LMFDB - Embedded newform 1323.4.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504 $ x^7 - x^6 - 43x^5 + 10x^4 + 513x^3 + 258x^2 - 936*x - 504
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.49279$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.49279 q^{2} +12.1851 q^{4} -8.19814 q^{5} -18.8030 q^{8} +O(q^{10})$	$q-4.49279 q^{2} +12.1851 q^{4} -8.19814 q^{5} -18.8030 q^{8} +36.8325 q^{10} +33.3981 q^{11} -54.5961 q^{13} -13.0034 q^{16} -1.19885 q^{17} -124.289 q^{19} -99.8955 q^{20} -150.051 q^{22} +133.996 q^{23} -57.7905 q^{25} +245.289 q^{26} +145.699 q^{29} +78.1442 q^{31} +208.845 q^{32} +5.38620 q^{34} -134.807 q^{37} +558.403 q^{38} +154.149 q^{40} -178.303 q^{41} +211.077 q^{43} +406.961 q^{44} -602.014 q^{46} -518.128 q^{47} +259.641 q^{50} -665.261 q^{52} -269.732 q^{53} -273.802 q^{55} -654.593 q^{58} -306.382 q^{59} +38.0209 q^{61} -351.085 q^{62} -834.270 q^{64} +447.586 q^{65} +610.708 q^{67} -14.6082 q^{68} +1084.41 q^{71} +1083.74 q^{73} +605.659 q^{74} -1514.48 q^{76} -1198.84 q^{79} +106.603 q^{80} +801.077 q^{82} -150.087 q^{83} +9.82838 q^{85} -948.323 q^{86} -627.984 q^{88} +221.463 q^{89} +1632.76 q^{92} +2327.84 q^{94} +1018.94 q^{95} -1606.63 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8}+O(q^{10}) $ 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8	$ 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8} + 12 q^{10} + 98 q^{11} - 124 q^{13} + 139 q^{16} + 30 q^{17} + 182 q^{19} - 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} - 245 q^{26} + 323 q^{29} + 26 q^{31} + 398 q^{32} + 114 q^{34} - 112 q^{37} + 1015 q^{38} - 147 q^{40} - 524 q^{41} + 8 q^{43} + 937 q^{44} - 339 q^{46} + 288 q^{47} + 2576 q^{50} - 1075 q^{52} + 1353 q^{53} + 156 q^{55} - 81 q^{58} + 165 q^{59} + 56 q^{61} - 1215 q^{62} - 1706 q^{64} + 1694 q^{65} - 988 q^{67} + 2625 q^{68} + 792 q^{71} + 1487 q^{73} + 2736 q^{74} + 1952 q^{76} - 1273 q^{79} - 2501 q^{80} - 2049 q^{82} - 1170 q^{83} + 216 q^{85} - 160 q^{86} + 9 q^{88} + 1058 q^{89} + 3834 q^{92} + 1653 q^{94} + 3260 q^{95} - 3730 q^{97}+O(q^{100}) $ 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8 + 12 * q^10 + 98 * q^11 - 124 * q^13 + 139 * q^16 + 30 * q^17 + 182 * q^19 - 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 - 245 * q^26 + 323 * q^29 + 26 * q^31 + 398 * q^32 + 114 * q^34 - 112 * q^37 + 1015 * q^38 - 147 * q^40 - 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 + 288 * q^47 + 2576 * q^50 - 1075 * q^52 + 1353 * q^53 + 156 * q^55 - 81 * q^58 + 165 * q^59 + 56 * q^61 - 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 + 2625 * q^68 + 792 * q^71 + 1487 * q^73 + 2736 * q^74 + 1952 * q^76 - 1273 * q^79 - 2501 * q^80 - 2049 * q^82 - 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 + 1058 * q^89 + 3834 * q^92 + 1653 * q^94 + 3260 * q^95 - 3730 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.49279	−1.58844	−0.794220	−	0.607630i	$-0.792120\pi$
$2$	−4.49279	−1.58844	−0.794220	+	0.607630i	$0.792120\pi$
$3$	0	0
$3$	0	0
$4$	12.1851	1.52314
$5$	−8.19814	−0.733264	−0.366632	−	0.930366i	$-0.619489\pi$
$5$	−8.19814	−0.733264	−0.366632	+	0.930366i	$0.619489\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−18.8030	−0.830982
$9$	0	0
$10$	36.8325	1.16475
$11$	33.3981	0.915447	0.457723	−	0.889095i	$-0.348665\pi$
$11$	33.3981	0.915447	0.457723	+	0.889095i	$0.348665\pi$
$12$	0	0
$13$	−54.5961	−1.16479	−0.582393	−	0.812907i	$-0.697884\pi$
$13$	−54.5961	−1.16479	−0.582393	+	0.812907i	$0.697884\pi$
$14$	0	0
$15$	0	0
$16$	−13.0034	−0.203177
$17$	−1.19885	−0.0171038	−0.00855191	−	0.999963i	$-0.502722\pi$
$17$	−1.19885	−0.0171038	−0.00855191	+	0.999963i	$0.502722\pi$
$18$	0	0
$19$	−124.289	−1.50073	−0.750363	−	0.661026i	$-0.770121\pi$
$19$	−124.289	−1.50073	−0.750363	+	0.661026i	$0.770121\pi$
$20$	−99.8955	−1.11687
$21$	0	0
$22$	−150.051	−1.45413
$23$	133.996	1.21478	0.607391	−	0.794403i	$-0.292216\pi$
$23$	133.996	1.21478	0.607391	+	0.794403i	$0.292216\pi$
$24$	0	0
$25$	−57.7905	−0.462324
$26$	245.289	1.85019
$27$	0	0
$28$	0	0
$29$	145.699	0.932949	0.466475	−	0.884535i	$-0.345524\pi$
$29$	145.699	0.932949	0.466475	+	0.884535i	$0.345524\pi$
$30$	0	0
$31$	78.1442	0.452745	0.226373	−	0.974041i	$-0.427313\pi$
$31$	78.1442	0.452745	0.226373	+	0.974041i	$0.427313\pi$
$32$	208.845	1.15372
$33$	0	0
$34$	5.38620	0.0271684
$35$	0	0
$36$	0	0
$37$	−134.807	−0.598976	−0.299488	−	0.954100i	$-0.596816\pi$
$37$	−134.807	−0.598976	−0.299488	+	0.954100i	$0.596816\pi$
$38$	558.403	2.38381
$39$	0	0
$40$	154.149	0.609329
$41$	−178.303	−0.679177	−0.339588	−	0.940574i	$-0.610288\pi$
$41$	−178.303	−0.679177	−0.339588	+	0.940574i	$0.610288\pi$
$42$	0	0
$43$	211.077	0.748579	0.374289	−	0.927312i	$-0.377887\pi$
$43$	211.077	0.748579	0.374289	+	0.927312i	$0.377887\pi$
$44$	406.961	1.39436
$45$	0	0
$46$	−602.014	−1.92961
$47$	−518.128	−1.60801	−0.804007	−	0.594620i	$-0.797303\pi$
$47$	−518.128	−1.60801	−0.804007	+	0.594620i	$0.797303\pi$
$48$	0	0
$49$	0	0
$50$	259.641	0.734374
$51$	0	0
