LMFDB - Embedded newform 1323.4.a.bo.1.4

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			1.4
Root			$-0.256959$ 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)

$f(q)$	$=$	$q-1.76924 q^{2} -4.86977 q^{4} +13.0512 q^{5} +22.7698 q^{8} +O(q^{10})$	$q-1.76924 q^{2} -4.86977 q^{4} +13.0512 q^{5} +22.7698 q^{8} -23.0908 q^{10} +52.9410 q^{11} -71.7001 q^{13} -1.32714 q^{16} +106.573 q^{17} +53.9285 q^{19} -63.5565 q^{20} -93.6656 q^{22} +18.6589 q^{23} +45.3345 q^{25} +126.855 q^{26} +261.806 q^{29} -122.326 q^{31} -179.810 q^{32} -188.554 q^{34} +278.137 q^{37} -95.4127 q^{38} +297.174 q^{40} -31.3788 q^{41} -347.239 q^{43} -257.811 q^{44} -33.0121 q^{46} +542.218 q^{47} -80.2078 q^{50} +349.163 q^{52} -257.057 q^{53} +690.945 q^{55} -463.198 q^{58} +315.620 q^{59} +138.804 q^{61} +216.425 q^{62} +328.745 q^{64} -935.775 q^{65} +397.901 q^{67} -518.987 q^{68} -843.419 q^{71} -872.281 q^{73} -492.093 q^{74} -262.620 q^{76} -554.912 q^{79} -17.3208 q^{80} +55.5168 q^{82} -297.114 q^{83} +1390.91 q^{85} +614.352 q^{86} +1205.46 q^{88} +102.484 q^{89} -90.8644 q^{92} -959.317 q^{94} +703.833 q^{95} -515.437 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 48 q^{4}+O(q^{10}) $ 8 * q + 48 * q^4	$ 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}+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

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 $

$2$	−1.76924	−0.625523	−0.312761	−	0.949832i	$-0.601254\pi$
$2$	−1.76924	−0.625523	−0.312761	+	0.949832i	$0.601254\pi$
$3$	0	0
$3$	0	0
$4$	−4.86977	−0.608722
$5$	13.0512	1.16734	0.583669	−	0.811992i	$-0.301617\pi$
$5$	13.0512	1.16734	0.583669	+	0.811992i	$0.301617\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.7698	1.00629
$9$	0	0
$10$	−23.0908	−0.730196
$11$	52.9410	1.45112	0.725560	−	0.688159i	$-0.241581\pi$
$11$	52.9410	1.45112	0.725560	+	0.688159i	$0.241581\pi$
$12$	0	0
$13$	−71.7001	−1.52970	−0.764848	−	0.644211i	$-0.777186\pi$
$13$	−71.7001	−1.52970	−0.764848	+	0.644211i	$0.777186\pi$
$14$	0	0
$15$	0	0
$16$	−1.32714	−0.0207366
$17$	106.573	1.52046	0.760229	−	0.649655i	$-0.225087\pi$
$17$	106.573	1.52046	0.760229	+	0.649655i	$0.225087\pi$
$18$	0	0
$19$	53.9285	0.651160	0.325580	−	0.945514i	$-0.394440\pi$
$19$	53.9285	0.651160	0.325580	+	0.945514i	$0.394440\pi$
$20$	−63.5565	−0.710583
$21$	0	0
$22$	−93.6656	−0.907708
$23$	18.6589	0.169158	0.0845792	−	0.996417i	$-0.473045\pi$
$23$	18.6589	0.169158	0.0845792	+	0.996417i	$0.473045\pi$
$24$	0	0
$25$	45.3345	0.362676
$26$	126.855	0.956859
$27$	0	0
$28$	0	0
$29$	261.806	1.67642	0.838208	−	0.545350i	$-0.183603\pi$
$29$	261.806	1.67642	0.838208	+	0.545350i	$0.183603\pi$
$30$	0	0
$31$	−122.326	−0.708725	−0.354362	−	0.935108i	$-0.615302\pi$
$31$	−122.326	−0.708725	−0.354362	+	0.935108i	$0.615302\pi$
$32$	−179.810	−0.993320
$33$	0	0
$34$	−188.554	−0.951081
$35$	0	0
$36$	0	0
$37$	278.137	1.23582	0.617912	−	0.786247i	$-0.287979\pi$
$37$	278.137	1.23582	0.617912	+	0.786247i	$0.287979\pi$
$38$	−95.4127	−0.407315
$39$	0	0
$40$	297.174	1.17468
$41$	−31.3788	−0.119525	−0.0597627	−	0.998213i	$-0.519034\pi$
$41$	−31.3788	−0.119525	−0.0597627	+	0.998213i	$0.519034\pi$
$42$	0	0
$43$	−347.239	−1.23148	−0.615739	−	0.787951i	$-0.711142\pi$
$43$	−347.239	−1.23148	−0.615739	+	0.787951i	$0.711142\pi$
$44$	−257.811	−0.883328
$45$	0	0
$46$	−33.0121	−0.105812
$47$	542.218	1.68278	0.841390	−	0.540428i	$-0.181738\pi$
$47$	542.218	1.68278	0.841390	+	0.540428i	$0.181738\pi$
$48$	0	0
$49$	0	0
$50$	−80.2078	−0.226862
$51$	0	0
$52$	349.163	0.931158
$53$	−257.057	−0.666216	−0.333108	−	0.942889i	$-0.608097\pi$
$53$	−257.057	−0.666216	−0.333108	+	0.942889i	$0.608097\pi$
$54$	0	0
$55$	690.945	1.69395
$56$	0	0
$57$	0	0
$58$	−463.198	−1.04864
$59$	315.620	0.696445	0.348223	−	0.937412i	$-0.386785\pi$
$59$	315.620	0.696445	0.348223	+	0.937412i	$0.386785\pi$
$60$	0	0
$61$	138.804	0.291344	0.145672	−	0.989333i	$-0.453466\pi$
$61$	138.804	0.291344	0.145672	+	0.989333i	$0.453466\pi$
$62$	216.425	0.443323
$63$	0	0
$64$	328.745	0.642081
$65$	−935.775	−1.78567
$66$	0	0
$67$	397.901	0.725542	0.362771	−	0.931878i	$-0.381831\pi$
$67$	397.901	0.725542	0.362771	+	0.931878i	$0.381831\pi$
$68$	−518.987	−0.925535
$69$	0	0
$70$	0	0
$71$	−843.419	−1.40979	−0.704897	−	0.709309i	$-0.749007\pi$
$71$	−843.419	−1.40979	−0.704897	+	0.709309i	$0.749007\pi$
$72$	0	0
$73$	−872.281	−1.39853	−0.699265	−	0.714862i	$-0.746489\pi$
$73$	−872.281	−1.39853	−0.699265	+	0.714862i	$0.746489\pi$
$74$	−492.093	−0.773036
$75$	0	0
$76$	−262.620	−0.396375
$77$	0	0
$78$	0	0
$79$	−554.912	−0.790285	−0.395143	−	0.918620i	$-0.629305\pi$
$79$	−554.912	−0.790285	−0.395143	+	0.918620i	$0.629305\pi$
$80$	−17.3208	−0.0242065
$81$	0	0
