LMFDB - Embedded newform 1323.4.a.bc.1.3

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:	$6$
Coefficient field:	6.6.346909504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 $ x^6 - x^5 - 12x^4 + 2x^3 + 39x^2 + 25x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.673097$ 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)

$f(q)$	$=$	$q-1.57109 q^{2} -5.53167 q^{4} -6.97204 q^{5} +21.2595 q^{8} +O(q^{10})$	$q-1.57109 q^{2} -5.53167 q^{4} -6.97204 q^{5} +21.2595 q^{8} +10.9537 q^{10} +32.5591 q^{11} +19.3425 q^{13} +10.8528 q^{16} -22.2459 q^{17} +155.215 q^{19} +38.5671 q^{20} -51.1534 q^{22} +76.9512 q^{23} -76.3906 q^{25} -30.3888 q^{26} -122.079 q^{29} -164.774 q^{31} -187.127 q^{32} +34.9504 q^{34} +66.2329 q^{37} -243.858 q^{38} -148.222 q^{40} +231.844 q^{41} -210.361 q^{43} -180.106 q^{44} -120.897 q^{46} -206.307 q^{47} +120.017 q^{50} -106.996 q^{52} +419.641 q^{53} -227.004 q^{55} +191.797 q^{58} -300.800 q^{59} -24.2225 q^{61} +258.875 q^{62} +207.171 q^{64} -134.857 q^{65} -274.603 q^{67} +123.057 q^{68} +336.736 q^{71} -693.601 q^{73} -104.058 q^{74} -858.601 q^{76} -584.526 q^{79} -75.6661 q^{80} -364.248 q^{82} +1209.75 q^{83} +155.100 q^{85} +330.497 q^{86} +692.191 q^{88} +1258.68 q^{89} -425.669 q^{92} +324.127 q^{94} -1082.17 q^{95} -1085.60 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{4} + 24 q^{5}+O(q^{10}) $ 6 * q + 18 * q^4 + 24 * q^5	$ 6 q + 18 q^{4} + 24 q^{5} - 30 q^{16} + 42 q^{17} + 12 q^{20} + 132 q^{22} + 222 q^{25} + 366 q^{26} - 312 q^{37} + 336 q^{38} + 360 q^{41} + 654 q^{43} + 774 q^{46} + 1812 q^{47} - 378 q^{58} - 6 q^{59} + 2058 q^{62} + 66 q^{64} + 42 q^{67} + 2910 q^{68} + 1956 q^{79} - 2868 q^{80} + 2892 q^{83} - 1944 q^{85} - 2532 q^{88} + 1518 q^{89}+O(q^{100}) $ 6 * q + 18 * q^4 + 24 * q^5 - 30 * q^16 + 42 * q^17 + 12 * q^20 + 132 * q^22 + 222 * q^25 + 366 * q^26 - 312 * q^37 + 336 * q^38 + 360 * q^41 + 654 * q^43 + 774 * q^46 + 1812 * q^47 - 378 * q^58 - 6 * q^59 + 2058 * q^62 + 66 * q^64 + 42 * q^67 + 2910 * q^68 + 1956 * q^79 - 2868 * q^80 + 2892 * q^83 - 1944 * q^85 - 2532 * q^88 + 1518 * q^89

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 $

$2$	−1.57109	−0.555465	−0.277732	−	0.960659i	$-0.589583\pi$
$2$	−1.57109	−0.555465	−0.277732	+	0.960659i	$0.589583\pi$
$3$	0	0
$3$	0	0
$4$	−5.53167	−0.691459
$5$	−6.97204	−0.623599	−0.311799	−	0.950148i	$-0.600932\pi$
$5$	−6.97204	−0.623599	−0.311799	+	0.950148i	$0.600932\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.2595	0.939546
$9$	0	0
$10$	10.9537	0.346387
$11$	32.5591	0.892450	0.446225	−	0.894921i	$-0.352768\pi$
$11$	32.5591	0.892450	0.446225	+	0.894921i	$0.352768\pi$
$12$	0	0
$13$	19.3425	0.412664	0.206332	−	0.978482i	$-0.433847\pi$
$13$	19.3425	0.412664	0.206332	+	0.978482i	$0.433847\pi$
$14$	0	0
$15$	0	0
$16$	10.8528	0.169575
$17$	−22.2459	−0.317379	−0.158689	−	0.987329i	$-0.550727\pi$
$17$	−22.2459	−0.317379	−0.158689	+	0.987329i	$0.550727\pi$
$18$	0	0
$19$	155.215	1.87415	0.937075	−	0.349127i	$-0.113522\pi$
$19$	155.215	1.87415	0.937075	+	0.349127i	$0.113522\pi$
$20$	38.5671	0.431193
$21$	0	0
$22$	−51.1534	−0.495724
$23$	76.9512	0.697628	0.348814	−	0.937192i	$-0.386585\pi$
$23$	76.9512	0.697628	0.348814	+	0.937192i	$0.386585\pi$
$24$	0	0
$25$	−76.3906	−0.611125
$26$	−30.3888	−0.229220
$27$	0	0
$28$	0	0
$29$	−122.079	−0.781707	−0.390854	−	0.920453i	$-0.627820\pi$
$29$	−122.079	−0.781707	−0.390854	+	0.920453i	$0.627820\pi$
$30$	0	0
$31$	−164.774	−0.954653	−0.477326	−	0.878726i	$-0.658394\pi$
$31$	−164.774	−0.954653	−0.477326	+	0.878726i	$0.658394\pi$
$32$	−187.127	−1.03374
$33$	0	0
$34$	34.9504	0.176293
$35$	0	0
$36$	0	0
$37$	66.2329	0.294287	0.147143	−	0.989115i	$-0.452992\pi$
$37$	66.2329	0.294287	0.147143	+	0.989115i	$0.452992\pi$
$38$	−243.858	−1.04102
$39$	0	0
$40$	−148.222	−0.585899
$41$	231.844	0.883121	0.441560	−	0.897232i	$-0.354425\pi$
$41$	231.844	0.883121	0.441560	+	0.897232i	$0.354425\pi$
$42$	0	0
$43$	−210.361	−0.746042	−0.373021	−	0.927823i	$-0.621678\pi$
$43$	−210.361	−0.746042	−0.373021	+	0.927823i	$0.621678\pi$
$44$	−180.106	−0.617093
$45$	0	0
$46$	−120.897	−0.387508
$47$	−206.307	−0.640276	−0.320138	−	0.947371i	$-0.603729\pi$
$47$	−206.307	−0.640276	−0.320138	+	0.947371i	$0.603729\pi$
$48$	0	0
$49$	0	0
$50$	120.017	0.339458
$51$	0	0
$52$	−106.996	−0.285340
$53$	419.641	1.08759	0.543794	−	0.839219i	$-0.316987\pi$
$53$	419.641	1.08759	0.543794	+	0.839219i	$0.316987\pi$
$54$	0	0
$55$	−227.004	−0.556530
$56$	0	0
$57$	0	0
$58$	191.797	0.434211
$59$	−300.800	−0.663742	−0.331871	−	0.943325i	$-0.607680\pi$
$59$	−300.800	−0.663742	−0.331871	+	0.943325i	$0.607680\pi$
$60$	0	0
$61$	−24.2225	−0.0508423	−0.0254211	−	0.999677i	$-0.508093\pi$
$61$	−24.2225	−0.0508423	−0.0254211	+	0.999677i	$0.508093\pi$
$62$	258.875	0.530276
$63$	0	0
