	0
$750$	0	0
$751$	15450.9	0.750747	0.375373	−	0.926874i	$-0.377515\pi$
$751$	15450.9	0.750747	0.375373	+	0.926874i	$0.377515\pi$
$752$	−64999.7	−3.15199
$753$	0	0
$754$	−41208.0	−1.99033
$755$	172.784	0.00832879
$756$	0	0
$757$	12434.7	0.597025	0.298513	−	0.954406i	$-0.403509\pi$
$757$	12434.7	0.597025	0.298513	+	0.954406i	$0.403509\pi$
$758$	−53057.6	−2.54240
$759$	0	0
$760$	318.164	0.0151855
$761$	−2318.62	−0.110447	−0.0552234	−	0.998474i	$-0.517587\pi$
$761$	−2318.62	−0.110447	−0.0552234	+	0.998474i	$0.517587\pi$
$762$	0	0
$763$	0	0
$764$	−39896.1	−1.88926
$765$	0	0
$766$	12093.2	0.570425
$767$	−22702.9	−1.06878
$768$	0	0
$769$	−23104.3	−1.08344	−0.541718	−	0.840560i	$-0.682226\pi$
$769$	−23104.3	−1.08344	−0.541718	+	0.840560i	$0.682226\pi$
$770$	0	0
$771$	0	0
$772$	−17950.2	−0.836841
$773$	−12433.6	−0.578532	−0.289266	−	0.957249i	$-0.593411\pi$
$773$	−12433.6	−0.578532	−0.289266	+	0.957249i	$0.593411\pi$
$774$	0	0
$775$	35206.6	1.63182
$776$	−63887.6	−2.95545
$777$	0	0
$778$	−41867.1	−1.92932
$779$	−2347.67	−0.107977
$780$	0	0
$781$	−2511.98	−0.115090
$782$	59144.5	2.70461
$783$	0	0
$784$	0	0
$785$	−1.17888	−5.36002e−5 0
$786$	0	0
$787$	24085.4	1.09092	0.545458	−	0.838138i	$-0.316356\pi$
$787$	24085.4	1.09092	0.545458	+	0.838138i	$0.316356\pi$
$788$	−106386.	−4.80943
$789$	0	0
$790$	−396.570	−0.0178599
$791$	0	0
$792$	0	0
$793$	−12055.0	−0.539831
$794$	−27189.8	−1.21528
$795$	0	0
$796$	32358.6	1.44085
$797$	36596.8	1.62651	0.813253	−	0.581910i	$-0.197694\pi$
$797$	36596.8	1.62651	0.813253	+	0.581910i	$0.197694\pi$
$798$	0	0
$799$	−31547.6	−1.39684
$800$	−86882.4	−3.83970
$801$	0	0
$802$	−26053.8	−1.14712
$803$	22825.0	1.00308
$804$	0	0
$805$	0	0
$806$	50135.9	2.19102
$807$	0	0
$808$	11856.5	0.516226
$809$	−20746.3	−0.901610	−0.450805	−	0.892622i	$-0.648863\pi$
$809$	−20746.3	−0.901610	−0.450805	+	0.892622i	$0.648863\pi$
$810$	0	0
$811$	8538.03	0.369680	0.184840	−	0.982769i	$-0.440823\pi$
$811$	8538.03	0.369680	0.184840	+	0.982769i	$0.440823\pi$
$812$	0	0
$813$	0	0
$814$	−22751.4	−0.979650
$815$	165.872	0.00712915
$816$	0	0
$817$	−8298.42	−0.355355
$818$	3262.09	0.139433
$819$	0	0
$820$	−482.562	−0.0205510
$821$	20949.8	0.890564	0.445282	−	0.895390i	$-0.353103\pi$
$821$	20949.8	0.890564	0.445282	+	0.895390i	$0.353103\pi$
$822$	0	0
$823$	−10130.5	−0.429073	−0.214536	−	0.976716i	$-0.568824\pi$
$823$	−10130.5	−0.429073	−0.214536	+	0.976716i	$0.568824\pi$
$824$	10525.0	0.444969
$825$	0	0
$826$	0	0
$827$	25327.3	1.06495	0.532476	−	0.846445i	$-0.321262\pi$