$82$	55.5168	0.0747659
$83$	−297.114	−0.392921	−0.196461	−	0.980512i	$-0.562945\pi$
$83$	−297.114	−0.392921	−0.196461	+	0.980512i	$0.562945\pi$
$84$	0	0
$85$	1390.91	1.77489
$86$	614.352	0.770317
$87$	0	0
$88$	1205.46	1.46025
$89$	102.484	0.122059	0.0610295	−	0.998136i	$-0.480562\pi$
$89$	102.484	0.122059	0.0610295	+	0.998136i	$0.480562\pi$
$90$	0	0
$91$	0	0
$92$	−90.8644	−0.102970
$93$	0	0
$94$	−959.317	−1.05262
$95$	703.833	0.760124
$96$	0	0
$97$	−515.437	−0.539533	−0.269766	−	0.962926i	$-0.586947\pi$
$97$	−515.437	−0.539533	−0.269766	+	0.962926i	$0.586947\pi$
$98$	0	0
$99$	0	0
$100$	−220.769	−0.220769
$101$	1101.95	1.08563	0.542813	−	0.839854i	$-0.317359\pi$
$101$	1101.95	1.08563	0.542813	+	0.839854i	$0.317359\pi$
$102$	0	0
$103$	−353.628	−0.338291	−0.169146	−	0.985591i	$-0.554101\pi$
$103$	−353.628	−0.338291	−0.169146	+	0.985591i	$0.554101\pi$
$104$	−1632.60	−1.53932
$105$	0	0
$106$	454.796	0.416733
$107$	−310.663	−0.280682	−0.140341	−	0.990103i	$-0.544820\pi$
$107$	−310.663	−0.280682	−0.140341	+	0.990103i	$0.544820\pi$
$108$	0	0
$109$	−514.984	−0.452537	−0.226268	−	0.974065i	$-0.572653\pi$
$109$	−514.984	−0.452537	−0.226268	+	0.974065i	$0.572653\pi$
$110$	−1222.45	−1.05960
$111$	0	0
$112$	0	0
$113$	1106.59	0.921233	0.460616	−	0.887599i	$-0.347628\pi$
$113$	1106.59	0.921233	0.460616	+	0.887599i	$0.347628\pi$
$114$	0	0
$115$	243.521	0.197465
$116$	−1274.93	−1.02047
$117$	0	0
$118$	−558.410	−0.435642
$119$	0	0
$120$	0	0
$121$	1471.75	1.10575
$122$	−245.578	−0.182242
$123$	0	0
$124$	595.701	0.431416
$125$	−1039.73	−0.743972
$126$	0	0
$127$	1591.62	1.11207	0.556037	−	0.831158i	$-0.312321\pi$
$127$	1591.62	1.11207	0.556037	+	0.831158i	$0.312321\pi$
$128$	856.850	0.591684
$129$	0	0
$130$	1655.61	1.11698
$131$	−817.256	−0.545068	−0.272534	−	0.962146i	$-0.587862\pi$
$131$	−817.256	−0.545068	−0.272534	+	0.962146i	$0.587862\pi$
$132$	0	0
$133$	0	0
$134$	−703.984	−0.453843
$135$	0	0
$136$	2426.65	1.53002
$137$	1028.13	0.641164	0.320582	−	0.947221i	$-0.396122\pi$
$137$	1028.13	0.641164	0.320582	+	0.947221i	$0.396122\pi$
$138$	0	0
$139$	−607.307	−0.370583	−0.185292	−	0.982684i	$-0.559323\pi$
$139$	−607.307	−0.370583	−0.185292	+	0.982684i	$0.559323\pi$
$140$	0	0
$141$	0	0
$142$	1492.21	0.881858
$143$	−3795.88	−2.21977
$144$	0	0
$145$	3416.88	1.95694
$146$	1543.28	0.874813
$147$	0	0
$148$	−1354.47	−0.752273
$149$	−1220.06	−0.670814	−0.335407	−	0.942073i	$-0.608874\pi$
$149$	−1220.06	−0.670814	−0.335407	+	0.942073i	$0.608874\pi$
$150$	0	0
$151$	1953.44	1.05277	0.526386	−	0.850246i	$-0.323547\pi$
$151$	1953.44	1.05277	0.526386	+	0.850246i	$0.323547\pi$
$152$	1227.94	0.655257
$153$	0	0
$154$	0	0
$155$	−1596.51	−0.827320
$156$	0	0
$157$	−1530.79	−0.778154	−0.389077	−	0.921205i	$-0.627206\pi$
$157$	−1530.79	−0.778154	−0.389077	+	0.921205i	$0.627206\pi$
$158$	981.776	0.494341
$159$	0	0
$160$	−2346.74	−1.15954
$161$	0	0
$162$	0	0
$163$	2283.20	1.09714	0.548570	−	0.836105i	$-0.315173\pi$
$163$	2283.20	1.09714	0.548570	+	0.836105i	$0.315173\pi$
$164$	152.808	0.0727577
$165$	0	0
$166$	525.667	0.245781
$167$	1071.13	0.496325	0.248162	−	0.968718i	$-0.420173\pi$
$167$	1071.13	0.496325	0.248162	+	0.968718i	$0.420173\pi$
$168$	0	0
$169$	2943.91	1.33997
$170$	−2460.86	−1.11023
$171$	0	0
$172$	1690.98	0.749627
$173$	3658.87	1.60797	0.803985	−	0.594649i	$-0.202709\pi$
$173$	3658.87	1.60797	0.803985	+	0.594649i	$0.202709\pi$
$174$	0	0
$175$	0	0
$176$	−70.2601	−0.0300912
$177$	0	0
$178$	−181.319	−0.0763507
$179$	810.233	0.338322	0.169161	−	0.985588i	$-0.445894\pi$
$179$	810.233	0.338322	0.169161	+	0.985588i	$0.445894\pi$
$180$	0	0
$181$	−1730.05	−0.710462	−0.355231	−	0.934779i	$-0.615598\pi$
$181$	−1730.05	−0.710462	−0.355231	+	0.934779i	$0.615598\pi$
$182$	0	0
$183$	0	0
$184$	424.858	0.170223
$185$	3630.03	1.44262
$186$	0	0
$187$	5642.09	2.20637
$188$	−2640.48	−1.02434
$189$	0	0
$190$	−1245.25	−0.475474
$191$	1075.60	0.407474	0.203737	−	0.979026i	$-0.434691\pi$
$191$	1075.60	0.407474	0.203737	+	0.979026i	$0.434691\pi$
$192$	0	0
$193$	4493.72	1.67598	0.837992	−	0.545683i	$-0.183730\pi$
$193$	4493.72	1.67598	0.837992	+	0.545683i	$0.183730\pi$
$194$	911.934	0.337490
$195$	0	0
$196$	0	0
$197$	−101.961	−0.0368753	−0.0184377	−	0.999830i	$-0.505869\pi$
$197$	−101.961	−0.0368753	−0.0184377	+	0.999830i	$0.505869\pi$
$198$	0	0
$199$	−535.588	−0.190788	−0.0953940	−	0.995440i	$-0.530411\pi$
$199$	−535.588	−0.190788	−0.0953940	+	0.995440i	$0.530411\pi$
$200$	1032.26	0.364958
$201$	0	0
$202$	−1949.62	−0.679083
$203$	0	0
$204$	0	0
$205$	−409.532	−0.139527
$206$	625.654	0.211609
$207$	0	0
$208$	95.1561	0.0317206
$209$	2855.03	0.944912
$210$	0	0
$211$	1247.47	0.407012	0.203506	−	0.979074i	$-0.434766\pi$