$64$	207.171	0.404630
$65$	−134.857	−0.257337
$66$	0	0
$67$	−274.603	−0.500718	−0.250359	−	0.968153i	$-0.580549\pi$
$67$	−274.603	−0.500718	−0.250359	+	0.968153i	$0.580549\pi$
$68$	123.057	0.219454
$69$	0	0
$70$	0	0
$71$	336.736	0.562863	0.281431	−	0.959581i	$-0.409191\pi$
$71$	336.736	0.562863	0.281431	+	0.959581i	$0.409191\pi$
$72$	0	0
$73$	−693.601	−1.11205	−0.556027	−	0.831164i	$-0.687675\pi$
$73$	−693.601	−1.11205	−0.556027	+	0.831164i	$0.687675\pi$
$74$	−104.058	−0.163466
$75$	0	0
$76$	−858.601	−1.29590
$77$	0	0
$78$	0	0
$79$	−584.526	−0.832459	−0.416230	−	0.909260i	$-0.636649\pi$
$79$	−584.526	−0.832459	−0.416230	+	0.909260i	$0.636649\pi$
$80$	−75.6661	−0.105747
$81$	0	0
$82$	−364.248	−0.490542
$83$	1209.75	1.59984	0.799922	−	0.600104i	$-0.204874\pi$
$83$	1209.75	1.59984	0.799922	+	0.600104i	$0.204874\pi$
$84$	0	0
$85$	155.100	0.197917
$86$	330.497	0.414400
$87$	0	0
$88$	692.191	0.838497
$89$	1258.68	1.49910	0.749550	−	0.661948i	$-0.230270\pi$
$89$	1258.68	1.49910	0.749550	+	0.661948i	$0.230270\pi$
$90$	0	0
$91$	0	0
$92$	−425.669	−0.482381
$93$	0	0
$94$	324.127	0.355651
$95$	−1082.17	−1.16872
$96$	0	0
$97$	−1085.60	−1.13636	−0.568178	−	0.822906i	$-0.692351\pi$
$97$	−1085.60	−1.13636	−0.568178	+	0.822906i	$0.692351\pi$
$98$	0	0
$99$	0	0
$100$	422.568	0.422568
$101$	−844.120	−0.831615	−0.415807	−	0.909453i	$-0.636501\pi$
$101$	−844.120	−0.831615	−0.415807	+	0.909453i	$0.636501\pi$
$102$	0	0
$103$	1290.47	1.23450	0.617250	−	0.786767i	$-0.288247\pi$
$103$	1290.47	1.23450	0.617250	+	0.786767i	$0.288247\pi$
$104$	411.211	0.387717
$105$	0	0
$106$	−659.295	−0.604117
$107$	212.075	0.191608	0.0958039	−	0.995400i	$-0.469458\pi$
$107$	212.075	0.191608	0.0958039	+	0.995400i	$0.469458\pi$
$108$	0	0
$109$	524.864	0.461219	0.230609	−	0.973046i	$-0.425928\pi$
$109$	524.864	0.461219	0.230609	+	0.973046i	$0.425928\pi$
$110$	356.644	0.309133
$111$	0	0
$112$	0	0
$113$	−590.941	−0.491956	−0.245978	−	0.969275i	$-0.579109\pi$
$113$	−590.941	−0.491956	−0.245978	+	0.969275i	$0.579109\pi$
$114$	0	0
$115$	−536.507	−0.435040
$116$	675.302	0.540519
$117$	0	0
$118$	472.584	0.368685
$119$	0	0
$120$	0	0
$121$	−270.903	−0.203533
$122$	38.0558	0.0282411
$123$	0	0
$124$	911.474	0.660103
$125$	1404.10	1.00470
$126$	0	0
$127$	2374.15	1.65883	0.829415	−	0.558633i	$-0.188674\pi$
$127$	2374.15	1.65883	0.829415	+	0.558633i	$0.188674\pi$
$128$	1171.53	0.808981
$129$	0	0
$130$	211.872	0.142941
$131$	2661.28	1.77494	0.887469	−	0.460868i	$-0.152462\pi$
$131$	2661.28	1.77494	0.887469	+	0.460868i	$0.152462\pi$
$132$	0	0
$133$	0	0
$134$	431.426	0.278131
$135$	0	0
$136$	−472.938	−0.298192
$137$	1405.23	0.876327	0.438164	−	0.898895i	$-0.355629\pi$
$137$	1405.23	0.876327	0.438164	+	0.898895i	$0.355629\pi$
$138$	0	0
$139$	858.529	0.523881	0.261941	−	0.965084i	$-0.415638\pi$
$139$	858.529	0.523881	0.261941	+	0.965084i	$0.415638\pi$
$140$	0	0
$141$	0	0
$142$	−529.044	−0.312650
$143$	629.774	0.368282
$144$	0	0
$145$	851.141	0.487472
$146$	1089.71	0.617706
$147$	0	0
$148$	−366.378	−0.203487
$149$	−3114.11	−1.71220	−0.856101	−	0.516808i	$-0.827120\pi$
$149$	−3114.11	−1.71220	−0.856101	+	0.516808i	$0.827120\pi$
$150$	0	0
$151$	−26.7635	−0.0144237	−0.00721186	−	0.999974i	$-0.502296\pi$
$151$	−26.7635	−0.0144237	−0.00721186	+	0.999974i	$0.502296\pi$
$152$	3299.80	1.76085
$153$	0	0
$154$	0	0
$155$	1148.81	0.595320
$156$	0	0
$157$	−2362.94	−1.20117	−0.600583	−	0.799562i	$-0.705065\pi$
$157$	−2362.94	−1.20117	−0.600583	+	0.799562i	$0.705065\pi$
$158$	918.343	0.462402
$159$	0	0
$160$	1304.66	0.644638
$161$	0	0
$162$	0	0
$163$	−284.500	−0.136710	−0.0683550	−	0.997661i	$-0.521775\pi$
$163$	−284.500	−0.136710	−0.0683550	+	0.997661i	$0.521775\pi$
$164$	−1282.49	−0.610642
$165$	0	0
$166$	−1900.62	−0.888657
$167$	3642.66	1.68789	0.843946	−	0.536429i	$-0.180227\pi$
$167$	3642.66	1.68789	0.843946	+	0.536429i	$0.180227\pi$
$168$	0	0
$169$	−1822.87	−0.829708
$170$	−243.676	−0.109936
$171$	0	0
$172$	1163.65	0.515857
$173$	2613.40	1.14852	0.574258	−	0.818674i	$-0.305291\pi$
$173$	2613.40	1.14852	0.574258	+	0.818674i	$0.305291\pi$
$174$	0	0
$175$	0	0
$176$	353.357	0.151337
$177$	0	0
$178$	−1977.50	−0.832697
$179$	−2757.44	−1.15140	−0.575700	−	0.817661i	$-0.695270\pi$
$179$	−2757.44	−1.15140	−0.575700	+	0.817661i	$0.695270\pi$
$180$	0	0
$181$	−2204.50	−0.905298	−0.452649	−	0.891689i	$-0.649521\pi$
$181$	−2204.50	−0.905298	−0.452649	+	0.891689i	$0.649521\pi$
$182$	0	0
$183$	0	0
$184$	1635.94	0.655453
$185$	−461.778	−0.183517
$186$	0	0
$187$	−724.309	−0.283244
$188$	1141.22	0.442725
$189$	0	0
$190$	1700.19	0.649181
$191$	3590.90	1.36036	0.680179	−	0.733046i	$-0.261902\pi$
$191$	3590.90	1.36036	0.680179	+	0.733046i	$0.261902\pi$
$192$	0	0
$193$	−3333.70	−1.24334	−0.621671	−	0.783278i	$-0.713546\pi$
$193$	−3333.70	−1.24334	−0.621671	+	0.783278i	$0.713546\pi$