$52$	−665.261	−1.77414
$53$	−269.732	−0.699067	−0.349533	−	0.936924i	$-0.613660\pi$
$53$	−269.732	−0.699067	−0.349533	+	0.936924i	$0.613660\pi$
$54$	0	0
$55$	−273.802	−0.671264
$56$	0	0
$57$	0	0
$58$	−654.593	−1.48193
$59$	−306.382	−0.676060	−0.338030	−	0.941135i	$-0.609760\pi$
$59$	−306.382	−0.676060	−0.338030	+	0.941135i	$0.609760\pi$
$60$	0	0
$61$	38.0209	0.0798045	0.0399023	−	0.999204i	$-0.487295\pi$
$61$	38.0209	0.0798045	0.0399023	+	0.999204i	$0.487295\pi$
$62$	−351.085	−0.719159
$63$	0	0
$64$	−834.270	−1.62943
$65$	447.586	0.854096
$66$	0	0
$67$	610.708	1.11358	0.556790	−	0.830653i	$-0.312033\pi$
$67$	610.708	1.11358	0.556790	+	0.830653i	$0.312033\pi$
$68$	−14.6082	−0.0260516
$69$	0	0
$70$	0	0
$71$	1084.41	1.81261	0.906305	−	0.422624i	$-0.138891\pi$
$71$	1084.41	1.81261	0.906305	+	0.422624i	$0.138891\pi$
$72$	0	0
$73$	1083.74	1.73757	0.868785	−	0.495190i	$-0.164901\pi$
$73$	1083.74	1.73757	0.868785	+	0.495190i	$0.164901\pi$
$74$	605.659	0.951438
$75$	0	0
$76$	−1514.48	−2.28582
$77$	0	0
$78$	0	0
$79$	−1198.84	−1.70734	−0.853671	−	0.520813i	$-0.825629\pi$
$79$	−1198.84	−1.70734	−0.853671	+	0.520813i	$0.825629\pi$
$80$	106.603	0.148983
$81$	0	0
$82$	801.077	1.07883
$83$	−150.087	−0.198484	−0.0992418	−	0.995063i	$-0.531642\pi$
$83$	−150.087	−0.198484	−0.0992418	+	0.995063i	$0.531642\pi$
$84$	0	0
$85$	9.82838	0.0125416
$86$	−948.323	−1.18907
$87$	0	0
$88$	−627.984	−0.760720
$89$	221.463	0.263764	0.131882	−	0.991265i	$-0.457898\pi$
$89$	221.463	0.263764	0.131882	+	0.991265i	$0.457898\pi$
$90$	0	0
$91$	0	0
$92$	1632.76	1.85029
$93$	0	0
$94$	2327.84	2.55424
$95$	1018.94	1.10043
$96$	0	0
$97$	−1606.63	−1.68174	−0.840869	−	0.541239i	$-0.817955\pi$
$97$	−1606.63	−1.68174	−0.840869	+	0.541239i	$0.817955\pi$
$98$	0	0
$99$	0	0
$100$	−704.186	−0.704186
$101$	−1159.68	−1.14250	−0.571249	−	0.820777i	$-0.693541\pi$
$101$	−1159.68	−1.14250	−0.571249	+	0.820777i	$0.693541\pi$
$102$	0	0
$103$	791.398	0.757075	0.378538	−	0.925586i	$-0.376427\pi$
$103$	791.398	0.757075	0.378538	+	0.925586i	$0.376427\pi$
$104$	1026.57	0.967917
$105$	0	0
$106$	1211.85	1.11043
$107$	−1876.54	−1.69544	−0.847721	−	0.530442i	$-0.822026\pi$
$107$	−1876.54	−1.69544	−0.847721	+	0.530442i	$0.822026\pi$
$108$	0	0
$109$	−1428.37	−1.25516	−0.627582	−	0.778551i	$-0.715955\pi$
$109$	−1428.37	−1.25516	−0.627582	+	0.778551i	$0.715955\pi$
$110$	1230.14	1.06626
$111$	0	0
$112$	0	0
$113$	−1016.39	−0.846144	−0.423072	−	0.906096i	$-0.639048\pi$
$113$	−1016.39	−0.846144	−0.423072	+	0.906096i	$0.639048\pi$
$114$	0	0
$115$	−1098.51	−0.890756
$116$	1775.36	1.42102
$117$	0	0
$118$	1376.51	1.07388
$119$	0	0
$120$	0	0
$121$	−215.565	−0.161957
$122$	−170.820	−0.126765
$123$	0	0
$124$	952.198	0.689596
$125$	1498.54	1.07227
$126$	0	0
$127$	505.178	0.352971	0.176485	−	0.984303i	$-0.443527\pi$
$127$	505.178	0.352971	0.176485	+	0.984303i	$0.443527\pi$
$128$	2077.44	1.43454
$129$	0	0
$130$	−2010.91	−1.35668
$131$	2511.19	1.67483	0.837417	−	0.546564i	$-0.184065\pi$
$131$	2511.19	1.67483	0.837417	+	0.546564i	$0.184065\pi$
$132$	0	0
$133$	0	0
$134$	−2743.78	−1.76885
$135$	0	0
$136$	22.5420	0.0142130
$137$	−2152.31	−1.34222	−0.671112	−	0.741356i	$-0.734183\pi$
$137$	−2152.31	−1.34222	−0.671112	+	0.741356i	$0.734183\pi$
$138$	0	0
$139$	−68.0412	−0.0415193	−0.0207596	−	0.999784i	$-0.506608\pi$
$139$	−68.0412	−0.0415193	−0.0207596	+	0.999784i	$0.506608\pi$
$140$	0	0
$141$	0	0
$142$	−4872.01	−2.87922
$143$	−1823.41	−1.06630
$144$	0	0
$145$	−1194.46	−0.684098
$146$	−4869.03	−2.76003
$147$	0	0
$148$	−1642.64	−0.912327
$149$	806.001	0.443155	0.221578	−	0.975143i	$-0.428879\pi$
$149$	806.001	0.443155	0.221578	+	0.975143i	$0.428879\pi$
$150$	0	0
$151$	−2419.94	−1.30418	−0.652092	−	0.758139i	$-0.726109\pi$
$151$	−2419.94	−1.30418	−0.652092	+	0.758139i	$0.726109\pi$
$152$	2337.00	1.24708
$153$	0	0
$154$	0	0
$155$	−640.637	−0.331982
$156$	0	0
$157$	309.080	0.157116	0.0785582	−	0.996910i	$-0.474968\pi$
$157$	309.080	0.157116	0.0785582	+	0.996910i	$0.474968\pi$
$158$	5386.13	2.71201
$159$	0	0
$160$	−1712.14	−0.845980
$161$	0	0
$162$	0	0
$163$	3859.17	1.85444	0.927219	−	0.374520i	$-0.122193\pi$
$163$	3859.17	1.85444	0.927219	+	0.374520i	$0.122193\pi$
$164$	−2172.65	−1.03448
$165$	0	0
$166$	674.307	0.315280
$167$	2907.66	1.34731	0.673656	−	0.739045i	$-0.264723\pi$
$167$	2907.66	1.34731	0.673656	+	0.739045i	$0.264723\pi$
$168$	0	0
$169$	783.732	0.356728
$170$	−44.1568	−0.0199216
$171$	0	0
$172$	2572.00	1.14019
$173$	117.665	0.0517105	0.0258552	−	0.999666i	$-0.491769\pi$
$173$	117.665	0.0517105	0.0258552	+	0.999666i	$0.491769\pi$
$174$	0	0
$175$	0	0
$176$	−434.288	−0.185998
$177$	0	0
$178$	−994.985	−0.418973
$179$	770.254	0.321628	0.160814	−	0.986985i	$-0.448588\pi$
$179$	770.254	0.321628	0.160814	+	0.986985i	$0.448588\pi$
$180$	0	0
$181$	−1669.12	−0.685442	−0.342721	−	0.939437i	$-0.611349\pi$
$181$	−1669.12	−0.685442	−0.342721	+	0.939437i	$0.611349\pi$
$182$	0	0
$183$	0	0
$184$	−2519.52	−1.00946
$185$	1105.17	0.439208
$186$	0	0
$187$	−40.0395	−0.0156576
$188$	−6313.46	−2.44924