$211$	1247.47	0.407012	0.203506	+	0.979074i	$0.434766\pi$
$212$	1251.81	0.405540
$213$	0	0
$214$	549.639	0.175573
$215$	−4531.90	−1.43755
$216$	0	0
$217$	0	0
$218$	911.133	0.283072
$219$	0	0
$220$	−3364.75	−1.03114
$221$	−7641.31	−2.32584
$222$	0	0
$223$	2573.28	0.772734	0.386367	−	0.922345i	$-0.373730\pi$
$223$	2573.28	0.772734	0.386367	+	0.922345i	$0.373730\pi$
$224$	0	0
$225$	0	0
$226$	−1957.83	−0.576252
$227$	1488.83	0.435318	0.217659	−	0.976025i	$-0.430158\pi$
$227$	1488.83	0.435318	0.217659	+	0.976025i	$0.430158\pi$
$228$	0	0
$229$	−3210.39	−0.926413	−0.463207	−	0.886250i	$-0.653301\pi$
$229$	−3210.39	−0.926413	−0.463207	+	0.886250i	$0.653301\pi$
$230$	−430.848	−0.123519
$231$	0	0
$232$	5961.26	1.68696
$233$	2927.69	0.823172	0.411586	−	0.911371i	$-0.364975\pi$
$233$	2927.69	0.823172	0.411586	+	0.911371i	$0.364975\pi$
$234$	0	0
$235$	7076.61	1.96437
$236$	−1537.00	−0.423941
$237$	0	0
$238$	0	0
$239$	4263.99	1.15404	0.577019	−	0.816731i	$-0.304216\pi$
$239$	4263.99	1.15404	0.577019	+	0.816731i	$0.304216\pi$
$240$	0	0
$241$	−4316.09	−1.15363	−0.576813	−	0.816876i	$-0.695704\pi$
$241$	−4316.09	−1.15363	−0.576813	+	0.816876i	$0.695704\pi$
$242$	−2603.89	−0.691671
$243$	0	0
$244$	−675.943	−0.177348
$245$	0	0
$246$	0	0
$247$	−3866.68	−0.996077
$248$	−2785.34	−0.713184
$249$	0	0
$250$	1839.54	0.465371
$251$	2107.60	0.530001	0.265001	−	0.964248i	$-0.414628\pi$
$251$	2107.60	0.530001	0.265001	+	0.964248i	$0.414628\pi$
$252$	0	0
$253$	987.819	0.245469
$254$	−2815.96	−0.695627
$255$	0	0
$256$	−4145.94	−1.01219
$257$	2820.63	0.684614	0.342307	−	0.939588i	$-0.388792\pi$
$257$	2820.63	0.684614	0.342307	+	0.939588i	$0.388792\pi$
$258$	0	0
$259$	0	0
$260$	4557.01	1.08698
$261$	0	0
$262$	1445.93	0.340953
$263$	5952.43	1.39560	0.697799	−	0.716293i	$-0.254163\pi$
$263$	5952.43	1.39560	0.697799	+	0.716293i	$0.254163\pi$
$264$	0	0
$265$	−3354.90	−0.777698
$266$	0	0
$267$	0	0
$268$	−1937.69	−0.441653
$269$	6643.01	1.50569	0.752847	−	0.658195i	$-0.228680\pi$
$269$	6643.01	1.50569	0.752847	+	0.658195i	$0.228680\pi$
$270$	0	0
$271$	961.827	0.215597	0.107799	−	0.994173i	$-0.465620\pi$
$271$	961.827	0.215597	0.107799	+	0.994173i	$0.465620\pi$
$272$	−141.437	−0.0315291
$273$	0	0
$274$	−1819.02	−0.401062
$275$	2400.05	0.526286
$276$	0	0
$277$	−5600.12	−1.21472	−0.607362	−	0.794425i	$-0.707772\pi$
$277$	−5600.12	−1.21472	−0.607362	+	0.794425i	$0.707772\pi$
$278$	1074.47	0.231808
$279$	0	0
$280$	0	0
$281$	−1044.12	−0.221661	−0.110831	−	0.993839i	$-0.535351\pi$
$281$	−1044.12	−0.221661	−0.110831	+	0.993839i	$0.535351\pi$
$282$	0	0
$283$	2323.98	0.488149	0.244074	−	0.969756i	$-0.421516\pi$
$283$	2323.98	0.488149	0.244074	+	0.969756i	$0.421516\pi$
$284$	4107.26	0.858172
$285$	0	0
$286$	6715.84	1.38852
$287$	0	0
$288$	0	0
$289$	6444.83	1.31179
$290$	−6045.31	−1.22411
$291$	0	0
$292$	4247.81	0.851316
$293$	4140.37	0.825540	0.412770	−	0.910835i	$-0.364561\pi$
$293$	4140.37	0.825540	0.412770	+	0.910835i	$0.364561\pi$
$294$	0	0
$295$	4119.23	0.812986
$296$	6333.13	1.24360
$297$	0	0
$298$	2158.59	0.419609
$299$	−1337.84	−0.258761
$300$	0	0
$301$	0	0
$302$	−3456.11	−0.658532
$303$	0	0
$304$	−71.5706	−0.0135028
$305$	1811.56	0.340097
$306$	0	0
$307$	3074.68	0.571600	0.285800	−	0.958289i	$-0.407741\pi$
$307$	3074.68	0.571600	0.285800	+	0.958289i	$0.407741\pi$
$308$	0	0
$309$	0	0
$310$	2824.62	0.517508
$311$	−2318.34	−0.422704	−0.211352	−	0.977410i	$-0.567787\pi$
$311$	−2318.34	−0.422704	−0.211352	+	0.977410i	$0.567787\pi$
$312$	0	0
$313$	1895.91	0.342374	0.171187	−	0.985239i	$-0.445240\pi$
$313$	1895.91	0.342374	0.171187	+	0.985239i	$0.445240\pi$
$314$	2708.34	0.486753
$315$	0	0
$316$	2702.30	0.481064
$317$	−972.308	−0.172272	−0.0861360	−	0.996283i	$-0.527452\pi$
$317$	−972.308	−0.172272	−0.0861360	+	0.996283i	$0.527452\pi$
$318$	0	0
$319$	13860.3	2.43268
$320$	4290.53	0.749525
$321$	0	0
$322$	0	0
$323$	5747.33	0.990062
$324$	0	0
$325$	−3250.49	−0.554783
$326$	−4039.53	−0.686286
$327$	0	0
$328$	−714.488	−0.120277
$329$	0	0
$330$	0	0
$331$	−4469.64	−0.742217	−0.371108	−	0.928590i	$-0.621022\pi$
$331$	−4469.64	−0.742217	−0.371108	+	0.928590i	$0.621022\pi$
$332$	1446.88	0.239180
$333$	0	0
$334$	−1895.08	−0.310462
$335$	5193.09	0.846952
$336$	0	0
$337$	−8494.45	−1.37306	−0.686532	−	0.727100i	$-0.740868\pi$
$337$	−8494.45	−1.37306	−0.686532	+	0.727100i	$0.740868\pi$
$338$	−5208.50	−0.838180
$339$	0	0
$340$	−6773.42	−1.08041
$341$	−6476.08	−1.02844
$342$	0	0
$343$	0	0
$344$	−7906.56	−1.23923
$345$	0	0
$346$	−6473.44	−1.00582
$347$	8461.30	1.30901	0.654505	−	0.756058i	$-0.272877\pi$
$347$	8461.30	1.30901	0.654505	+	0.756058i	$0.272877\pi$
$348$	0	0
$349$	−5910.68	−0.906566	−0.453283	−	0.891367i	$-0.649747\pi$