$194$	1705.58	0.631205
$195$	0	0
$196$	0	0
$197$	−3125.50	−1.13037	−0.565185	−	0.824964i	$-0.691195\pi$
$197$	−3125.50	−1.13037	−0.565185	+	0.824964i	$0.691195\pi$
$198$	0	0
$199$	4384.56	1.56188	0.780938	−	0.624608i	$-0.214741\pi$
$199$	4384.56	1.56188	0.780938	+	0.624608i	$0.214741\pi$
$200$	−1624.03	−0.574180
$201$	0	0
$202$	1326.19	0.461933
$203$	0	0
$204$	0	0
$205$	−1616.43	−0.550713
$206$	−2027.44	−0.685721
$207$	0	0
$208$	209.920	0.0699774
$209$	5053.68	1.67259
$210$	0	0
$211$	−2333.73	−0.761423	−0.380712	−	0.924694i	$-0.624321\pi$
$211$	−2333.73	−0.761423	−0.380712	+	0.924694i	$0.624321\pi$
$212$	−2321.32	−0.752023
$213$	0	0
$214$	−333.189	−0.106431
$215$	1466.65	0.465231
$216$	0	0
$217$	0	0
$218$	−824.609	−0.256191
$219$	0	0
$220$	1255.71	0.384818
$221$	−430.292	−0.130971
$222$	0	0
$223$	−226.956	−0.0681528	−0.0340764	−	0.999419i	$-0.510849\pi$
$223$	−226.956	−0.0681528	−0.0340764	+	0.999419i	$0.510849\pi$
$224$	0	0
$225$	0	0
$226$	928.422	0.273264
$227$	−1298.84	−0.379767	−0.189884	−	0.981807i	$-0.560811\pi$
$227$	−1298.84	−0.379767	−0.189884	+	0.981807i	$0.560811\pi$
$228$	0	0
$229$	−445.975	−0.128694	−0.0643469	−	0.997928i	$-0.520496\pi$
$229$	−445.975	−0.128694	−0.0643469	+	0.997928i	$0.520496\pi$
$230$	842.902	0.241649
$231$	0	0
$232$	−2595.34	−0.734450
$233$	3599.79	1.01215	0.506073	−	0.862490i	$-0.331097\pi$
$233$	3599.79	1.01215	0.506073	+	0.862490i	$0.331097\pi$
$234$	0	0
$235$	1438.38	0.399275
$236$	1663.93	0.458951
$237$	0	0
$238$	0	0
$239$	5201.62	1.40780	0.703902	−	0.710298i	$-0.251440\pi$
$239$	5201.62	1.40780	0.703902	+	0.710298i	$0.251440\pi$
$240$	0	0
$241$	5929.56	1.58488	0.792440	−	0.609949i	$-0.208810\pi$
$241$	5929.56	1.58488	0.792440	+	0.609949i	$0.208810\pi$
$242$	425.613	0.113055
$243$	0	0
$244$	133.991	0.0351553
$245$	0	0
$246$	0	0
$247$	3002.25	0.773395
$248$	−3503.01	−0.896940
$249$	0	0
$250$	−2205.98	−0.558073
$251$	5101.02	1.28276	0.641381	−	0.767222i	$-0.278362\pi$
$251$	5101.02	1.28276	0.641381	+	0.767222i	$0.278362\pi$
$252$	0	0
$253$	2505.47	0.622598
$254$	−3730.00	−0.921421
$255$	0	0
$256$	−3497.94	−0.853990
$257$	1441.05	0.349768	0.174884	−	0.984589i	$-0.444045\pi$
$257$	1441.05	0.349768	0.174884	+	0.984589i	$0.444045\pi$
$258$	0	0
$259$	0	0
$260$	745.982	0.177938
$261$	0	0
$262$	−4181.11	−0.985915
$263$	7287.63	1.70865	0.854325	−	0.519740i	$-0.173971\pi$
$263$	7287.63	1.70865	0.854325	+	0.519740i	$0.173971\pi$
$264$	0	0
$265$	−2925.76	−0.678218
$266$	0	0
$267$	0	0
$268$	1519.01	0.346226
$269$	−1964.67	−0.445310	−0.222655	−	0.974897i	$-0.571472\pi$
$269$	−1964.67	−0.445310	−0.222655	+	0.974897i	$0.571472\pi$
$270$	0	0
$271$	−2304.66	−0.516599	−0.258299	−	0.966065i	$-0.583162\pi$
$271$	−2304.66	−0.516599	−0.258299	+	0.966065i	$0.583162\pi$
$272$	−241.430	−0.0538194
$273$	0	0
$274$	−2207.74	−0.486769
$275$	−2487.21	−0.545398
$276$	0	0
$277$	−1685.97	−0.365704	−0.182852	−	0.983140i	$-0.558533\pi$
$277$	−1685.97	−0.365704	−0.182852	+	0.983140i	$0.558533\pi$
$278$	−1348.83	−0.290997
$279$	0	0
$280$	0	0
$281$	−2841.43	−0.603223	−0.301612	−	0.953431i	$-0.597525\pi$
$281$	−2841.43	−0.603223	−0.301612	+	0.953431i	$0.597525\pi$
$282$	0	0
$283$	7307.50	1.53493	0.767466	−	0.641090i	$-0.221517\pi$
$283$	7307.50	1.53493	0.767466	+	0.641090i	$0.221517\pi$
$284$	−1862.72	−0.389197
$285$	0	0
$286$	−989.432	−0.204568
$287$	0	0
$288$	0	0
$289$	−4418.12	−0.899271
$290$	−1337.22	−0.270773
$291$	0	0
$292$	3836.78	0.768940
$293$	6778.11	1.35147	0.675736	−	0.737143i	$-0.263826\pi$
$293$	6778.11	1.35147	0.675736	+	0.737143i	$0.263826\pi$
$294$	0	0
$295$	2097.19	0.413909
$296$	1408.08	0.276496
$297$	0	0
$298$	4892.56	0.951068
$299$	1488.43	0.287886
$300$	0	0
$301$	0	0
$302$	42.0479	0.00801187
$303$	0	0
$304$	1684.52	0.317809
$305$	168.881	0.0317052
$306$	0	0
$307$	8157.51	1.51653	0.758263	−	0.651949i	$-0.226048\pi$
$307$	8157.51	1.51653	0.758263	+	0.651949i	$0.226048\pi$
$308$	0	0
$309$	0	0
$310$	−1804.88	−0.330679
$311$	−2545.26	−0.464078	−0.232039	−	0.972706i	$-0.574540\pi$
$311$	−2545.26	−0.464078	−0.232039	+	0.972706i	$0.574540\pi$
$312$	0	0
$313$	2777.18	0.501519	0.250760	−	0.968049i	$-0.419320\pi$
$313$	2777.18	0.501519	0.250760	+	0.968049i	$0.419320\pi$
$314$	3712.39	0.667205
$315$	0	0
$316$	3233.40	0.575611
$317$	4832.35	0.856188	0.428094	−	0.903734i	$-0.359185\pi$
$317$	4832.35	0.856188	0.428094	+	0.903734i	$0.359185\pi$
$318$	0	0
$319$	−3974.79	−0.697635
$320$	−1444.40	−0.252327
$321$	0	0
$322$	0	0
$323$	−3452.91	−0.594815
$324$	0	0
$325$	−1477.58	−0.252189
$326$	446.975	0.0759376
$327$	0	0
$328$	4928.89	0.829732
$329$	0	0
$330$	0	0
$331$	2609.24	0.433284	0.216642	−	0.976251i	$-0.430490\pi$
$331$	2609.24	0.433284	0.216642	+	0.976251i	$0.430490\pi$
$332$	−6691.93	−1.10623
$333$	0	0
$334$	−5722.96	−0.937564
$335$	1914.54	0.312247
$336$	0	0