$189$	0	0
$190$	−4577.86	−1.74796
$191$	234.737	0.0889266	0.0444633	−	0.999011i	$-0.485842\pi$
$191$	234.737	0.0889266	0.0444633	+	0.999011i	$0.485842\pi$
$192$	0	0
$193$	−1405.12	−0.524056	−0.262028	−	0.965060i	$-0.584391\pi$
$193$	−1405.12	−0.524056	−0.262028	+	0.965060i	$0.584391\pi$
$194$	7218.25	2.67134
$195$	0	0
$196$	0	0
$197$	2361.35	0.854006	0.427003	−	0.904250i	$-0.359569\pi$
$197$	2361.35	0.854006	0.427003	+	0.904250i	$0.359569\pi$
$198$	0	0
$199$	4376.07	1.55885	0.779426	−	0.626494i	$-0.215511\pi$
$199$	4376.07	1.55885	0.779426	+	0.626494i	$0.215511\pi$
$200$	1086.63	0.384183
$201$	0	0
$202$	5210.19	1.81479
$203$	0	0
$204$	0	0
$205$	1461.75	0.498016
$206$	−3555.58	−1.20257
$207$	0	0
$208$	709.932	0.236658
$209$	−4151.01	−1.37383
$210$	0	0
$211$	2545.66	0.830573	0.415286	−	0.909691i	$-0.363681\pi$
$211$	2545.66	0.830573	0.415286	+	0.909691i	$0.363681\pi$
$212$	−3286.72	−1.06478
$213$	0	0
$214$	8430.91	2.69311
$215$	−1730.44	−0.548906
$216$	0	0
$217$	0	0
$218$	6417.35	1.99375
$219$	0	0
$220$	−3336.32	−1.02243
$221$	65.4528	0.0199223
$222$	0	0
$223$	−2022.14	−0.607231	−0.303615	−	0.952795i	$-0.598194\pi$
$223$	−2022.14	−0.607231	−0.303615	+	0.952795i	$0.598194\pi$
$224$	0	0
$225$	0	0
$226$	4566.44	1.34405
$227$	432.266	0.126390	0.0631949	−	0.998001i	$-0.479871\pi$
$227$	432.266	0.126390	0.0631949	+	0.998001i	$0.479871\pi$
$228$	0	0
$229$	−2660.05	−0.767603	−0.383801	−	0.923416i	$-0.625385\pi$
$229$	−2660.05	−0.767603	−0.383801	+	0.923416i	$0.625385\pi$
$230$	4935.39	1.41491
$231$	0	0
$232$	−2739.57	−0.775264
$233$	6823.98	1.91869	0.959343	−	0.282242i	$-0.0910780\pi$
$233$	6823.98	1.91869	0.959343	+	0.282242i	$0.0910780\pi$
$234$	0	0
$235$	4247.68	1.17910
$236$	−3733.31	−1.02974
$237$	0	0
$238$	0	0
$239$	−325.438	−0.0880787	−0.0440393	−	0.999030i	$-0.514023\pi$
$239$	−325.438	−0.0880787	−0.0440393	+	0.999030i	$0.514023\pi$
$240$	0	0
$241$	−1806.27	−0.482790	−0.241395	−	0.970427i	$-0.577605\pi$
$241$	−1806.27	−0.482790	−0.241395	+	0.970427i	$0.577605\pi$
$242$	968.490	0.257260
$243$	0	0
$244$	463.290	0.121554
$245$	0	0
$246$	0	0
$247$	6785.67	1.74802
$248$	−1469.34	−0.376223
$249$	0	0
$250$	−6732.63	−1.70324
$251$	1606.29	0.403937	0.201969	−	0.979392i	$-0.435266\pi$
$251$	1606.29	0.403937	0.201969	+	0.979392i	$0.435266\pi$
$252$	0	0
$253$	4475.20	1.11207
$254$	−2269.66	−0.560673
$255$	0	0
$256$	−2659.33	−0.649250
$257$	2166.19	0.525772	0.262886	−	0.964827i	$-0.415326\pi$
$257$	2166.19	0.525772	0.262886	+	0.964827i	$0.415326\pi$
$258$	0	0
$259$	0	0
$260$	5453.91	1.30091
$261$	0	0
$262$	−11282.2	−2.66038
$263$	5791.63	1.35790	0.678949	−	0.734185i	$-0.262436\pi$
$263$	5791.63	1.35790	0.678949	+	0.734185i	$0.262436\pi$
$264$	0	0
$265$	2211.30	0.512600
$266$	0	0
$267$	0	0
$268$	7441.56	1.69614
$269$	3492.17	0.791529	0.395764	−	0.918352i	$-0.370480\pi$
$269$	3492.17	0.791529	0.395764	+	0.918352i	$0.370480\pi$
$270$	0	0
$271$	−5533.06	−1.24026	−0.620128	−	0.784501i	$-0.712919\pi$
$271$	−5533.06	−1.24026	−0.620128	+	0.784501i	$0.712919\pi$
$272$	15.5891	0.00347511
$273$	0	0
$274$	9669.89	2.13204
$275$	−1930.09	−0.423233
$276$	0	0
$277$	2674.56	0.580140	0.290070	−	0.957005i	$-0.406321\pi$
$277$	2674.56	0.580140	0.290070	+	0.957005i	$0.406321\pi$
$278$	305.695	0.0659509
$279$	0	0
$280$	0	0
$281$	4545.05	0.964893	0.482446	−	0.875926i	$-0.339748\pi$
$281$	4545.05	0.964893	0.482446	+	0.875926i	$0.339748\pi$
$282$	0	0
$283$	−29.1148	−0.00611553	−0.00305777	−	0.999995i	$-0.500973\pi$
$283$	−29.1148	−0.00611553	−0.00305777	+	0.999995i	$0.500973\pi$
$284$	13213.6	2.76087
$285$	0	0
$286$	8192.18	1.69375
$287$	0	0
$288$	0	0
$289$	−4911.56	−0.999707
$290$	5366.44	1.08665
$291$	0	0
$292$	13205.6	2.64657
$293$	2605.05	0.519415	0.259708	−	0.965687i	$-0.416374\pi$
$293$	2605.05	0.519415	0.259708	+	0.965687i	$0.416374\pi$
$294$	0	0
$295$	2511.76	0.495730
$296$	2534.77	0.497739
$297$	0	0
$298$	−3621.19	−0.703926
$299$	−7315.63	−1.41496
$300$	0	0
$301$	0	0
$302$	10872.3	2.07162
$303$	0	0
$304$	1616.17	0.304914
$305$	−311.701	−0.0585178
$306$	0	0
$307$	7800.91	1.45023	0.725116	−	0.688626i	$-0.241786\pi$
$307$	7800.91	1.45023	0.725116	+	0.688626i	$0.241786\pi$
$308$	0	0
$309$	0	0
$310$	2878.25	0.527334
$311$	7083.86	1.29160	0.645802	−	0.763505i	$-0.276523\pi$
$311$	7083.86	1.29160	0.645802	+	0.763505i	$0.276523\pi$
$312$	0	0
$313$	2684.54	0.484790	0.242395	−	0.970178i	$-0.422067\pi$
$313$	2684.54	0.484790	0.242395	+	0.970178i	$0.422067\pi$
$314$	−1388.63	−0.249570
$315$	0	0
$316$	−14608.0	−2.60053
$317$	7429.11	1.31628	0.658139	−	0.752896i	$-0.271344\pi$
$317$	7429.11	1.31628	0.658139	+	0.752896i	$0.271344\pi$
$318$	0	0
$319$	4866.06	0.854065
$320$	6839.47	1.19481
$321$	0	0
$322$	0	0
$323$	149.004	0.0256681
$324$	0	0
$325$	3155.14	0.538509
$326$	−17338.4	−2.94566
$327$	0	0
$328$	3352.63	0.564384
$329$	0	0
$330$	0	0
$331$	7566.66	1.25650	0.628250	−	0.778012i	$-0.283772\pi$
$331$	7566.66	1.25650	0.628250	+	0.778012i	$0.283772\pi$
$332$	−1828.83	−0.302319
$333$	0	0
$334$	−13063.5	−2.14013