$349$	−5910.68	−0.906566	−0.453283	+	0.891367i	$0.649747\pi$
$350$	0	0
$351$	0	0
$352$	−9519.34	−1.44143
$353$	−6046.17	−0.911630	−0.455815	−	0.890075i	$-0.650652\pi$
$353$	−6046.17	−0.911630	−0.455815	+	0.890075i	$0.650652\pi$
$354$	0	0
$355$	−11007.6	−1.64571
$356$	−499.073	−0.0743000
$357$	0	0
$358$	−1433.50	−0.211628
$359$	10896.7	1.60196	0.800982	−	0.598688i	$-0.204311\pi$
$359$	10896.7	1.60196	0.800982	+	0.598688i	$0.204311\pi$
$360$	0	0
$361$	−3950.72	−0.575990
$362$	3060.88	0.444410
$363$	0	0
$364$	0	0
$365$	−11384.3	−1.63256
$366$	0	0
$367$	7775.66	1.10596	0.552979	−	0.833195i	$-0.313491\pi$
$367$	7775.66	1.10596	0.552979	+	0.833195i	$0.313491\pi$
$368$	−24.7629	−0.00350776
$369$	0	0
$370$	−6422.42	−0.902394
$371$	0	0
$372$	0	0
$373$	11929.1	1.65594	0.827969	−	0.560774i	$-0.189496\pi$
$373$	11929.1	1.65594	0.827969	+	0.560774i	$0.189496\pi$
$374$	−9982.24	−1.38013
$375$	0	0
$376$	12346.2	1.69337
$377$	−18771.5	−2.56441
$378$	0	0
$379$	−8130.56	−1.10195	−0.550975	−	0.834522i	$-0.685744\pi$
$379$	−8130.56	−1.10195	−0.550975	+	0.834522i	$0.685744\pi$
$380$	−3427.51	−0.462704
$381$	0	0
$382$	−1903.00	−0.254884
$383$	−10851.8	−1.44778	−0.723889	−	0.689916i	$-0.757647\pi$
$383$	−10851.8	−1.44778	−0.723889	+	0.689916i	$0.757647\pi$
$384$	0	0
$385$	0	0
$386$	−7950.49	−1.04837
$387$	0	0
$388$	2510.06	0.328425
$389$	11769.7	1.53406	0.767031	−	0.641610i	$-0.221733\pi$
$389$	11769.7	1.53406	0.767031	+	0.641610i	$0.221733\pi$
$390$	0	0
$391$	1988.53	0.257198
$392$	0	0
$393$	0	0
$394$	180.394	0.0230663
$395$	−7242.29	−0.922529
$396$	0	0
$397$	10132.5	1.28094	0.640472	−	0.767981i	$-0.278739\pi$
$397$	10132.5	1.28094	0.640472	+	0.767981i	$0.278739\pi$
$398$	947.586	0.119342
$399$	0	0
$400$	−60.1652	−0.00752064
$401$	4804.24	0.598285	0.299143	−	0.954208i	$-0.403299\pi$
$401$	4804.24	0.598285	0.299143	+	0.954208i	$0.403299\pi$
$402$	0	0
$403$	8770.82	1.08413
$404$	−5366.25	−0.660843
$405$	0	0
$406$	0	0
$407$	14724.9	1.79333
$408$	0	0
$409$	−8837.50	−1.06843	−0.534213	−	0.845350i	$-0.679392\pi$
$409$	−8837.50	−1.06843	−0.534213	+	0.845350i	$0.679392\pi$
$410$	724.562	0.0872770
$411$	0	0
$412$	1722.09	0.205925
$413$	0	0
$414$	0	0
$415$	−3877.70	−0.458671
$416$	12892.4	1.51948
$417$	0	0
$418$	−5051.25	−0.591064
$419$	−5730.73	−0.668173	−0.334086	−	0.942542i	$-0.608428\pi$
$419$	−5730.73	−0.668173	−0.334086	+	0.942542i	$0.608428\pi$
$420$	0	0
$421$	1841.98	0.213236	0.106618	−	0.994300i	$-0.465998\pi$
$421$	1841.98	0.213236	0.106618	+	0.994300i	$0.465998\pi$
$422$	−2207.08	−0.254595
$423$	0	0
$424$	−5853.12	−0.670407
$425$	4831.44	0.551433
$426$	0	0
$427$	0	0
$428$	1512.86	0.170857
$429$	0	0
$430$	8018.04	0.899219
$431$	−9147.01	−1.02226	−0.511132	−	0.859502i	$-0.670774\pi$
$431$	−9147.01	−1.02226	−0.511132	+	0.859502i	$0.670774\pi$
$432$	0	0
$433$	−2469.71	−0.274103	−0.137052	−	0.990564i	$-0.543763\pi$
$433$	−2469.71	−0.274103	−0.137052	+	0.990564i	$0.543763\pi$
$434$	0	0
$435$	0	0
$436$	2507.86	0.275469
$437$	1006.24	0.110149
$438$	0	0
$439$	−3704.39	−0.402736	−0.201368	−	0.979516i	$-0.564539\pi$
$439$	−3704.39	−0.402736	−0.201368	+	0.979516i	$0.564539\pi$
$440$	15732.7	1.70460
$441$	0	0
$442$	13519.3	1.45486
$443$	1351.39	0.144935	0.0724676	−	0.997371i	$-0.476913\pi$
$443$	1351.39	0.144935	0.0724676	+	0.997371i	$0.476913\pi$
$444$	0	0
$445$	1337.54	0.142484
$446$	−4552.76	−0.483362
$447$	0	0
$448$	0	0
$449$	15461.0	1.62505	0.812526	−	0.582926i	$-0.198092\pi$
$449$	15461.0	1.62505	0.812526	+	0.582926i	$0.198092\pi$
$450$	0	0
$451$	−1661.23	−0.173446
$452$	−5388.85	−0.560774
$453$	0	0
$454$	−2634.11	−0.272301
$455$	0	0
$456$	0	0
$457$	13569.6	1.38897	0.694486	−	0.719506i	$-0.255632\pi$
$457$	13569.6	1.38897	0.694486	+	0.719506i	$0.255632\pi$
$458$	5679.97	0.579492
$459$	0	0
$460$	−1185.89	−0.120201
$461$	7979.72	0.806188	0.403094	−	0.915159i	$-0.367935\pi$
$461$	7979.72	0.806188	0.403094	+	0.915159i	$0.367935\pi$
$462$	0	0
$463$	−18086.2	−1.81542	−0.907709	−	0.419600i	$-0.862170\pi$
$463$	−18086.2	−1.81542	−0.907709	+	0.419600i	$0.862170\pi$
$464$	−347.453	−0.0347631
$465$	0	0
$466$	−5179.79	−0.514913
$467$	5804.55	0.575166	0.287583	−	0.957756i	$-0.407148\pi$
$467$	5804.55	0.575166	0.287583	+	0.957756i	$0.407148\pi$
$468$	0	0
$469$	0	0
$470$	−12520.3	−1.22876
$471$	0	0
$472$	7186.61	0.700827
$473$	−18383.2	−1.78702
$474$	0	0
$475$	2444.82	0.236160
$476$	0	0
$477$	0	0
$478$	−7544.05	−0.721876
$479$	4345.88	0.414548	0.207274	−	0.978283i	$-0.433541\pi$
$479$	4345.88	0.414548	0.207274	+	0.978283i	$0.433541\pi$
$480$	0	0
$481$	−19942.5	−1.89043
$482$	7636.22	0.721619
$483$	0	0
$484$	−7167.10	−0.673093
$485$	−6727.08	−0.629817
$486$	0	0
$487$	−18781.1	−1.74754	−0.873771	−	0.486337i	$-0.838333\pi$