$337$	−8134.50	−1.31488	−0.657440	−	0.753507i	$-0.728360\pi$
$337$	−8134.50	−1.31488	−0.657440	+	0.753507i	$0.728360\pi$
$338$	2863.89	0.460874
$339$	0	0
$340$	−857.961	−0.136851
$341$	−5364.89	−0.851980
$342$	0	0
$343$	0	0
$344$	−4472.17	−0.700940
$345$	0	0
$346$	−4105.89	−0.637960
$347$	−10500.5	−1.62449	−0.812246	−	0.583314i	$-0.801756\pi$
$347$	−10500.5	−1.62449	−0.812246	+	0.583314i	$0.801756\pi$
$348$	0	0
$349$	10622.4	1.62924	0.814618	−	0.579998i	$-0.196947\pi$
$349$	10622.4	1.62924	0.814618	+	0.579998i	$0.196947\pi$
$350$	0	0
$351$	0	0
$352$	−6092.68	−0.922560
$353$	7575.12	1.14216	0.571081	−	0.820894i	$-0.306524\pi$
$353$	7575.12	1.14216	0.571081	+	0.820894i	$0.306524\pi$
$354$	0	0
$355$	−2347.74	−0.351000
$356$	−6962.61	−1.03657
$357$	0	0
$358$	4332.18	0.639562
$359$	−9131.59	−1.34247	−0.671235	−	0.741244i	$-0.734236\pi$
$359$	−9131.59	−1.34247	−0.671235	+	0.741244i	$0.734236\pi$
$360$	0	0
$361$	17232.8	2.51244
$362$	3463.47	0.502861
$363$	0	0
$364$	0	0
$365$	4835.82	0.693475
$366$	0	0
$367$	−3582.52	−0.509554	−0.254777	−	0.967000i	$-0.582002\pi$
$367$	−3582.52	−0.509554	−0.254777	+	0.967000i	$0.582002\pi$
$368$	835.135	0.118300
$369$	0	0
$370$	725.496	0.101937
$371$	0	0
$372$	0	0
$373$	5192.26	0.720765	0.360382	−	0.932805i	$-0.382646\pi$
$373$	5192.26	0.720765	0.360382	+	0.932805i	$0.382646\pi$
$374$	1137.96	0.157332
$375$	0	0
$376$	−4385.99	−0.601569
$377$	−2361.31	−0.322583
$378$	0	0
$379$	10012.2	1.35696	0.678482	−	0.734617i	$-0.262638\pi$
$379$	10012.2	1.35696	0.678482	+	0.734617i	$0.262638\pi$
$380$	5986.20	0.808120
$381$	0	0
$382$	−5641.63	−0.755631
$383$	−3347.99	−0.446670	−0.223335	−	0.974742i	$-0.571694\pi$
$383$	−3347.99	−0.446670	−0.223335	+	0.974742i	$0.571694\pi$
$384$	0	0
$385$	0	0
$386$	5237.55	0.690633
$387$	0	0
$388$	6005.21	0.785743
$389$	−510.839	−0.0665824	−0.0332912	−	0.999446i	$-0.510599\pi$
$389$	−510.839	−0.0665824	−0.0332912	+	0.999446i	$0.510599\pi$
$390$	0	0
$391$	−1711.85	−0.221412
$392$	0	0
$393$	0	0
$394$	4910.45	0.627880
$395$	4075.34	0.519120
$396$	0	0
$397$	11828.7	1.49538	0.747691	−	0.664046i	$-0.231162\pi$
$397$	11828.7	1.49538	0.747691	+	0.664046i	$0.231162\pi$
$398$	−6888.55	−0.867567
$399$	0	0
$400$	−829.051	−0.103631
$401$	−7881.42	−0.981494	−0.490747	−	0.871302i	$-0.663276\pi$
$401$	−7881.42	−0.981494	−0.490747	+	0.871302i	$0.663276\pi$
$402$	0	0
$403$	−3187.13	−0.393951
$404$	4669.40	0.575028
$405$	0	0
$406$	0	0
$407$	2156.48	0.262636
$408$	0	0
$409$	1579.66	0.190976	0.0954881	−	0.995431i	$-0.469559\pi$
$409$	1579.66	0.190976	0.0954881	+	0.995431i	$0.469559\pi$
$410$	2539.55	0.305901
$411$	0	0
$412$	−7138.44	−0.853606
$413$	0	0
$414$	0	0
$415$	−8434.42	−0.997661
$416$	−3619.49	−0.426587
$417$	0	0
$418$	−7939.79	−0.929062
$419$	433.887	0.0505889	0.0252944	−	0.999680i	$-0.491948\pi$
$419$	433.887	0.0505889	0.0252944	+	0.999680i	$0.491948\pi$
$420$	0	0
$421$	813.228	0.0941432	0.0470716	−	0.998892i	$-0.485011\pi$
$421$	813.228	0.0941432	0.0470716	+	0.998892i	$0.485011\pi$
$422$	3666.50	0.422944
$423$	0	0
$424$	8921.36	1.02184
$425$	1699.38	0.193958
$426$	0	0
$427$	0	0
$428$	−1173.13	−0.132489
$429$	0	0
$430$	−2304.24	−0.258419
$431$	12624.4	1.41090	0.705450	−	0.708759i	$-0.250745\pi$
$431$	12624.4	1.41090	0.705450	+	0.708759i	$0.250745\pi$
$432$	0	0
$433$	−8037.33	−0.892031	−0.446016	−	0.895025i	$-0.647157\pi$
$433$	−8037.33	−0.892031	−0.446016	+	0.895025i	$0.647157\pi$
$434$	0	0
$435$	0	0
$436$	−2903.38	−0.318914
$437$	11944.0	1.30746
$438$	0	0
$439$	6820.72	0.741538	0.370769	−	0.928725i	$-0.379094\pi$
$439$	6820.72	0.741538	0.370769	+	0.928725i	$0.379094\pi$
$440$	−4825.98	−0.522886
$441$	0	0
$442$	676.027	0.0727496
$443$	−678.956	−0.0728175	−0.0364088	−	0.999337i	$-0.511592\pi$
$443$	−678.956	−0.0728175	−0.0364088	+	0.999337i	$0.511592\pi$
$444$	0	0
$445$	−8775.58	−0.934837
$446$	356.568	0.0378565
$447$	0	0
$448$	0	0
$449$	9068.89	0.953201	0.476601	−	0.879120i	$-0.341869\pi$
$449$	9068.89	0.953201	0.476601	+	0.879120i	$0.341869\pi$
$450$	0	0
$451$	7548.64	0.788141
$452$	3268.89	0.340168
$453$	0	0
$454$	2040.60	0.210947
$455$	0	0
$456$	0	0
$457$	4566.21	0.467392	0.233696	−	0.972310i	$-0.424918\pi$
$457$	4566.21	0.467392	0.233696	+	0.972310i	$0.424918\pi$
$458$	700.668	0.0714848
$459$	0	0
$460$	2967.78	0.300812
$461$	−7604.79	−0.768308	−0.384154	−	0.923269i	$-0.625507\pi$
$461$	−7604.79	−0.768308	−0.384154	+	0.923269i	$0.625507\pi$
$462$	0	0
$463$	−10608.6	−1.06484	−0.532422	−	0.846479i	$-0.678718\pi$
$463$	−10608.6	−1.06484	−0.532422	+	0.846479i	$0.678718\pi$
$464$	−1324.90	−0.132558
$465$	0	0
$466$	−5655.60	−0.562212
$467$	−13597.3	−1.34734	−0.673668	−	0.739034i	$-0.735282\pi$
$467$	−13597.3	−1.34734	−0.673668	+	0.739034i	$0.735282\pi$
$468$	0	0
$469$	0	0
$470$	−2259.83	−0.221783
$471$	0	0
$472$	−6394.85	−0.623616
$473$	−6849.18	−0.665805