$335$	−5006.67	−0.816548
$336$	0	0
$337$	3086.93	0.498979	0.249490	−	0.968377i	$-0.419737\pi$
$337$	3086.93	0.498979	0.249490	+	0.968377i	$0.419737\pi$
$338$	−3521.14	−0.566642
$339$	0	0
$340$	119.760	0.0191027
$341$	2609.87	0.414464
$342$	0	0
$343$	0	0
$344$	−3968.87	−0.622056
$345$	0	0
$346$	−528.644	−0.0821390
$347$	7922.81	1.22570	0.612851	−	0.790198i	$-0.290023\pi$
$347$	7922.81	1.22570	0.612851	+	0.790198i	$0.290023\pi$
$348$	0	0
$349$	200.244	0.0307129	0.0153564	−	0.999882i	$-0.495112\pi$
$349$	200.244	0.0307129	0.0153564	+	0.999882i	$0.495112\pi$
$350$	0	0
$351$	0	0
$352$	6975.04	1.05617
$353$	−8161.49	−1.23057	−0.615286	−	0.788304i	$-0.710960\pi$
$353$	−8161.49	−1.23057	−0.615286	+	0.788304i	$0.710960\pi$
$354$	0	0
$355$	−8890.11	−1.32912
$356$	2698.55	0.401750
$357$	0	0
$358$	−3460.59	−0.510888
$359$	−1990.50	−0.292631	−0.146315	−	0.989238i	$-0.546741\pi$
$359$	−1990.50	−0.292631	−0.146315	+	0.989238i	$0.546741\pi$
$360$	0	0
$361$	8588.68	1.25218
$362$	7499.02	1.08878
$363$	0	0
$364$	0	0
$365$	−8884.68	−1.27410
$366$	0	0
$367$	11008.8	1.56581	0.782907	−	0.622139i	$-0.213736\pi$
$367$	11008.8	1.56581	0.782907	+	0.622139i	$0.213736\pi$
$368$	−1742.39	−0.246816
$369$	0	0
$370$	−4965.28	−0.697655
$371$	0	0
$372$	0	0
$373$	−449.248	−0.0623625	−0.0311812	−	0.999514i	$-0.509927\pi$
$373$	−449.248	−0.0623625	−0.0311812	+	0.999514i	$0.509927\pi$
$374$	179.889	0.0248712
$375$	0	0
$376$	9742.34	1.33623
$377$	−7954.57	−1.08669
$378$	0	0
$379$	958.936	0.129966	0.0649832	−	0.997886i	$-0.479301\pi$
$379$	958.936	0.129966	0.0649832	+	0.997886i	$0.479301\pi$
$380$	12415.9	1.67611
$381$	0	0
$382$	−1054.62	−0.141255
$383$	−6854.48	−0.914485	−0.457242	−	0.889342i	$-0.651163\pi$
$383$	−6854.48	−0.914485	−0.457242	+	0.889342i	$0.651163\pi$
$384$	0	0
$385$	0	0
$386$	6312.91	0.832432
$387$	0	0
$388$	−19577.0	−2.56153
$389$	−3318.57	−0.432540	−0.216270	−	0.976334i	$-0.569389\pi$
$389$	−3318.57	−0.432540	−0.216270	+	0.976334i	$0.569389\pi$
$390$	0	0
$391$	−160.641	−0.0207774
$392$	0	0
$393$	0	0
$394$	−10609.0	−1.35654
$395$	9828.25	1.25193
$396$	0	0
$397$	−5813.51	−0.734942	−0.367471	−	0.930035i	$-0.619776\pi$
$397$	−5813.51	−0.734942	−0.367471	+	0.930035i	$0.619776\pi$
$398$	−19660.8	−2.47614
$399$	0	0
$400$	751.471	0.0939338
$401$	−860.524	−0.107163	−0.0535817	−	0.998563i	$-0.517064\pi$
$401$	−860.524	−0.107163	−0.0535817	+	0.998563i	$0.517064\pi$
$402$	0	0
$403$	−4266.37	−0.527352
$404$	−14130.8	−1.74019
$405$	0	0
$406$	0	0
$407$	−4502.30	−0.548331
$408$	0	0
$409$	7205.66	0.871142	0.435571	−	0.900154i	$-0.356547\pi$
$409$	7205.66	0.871142	0.435571	+	0.900154i	$0.356547\pi$
$410$	−6567.34	−0.791068
$411$	0	0
$412$	9643.30	1.15313
$413$	0	0
$414$	0	0
$415$	1230.43	0.145541
$416$	−11402.1	−1.34384
$417$	0	0
$418$	18649.6	2.18225
$419$	12568.5	1.46542	0.732711	−	0.680540i	$-0.238255\pi$
$419$	12568.5	1.46542	0.732711	+	0.680540i	$0.238255\pi$
$420$	0	0
$421$	7819.34	0.905205	0.452603	−	0.891712i	$-0.350496\pi$
$421$	7819.34	0.905205	0.452603	+	0.891712i	$0.350496\pi$
$422$	−11437.1	−1.31932
$423$	0	0
$424$	5071.77	0.580912
$425$	69.2824	0.00790751
$426$	0	0
$427$	0	0
$428$	−22866.0	−2.58240
$429$	0	0
$430$	7774.48	0.871904
$431$	11713.1	1.30905	0.654526	−	0.756039i	$-0.272868\pi$
$431$	11713.1	1.30905	0.654526	+	0.756039i	$0.272868\pi$
$432$	0	0
$433$	−12766.0	−1.41685	−0.708426	−	0.705785i	$-0.750594\pi$
$433$	−12766.0	−1.41685	−0.708426	+	0.705785i	$0.750594\pi$
$434$	0	0
$435$	0	0
$436$	−17404.9	−1.91179
$437$	−16654.1	−1.82306
$438$	0	0
$439$	7106.41	0.772598	0.386299	−	0.922374i	$-0.373753\pi$
$439$	7106.41	0.772598	0.386299	+	0.922374i	$0.373753\pi$
$440$	5148.30	0.557808
$441$	0	0
$442$	−294.065	−0.0316454
$443$	−7723.24	−0.828313	−0.414156	−	0.910206i	$-0.635923\pi$
$443$	−7723.24	−0.828313	−0.414156	+	0.910206i	$0.635923\pi$
$444$	0	0
$445$	−1815.58	−0.193409
$446$	9085.05	0.964550
$447$	0	0
$448$	0	0
$449$	−5221.96	−0.548863	−0.274431	−	0.961607i	$-0.588490\pi$
$449$	−5221.96	−0.548863	−0.274431	+	0.961607i	$0.588490\pi$
$450$	0	0
$451$	−5954.98	−0.621750
$452$	−12384.9	−1.28880
$453$	0	0
$454$	−1942.08	−0.200763
$455$	0	0
$456$	0	0
$457$	17829.6	1.82502	0.912511	−	0.409053i	$-0.134141\pi$
$457$	17829.6	1.82502	0.912511	+	0.409053i	$0.134141\pi$
$458$	11951.0	1.21929
$459$	0	0
$460$	−13385.6	−1.35675
$461$	5283.56	0.533796	0.266898	−	0.963725i	$-0.414001\pi$
$461$	5283.56	0.533796	0.266898	+	0.963725i	$0.414001\pi$
$462$	0	0
$463$	16885.1	1.69485	0.847427	−	0.530912i	$-0.178151\pi$
$463$	16885.1	1.69485	0.847427	+	0.530912i	$0.178151\pi$
$464$	−1894.57	−0.189554
$465$	0	0
$466$	−30658.7	−3.04772
$467$	11603.4	1.14976	0.574882	−	0.818236i	$-0.305048\pi$
$467$	11603.4	1.14976	0.574882	+	0.818236i	$0.305048\pi$
$468$	0	0
$469$	0	0
$470$	−19083.9	−1.87293
$471$	0	0
$472$	5760.89	0.561794
$473$	7049.56	0.685284
$474$	0	0
$475$	7182.71	0.693821
$476$	0	0
$477$	0	0
$478$	1462.12	0.139908
$479$	−10885.0	−1.03830	−0.519151	−	0.854683i	$-0.673752\pi$
$479$	−10885.0	−1.03830	−0.519151	+	0.854683i	$0.673752\pi$