$487$	−18781.1	−1.74754	−0.873771	+	0.486337i	$0.838333\pi$
$488$	3160.53	0.293177
$489$	0	0
$490$	0	0
$491$	−7552.56	−0.694180	−0.347090	−	0.937832i	$-0.612830\pi$
$491$	−7552.56	−0.694180	−0.347090	+	0.937832i	$0.612830\pi$
$492$	0	0
$493$	27901.5	2.54892
$494$	6841.11	0.623069
$495$	0	0
$496$	162.344	0.0146965
$497$	0	0
$498$	0	0
$499$	8901.90	0.798605	0.399302	−	0.916819i	$-0.369252\pi$
$499$	8901.90	0.798605	0.399302	+	0.916819i	$0.369252\pi$
$500$	5063.26	0.452872
$501$	0	0
$502$	−3728.86	−0.331528
$503$	−18241.5	−1.61700	−0.808498	−	0.588500i	$-0.799719\pi$
$503$	−18241.5	−1.61700	−0.808498	+	0.588500i	$0.799719\pi$
$504$	0	0
$505$	14381.8	1.26729
$506$	−1747.69	−0.153546
$507$	0	0
$508$	−7750.82	−0.676943
$509$	−8984.77	−0.782403	−0.391201	−	0.920305i	$-0.627940\pi$
$509$	−8984.77	−0.782403	−0.391201	+	0.920305i	$0.627940\pi$
$510$	0	0
$511$	0	0
$512$	480.383	0.0414651
$513$	0	0
$514$	−4990.38	−0.428241
$515$	−4615.28	−0.394900
$516$	0	0
$517$	28705.6	2.44192
$518$	0	0
$519$	0	0
$520$	−21307.4	−1.79690
$521$	8888.73	0.747452	0.373726	−	0.927539i	$-0.378080\pi$
$521$	8888.73	0.747452	0.373726	+	0.927539i	$0.378080\pi$
$522$	0	0
$523$	−21038.9	−1.75902	−0.879510	−	0.475880i	$-0.842130\pi$
$523$	−21038.9	−1.75902	−0.879510	+	0.475880i	$0.842130\pi$
$524$	3979.85	0.331795
$525$	0	0
$526$	−10531.3	−0.872978
$527$	−13036.7	−1.07759
$528$	0	0
$529$	−11818.8	−0.971385
$530$	5935.65	0.486468
$531$	0	0
$532$	0	0
$533$	2249.86	0.182838
$534$	0	0
$535$	−4054.53	−0.327650
$536$	9060.11	0.730107
$537$	0	0
$538$	−11753.1	−0.941846
$539$	0	0
$540$	0	0
$541$	8006.13	0.636249	0.318124	−	0.948049i	$-0.396947\pi$
$541$	8006.13	0.636249	0.318124	+	0.948049i	$0.396947\pi$
$542$	−1701.71	−0.134861
$543$	0	0
$544$	−19162.9	−1.51030
$545$	−6721.17	−0.528263
$546$	0	0
$547$	22259.4	1.73993	0.869965	−	0.493114i	$-0.164141\pi$
$547$	22259.4	1.73993	0.869965	+	0.493114i	$0.164141\pi$
$548$	−5006.78	−0.390290
$549$	0	0
$550$	−4246.28	−0.329204
$551$	14118.8	1.09162
$552$	0	0
$553$	0	0
$554$	9907.98	0.759837
$555$	0	0
$556$	2957.44	0.225582
$557$	−7483.92	−0.569307	−0.284653	−	0.958630i	$-0.591879\pi$
$557$	−7483.92	−0.569307	−0.284653	+	0.958630i	$0.591879\pi$
$558$	0	0
$559$	24897.1	1.88378
$560$	0	0
$561$	0	0
$562$	1847.30	0.138654
$563$	6292.69	0.471057	0.235529	−	0.971867i	$-0.424318\pi$
$563$	6292.69	0.471057	0.235529	+	0.971867i	$0.424318\pi$
$564$	0	0
$565$	14442.4	1.07539
$566$	−4111.69	−0.305348
$567$	0	0
$568$	−19204.5	−1.41866
$569$	10338.4	0.761705	0.380853	−	0.924636i	$-0.375631\pi$
$569$	10338.4	0.761705	0.380853	+	0.924636i	$0.375631\pi$
$570$	0	0
$571$	−14539.6	−1.06561	−0.532806	−	0.846238i	$-0.678862\pi$
$571$	−14539.6	−1.06561	−0.532806	+	0.846238i	$0.678862\pi$
$572$	18485.1	1.35122
$573$	0	0
$574$	0	0
$575$	845.889	0.0613496
$576$	0	0
$577$	−11945.1	−0.861837	−0.430919	−	0.902391i	$-0.641810\pi$
$577$	−11945.1	−0.861837	−0.430919	+	0.902391i	$0.641810\pi$
$578$	−11402.5	−0.820556
$579$	0	0
$580$	−16639.4	−1.19123
$581$	0	0
$582$	0	0
$583$	−13608.8	−0.966759
$584$	−19861.6	−1.40733
$585$	0	0
$586$	−7325.34	−0.516394
$587$	−27431.6	−1.92883	−0.964414	−	0.264398i	$-0.914827\pi$
$587$	−27431.6	−1.92883	−0.964414	+	0.264398i	$0.914827\pi$
$588$	0	0
$589$	−6596.88	−0.461493
$590$	−7287.93	−0.508541
$591$	0	0
$592$	−369.127	−0.0256267
$593$	3648.83	0.252680	0.126340	−	0.991987i	$-0.459677\pi$
$593$	3648.83	0.252680	0.126340	+	0.991987i	$0.459677\pi$
$594$	0	0
$595$	0	0
$596$	5941.42	0.408339
$597$	0	0
$598$	2366.97	0.161861
$599$	−10849.7	−0.740078	−0.370039	−	0.929016i	$-0.620656\pi$
$599$	−10849.7	−0.740078	−0.370039	+	0.929016i	$0.620656\pi$
$600$	0	0
$601$	23865.9	1.61982	0.809908	−	0.586557i	$-0.199517\pi$
$601$	23865.9	1.61982	0.809908	+	0.586557i	$0.199517\pi$
$602$	0	0
$603$	0	0
$604$	−9512.79	−0.640844
$605$	19208.2	1.29078
$606$	0	0
$607$	−21519.9	−1.43899	−0.719493	−	0.694500i	$-0.755626\pi$
$607$	−21519.9	−1.43899	−0.719493	+	0.694500i	$0.755626\pi$
$608$	−9696.89	−0.646811
$609$	0	0
$610$	−3205.09	−0.212738
$611$	−38877.1	−2.57414
$612$	0	0
$613$	6178.96	0.407122	0.203561	−	0.979062i	$-0.434749\pi$
$613$	6178.96	0.407122	0.203561	+	0.979062i	$0.434749\pi$
$614$	−5439.86	−0.357549
$615$	0	0
$616$	0	0
$617$	1011.92	0.0660267	0.0330134	−	0.999455i	$-0.489490\pi$
$617$	1011.92	0.0660267	0.0330134	+	0.999455i	$0.489490\pi$
$618$	0	0
$619$	−3968.54	−0.257689	−0.128844	−	0.991665i	$-0.541127\pi$
$619$	−3968.54	−0.257689	−0.128844	+	0.991665i	$0.541127\pi$
$620$	7774.63	0.503608
$621$	0	0
$622$	4101.71	0.264411
$623$	0	0
$624$	0	0
$625$	−19236.6	−1.23114
$626$	−3354.32	−0.214162
$627$	0	0
$628$	7454.59	0.473679
$629$	29642.0	1.87902
$630$	0	0