$827$	25327.3	1.06495	0.532476	+	0.846445i	$0.321262\pi$
$828$	0	0
$829$	19975.9	0.836901	0.418451	−	0.908240i	$-0.362573\pi$
$829$	19975.9	0.836901	0.418451	+	0.908240i	$0.362573\pi$
$830$	1002.46	0.0419227
$831$	0	0
$832$	−61704.6	−2.57118
$833$	0	0
$834$	0	0
$835$	−112.870	−0.00467787
$836$	−13123.8	−0.542939
$837$	0	0
$838$	−52279.6	−2.15510
$839$	−3888.98	−0.160027	−0.0800134	−	0.996794i	$-0.525496\pi$
$839$	−3888.98	−0.160027	−0.0800134	+	0.996794i	$0.525496\pi$
$840$	0	0
$841$	29236.1	1.19874
$842$	−32521.3	−1.33107
$843$	0	0
$844$	−95166.2	−3.88123
$845$	226.116	0.00920547
$846$	0	0
$847$	0	0
$848$	−80998.1	−3.28006
$849$	0	0
$850$	−78821.2	−3.18064
$851$	−13741.7	−0.553537
$852$	0	0
$853$	32093.5	1.28823	0.644116	−	0.764928i	$-0.277225\pi$
$853$	32093.5	1.28823	0.644116	+	0.764928i	$0.277225\pi$
$854$	0	0
$855$	0	0
$856$	−31386.1	−1.25322
$857$	−29879.8	−1.19098	−0.595492	−	0.803361i	$-0.703043\pi$
$857$	−29879.8	−1.19098	−0.595492	+	0.803361i	$0.703043\pi$
$858$	0	0
$859$	−32988.3	−1.31030	−0.655150	−	0.755499i	$-0.727394\pi$
$859$	−32988.3	−1.31030	−0.655150	+	0.755499i	$0.727394\pi$
$860$	−1705.73	−0.0676337
$861$	0	0
$862$	−13891.7	−0.548903
$863$	26716.8	1.05382	0.526912	−	0.849920i	$-0.323350\pi$
$863$	26716.8	1.05382	0.526912	+	0.849920i	$0.323350\pi$
$864$	0	0
$865$	−821.590	−0.0322947
$866$	−53422.1	−2.09626
$867$	0	0
$868$	0	0
$869$	10363.6	0.404558
$870$	0	0
$871$	−2838.98	−0.110442
$872$	−63763.9	−2.47628
$873$	0	0
$874$	−10831.5	−0.419199
$875$	0	0
$876$	0	0
$877$	−17965.7	−0.691742	−0.345871	−	0.938282i	$-0.612417\pi$
$877$	−17965.7	−0.691742	−0.345871	+	0.938282i	$0.612417\pi$
$878$	91868.6	3.53122
$879$	0	0
$880$	−1346.81	−0.0515921
$881$	−15852.9	−0.606239	−0.303119	−	0.952953i	$-0.598028\pi$
$881$	−15852.9	−0.606239	−0.303119	+	0.952953i	$0.598028\pi$
$882$	0	0
$883$	−22050.7	−0.840392	−0.420196	−	0.907433i	$-0.638039\pi$
$883$	−22050.7	−0.840392	−0.420196	+	0.907433i	$0.638039\pi$
$884$	−82143.9	−3.12534
$885$	0	0
$886$	20354.7	0.771817
$887$	−27772.8	−1.05132	−0.525658	−	0.850696i	$-0.676181\pi$
$887$	−27772.8	−1.05132	−0.525658	+	0.850696i	$0.676181\pi$
$888$	0	0
$889$	0	0
$890$	−229.922	−0.00865956
$891$	0	0
$892$	72320.5	2.71465
$893$	5777.50	0.216502
$894$	0	0
$895$	−184.728	−0.00689918
$896$	0	0
$897$	0	0
$898$	−50723.8	−1.88494
$899$	−65243.2	−2.42045
$900$	0	0
$901$	−39312.4	−1.45359
$902$	17232.1	0.636103
$903$	0	0
$904$	−101394.	−3.73043
$905$	241.288	0.00886262