$480$	0	0
$481$	7359.93	0.697679
$482$	8115.21	0.766883
$483$	0	0
$484$	−2626.70	−0.246684
$485$	13171.4	1.23316
$486$	0	0
$487$	6080.50	0.565778	0.282889	−	0.959153i	$-0.408707\pi$
$487$	6080.50	0.565778	0.282889	+	0.959153i	$0.408707\pi$
$488$	−714.906	−0.0663162
$489$	0	0
$490$	0	0
$491$	2787.59	0.256216	0.128108	−	0.991760i	$-0.459110\pi$
$491$	2787.59	0.256216	0.128108	+	0.991760i	$0.459110\pi$
$492$	0	0
$493$	−174.671	−0.0159570
$494$	−30486.6	−2.77663
$495$	0	0
$496$	−1016.14	−0.0919877
$497$	0	0
$498$	0	0
$499$	−797.710	−0.0715640	−0.0357820	−	0.999360i	$-0.511392\pi$
$499$	−797.710	−0.0715640	−0.0357820	+	0.999360i	$0.511392\pi$
$500$	18260.0	1.63322
$501$	0	0
$502$	−7216.73	−0.641630
$503$	11880.8	1.05316	0.526581	−	0.850125i	$-0.323474\pi$
$503$	11880.8	1.05316	0.526581	+	0.850125i	$0.323474\pi$
$504$	0	0
$505$	9507.20	0.837752
$506$	−20106.1	−1.76646
$507$	0	0
$508$	6155.67	0.537625
$509$	−3817.45	−0.332428	−0.166214	−	0.986090i	$-0.553154\pi$
$509$	−3817.45	−0.332428	−0.166214	+	0.986090i	$0.553154\pi$
$510$	0	0
$511$	0	0
$512$	−4671.70	−0.403246
$513$	0	0
$514$	−9732.24	−0.835157
$515$	−6487.99	−0.555136
$516$	0	0
$517$	−17304.5	−1.47205
$518$	0	0
$519$	0	0
$520$	−8415.96	−0.709739
$521$	−14854.6	−1.24912	−0.624559	−	0.780978i	$-0.714721\pi$
$521$	−14854.6	−1.24912	−0.624559	+	0.780978i	$0.714721\pi$
$522$	0	0
$523$	4929.02	0.412105	0.206052	−	0.978541i	$-0.433938\pi$
$523$	4929.02	0.412105	0.206052	+	0.978541i	$0.433938\pi$
$524$	30599.2	2.55101
$525$	0	0
$526$	−26020.6	−2.15694
$527$	−93.6835	−0.00774368
$528$	0	0
$529$	5787.81	0.475697
$530$	−9934.90	−0.814235
$531$	0	0
$532$	0	0
$533$	9734.64	0.791096
$534$	0	0
$535$	15384.2	1.24321
$536$	−11483.1	−0.925365
$537$	0	0
$538$	−15689.6	−1.25730
$539$	0	0
$540$	0	0
$541$	−17040.3	−1.35419	−0.677097	−	0.735894i	$-0.736762\pi$
$541$	−17040.3	−1.35419	−0.677097	+	0.735894i	$0.736762\pi$
$542$	24858.9	1.97007
$543$	0	0
$544$	−250.375	−0.0197330
$545$	11710.0	0.920366
$546$	0	0
$547$	−11566.3	−0.904096	−0.452048	−	0.891994i	$-0.649306\pi$
$547$	−11566.3	−0.904096	−0.452048	+	0.891994i	$0.649306\pi$
$548$	−26226.3	−2.04440
$549$	0	0
$550$	8671.51	0.672280
$551$	−18108.7	−1.40010
$552$	0	0
$553$	0	0
$554$	−12016.2	−0.921517
$555$	0	0
$556$	−829.092	−0.0632398
$557$	−7632.22	−0.580588	−0.290294	−	0.956938i	$-0.593753\pi$
$557$	−7632.22	−0.580588	−0.290294	+	0.956938i	$0.593753\pi$
$558$	0	0
$559$	−11524.0	−0.871935
$560$	0	0
$561$	0	0
$562$	−20419.9	−1.53267
$563$	8856.14	0.662951	0.331476	−	0.943464i	$-0.392454\pi$
$563$	8856.14	0.662951	0.331476	+	0.943464i	$0.392454\pi$
$564$	0	0
$565$	8332.54	0.620447
$566$	130.807	0.00971416
$567$	0	0
$568$	−20390.1	−1.50625
$569$	−9455.97	−0.696687	−0.348343	−	0.937367i	$-0.613256\pi$
$569$	−9455.97	−0.696687	−0.348343	+	0.937367i	$0.613256\pi$
$570$	0	0
$571$	−11895.7	−0.871841	−0.435920	−	0.899985i	$-0.643577\pi$
$571$	−11895.7	−0.871841	−0.435920	+	0.899985i	$0.643577\pi$
$572$	−22218.5	−1.62413
$573$	0	0
$574$	0	0
$575$	−7743.67	−0.561623
$576$	0	0
$577$	−12231.2	−0.882482	−0.441241	−	0.897389i	$-0.645462\pi$
$577$	−12231.2	−0.882482	−0.441241	+	0.897389i	$0.645462\pi$
$578$	22066.6	1.58798
$579$	0	0
$580$	−14554.6	−1.04198
$581$	0	0
$582$	0	0
$583$	−9008.54	−0.639958
$584$	−20377.6	−1.44389
$585$	0	0
$586$	−11703.9	−0.825061
$587$	−3450.15	−0.242595	−0.121297	−	0.992616i	$-0.538705\pi$
$587$	−3450.15	−0.242595	−0.121297	+	0.992616i	$0.538705\pi$
$588$	0	0
$589$	−9712.44	−0.679447
$590$	−11284.8	−0.787438
$591$	0	0
$592$	1752.94	0.121698
$593$	3550.47	0.245869	0.122935	−	0.992415i	$-0.460769\pi$
$593$	3550.47	0.245869	0.122935	+	0.992415i	$0.460769\pi$
$594$	0	0
$595$	0	0
$596$	9821.24	0.674989
$597$	0	0
$598$	32867.6	2.24758
$599$	−20992.1	−1.43191	−0.715956	−	0.698146i	$-0.754009\pi$
$599$	−20992.1	−1.43191	−0.715956	+	0.698146i	$0.754009\pi$
$600$	0	0
$601$	6836.28	0.463990	0.231995	−	0.972717i	$-0.425475\pi$
$601$	6836.28	0.463990	0.231995	+	0.972717i	$0.425475\pi$
$602$	0	0
$603$	0	0
$604$	−29487.3	−1.98646
$605$	1767.24	0.118758
$606$	0	0
$607$	6273.64	0.419505	0.209752	−	0.977755i	$-0.432734\pi$
$607$	6273.64	0.419505	0.209752	+	0.977755i	$0.432734\pi$
$608$	−25957.1	−1.73141
$609$	0	0
$610$	1400.40	0.0929520
$611$	28287.7	1.87299
$612$	0	0
$613$	−14473.3	−0.953624	−0.476812	−	0.879005i	$-0.658208\pi$
$613$	−14473.3	−0.953624	−0.476812	+	0.879005i	$0.658208\pi$
$614$	−35047.8	−2.30361
$615$	0	0
$616$	0	0
$617$	21999.9	1.43546	0.717732	−	0.696319i	$-0.245180\pi$
$617$	21999.9	1.43546	0.717732	+	0.696319i	$0.245180\pi$
$618$	0	0
$619$	16852.1	1.09426	0.547128	−	0.837049i	$-0.315721\pi$
$619$	16852.1	1.09426	0.547128	+	0.837049i	$0.315721\pi$
$620$	−7806.26	−0.505656
$621$	0	0
$622$	−31826.3	−2.05164
$623$	0	0
$624$	0	0
$625$	−5061.44	−0.323932
$626$	−12061.1	−0.770060
$627$	0	0
$628$	3766.18	0.239311
$629$	161.614	0.0102448
$630$	0	0
$631$	−21296.7	−1.34359	−0.671796	−	0.740736i	$-0.734477\pi$
$631$	−21296.7	−1.34359	−0.671796	+	0.740736i	$0.734477\pi$