$631$	9288.00	0.585974	0.292987	−	0.956116i	$-0.405351\pi$
$631$	9288.00	0.585974	0.292987	+	0.956116i	$0.405351\pi$
$632$	−12635.2	−0.795257
$633$	0	0
$634$	1720.25	0.107760
$635$	20772.6	1.29816
$636$	0	0
$637$	0	0
$638$	−24522.2	−1.52170
$639$	0	0
$640$	11182.9	0.690695
$641$	−3258.86	−0.200807	−0.100404	−	0.994947i	$-0.532013\pi$
$641$	−3258.86	−0.200807	−0.100404	+	0.994947i	$0.532013\pi$
$642$	0	0
$643$	−17217.9	−1.05600	−0.528001	−	0.849244i	$-0.677058\pi$
$643$	−17217.9	−1.05600	−0.528001	+	0.849244i	$0.677058\pi$
$644$	0	0
$645$	0	0
$646$	−10168.4	−0.619306
$647$	−10030.7	−0.609500	−0.304750	−	0.952432i	$-0.598573\pi$
$647$	−10030.7	−0.609500	−0.304750	+	0.952432i	$0.598573\pi$
$648$	0	0
$649$	16709.3	1.01063
$650$	5750.91	0.347030
$651$	0	0
$652$	−11118.6	−0.667852
$653$	31203.6	1.86997	0.934986	−	0.354684i	$-0.115411\pi$
$653$	31203.6	1.86997	0.934986	+	0.354684i	$0.115411\pi$
$654$	0	0
$655$	−10666.2	−0.636278
$656$	41.6440	0.00247855
$657$	0	0
$658$	0	0
$659$	−2666.60	−0.157627	−0.0788133	−	0.996889i	$-0.525113\pi$
$659$	−2666.60	−0.157627	−0.0788133	+	0.996889i	$0.525113\pi$
$660$	0	0
$661$	−20322.3	−1.19583	−0.597917	−	0.801558i	$-0.704005\pi$
$661$	−20322.3	−1.19583	−0.597917	+	0.801558i	$0.704005\pi$
$662$	7907.89	0.464273
$663$	0	0
$664$	−6765.21	−0.395393
$665$	0	0
$666$	0	0
$667$	4884.99	0.283580
$668$	−5216.14	−0.302124
$669$	0	0
$670$	−9187.85	−0.529788
$671$	7348.42	0.422776
$672$	0	0
$673$	−16480.1	−0.943922	−0.471961	−	0.881619i	$-0.656454\pi$
$673$	−16480.1	−0.943922	−0.471961	+	0.881619i	$0.656454\pi$
$674$	15028.8	0.858882
$675$	0	0
$676$	−14336.2	−0.815667
$677$	−19664.0	−1.11632	−0.558161	−	0.829732i	$-0.688493\pi$
$677$	−19664.0	−1.11632	−0.558161	+	0.829732i	$0.688493\pi$
$678$	0	0
$679$	0	0
$680$	31670.7	1.78605
$681$	0	0
$682$	11457.8	0.643315
$683$	12069.8	0.676192	0.338096	−	0.941112i	$-0.390217\pi$
$683$	12069.8	0.676192	0.338096	+	0.941112i	$0.390217\pi$
$684$	0	0
$685$	13418.4	0.748454
$686$	0	0
$687$	0	0
$688$	460.835	0.0255366
$689$	18431.0	1.01911
$690$	0	0
$691$	−1083.55	−0.0596531	−0.0298265	−	0.999555i	$-0.509495\pi$
$691$	−1083.55	−0.0596531	−0.0298265	+	0.999555i	$0.509495\pi$
$692$	−17817.9	−0.978806
$693$	0	0
$694$	−14970.1	−0.818815
$695$	−7926.09	−0.432596
$696$	0	0
$697$	−3344.14	−0.181733
$698$	10457.4	0.567078
$699$	0	0
$700$	0	0
$701$	−19478.9	−1.04951	−0.524755	−	0.851253i	$-0.675843\pi$
$701$	−19478.9	−1.04951	−0.524755	+	0.851253i	$0.675843\pi$
$702$	0	0
$703$	14999.5	0.804720
$704$	17404.1	0.931736
$705$	0	0
$706$	10697.2	0.570245
$707$	0	0
$708$	0	0
$709$	27626.7	1.46339	0.731695	−	0.681632i	$-0.238729\pi$
$709$	27626.7	1.46339	0.731695	+	0.681632i	$0.238729\pi$
$710$	19475.2	1.02943
$711$	0	0
$712$	2333.53	0.122827
$713$	−2282.47	−0.119887
$714$	0	0
$715$	−49540.9	−2.59122
$716$	−3945.65	−0.205944
$717$	0	0
$718$	−19278.9	−1.00206
$719$	2134.94	0.110737	0.0553684	−	0.998466i	$-0.482367\pi$
$719$	2134.94	0.110737	0.0553684	+	0.998466i	$0.482367\pi$
$720$	0	0
$721$	0	0
$722$	6989.79	0.360295
$723$	0	0
$724$	8424.95	0.432473
$725$	11868.8	0.607996
$726$	0	0
$727$	12294.4	0.627198	0.313599	−	0.949556i	$-0.398465\pi$
$727$	12294.4	0.627198	0.313599	+	0.949556i	$0.398465\pi$
$728$	0	0
$729$	0	0
$730$	20141.7	1.02120
$731$	−37006.4	−1.87241
$732$	0	0
$733$	−18207.7	−0.917487	−0.458744	−	0.888569i	$-0.651700\pi$
$733$	−18207.7	−0.917487	−0.458744	+	0.888569i	$0.651700\pi$
$734$	−13757.1	−0.691801
$735$	0	0
$736$	−3355.05	−0.168028
$737$	21065.3	1.05285
$738$	0	0
$739$	2537.36	0.126304	0.0631519	−	0.998004i	$-0.479885\pi$
$739$	2537.36	0.126304	0.0631519	+	0.998004i	$0.479885\pi$
$740$	−17677.4	−0.878156
$741$	0	0
$742$	0	0
$743$	20574.2	1.01587	0.507937	−	0.861394i	$-0.330408\pi$
$743$	20574.2	1.01587	0.507937	+	0.861394i	$0.330408\pi$
$744$	0	0
$745$	−15923.3	−0.783066
$746$	−21105.5	−1.03583
$747$	0	0
$748$	−27475.7	−1.34306
$749$	0	0
$750$	0	0
$751$	24946.6	1.21213	0.606067	−	0.795414i	$-0.292746\pi$
$751$	24946.6	1.21213	0.606067	+	0.795414i	$0.292746\pi$
$752$	−719.599	−0.0348951
$753$	0	0
$754$	33211.4	1.60409
$755$	25494.7	1.22894
$756$	0	0
$757$	−24686.4	−1.18526	−0.592631	−	0.805474i	$-0.701911\pi$
$757$	−24686.4	−1.18526	−0.592631	+	0.805474i	$0.701911\pi$
$758$	14384.9	0.689294
$759$	0	0
$760$	16026.1	0.764906
$761$	−39431.3	−1.87830	−0.939148	−	0.343512i	$-0.888383\pi$
$761$	−39431.3	−1.87830	−0.939148	+	0.343512i	$0.888383\pi$
$762$	0	0
$763$	0	0
$764$	−5237.91	−0.248038
$765$	0	0
$766$	19199.4	0.905618
$767$	−22630.0	−1.06535
$768$	0	0
$769$	−2762.60	−0.129547	−0.0647736	−	0.997900i	$-0.520633\pi$
$769$	−2762.60	−0.129547	−0.0647736	+	0.997900i	$0.520633\pi$
$770$	0	0
$771$	0	0
$772$	−21883.4	−1.02021