$474$	0	0
$475$	−11857.0	−1.14534
$476$	0	0
$477$	0	0
$478$	−8172.22	−0.781985
$479$	11950.4	1.13993	0.569965	−	0.821669i	$-0.306957\pi$
$479$	11950.4	1.13993	0.569965	+	0.821669i	$0.306957\pi$
$480$	0	0
$481$	1281.11	0.121442
$482$	−9315.87	−0.880345
$483$	0	0
$484$	1498.54	0.140735
$485$	7568.88	0.708629
$486$	0	0
$487$	16336.6	1.52009	0.760044	−	0.649872i	$-0.225178\pi$
$487$	16336.6	1.52009	0.760044	+	0.649872i	$0.225178\pi$
$488$	−514.959	−0.0477686
$489$	0	0
$490$	0	0
$491$	−21307.1	−1.95841	−0.979204	−	0.202880i	$-0.934970\pi$
$491$	−21307.1	−1.95841	−0.979204	+	0.202880i	$0.934970\pi$
$492$	0	0
$493$	2715.77	0.248097
$494$	−4716.81	−0.429594
$495$	0	0
$496$	−1788.25	−0.161885
$497$	0	0
$498$	0	0
$499$	16006.8	1.43600	0.718000	−	0.696043i	$-0.245058\pi$
$499$	16006.8	1.43600	0.718000	+	0.696043i	$0.245058\pi$
$500$	−7767.04	−0.694706
$501$	0	0
$502$	−8014.16	−0.712529
$503$	−14633.8	−1.29719	−0.648597	−	0.761132i	$-0.724644\pi$
$503$	−14633.8	−1.29719	−0.648597	+	0.761132i	$0.724644\pi$
$504$	0	0
$505$	5885.24	0.518594
$506$	−3936.31	−0.345831
$507$	0	0
$508$	−13133.0	−1.14701
$509$	−3992.22	−0.347646	−0.173823	−	0.984777i	$-0.555612\pi$
$509$	−3992.22	−0.347646	−0.173823	+	0.984777i	$0.555612\pi$
$510$	0	0
$511$	0	0
$512$	−3876.64	−0.334619
$513$	0	0
$514$	−2264.03	−0.194284
$515$	−8997.19	−0.769832
$516$	0	0
$517$	−6717.18	−0.571415
$518$	0	0
$519$	0	0
$520$	−2866.98	−0.241780
$521$	21464.7	1.80496	0.902480	−	0.430731i	$-0.141744\pi$
$521$	21464.7	1.80496	0.902480	+	0.430731i	$0.141744\pi$
$522$	0	0
$523$	−5217.23	−0.436201	−0.218101	−	0.975926i	$-0.569986\pi$
$523$	−5217.23	−0.436201	−0.218101	+	0.975926i	$0.569986\pi$
$524$	−14721.3	−1.22730
$525$	0	0
$526$	−11449.5	−0.949094
$527$	3665.55	0.302986
$528$	0	0
$529$	−6245.51	−0.513316
$530$	4596.63	0.376726
$531$	0	0
$532$	0	0
$533$	4484.44	0.364432
$534$	0	0
$535$	−1478.59	−0.119486
$536$	−5837.92	−0.470447
$537$	0	0
$538$	3086.68	0.247354
$539$	0	0
$540$	0	0
$541$	−9563.03	−0.759976	−0.379988	−	0.924991i	$-0.624072\pi$
$541$	−9563.03	−0.759976	−0.379988	+	0.924991i	$0.624072\pi$
$542$	3620.83	0.286952
$543$	0	0
$544$	4162.81	0.328086
$545$	−3659.37	−0.287615
$546$	0	0
$547$	2206.54	0.172477	0.0862383	−	0.996275i	$-0.472515\pi$
$547$	2206.54	0.172477	0.0862383	+	0.996275i	$0.472515\pi$
$548$	−7773.27	−0.605944
$549$	0	0
$550$	3907.64	0.302949
$551$	−18948.6	−1.46504
$552$	0	0
$553$	0	0
$554$	2648.81	0.203136
$555$	0	0
$556$	−4749.10	−0.362242
$557$	4723.55	0.359324	0.179662	−	0.983728i	$-0.442500\pi$
$557$	4723.55	0.359324	0.179662	+	0.983728i	$0.442500\pi$
$558$	0	0
$559$	−4068.91	−0.307865
$560$	0	0
$561$	0	0
$562$	4464.15	0.335069
$563$	14140.9	1.05856	0.529280	−	0.848447i	$-0.322462\pi$
$563$	14140.9	1.05856	0.529280	+	0.848447i	$0.322462\pi$
$564$	0	0
$565$	4120.07	0.306783
$566$	−11480.7	−0.852600
$567$	0	0
$568$	7158.85	0.528835
$569$	14817.9	1.09174	0.545868	−	0.837871i	$-0.316200\pi$
$569$	14817.9	1.09174	0.545868	+	0.837871i	$0.316200\pi$
$570$	0	0
$571$	−20877.9	−1.53015	−0.765073	−	0.643944i	$-0.777297\pi$
$571$	−20877.9	−1.53015	−0.765073	+	0.643944i	$0.777297\pi$
$572$	−3483.70	−0.254652
$573$	0	0
$574$	0	0
$575$	−5878.35	−0.426338
$576$	0	0
$577$	−25816.5	−1.86266	−0.931330	−	0.364176i	$-0.881351\pi$
$577$	−25816.5	−1.86266	−0.931330	+	0.364176i	$0.881351\pi$
$578$	6941.27	0.499513
$579$	0	0
$580$	−4708.23	−0.337067
$581$	0	0
$582$	0	0
$583$	13663.2	0.970618
$584$	−14745.6	−1.04483
$585$	0	0
$586$	−10649.0	−0.750695
$587$	1727.78	0.121487	0.0607435	−	0.998153i	$-0.480653\pi$
$587$	1727.78	0.121487	0.0607435	+	0.998153i	$0.480653\pi$
$588$	0	0
$589$	−25575.4	−1.78916
$590$	−3294.88	−0.229912
$591$	0	0
$592$	718.811	0.0499036
$593$	−14580.1	−1.00966	−0.504832	−	0.863217i	$-0.668446\pi$
$593$	−14580.1	−1.00966	−0.504832	+	0.863217i	$0.668446\pi$
$594$	0	0
$595$	0	0
$596$	17226.3	1.18392
$597$	0	0
$598$	−2338.45	−0.159911
$599$	27691.5	1.88889	0.944444	−	0.328673i	$-0.106602\pi$
$599$	27691.5	1.88889	0.944444	+	0.328673i	$0.106602\pi$
$600$	0	0
$601$	783.554	0.0531811	0.0265906	−	0.999646i	$-0.491535\pi$
$601$	783.554	0.0531811	0.0265906	+	0.999646i	$0.491535\pi$
$602$	0	0
$603$	0	0
$604$	148.047	0.00997342
$605$	1888.74	0.126923
$606$	0	0
$607$	23576.1	1.57648	0.788241	−	0.615366i	$-0.210992\pi$
$607$	23576.1	1.57648	0.788241	+	0.615366i	$0.210992\pi$
$608$	−29044.9	−1.93738
$609$	0	0
$610$	−265.327	−0.0176111
$611$	−3990.49	−0.264219
$612$	0	0
$613$	25158.3	1.65764	0.828820	−	0.559515i	$-0.189012\pi$
$613$	25158.3	1.65764	0.828820	+	0.559515i	$0.189012\pi$
$614$	−12816.2	−0.842377
$615$	0	0
$616$	0	0
$617$	12873.3	0.839968	0.419984	−	0.907532i	$-0.362036\pi$
$617$	12873.3	0.839968	0.419984	+	0.907532i	$0.362036\pi$
$618$	0	0
$619$	−14591.6	−0.947476	−0.473738	−	0.880666i	$-0.657096\pi$