$632$	22541.8	1.41877
$633$	0	0
$634$	−33377.4	−2.09083
$635$	−4141.52	−0.258821
$636$	0	0
$637$	0	0
$638$	−21862.2	−1.35663
$639$	0	0
$640$	−17031.1	−1.05190
$641$	6878.90	0.423869	0.211935	−	0.977284i	$-0.432024\pi$
$641$	6878.90	0.423869	0.211935	+	0.977284i	$0.432024\pi$
$642$	0	0
$643$	7514.99	0.460906	0.230453	−	0.973084i	$-0.425979\pi$
$643$	7514.99	0.460906	0.230453	+	0.973084i	$0.425979\pi$
$644$	0	0
$645$	0	0
$646$	−669.444	−0.0407723
$647$	−5992.65	−0.364135	−0.182067	−	0.983286i	$-0.558279\pi$
$647$	−5992.65	−0.364135	−0.182067	+	0.983286i	$0.558279\pi$
$648$	0	0
$649$	−10232.6	−0.618896
$650$	−14175.4	−0.855389
$651$	0	0
$652$	47024.5	2.82457
$653$	18811.8	1.12736	0.563678	−	0.825995i	$-0.309386\pi$
$653$	18811.8	1.12736	0.563678	+	0.825995i	$0.309386\pi$
$654$	0	0
$655$	−20587.1	−1.22810
$656$	2318.54	0.137993
$657$	0	0
$658$	0	0
$659$	19824.0	1.17182	0.585912	−	0.810375i	$-0.300737\pi$
$659$	19824.0	1.17182	0.585912	+	0.810375i	$0.300737\pi$
$660$	0	0
$661$	605.232	0.0356139	0.0178070	−	0.999841i	$-0.494332\pi$
$661$	605.232	0.0356139	0.0178070	+	0.999841i	$0.494332\pi$
$662$	−33995.4	−1.99587
$663$	0	0
$664$	2822.08	0.164936
$665$	0	0
$666$	0	0
$667$	19523.0	1.13333
$668$	35430.2	2.05215
$669$	0	0
$670$	22493.9	1.29704
$671$	1269.83	0.0730568
$672$	0	0
$673$	9474.72	0.542680	0.271340	−	0.962484i	$-0.412533\pi$
$673$	9474.72	0.542680	0.271340	+	0.962484i	$0.412533\pi$
$674$	−13868.9	−0.792599
$675$	0	0
$676$	9549.89	0.543348
$677$	15227.0	0.864430	0.432215	−	0.901771i	$-0.357732\pi$
$677$	15227.0	0.864430	0.432215	+	0.901771i	$0.357732\pi$
$678$	0	0
$679$	0	0
$680$	−184.803	−0.0104219
$681$	0	0
$682$	−11725.6	−0.658352
$683$	13326.9	0.746617	0.373309	−	0.927707i	$-0.378223\pi$
$683$	13326.9	0.746617	0.373309	+	0.927707i	$0.378223\pi$
$684$	0	0
$685$	17645.0	0.984204
$686$	0	0
$687$	0	0
$688$	−2744.71	−0.152094
$689$	14726.3	0.814264
$690$	0	0
$691$	−11465.0	−0.631188	−0.315594	−	0.948894i	$-0.602204\pi$
$691$	−11465.0	−0.631188	−0.315594	+	0.948894i	$0.602204\pi$
$692$	1433.77	0.0787624
$693$	0	0
$694$	−35595.5	−1.94695
$695$	557.811	0.0304446
$696$	0	0
$697$	213.759	0.0116165
$698$	−899.653	−0.0487856
$699$	0	0
$700$	0	0
$701$	3022.89	0.162872	0.0814358	−	0.996679i	$-0.474049\pi$
$701$	3022.89	0.162872	0.0814358	+	0.996679i	$0.474049\pi$
$702$	0	0
$703$	16755.0	0.898899
$704$	−27863.1	−1.49166
$705$	0	0
$706$	36667.8	1.95469
$707$	0	0
$708$	0	0
$709$	−8454.31	−0.447826	−0.223913	−	0.974609i	$-0.571883\pi$
$709$	−8454.31	−0.447826	−0.223913	+	0.974609i	$0.571883\pi$
$710$	39941.4	2.11123
$711$	0	0
$712$	−4164.16	−0.219183
$713$	10471.0	0.549987
$714$	0	0
$715$	14948.5	0.781879
$716$	9385.66	0.489886
$717$	0	0
$718$	8942.89	0.464827
$719$	−30109.8	−1.56176	−0.780881	−	0.624680i	$-0.785229\pi$
$719$	−30109.8	−1.56176	−0.780881	+	0.624680i	$0.785229\pi$
$720$	0	0
$721$	0	0
$722$	−38587.1	−1.98901
$723$	0	0
$724$	−20338.5	−1.04403
$725$	−8419.99	−0.431325
$726$	0	0
$727$	−5605.16	−0.285947	−0.142974	−	0.989726i	$-0.545666\pi$
$727$	−5605.16	−0.285947	−0.142974	+	0.989726i	$0.545666\pi$
$728$	0	0
$729$	0	0
$730$	39917.0	2.02383
$731$	−253.050	−0.0128036
$732$	0	0
$733$	−12654.2	−0.637647	−0.318824	−	0.947814i	$-0.603288\pi$
$733$	−12654.2	−0.637647	−0.318824	+	0.947814i	$0.603288\pi$
$734$	−49460.1	−2.48720
$735$	0	0
$736$	27984.3	1.40152
$737$	20396.5	1.01942
$738$	0	0
$739$	−1082.43	−0.0538805	−0.0269403	−	0.999637i	$-0.508576\pi$
$739$	−1082.43	−0.0538805	−0.0269403	+	0.999637i	$0.508576\pi$
$740$	13466.6	0.668976
$741$	0	0
$742$	0	0
$743$	11054.5	0.545830	0.272915	−	0.962038i	$-0.412012\pi$
$743$	11054.5	0.545830	0.272915	+	0.962038i	$0.412012\pi$
$744$	0	0
$745$	−6607.71	−0.324950
$746$	2018.38	0.0990591
$747$	0	0
$748$	−487.887	−0.0238488
$749$	0	0
$750$	0	0
$751$	−13634.8	−0.662505	−0.331252	−	0.943542i	$-0.607471\pi$
$751$	−13634.8	−0.662505	−0.331252	+	0.943542i	$0.607471\pi$
$752$	6737.40	0.326712
$753$	0	0
$754$	35738.2	1.72614
$755$	19839.0	0.956312
$756$	0	0
$757$	6016.97	0.288891	0.144446	−	0.989513i	$-0.453860\pi$
$757$	6016.97	0.288891	0.144446	+	0.989513i	$0.453860\pi$
$758$	−4308.30	−0.206444
$759$	0	0
$760$	−19159.0	−0.914436
$761$	−10177.7	−0.484810	−0.242405	−	0.970175i	$-0.577936\pi$
$761$	−10177.7	−0.484810	−0.242405	+	0.970175i	$0.577936\pi$
$762$	0	0
$763$	0	0
$764$	2860.31	0.135448
$765$	0	0
$766$	30795.7	1.45260
$767$	16727.2	0.787465
$768$	0	0
$769$	−27107.4	−1.27116	−0.635578	−	0.772037i	$-0.719238\pi$
$769$	−27107.4	−1.27116	−0.635578	+	0.772037i	$0.719238\pi$
$770$	0	0
$771$	0	0
$772$	−17121.6	−0.798212
$773$	27976.4	1.30174	0.650868	−	0.759191i	$-0.274405\pi$
$773$	27976.4	1.30174	0.650868	+	0.759191i	$0.274405\pi$
$774$	0	0
$775$	−4515.99	−0.209315
$776$	30209.4	1.39749
$777$	0	0
$778$	14909.6	0.687065
$779$	22161.0	1.01926
$780$	0	0
$781$	36217.1	1.65935
$782$	721.727	0.0330037
$783$	0	0
$784$	0	0
$785$	−2533.88	−0.115208
$786$	0	0
$787$	−13196.6	−0.597723	−0.298862	−	0.954296i	$-0.596607\pi$
$787$	−13196.6	−0.597723	−0.298862	+	0.954296i	$0.596607\pi$