$773$	742.885	0.0345662	0.0172831	−	0.999851i	$-0.494498\pi$
$773$	742.885	0.0345662	0.0172831	+	0.999851i	$0.494498\pi$
$774$	0	0
$775$	−5545.60	−0.257037
$776$	−11736.4	−0.542927
$777$	0	0
$778$	−20823.6	−0.959590
$779$	−1692.21	−0.0778303
$780$	0	0
$781$	−44651.5	−2.04578
$782$	−3518.20	−0.160883
$783$	0	0
$784$	0	0
$785$	−19978.7	−0.908368
$786$	0	0
$787$	−35710.9	−1.61748	−0.808739	−	0.588167i	$-0.799850\pi$
$787$	−35710.9	−1.61748	−0.808739	+	0.588167i	$0.799850\pi$
$788$	496.528	0.0224468
$789$	0	0
$790$	12813.4	0.577063
$791$	0	0
$792$	0	0
$793$	−9952.25	−0.445668
$794$	−17926.9	−0.801260
$795$	0	0
$796$	2608.19	0.116137
$797$	−3201.08	−0.142268	−0.0711342	−	0.997467i	$-0.522662\pi$
$797$	−3201.08	−0.142268	−0.0711342	+	0.997467i	$0.522662\pi$
$798$	0	0
$799$	57785.9	2.55860
$800$	−8151.60	−0.360253
$801$	0	0
$802$	−8499.88	−0.374241
$803$	−46179.4	−2.02944
$804$	0	0
$805$	0	0
$806$	−15517.7	−0.678149
$807$	0	0
$808$	25091.2	1.09246
$809$	2163.10	0.0940056	0.0470028	−	0.998895i	$-0.485033\pi$
$809$	2163.10	0.0940056	0.0470028	+	0.998895i	$0.485033\pi$
$810$	0	0
$811$	−11547.5	−0.499983	−0.249991	−	0.968248i	$-0.580428\pi$
$811$	−11547.5	−0.499983	−0.249991	+	0.968248i	$0.580428\pi$
$812$	0	0
$813$	0	0
$814$	−26051.9	−1.12177
$815$	29798.5	1.28073
$816$	0	0
$817$	−18726.1	−0.801889
$818$	15635.7	0.668324
$819$	0	0
$820$	1994.33	0.0849328
$821$	6148.38	0.261364	0.130682	−	0.991424i	$-0.458283\pi$
$821$	6148.38	0.261364	0.130682	+	0.991424i	$0.458283\pi$
$822$	0	0
$823$	24957.6	1.05707	0.528534	−	0.848912i	$-0.322742\pi$
$823$	24957.6	1.05707	0.528534	+	0.848912i	$0.322742\pi$
$824$	−8052.03	−0.340419
$825$	0	0
$826$	0	0
$827$	−28221.3	−1.18664	−0.593320	−	0.804967i	$-0.702183\pi$
$827$	−28221.3	−1.18664	−0.593320	+	0.804967i	$0.702183\pi$
$828$	0	0
$829$	29913.9	1.25326	0.626630	−	0.779317i	$-0.284434\pi$
$829$	29913.9	1.25326	0.626630	+	0.779317i	$0.284434\pi$
$830$	6860.59	0.286909
$831$	0	0
$832$	−23571.1	−0.982188
$833$	0	0
$834$	0	0
$835$	13979.5	0.579378
$836$	−13903.3	−0.575188
$837$	0	0
$838$	10139.1	0.417957
$839$	−45271.0	−1.86285	−0.931424	−	0.363935i	$-0.881433\pi$
$839$	−45271.0	−1.86285	−0.931424	+	0.363935i	$0.881433\pi$
$840$	0	0
$841$	44153.2	1.81037
$842$	−3258.91	−0.133384
$843$	0	0
$844$	−6074.90	−0.247757
$845$	38421.6	1.56419
$846$	0	0
$847$	0	0
$848$	341.150	0.0138150
$849$	0	0
$850$	−8548.00	−0.344934
$851$	5189.73	0.209050
$852$	0	0
$853$	8754.63	0.351410	0.175705	−	0.984443i	$-0.443780\pi$
$853$	8754.63	0.351410	0.175705	+	0.984443i	$0.443780\pi$
$854$	0	0
$855$	0	0
$856$	−7073.73	−0.282448
$857$	−3502.27	−0.139598	−0.0697988	−	0.997561i	$-0.522236\pi$
$857$	−3502.27	−0.139598	−0.0697988	+	0.997561i	$0.522236\pi$
$858$	0	0
$859$	−2953.62	−0.117318	−0.0586591	−	0.998278i	$-0.518683\pi$
$859$	−2953.62	−0.117318	−0.0586591	+	0.998278i	$0.518683\pi$
$860$	22069.3	0.875067
$861$	0	0
$862$	16183.3	0.639449
$863$	−15352.8	−0.605578	−0.302789	−	0.953058i	$-0.597918\pi$
$863$	−15352.8	−0.605578	−0.302789	+	0.953058i	$0.597918\pi$
$864$	0	0
$865$	47752.8	1.87704
$866$	4369.53	0.171458
$867$	0	0
$868$	0	0
$869$	−29377.6	−1.14680
$870$	0	0
$871$	−28529.5	−1.10986
$872$	−11726.1	−0.455384
$873$	0	0
$874$	−1780.29	−0.0689008
$875$	0	0
$876$	0	0
$877$	43246.5	1.66514	0.832572	−	0.553916i	$-0.186867\pi$
$877$	43246.5	1.66514	0.832572	+	0.553916i	$0.186867\pi$
$878$	6553.97	0.251920
$879$	0	0
$880$	−916.981	−0.0351266
$881$	20539.6	0.785467	0.392733	−	0.919652i	$-0.371529\pi$
$881$	20539.6	0.785467	0.392733	+	0.919652i	$0.371529\pi$
$882$	0	0
$883$	−21156.9	−0.806325	−0.403163	−	0.915128i	$-0.632089\pi$
$883$	−21156.9	−0.806325	−0.403163	+	0.915128i	$0.632089\pi$
$884$	37211.4	1.41579
$885$	0	0
$886$	−2390.93	−0.0906602
$887$	−9397.58	−0.355738	−0.177869	−	0.984054i	$-0.556920\pi$
$887$	−9397.58	−0.355738	−0.177869	+	0.984054i	$0.556920\pi$
$888$	0	0
$889$	0	0
$890$	−2366.43	−0.0891270
$891$	0	0
$892$	−12531.3	−0.470380
$893$	29241.0	1.09576
$894$	0	0
$895$	10574.5	0.394936
$896$	0	0
$897$	0	0
$898$	−27354.2	−1.01651
$899$	−32025.7	−1.18812
$900$	0	0
$901$	−27395.3	−1.01295
$902$	2939.11	0.108494
$903$	0	0
$904$	25196.8	0.927029
$905$	−22579.3	−0.829348
$906$	0	0
$907$	11590.9	0.424332	0.212166	−	0.977234i	$-0.431948\pi$
$907$	11590.9	0.424332	0.212166	+	0.977234i	$0.431948\pi$
$908$	−7250.27	−0.264988
$909$	0	0
$910$	0	0
$911$	30564.4	1.11158	0.555788	−	0.831324i	$-0.312417\pi$
$911$	30564.4	1.11158	0.555788	+	0.831324i	$0.312417\pi$
$912$	0	0
$913$	−15729.5	−0.570176
$914$	−24008.0	−0.868833
$915$	0	0
$916$	15633.9	0.563928
$917$	0	0
$918$	0	0
$919$	−28321.3	−1.01658	−0.508288	−	0.861187i	$-0.669721\pi$
$919$	−28321.3	−1.01658	−0.508288	+	0.861187i	$0.669721\pi$