$619$	−14591.6	−0.947476	−0.473738	+	0.880666i	$0.657096\pi$
$620$	−6354.84	−0.411640
$621$	0	0
$622$	3998.83	0.257779
$623$	0	0
$624$	0	0
$625$	−240.648	−0.0154015
$626$	−4363.20	−0.278576
$627$	0	0
$628$	13071.0	0.830557
$629$	−1473.41	−0.0934003
$630$	0	0
$631$	20163.1	1.27208	0.636039	−	0.771657i	$-0.280572\pi$
$631$	20163.1	1.27208	0.636039	+	0.771657i	$0.280572\pi$
$632$	−12426.7	−0.782133
$633$	0	0
$634$	−7592.05	−0.475582
$635$	−16552.6	−1.03444
$636$	0	0
$637$	0	0
$638$	6244.76	0.387511
$639$	0	0
$640$	−8167.95	−0.504479
$641$	−10687.8	−0.658572	−0.329286	−	0.944230i	$-0.606808\pi$
$641$	−10687.8	−0.658572	−0.329286	+	0.944230i	$0.606808\pi$
$642$	0	0
$643$	9248.53	0.567226	0.283613	−	0.958939i	$-0.408467\pi$
$643$	9248.53	0.567226	0.283613	+	0.958939i	$0.408467\pi$
$644$	0	0
$645$	0	0
$646$	5424.84	0.330399
$647$	14521.5	0.882380	0.441190	−	0.897414i	$-0.354556\pi$
$647$	14521.5	0.882380	0.441190	+	0.897414i	$0.354556\pi$
$648$	0	0
$649$	−9793.78	−0.592357
$650$	2321.42	0.140082
$651$	0	0
$652$	1573.76	0.0945294
$653$	−5894.47	−0.353244	−0.176622	−	0.984279i	$-0.556517\pi$
$653$	−5894.47	−0.353244	−0.176622	+	0.984279i	$0.556517\pi$
$654$	0	0
$655$	−18554.5	−1.10685
$656$	2516.15	0.149755
$657$	0	0
$658$	0	0
$659$	−8398.63	−0.496455	−0.248228	−	0.968702i	$-0.579848\pi$
$659$	−8398.63	−0.496455	−0.248228	+	0.968702i	$0.579848\pi$
$660$	0	0
$661$	−3513.82	−0.206765	−0.103383	−	0.994642i	$-0.532967\pi$
$661$	−3513.82	−0.206765	−0.103383	+	0.994642i	$0.532967\pi$
$662$	−4099.36	−0.240674
$663$	0	0
$664$	25718.6	1.50313
$665$	0	0
$666$	0	0
$667$	−9394.13	−0.545341
$668$	−20150.0	−1.16711
$669$	0	0
$670$	−3007.92	−0.173442
$671$	−788.665	−0.0453742
$672$	0	0
$673$	−14499.1	−0.830457	−0.415229	−	0.909717i	$-0.636298\pi$
$673$	−14499.1	−0.830457	−0.415229	+	0.909717i	$0.636298\pi$
$674$	12780.0	0.730369
$675$	0	0
$676$	10083.5	0.573709
$677$	−26741.4	−1.51810	−0.759051	−	0.651031i	$-0.774337\pi$
$677$	−26741.4	−1.51810	−0.759051	+	0.651031i	$0.774337\pi$
$678$	0	0
$679$	0	0
$680$	3297.34	0.185952
$681$	0	0
$682$	8428.73	0.473245
$683$	−25949.3	−1.45377	−0.726883	−	0.686761i	$-0.759032\pi$
$683$	−25949.3	−1.45377	−0.726883	+	0.686761i	$0.759032\pi$
$684$	0	0
$685$	−9797.32	−0.546476
$686$	0	0
$687$	0	0
$688$	−2283.01	−0.126510
$689$	8116.90	0.448809
$690$	0	0
$691$	−3583.56	−0.197286	−0.0986432	−	0.995123i	$-0.531450\pi$
$691$	−3583.56	−0.197286	−0.0986432	+	0.995123i	$0.531450\pi$
$692$	−14456.5	−0.794152
$693$	0	0
$694$	16497.3	0.902348
$695$	−5985.70	−0.326691
$696$	0	0
$697$	−5157.59	−0.280284
$698$	−16688.7	−0.904983
$699$	0	0
$700$	0	0
$701$	−15096.1	−0.813367	−0.406683	−	0.913569i	$-0.633315\pi$
$701$	−15096.1	−0.813367	−0.406683	+	0.913569i	$0.633315\pi$
$702$	0	0
$703$	10280.4	0.551538
$704$	6745.30	0.361112
$705$	0	0
$706$	−11901.2	−0.634430
$707$	0	0
$708$	0	0
$709$	−29170.5	−1.54517	−0.772583	−	0.634914i	$-0.781036\pi$
$709$	−29170.5	−1.54517	−0.772583	+	0.634914i	$0.781036\pi$
$710$	3688.52	0.194968
$711$	0	0
$712$	26758.9	1.40847
$713$	−12679.5	−0.665992
$714$	0	0
$715$	−4390.81	−0.229660
$716$	15253.2	0.796146
$717$	0	0
$718$	14346.6	0.745695
$719$	−9145.94	−0.474390	−0.237195	−	0.971462i	$-0.576228\pi$
$719$	−9145.94	−0.474390	−0.237195	+	0.971462i	$0.576228\pi$
$720$	0	0
$721$	0	0
$722$	−27074.4	−1.39557
$723$	0	0
$724$	12194.6	0.625977
$725$	9325.70	0.477721
$726$	0	0
$727$	−35509.8	−1.81154	−0.905768	−	0.423773i	$-0.860705\pi$
$727$	−35509.8	−1.81154	−0.905768	+	0.423773i	$0.860705\pi$
$728$	0	0
$729$	0	0
$730$	−7597.51	−0.385201
$731$	4679.69	0.236778
$732$	0	0
$733$	−6476.19	−0.326335	−0.163168	−	0.986598i	$-0.552171\pi$
$733$	−6476.19	−0.326335	−0.163168	+	0.986598i	$0.552171\pi$
$734$	5628.47	0.283039
$735$	0	0
$736$	−14399.6	−0.721165
$737$	−8940.83	−0.446866
$738$	0	0
$739$	31421.5	1.56408	0.782042	−	0.623226i	$-0.214178\pi$
$739$	31421.5	1.56408	0.782042	+	0.623226i	$0.214178\pi$
$740$	2554.41	0.126894
$741$	0	0
$742$	0	0
$743$	12845.7	0.634268	0.317134	−	0.948381i	$-0.397279\pi$
$743$	12845.7	0.634268	0.317134	+	0.948381i	$0.397279\pi$
$744$	0	0
$745$	21711.7	1.06773
$746$	−8157.52	−0.400359
$747$	0	0
$748$	4006.64	0.195852
$749$	0	0
$750$	0	0
$751$	24892.6	1.20951	0.604756	−	0.796411i	$-0.293271\pi$
$751$	24892.6	1.20951	0.604756	+	0.796411i	$0.293271\pi$
$752$	−2239.01	−0.108575
$753$	0	0
$754$	3709.83	0.179183
$755$	186.596	0.00899461
$756$	0	0
$757$	34261.1	1.64497	0.822483	−	0.568789i	$-0.192588\pi$
$757$	34261.1	1.64497	0.822483	+	0.568789i	$0.192588\pi$
$758$	−15730.0	−0.753746
$759$	0	0
$760$	−23006.4	−1.09806
$761$	−8054.98	−0.383696	−0.191848	−	0.981425i	$-0.561448\pi$
$761$	−8054.98	−0.383696	−0.191848	+	0.981425i	$0.561448\pi$
$762$	0	0
$763$	0	0
$764$	−19863.7	−0.940632
$765$	0	0
$766$	5260.00	0.248109
$767$	−5818.21	−0.273903
$768$	0	0