$788$	28773.4	1.30077
$789$	0	0
$790$	−44156.3	−1.98862
$791$	0	0
$792$	0	0
$793$	−2075.79	−0.0929553
$794$	26118.9	1.16741
$795$	0	0
$796$	53323.1	2.37436
$797$	15109.4	0.671523	0.335761	−	0.941947i	$-0.391006\pi$
$797$	15109.4	0.671523	0.335761	+	0.941947i	$0.391006\pi$
$798$	0	0
$799$	621.160	0.0275032
$800$	−12069.3	−0.533391
$801$	0	0
$802$	3866.15	0.170223
$803$	36195.0	1.59065
$804$	0	0
$805$	0	0
$806$	19167.9	0.837667
$807$	0	0
$808$	21805.4	0.949395
$809$	−1786.54	−0.0776410	−0.0388205	−	0.999246i	$-0.512360\pi$
$809$	−1786.54	−0.0776410	−0.0388205	+	0.999246i	$0.512360\pi$
$810$	0	0
$811$	−3985.96	−0.172584	−0.0862921	−	0.996270i	$-0.527502\pi$
$811$	−3985.96	−0.172584	−0.0862921	+	0.996270i	$0.527502\pi$
$812$	0	0
$813$	0	0
$814$	20227.9	0.870991
$815$	−31638.0	−1.35979
$816$	0	0
$817$	−26234.4	−1.12341
$818$	−32373.5	−1.38376
$819$	0	0
$820$	17811.7	0.758549
$821$	11435.7	0.486127	0.243064	−	0.970010i	$-0.421848\pi$
$821$	11435.7	0.486127	0.243064	+	0.970010i	$0.421848\pi$
$822$	0	0
$823$	16083.0	0.681187	0.340593	−	0.940211i	$-0.389372\pi$
$823$	16083.0	0.681187	0.340593	+	0.940211i	$0.389372\pi$
$824$	−14880.6	−0.629116
$825$	0	0
$826$	0	0
$827$	−24003.6	−1.00929	−0.504647	−	0.863326i	$-0.668378\pi$
$827$	−24003.6	−1.00929	−0.504647	+	0.863326i	$0.668378\pi$
$828$	0	0
$829$	10329.8	0.432773	0.216386	−	0.976308i	$-0.430573\pi$
$829$	10329.8	0.432773	0.216386	+	0.976308i	$0.430573\pi$
$830$	−5528.07	−0.231183
$831$	0	0
$832$	45547.9	1.89794
$833$	0	0
$834$	0	0
$835$	−23837.4	−0.987935
$836$	−50580.7	−2.09255
$837$	0	0
$838$	−56467.6	−2.32774
$839$	29488.6	1.21342	0.606710	−	0.794923i	$-0.292489\pi$
$839$	29488.6	1.21342	0.606710	+	0.794923i	$0.292489\pi$
$840$	0	0
$841$	−3160.94	−0.129605
$842$	−35130.6	−1.43786
$843$	0	0
$844$	31019.3	1.26508
$845$	−6425.14	−0.261576
$846$	0	0
$847$	0	0
$848$	3507.42	0.142035
$849$	0	0
$850$	−311.271	−0.0125606
$851$	−18063.5	−0.727626
$852$	0	0
$853$	9482.29	0.380618	0.190309	−	0.981724i	$-0.439051\pi$
$853$	9482.29	0.380618	0.190309	+	0.981724i	$0.439051\pi$
$854$	0	0
$855$	0	0
$856$	35284.6	1.40888
$857$	−47038.7	−1.87492	−0.937462	−	0.348087i	$-0.886831\pi$
$857$	−47038.7	−1.87492	−0.937462	+	0.348087i	$0.886831\pi$
$858$	0	0
$859$	31374.3	1.24619	0.623096	−	0.782146i	$-0.285875\pi$
$859$	31374.3	1.24619	0.623096	+	0.782146i	$0.285875\pi$
$860$	−21085.6	−0.836062
$861$	0	0
$862$	−52624.6	−2.07935
$863$	−17338.9	−0.683922	−0.341961	−	0.939714i	$-0.611091\pi$
$863$	−17338.9	−0.683922	−0.341961	+	0.939714i	$0.611091\pi$
$864$	0	0
$865$	−964.635	−0.0379174
$866$	57355.1	2.25059
$867$	0	0
$868$	0	0
$869$	−40039.0	−1.56298
$870$	0	0
$871$	−33342.3	−1.29708
$872$	26857.6	1.04302
$873$	0	0
$874$	74823.5	2.89581
$875$	0	0
$876$	0	0
$877$	25006.5	0.962838	0.481419	−	0.876491i	$-0.340121\pi$
$877$	25006.5	0.962838	0.481419	+	0.876491i	$0.340121\pi$
$878$	−31927.6	−1.22723
$879$	0	0
$880$	3560.35	0.136386
$881$	38905.0	1.48779	0.743895	−	0.668296i	$-0.232976\pi$
$881$	38905.0	1.48779	0.743895	+	0.668296i	$0.232976\pi$
$882$	0	0
$883$	−15343.4	−0.584764	−0.292382	−	0.956302i	$-0.594448\pi$
$883$	−15343.4	−0.584764	−0.292382	+	0.956302i	$0.594448\pi$
$884$	797.552	0.0303445
$885$	0	0
$886$	34698.9	1.31573
$887$	−42542.6	−1.61042	−0.805210	−	0.592990i	$-0.797947\pi$
$887$	−42542.6	−1.61042	−0.805210	+	0.592990i	$0.797947\pi$
$888$	0	0
$889$	0	0
$890$	8157.02	0.307218
$891$	0	0
$892$	−24640.1	−0.924900
$893$	64397.4	2.41319
$894$	0	0
$895$	−6314.65	−0.235839
$896$	0	0
$897$	0	0
$898$	23461.2	0.871836
$899$	11385.5	0.422389
$900$	0	0
$901$	323.369	0.0119567
$902$	26754.5	0.987613
$903$	0	0
$904$	19111.2	0.703131
$905$	13683.7	0.502610
$906$	0	0
$907$	33699.3	1.23370	0.616850	−	0.787081i	$-0.288408\pi$
$907$	33699.3	1.23370	0.616850	+	0.787081i	$0.288408\pi$
$908$	5267.22	0.192510
$909$	0	0
$910$	0	0
$911$	654.806	0.0238141	0.0119071	−	0.999929i	$-0.496210\pi$
$911$	654.806	0.0238141	0.0119071	+	0.999929i	$0.496210\pi$
$912$	0	0
$913$	−5012.61	−0.181701
$914$	−80104.7	−2.89894
$915$	0	0
$916$	−32413.1	−1.16917
$917$	0	0
$918$	0	0
$919$	50222.2	1.80270	0.901348	−	0.433096i	$-0.142579\pi$
$919$	50222.2	1.80270	0.901348	+	0.433096i	$0.142579\pi$
$920$	20655.3	0.740203
$921$	0	0
$922$	−23737.9	−0.847903
$923$	−59204.3	−2.11130
$924$	0	0
$925$	7790.56	0.276921
$926$	−75861.2	−2.69217
$927$	0	0
$928$	30428.4	1.07636
$929$	−26946.4	−0.951650	−0.475825	−	0.879540i	$-0.657851\pi$
$929$	−26946.4	−0.951650	−0.475825	+	0.879540i	$0.657851\pi$
$930$	0	0
$931$	0	0
$932$	83151.2	2.92243
$933$	0	0
$934$	−52131.5	−1.82633
$935$	328.249	0.0114812
$936$	0	0
$937$	48370.5	1.68644	0.843220	−	0.537568i	$-0.180657\pi$
$937$	48370.5	1.68644	0.843220	+	0.537568i	$0.180657\pi$
$938$	0	0
$939$	0	0
$940$	51758.6	1.79594
$941$	23148.5	0.801934	0.400967	−	0.916092i	$-0.368674\pi$
$941$	23148.5	0.801934	0.400967	+	0.916092i	$0.368674\pi$
$942$	0	0
$943$	−23891.8	−0.825052
$944$	3983.99	0.137360
$945$	0	0
$946$	−31672.2	−1.08853