$920$	5544.92	0.198707
$921$	0	0
$922$	−14118.1	−0.504289
$923$	60473.2	2.15656
$924$	0	0
$925$	12609.2	0.448204
$926$	31999.0	1.13559
$927$	0	0
$928$	−47075.3	−1.66522
$929$	50804.0	1.79421	0.897107	−	0.441813i	$-0.145665\pi$
$929$	50804.0	1.79421	0.897107	+	0.441813i	$0.145665\pi$
$930$	0	0
$931$	0	0
$932$	−14257.2	−0.501083
$933$	0	0
$934$	−10269.7	−0.359779
$935$	73636.2	2.57557
$936$	0	0
$937$	15273.4	0.532507	0.266253	−	0.963903i	$-0.414214\pi$
$937$	15273.4	0.532507	0.266253	+	0.963903i	$0.414214\pi$
$938$	0	0
$939$	0	0
$940$	−34461.5	−1.19576
$941$	−55573.4	−1.92523	−0.962615	−	0.270873i	$-0.912688\pi$
$941$	−55573.4	−1.92523	−0.962615	+	0.270873i	$0.912688\pi$
$942$	0	0
$943$	−585.493	−0.0202187
$944$	−418.872	−0.0144419
$945$	0	0
$946$	32524.4	1.11782
$947$	−22070.1	−0.757321	−0.378660	−	0.925536i	$-0.623615\pi$
$947$	−22070.1	−0.757321	−0.378660	+	0.925536i	$0.623615\pi$
$948$	0	0
$949$	62542.7	2.13933
$950$	−4325.49	−0.147723
$951$	0	0
$952$	0	0
$953$	51073.6	1.73603	0.868014	−	0.496539i	$-0.165396\pi$
$953$	51073.6	1.73603	0.868014	+	0.496539i	$0.165396\pi$
$954$	0	0
$955$	14037.9	0.475659
$956$	−20764.7	−0.702487
$957$	0	0
$958$	−7688.93	−0.259309
$959$	0	0
$960$	0	0
$961$	−14827.3	−0.497709
$962$	35283.1	1.18251
$963$	0	0
$964$	21018.4	0.702237
$965$	58648.5	1.95644
$966$	0	0
$967$	−3212.76	−0.106841	−0.0534206	−	0.998572i	$-0.517012\pi$
$967$	−3212.76	−0.106841	−0.0534206	+	0.998572i	$0.517012\pi$
$968$	33511.5	1.11271
$969$	0	0
$970$	11901.9	0.393965
$971$	19188.8	0.634189	0.317095	−	0.948394i	$-0.397293\pi$
$971$	19188.8	0.634189	0.317095	+	0.948394i	$0.397293\pi$
$972$	0	0
$973$	0	0
$974$	33228.4	1.09313
$975$	0	0
$976$	−184.212	−0.00604148
$977$	13111.9	0.429363	0.214682	−	0.976684i	$-0.431129\pi$
$977$	13111.9	0.429363	0.214682	+	0.976684i	$0.431129\pi$
$978$	0	0
$979$	5425.60	0.177122
$980$	0	0
$981$	0	0
$982$	13362.3	0.434225
$983$	38542.1	1.25056	0.625281	−	0.780400i	$-0.284984\pi$
$983$	38542.1	1.25056	0.625281	+	0.780400i	$0.284984\pi$
$984$	0	0
$985$	−1330.72	−0.0430459
$986$	−49364.5	−1.59441
$987$	0	0
$988$	18829.9	0.606333
$989$	−6479.09	−0.208315
$990$	0	0
$991$	6663.59	0.213598	0.106799	−	0.994281i	$-0.465940\pi$
$991$	6663.59	0.213598	0.106799	+	0.994281i	$0.465940\pi$
$992$	21995.5	0.703991
$993$	0	0
$994$	0	0
$995$	−6990.07	−0.222714
$996$	0	0
$997$	9905.79	0.314664	0.157332	−	0.987546i	$-0.449711\pi$
$997$	9905.79	0.314664	0.157332	+	0.987546i	$0.449711\pi$
$998$	−15749.6	−0.499545
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bo.1.4		8
3.2	odd	2	inner	1323.4.a.bo.1.5		8
7.2	even	3		189.4.e.h.109.5	yes	16
7.4	even	3		189.4.e.h.163.5	yes	16
7.6	odd	2		1323.4.a.bn.1.4		8
21.2	odd	6		189.4.e.h.109.4	✓	16
21.11	odd	6		189.4.e.h.163.4	yes	16
21.20	even	2		1323.4.a.bn.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.h.109.4	✓	16	21.2	odd	6
189.4.e.h.109.5	yes	16	7.2	even	3
189.4.e.h.163.4	yes	16	21.11	odd	6
189.4.e.h.163.5	yes	16	7.4	even	3
1323.4.a.bn.1.4		8	7.6	odd	2
1323.4.a.bn.1.5		8	21.20	even	2
1323.4.a.bo.1.4		8	1.1	even	1	trivial
1323.4.a.bo.1.5		8	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)

\(f(q)\)	\(=\)	\(q-1.76924 q^{2} -4.86977 q^{4} +13.0512 q^{5} +22.7698 q^{8} +O(q^{10})\)	\(q-1.76924 q^{2} -4.86977 q^{4} +13.0512 q^{5} +22.7698 q^{8} -23.0908 q^{10} +52.9410 q^{11} -71.7001 q^{13} -1.32714 q^{16} +106.573 q^{17} +53.9285 q^{19} -63.5565 q^{20} -93.6656 q^{22} +18.6589 q^{23} +45.3345 q^{25} +126.855 q^{26} +261.806 q^{29} -122.326 q^{31} -179.810 q^{32} -188.554 q^{34} +278.137 q^{37} -95.4127 q^{38} +297.174 q^{40} -31.3788 q^{41} -347.239 q^{43} -257.811 q^{44} -33.0121 q^{46} +542.218 q^{47} -80.2078 q^{50} +349.163 q^{52} -257.057 q^{53} +690.945 q^{55} -463.198 q^{58} +315.620 q^{59} +138.804 q^{61} +216.425 q^{62} +328.745 q^{64} -935.775 q^{65} +397.901 q^{67} -518.987 q^{68} -843.419 q^{71} -872.281 q^{73} -492.093 q^{74} -262.620 q^{76} -554.912 q^{79} -17.3208 q^{80} +55.5168 q^{82} -297.114 q^{83} +1390.91 q^{85} +614.352 q^{86} +1205.46 q^{88} +102.484 q^{89} -90.8644 q^{92} -959.317 q^{94} +703.833 q^{95} -515.437 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 48 q^{4}+O(q^{10}) \) 8 * q + 48 * q^4	\( 8 q + 48 q^{4} + 44 q^{10} - 84 q^{13} + 156 q^{16} + 12 q^{19} + 224 q^{22} + 408 q^{25} + 800 q^{31} - 948 q^{34} + 692 q^{37} + 96 q^{40} + 1456 q^{43} + 1524 q^{46} + 1972 q^{52} - 1280 q^{55} + 2372 q^{58} + 216 q^{61} + 4964 q^{64} + 684 q^{67} - 4564 q^{73} - 380 q^{76} + 556 q^{79} + 3340 q^{82} + 1296 q^{85} + 6696 q^{88} - 492 q^{94} + 584 q^{97}+O(q^{100}) \) 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

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