$769$	1915.85	0.0898403	0.0449202	−	0.998991i	$-0.485697\pi$
$769$	1915.85	0.0898403	0.0449202	+	0.998991i	$0.485697\pi$
$770$	0	0
$771$	0	0
$772$	18441.0	0.859721
$773$	15431.6	0.718027	0.359014	−	0.933332i	$-0.383113\pi$
$773$	15431.6	0.718027	0.359014	+	0.933332i	$0.383113\pi$
$774$	0	0
$775$	12587.2	0.583412
$776$	−23079.4	−1.06766
$777$	0	0
$778$	802.574	0.0369842
$779$	35985.8	1.65510
$780$	0	0
$781$	10963.8	0.502327
$782$	2689.48	0.122987
$783$	0	0
$784$	0	0
$785$	16474.5	0.749046
$786$	0	0
$787$	−18487.3	−0.837358	−0.418679	−	0.908134i	$-0.637507\pi$
$787$	−18487.3	−0.837358	−0.418679	+	0.908134i	$0.637507\pi$
$788$	17289.3	0.781604
$789$	0	0
$790$	−6402.73	−0.288353
$791$	0	0
$792$	0	0
$793$	−468.524	−0.0209808
$794$	−18584.0	−0.830632
$795$	0	0
$796$	−24254.0	−1.07997
$797$	7017.35	0.311879	0.155939	−	0.987767i	$-0.450160\pi$
$797$	7017.35	0.311879	0.155939	+	0.987767i	$0.450160\pi$
$798$	0	0
$799$	4589.50	0.203210
$800$	14294.7	0.631743
$801$	0	0
$802$	12382.4	0.545185
$803$	−22583.1	−0.992452
$804$	0	0
$805$	0	0
$806$	5007.27	0.218826
$807$	0	0
$808$	−17945.6	−0.781340
$809$	13231.2	0.575012	0.287506	−	0.957779i	$-0.407174\pi$
$809$	13231.2	0.575012	0.287506	+	0.957779i	$0.407174\pi$
$810$	0	0
$811$	8930.40	0.386669	0.193335	−	0.981133i	$-0.438070\pi$
$811$	8930.40	0.386669	0.193335	+	0.981133i	$0.438070\pi$
$812$	0	0
$813$	0	0
$814$	−3388.03	−0.145885
$815$	1983.54	0.0852521
$816$	0	0
$817$	−32651.3	−1.39819
$818$	−2481.79	−0.106081
$819$	0	0
$820$	8941.54	0.380795
$821$	−35165.3	−1.49486	−0.747429	−	0.664341i	$-0.768712\pi$
$821$	−35165.3	−1.49486	−0.747429	+	0.664341i	$0.768712\pi$
$822$	0	0
$823$	−9236.94	−0.391227	−0.195613	−	0.980681i	$-0.562670\pi$
$823$	−9236.94	−0.391227	−0.195613	+	0.980681i	$0.562670\pi$
$824$	27434.7	1.15987
$825$	0	0
$826$	0	0
$827$	22176.8	0.932483	0.466241	−	0.884658i	$-0.345608\pi$
$827$	22176.8	0.932483	0.466241	+	0.884658i	$0.345608\pi$
$828$	0	0
$829$	36563.5	1.53185	0.765925	−	0.642930i	$-0.222281\pi$
$829$	36563.5	1.53185	0.765925	+	0.642930i	$0.222281\pi$
$830$	13251.2	0.554165
$831$	0	0
$832$	4007.19	0.166976
$833$	0	0
$834$	0	0
$835$	−25396.8	−1.05257
$836$	−27955.3	−1.15652
$837$	0	0
$838$	−681.675	−0.0281003
$839$	44231.4	1.82007	0.910035	−	0.414531i	$-0.136055\pi$
$839$	44231.4	1.82007	0.910035	+	0.414531i	$0.136055\pi$
$840$	0	0
$841$	−9485.70	−0.388933
$842$	−1277.65	−0.0522932
$843$	0	0
$844$	12909.4	0.526493
$845$	12709.1	0.517405
$846$	0	0
$847$	0	0
$848$	4554.28	0.184427
$849$	0	0
$850$	−2669.88	−0.107737
$851$	5096.70	0.205303
$852$	0	0
$853$	19451.0	0.780762	0.390381	−	0.920653i	$-0.372343\pi$
$853$	19451.0	0.780762	0.390381	+	0.920653i	$0.372343\pi$
$854$	0	0
$855$	0	0
$856$	4508.60	0.180024
$857$	26386.1	1.05173	0.525865	−	0.850568i	$-0.323742\pi$
$857$	26386.1	1.05173	0.525865	+	0.850568i	$0.323742\pi$
$858$	0	0
$859$	−27227.7	−1.08149	−0.540744	−	0.841187i	$-0.681857\pi$
$859$	−27227.7	−1.08149	−0.540744	+	0.841187i	$0.681857\pi$
$860$	−8113.02	−0.321688
$861$	0	0
$862$	−19834.2	−0.783705
$863$	−20901.0	−0.824425	−0.412212	−	0.911088i	$-0.635244\pi$
$863$	−20901.0	−0.824425	−0.412212	+	0.911088i	$0.635244\pi$
$864$	0	0
$865$	−18220.8	−0.716213
$866$	12627.4	0.495492
$867$	0	0
$868$	0	0
$869$	−19031.6	−0.742928
$870$	0	0
$871$	−5311.50	−0.206628
$872$	11158.3	0.433336
$873$	0	0
$874$	−18765.1	−0.726248
$875$	0	0
$876$	0	0
$877$	12516.3	0.481921	0.240961	−	0.970535i	$-0.422538\pi$
$877$	12516.3	0.481921	0.240961	+	0.970535i	$0.422538\pi$
$878$	−10716.0	−0.411898
$879$	0	0
$880$	−2463.62	−0.0943735
$881$	−1308.26	−0.0500301	−0.0250150	−	0.999687i	$-0.507963\pi$
$881$	−1308.26	−0.0500301	−0.0250150	+	0.999687i	$0.507963\pi$
$882$	0	0
$883$	19537.5	0.744609	0.372305	−	0.928111i	$-0.378568\pi$
$883$	19537.5	0.744609	0.372305	+	0.928111i	$0.378568\pi$
$884$	2380.23	0.0905609
$885$	0	0
$886$	1066.70	0.0404475
$887$	−286.574	−0.0108480	−0.00542402	−	0.999985i	$-0.501727\pi$
$887$	−286.574	−0.0108480	−0.00542402	+	0.999985i	$0.501727\pi$
$888$	0	0
$889$	0	0
$890$	13787.2	0.519269
$891$	0	0
$892$	1255.44	0.0471249
$893$	−32022.1	−1.19997
$894$	0	0
$895$	19225.0	0.718011
$896$	0	0
$897$	0	0
$898$	−14248.1	−0.529470
$899$	20115.4	0.746259
$900$	0	0
$901$	−9335.32	−0.345177
$902$	−11859.6	−0.437784
$903$	0	0
$904$	−12563.1	−0.462215
$905$	15369.9	0.564543
$906$	0	0
$907$	−7960.25	−0.291418	−0.145709	−	0.989328i	$-0.546546\pi$
$907$	−7960.25	−0.291418	−0.145709	+	0.989328i	$0.546546\pi$
$908$	7184.77	0.262594
$909$	0	0
$910$	0	0
$911$	38900.9	1.41476	0.707378	−	0.706835i	$-0.249878\pi$
$911$	38900.9	1.41476	0.707378	+	0.706835i	$0.249878\pi$
$912$	0	0
$913$	39388.4	1.42778
$914$	−7173.93	−0.259620
$915$	0	0
$916$	2466.99	0.0889865
$917$	0	0
$918$	0	0
$919$	1315.54	0.0472205	0.0236103	−	0.999721i	$-0.492484\pi$
$919$	1315.54	0.0472205	0.0236103	+	0.999721i	$0.492484\pi$