$947$	−34082.8	−1.16953	−0.584764	−	0.811203i	$-0.698813\pi$
$947$	−34082.8	−1.16953	−0.584764	+	0.811203i	$0.698813\pi$
$948$	0	0
$949$	−59168.1	−2.02390
$950$	−32270.4	−1.10209
$951$	0	0
$952$	0	0
$953$	27180.8	0.923895	0.461947	−	0.886907i	$-0.347151\pi$
$953$	27180.8	0.923895	0.461947	+	0.886907i	$0.347151\pi$
$954$	0	0
$955$	−1924.41	−0.0652067
$956$	−3965.51	−0.134156
$957$	0	0
$958$	48903.8	1.64928
$959$	0	0
$960$	0	0
$961$	−23684.5	−0.795022
$962$	−33066.6	−1.10822
$963$	0	0
$964$	−22009.7	−0.735358
$965$	11519.4	0.384271
$966$	0	0
$967$	5650.95	0.187924	0.0939619	−	0.995576i	$-0.470047\pi$
$967$	5650.95	0.187924	0.0939619	+	0.995576i	$0.470047\pi$
$968$	4053.27	0.134584
$969$	0	0
$970$	−59176.2	−1.95880
$971$	33830.8	1.11811	0.559054	−	0.829131i	$-0.311165\pi$
$971$	33830.8	1.11811	0.559054	+	0.829131i	$0.311165\pi$
$972$	0	0
$973$	0	0
$974$	−27318.4	−0.898704
$975$	0	0
$976$	−494.399	−0.0162145
$977$	9015.14	0.295210	0.147605	−	0.989046i	$-0.452844\pi$
$977$	9015.14	0.295210	0.147605	+	0.989046i	$0.452844\pi$
$978$	0	0
$979$	7396.43	0.241462
$980$	0	0
$981$	0	0
$982$	−12524.1	−0.406984
$983$	−57809.1	−1.87571	−0.937856	−	0.347025i	$-0.887192\pi$
$983$	−57809.1	−1.87571	−0.937856	+	0.347025i	$0.887192\pi$
$984$	0	0
$985$	−19358.7	−0.626212
$986$	784.762	0.0253468
$987$	0	0
$988$	82684.5	2.66249
$989$	28283.3	0.909361
$990$	0	0
$991$	−21178.9	−0.678879	−0.339440	−	0.940628i	$-0.610237\pi$
$991$	−21178.9	−0.678879	−0.339440	+	0.940628i	$0.610237\pi$
$992$	16320.0	0.522340
$993$	0	0
$994$	0	0
$995$	−35875.7	−1.14305
$996$	0	0
$997$	−30608.4	−0.972294	−0.486147	−	0.873877i	$-0.661598\pi$
$997$	−30608.4	−0.972294	−0.486147	+	0.873877i	$0.661598\pi$
$998$	3583.94	0.113675
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bj.1.1		7
3.2	odd	2		1323.4.a.bi.1.7		7
7.2	even	3		189.4.e.f.109.7	✓	14
7.4	even	3		189.4.e.f.163.7	yes	14
7.6	odd	2		1323.4.a.bk.1.1		7
21.2	odd	6		189.4.e.g.109.1	yes	14
21.11	odd	6		189.4.e.g.163.1	yes	14
21.20	even	2		1323.4.a.bh.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.f.109.7	✓	14	7.2	even	3
189.4.e.f.163.7	yes	14	7.4	even	3
189.4.e.g.109.1	yes	14	21.2	odd	6
189.4.e.g.163.1	yes	14	21.11	odd	6
1323.4.a.bh.1.7		7	21.20	even	2
1323.4.a.bi.1.7		7	3.2	odd	2
1323.4.a.bj.1.1		7	1.1	even	1	trivial
1323.4.a.bk.1.1		7	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)

\(f(q)\)	\(=\)	\(q-4.49279 q^{2} +12.1851 q^{4} -8.19814 q^{5} -18.8030 q^{8} +O(q^{10})\)	\(q-4.49279 q^{2} +12.1851 q^{4} -8.19814 q^{5} -18.8030 q^{8} +36.8325 q^{10} +33.3981 q^{11} -54.5961 q^{13} -13.0034 q^{16} -1.19885 q^{17} -124.289 q^{19} -99.8955 q^{20} -150.051 q^{22} +133.996 q^{23} -57.7905 q^{25} +245.289 q^{26} +145.699 q^{29} +78.1442 q^{31} +208.845 q^{32} +5.38620 q^{34} -134.807 q^{37} +558.403 q^{38} +154.149 q^{40} -178.303 q^{41} +211.077 q^{43} +406.961 q^{44} -602.014 q^{46} -518.128 q^{47} +259.641 q^{50} -665.261 q^{52} -269.732 q^{53} -273.802 q^{55} -654.593 q^{58} -306.382 q^{59} +38.0209 q^{61} -351.085 q^{62} -834.270 q^{64} +447.586 q^{65} +610.708 q^{67} -14.6082 q^{68} +1084.41 q^{71} +1083.74 q^{73} +605.659 q^{74} -1514.48 q^{76} -1198.84 q^{79} +106.603 q^{80} +801.077 q^{82} -150.087 q^{83} +9.82838 q^{85} -948.323 q^{86} -627.984 q^{88} +221.463 q^{89} +1632.76 q^{92} +2327.84 q^{94} +1018.94 q^{95} -1606.63 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8}+O(q^{10}) \) 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8	\( 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8} + 12 q^{10} + 98 q^{11} - 124 q^{13} + 139 q^{16} + 30 q^{17} + 182 q^{19} - 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} - 245 q^{26} + 323 q^{29} + 26 q^{31} + 398 q^{32} + 114 q^{34} - 112 q^{37} + 1015 q^{38} - 147 q^{40} - 524 q^{41} + 8 q^{43} + 937 q^{44} - 339 q^{46} + 288 q^{47} + 2576 q^{50} - 1075 q^{52} + 1353 q^{53} + 156 q^{55} - 81 q^{58} + 165 q^{59} + 56 q^{61} - 1215 q^{62} - 1706 q^{64} + 1694 q^{65} - 988 q^{67} + 2625 q^{68} + 792 q^{71} + 1487 q^{73} + 2736 q^{74} + 1952 q^{76} - 1273 q^{79} - 2501 q^{80} - 2049 q^{82} - 1170 q^{83} + 216 q^{85} - 160 q^{86} + 9 q^{88} + 1058 q^{89} + 3834 q^{92} + 1653 q^{94} + 3260 q^{95} - 3730 q^{97}+O(q^{100}) \) 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8 + 12 * q^10 + 98 * q^11 - 124 * q^13 + 139 * q^16 + 30 * q^17 + 182 * q^19 - 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 - 245 * q^26 + 323 * q^29 + 26 * q^31 + 398 * q^32 + 114 * q^34 - 112 * q^37 + 1015 * q^38 - 147 * q^40 - 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 + 288 * q^47 + 2576 * q^50 - 1075 * q^52 + 1353 * q^53 + 156 * q^55 - 81 * q^58 + 165 * q^59 + 56 * q^61 - 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 + 2625 * q^68 + 792 * q^71 + 1487 * q^73 + 2736 * q^74 + 1952 * q^76 - 1273 * q^79 - 2501 * q^80 - 2049 * q^82 - 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 + 1058 * q^89 + 3834 * q^92 + 1653 * q^94 + 3260 * q^95 - 3730 * q^97

Introduction

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