$906$	0	0
$907$	−5896.03	−0.215848	−0.107924	−	0.994159i	$-0.534420\pi$
$907$	−5896.03	−0.215848	−0.107924	+	0.994159i	$0.534420\pi$
$908$	17428.2	0.636978
$909$	0	0
$910$	0	0
$911$	42197.8	1.53466	0.767331	−	0.641251i	$-0.221584\pi$
$911$	42197.8	1.53466	0.767331	+	0.641251i	$0.221584\pi$
$912$	0	0
$913$	−26197.4	−0.949623
$914$	65647.4	2.37574
$915$	0	0
$916$	7748.17	0.279483
$917$	0	0
$918$	0	0
$919$	−40928.7	−1.46911	−0.734555	−	0.678549i	$-0.762609\pi$
$919$	−40928.7	−1.46911	−0.734555	+	0.678549i	$0.762609\pi$
$920$	−1410.54	−0.0505479
$921$	0	0
$922$	52278.6	1.86736
$923$	2879.41	0.102684
$924$	0	0
$925$	18313.4	0.650964
$926$	12371.7	0.439050
$927$	0	0
$928$	161007.	5.69537
$929$	21478.9	0.758558	0.379279	−	0.925282i	$-0.376172\pi$
$929$	21478.9	0.758558	0.379279	+	0.925282i	$0.376172\pi$
$930$	0	0
$931$	0	0
$932$	−97670.3	−3.43272
$933$	0	0
$934$	29061.9	1.01813
$935$	−653.675	−0.0228636
$936$	0	0
$937$	−19560.1	−0.681964	−0.340982	−	0.940070i	$-0.610759\pi$
$937$	−19560.1	−0.681964	−0.340982	+	0.940070i	$0.610759\pi$
$938$	0	0
$939$	0	0
$940$	1187.56	0.0412063
$941$	48602.4	1.68373	0.841867	−	0.539685i	$-0.181457\pi$
$941$	48602.4	1.68373	0.841867	+	0.539685i	$0.181457\pi$
$942$	0	0
$943$	10408.1	0.359421
$944$	165806.	5.71666
$945$	0	0
$946$	60911.0	2.09343
$947$	−28778.0	−0.987496	−0.493748	−	0.869605i	$-0.664373\pi$
$947$	−28778.0	−0.987496	−0.493748	+	0.869605i	$0.664373\pi$
$948$	0	0
$949$	−26163.7	−0.894951
$950$	14435.0	0.492982
$951$	0	0
$952$	0	0
$953$	−26009.2	−0.884072	−0.442036	−	0.896997i	$-0.645744\pi$
$953$	−26009.2	−0.884072	−0.442036	+	0.896997i	$0.645744\pi$
$954$	0	0
$955$	363.920	0.0123311
$956$	−32629.9	−1.10390
$957$	0	0
$958$	−20574.4	−0.693872
$959$	0	0
$960$	0	0
$961$	49587.5	1.66451
$962$	26079.3	0.874043
$963$	0	0
$964$	3424.97	0.114430
$965$	163.736	0.00546201
$966$	0	0
$967$	−13284.5	−0.441779	−0.220889	−	0.975299i	$-0.570896\pi$
$967$	−13284.5	−0.441779	−0.220889	+	0.975299i	$0.570896\pi$
$968$	−39517.2	−1.31212
$969$	0	0
$970$	919.836	0.0304476
$971$	−44248.6	−1.46242	−0.731208	−	0.682155i	$-0.761043\pi$
$971$	−44248.6	−1.46242	−0.731208	+	0.682155i	$0.761043\pi$
$972$	0	0
$973$	0	0
$974$	48380.0	1.59158
$975$	0	0
$976$	88041.2	2.88743
$977$	48967.4	1.60349	0.801744	−	0.597668i	$-0.203906\pi$
$977$	48967.4	1.60349	0.801744	+	0.597668i	$0.203906\pi$
$978$	0	0
$979$	6008.58	0.196154
$980$	0	0
$981$	0	0
$982$	−94549.7	−3.07251
$983$	−49856.3	−1.61767	−0.808835	−	0.588035i	$-0.799902\pi$