$920$	−11405.9	−0.408740
$921$	0	0
$922$	11947.8	0.426768
$923$	6513.31	0.232273
$924$	0	0
$925$	−5059.57	−0.179846
$926$	16667.1	0.591483
$927$	0	0
$928$	22844.2	0.808081
$929$	46155.4	1.63004	0.815021	−	0.579432i	$-0.196726\pi$
$929$	46155.4	1.63004	0.815021	+	0.579432i	$0.196726\pi$
$930$	0	0
$931$	0	0
$932$	−19912.9	−0.699858
$933$	0	0
$934$	21362.5	0.748397
$935$	5049.91	0.176631
$936$	0	0
$937$	41274.8	1.43905	0.719525	−	0.694466i	$-0.244359\pi$
$937$	41274.8	1.43905	0.719525	+	0.694466i	$0.244359\pi$
$938$	0	0
$939$	0	0
$940$	−7956.66	−0.276083
$941$	29642.6	1.02691	0.513454	−	0.858117i	$-0.328366\pi$
$941$	29642.6	1.02691	0.513454	+	0.858117i	$0.328366\pi$
$942$	0	0
$943$	17840.7	0.616090
$944$	−3264.52	−0.112554
$945$	0	0
$946$	10760.7	0.369831
$947$	24856.8	0.852944	0.426472	−	0.904501i	$-0.359756\pi$
$947$	24856.8	0.852944	0.426472	+	0.904501i	$0.359756\pi$
$948$	0	0
$949$	−13416.0	−0.458905
$950$	18628.4	0.636196
$951$	0	0
$952$	0	0
$953$	−36288.1	−1.23346	−0.616729	−	0.787175i	$-0.711543\pi$
$953$	−36288.1	−1.23346	−0.616729	+	0.787175i	$0.711543\pi$
$954$	0	0
$955$	−25035.9	−0.848317
$956$	−28773.7	−0.973438
$957$	0	0
$958$	−18775.1	−0.633190
$959$	0	0
$960$	0	0
$961$	−2640.61	−0.0886378
$962$	−2012.74	−0.0674566
$963$	0	0
$964$	−32800.4	−1.09588
$965$	23242.7	0.775347
$966$	0	0
$967$	−16773.7	−0.557815	−0.278908	−	0.960318i	$-0.589972\pi$
$967$	−16773.7	−0.557815	−0.278908	+	0.960318i	$0.589972\pi$
$968$	−5759.25	−0.191229
$969$	0	0
$970$	−11891.4	−0.393619
$971$	4458.85	0.147365	0.0736825	−	0.997282i	$-0.476525\pi$
$971$	4458.85	0.147365	0.0736825	+	0.997282i	$0.476525\pi$
$972$	0	0
$973$	0	0
$974$	−25666.3	−0.844355
$975$	0	0
$976$	−262.882	−0.00862156
$977$	4100.18	0.134265	0.0671323	−	0.997744i	$-0.478615\pi$
$977$	4100.18	0.134265	0.0671323	+	0.997744i	$0.478615\pi$
$978$	0	0
$979$	40981.6	1.33787
$980$	0	0
$981$	0	0
$982$	33475.5	1.08783
$983$	8704.08	0.282418	0.141209	−	0.989980i	$-0.454901\pi$
$983$	8704.08	0.282418	0.141209	+	0.989980i	$0.454901\pi$
$984$	0	0
$985$	21791.1	0.704897
$986$	−4266.71	−0.137809
$987$	0	0
$988$	−16607.5	−0.534771
$989$	−16187.6	−0.520460
$990$	0	0
$991$	−20466.1	−0.656030	−0.328015	−	0.944673i	$-0.606380\pi$
$991$	−20466.1	−0.656030	−0.328015	+	0.944673i	$0.606380\pi$
$992$	30833.6	0.986861
$993$	0	0
$994$	0	0
$995$	−30569.4	−0.973984
$996$	0	0
$997$	−9354.70	−0.297158	−0.148579	−	0.988901i	$-0.547470\pi$
$997$	−9354.70	−0.297158	−0.148579	+	0.988901i	$0.547470\pi$
$998$	−25148.2	−0.797647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bc.1.3	yes	6
3.2	odd	2		1323.4.a.bb.1.4	yes	6
7.6	odd	2		1323.4.a.bb.1.3	✓	6
21.20	even	2	inner	1323.4.a.bc.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bb.1.3	✓	6	7.6	odd	2
1323.4.a.bb.1.4	yes	6	3.2	odd	2
1323.4.a.bc.1.3	yes	6	1.1	even	1	trivial
1323.4.a.bc.1.4	yes	6	21.20	even	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)

\(f(q)\)	\(=\)	\(q-1.57109 q^{2} -5.53167 q^{4} -6.97204 q^{5} +21.2595 q^{8} +O(q^{10})\)	\(q-1.57109 q^{2} -5.53167 q^{4} -6.97204 q^{5} +21.2595 q^{8} +10.9537 q^{10} +32.5591 q^{11} +19.3425 q^{13} +10.8528 q^{16} -22.2459 q^{17} +155.215 q^{19} +38.5671 q^{20} -51.1534 q^{22} +76.9512 q^{23} -76.3906 q^{25} -30.3888 q^{26} -122.079 q^{29} -164.774 q^{31} -187.127 q^{32} +34.9504 q^{34} +66.2329 q^{37} -243.858 q^{38} -148.222 q^{40} +231.844 q^{41} -210.361 q^{43} -180.106 q^{44} -120.897 q^{46} -206.307 q^{47} +120.017 q^{50} -106.996 q^{52} +419.641 q^{53} -227.004 q^{55} +191.797 q^{58} -300.800 q^{59} -24.2225 q^{61} +258.875 q^{62} +207.171 q^{64} -134.857 q^{65} -274.603 q^{67} +123.057 q^{68} +336.736 q^{71} -693.601 q^{73} -104.058 q^{74} -858.601 q^{76} -584.526 q^{79} -75.6661 q^{80} -364.248 q^{82} +1209.75 q^{83} +155.100 q^{85} +330.497 q^{86} +692.191 q^{88} +1258.68 q^{89} -425.669 q^{92} +324.127 q^{94} -1082.17 q^{95} -1085.60 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{4} + 24 q^{5}+O(q^{10}) \) 6 * q + 18 * q^4 + 24 * q^5	\( 6 q + 18 q^{4} + 24 q^{5} - 30 q^{16} + 42 q^{17} + 12 q^{20} + 132 q^{22} + 222 q^{25} + 366 q^{26} - 312 q^{37} + 336 q^{38} + 360 q^{41} + 654 q^{43} + 774 q^{46} + 1812 q^{47} - 378 q^{58} - 6 q^{59} + 2058 q^{62} + 66 q^{64} + 42 q^{67} + 2910 q^{68} + 1956 q^{79} - 2868 q^{80} + 2892 q^{83} - 1944 q^{85} - 2532 q^{88} + 1518 q^{89}+O(q^{100}) \) 6 * q + 18 * q^4 + 24 * q^5 - 30 * q^16 + 42 * q^17 + 12 * q^20 + 132 * q^22 + 222 * q^25 + 366 * q^26 - 312 * q^37 + 336 * q^38 + 360 * q^41 + 654 * q^43 + 774 * q^46 + 1812 * q^47 - 378 * q^58 - 6 * q^59 + 2058 * q^62 + 66 * q^64 + 42 * q^67 + 2910 * q^68 + 1956 * q^79 - 2868 * q^80 + 2892 * q^83 - 1944 * q^85 - 2532 * q^88 + 1518 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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