$983$	−49856.3	−1.61767	−0.808835	+	0.588035i	$0.799902\pi$
$984$	0	0
$985$	970.414	0.0313908
$986$	146068.	4.71780
$987$	0	0
$988$	15043.5	0.484410
$989$	36790.0	1.18286
$990$	0	0
$991$	48648.1	1.55939	0.779696	−	0.626158i	$-0.215373\pi$
$991$	48648.1	1.55939	0.779696	+	0.626158i	$0.215373\pi$
$992$	−195889.	−6.26965
$993$	0	0
$994$	0	0
$995$	−295.165	−0.00940437
$996$	0	0
$997$	−9709.15	−0.308417	−0.154209	−	0.988038i	$-0.549283\pi$
$997$	−9709.15	−0.308417	−0.154209	+	0.988038i	$0.549283\pi$
$998$	−38398.8	−1.21793
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bn.1.8		8
3.2	odd	2	inner	1323.4.a.bn.1.1		8
7.3	odd	6		189.4.e.h.163.1	yes	16
7.5	odd	6		189.4.e.h.109.1	✓	16
7.6	odd	2		1323.4.a.bo.1.8		8
21.5	even	6		189.4.e.h.109.8	yes	16
21.17	even	6		189.4.e.h.163.8	yes	16
21.20	even	2		1323.4.a.bo.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.h.109.1	✓	16	7.5	odd	6
189.4.e.h.109.8	yes	16	21.5	even	6
189.4.e.h.163.1	yes	16	7.3	odd	6
189.4.e.h.163.8	yes	16	21.17	even	6
1323.4.a.bn.1.1		8	3.2	odd	2	inner
1323.4.a.bn.1.8		8	1.1	even	1	trivial
1323.4.a.bo.1.1		8	21.20	even	2
1323.4.a.bo.1.8		8	7.6	odd	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)

\(f(q)\)	\(=\)	\(q+5.46178 q^{2} +21.8311 q^{4} -0.199136 q^{5} +75.5424 q^{8} -1.08764 q^{10} +28.4233 q^{11} -32.5809 q^{13} +237.948 q^{16} +115.488 q^{17} -21.1500 q^{19} -4.34736 q^{20} +155.242 q^{22} +93.7656 q^{23} -124.960 q^{25} -177.950 q^{26} +231.571 q^{29} -281.742 q^{31} +695.280 q^{32} +630.770 q^{34} -146.554 q^{37} -115.517 q^{38} -15.0432 q^{40} +111.001 q^{41} +392.361 q^{43} +620.513 q^{44} +512.128 q^{46} -273.168 q^{47} -682.506 q^{50} -711.277 q^{52} -340.403 q^{53} -5.66011 q^{55} +1264.79 q^{58} +696.817 q^{59} +370.002 q^{61} -1538.81 q^{62} +1893.89 q^{64} +6.48804 q^{65} +87.1362 q^{67} +2521.23 q^{68} -88.3772 q^{71} +803.036 q^{73} -800.446 q^{74} -461.727 q^{76} +364.616 q^{79} -47.3840 q^{80} +606.264 q^{82} -921.684 q^{83} -22.9978 q^{85} +2142.99 q^{86} +2147.17 q^{88} +211.396 q^{89} +2047.01 q^{92} -1491.99 q^{94} +4.21172 q^{95} -845.718 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 48 q^{4} - 44 q^{10} + 84 q^{13} + 156 q^{16} - 12 q^{19} + 224 q^{22} + 408 q^{25} - 800 q^{31} + 948 q^{34} + 692 q^{37} - 96 q^{40} + 1456 q^{43} + 1524 q^{46} - 1972 q^{52} + 1280 q^{55} + 2372 q^{58}+ \cdots - 584 q^{97}+O(q^{100}) \) 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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