LMFDB - Embedded newform 1323.4.a.bj.1.7

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			1.7
Root			$5.09184$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.09184 q^{2} +17.9269 q^{4} -18.5985 q^{5} +50.5460 q^{8} +O(q^{10})$	$q+5.09184 q^{2} +17.9269 q^{4} -18.5985 q^{5} +50.5460 q^{8} -94.7009 q^{10} -1.05631 q^{11} -58.7684 q^{13} +113.957 q^{16} +43.7810 q^{17} +131.880 q^{19} -333.413 q^{20} -5.37857 q^{22} +161.410 q^{23} +220.906 q^{25} -299.239 q^{26} +64.0250 q^{29} +55.9772 q^{31} +175.885 q^{32} +222.926 q^{34} +296.995 q^{37} +671.513 q^{38} -940.082 q^{40} +80.6724 q^{41} -134.280 q^{43} -18.9363 q^{44} +821.874 q^{46} +233.366 q^{47} +1124.82 q^{50} -1053.53 q^{52} +387.066 q^{53} +19.6458 q^{55} +326.005 q^{58} +722.871 q^{59} -388.144 q^{61} +285.027 q^{62} -16.0817 q^{64} +1093.01 q^{65} -730.041 q^{67} +784.856 q^{68} -685.470 q^{71} +275.371 q^{73} +1512.25 q^{74} +2364.20 q^{76} +854.544 q^{79} -2119.44 q^{80} +410.771 q^{82} +922.479 q^{83} -814.263 q^{85} -683.734 q^{86} -53.3922 q^{88} -301.729 q^{89} +2893.57 q^{92} +1188.27 q^{94} -2452.78 q^{95} -913.706 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.09184	1.80024	0.900119	−	0.435644i	$-0.143479\pi$
$2$	5.09184	1.80024	0.900119	+	0.435644i	$0.143479\pi$
$3$	0	0
$3$	0	0
$4$	17.9269	2.24086
$5$	−18.5985	−1.66350	−0.831752	−	0.555147i	$-0.812662\pi$
$5$	−18.5985	−1.66350	−0.831752	+	0.555147i	$0.812662\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	50.5460	2.23384
$9$	0	0
$10$	−94.7009	−2.99470
$11$	−1.05631	−0.0289536	−0.0144768	−	0.999895i	$-0.504608\pi$
$11$	−1.05631	−0.0289536	−0.0144768	+	0.999895i	$0.504608\pi$
$12$	0	0
$13$	−58.7684	−1.25380	−0.626900	−	0.779099i	$-0.715677\pi$
$13$	−58.7684	−1.25380	−0.626900	+	0.779099i	$0.715677\pi$
$14$	0	0
$15$	0	0
$16$	113.957	1.78058
$17$	43.7810	0.624615	0.312308	−	0.949981i	$-0.398898\pi$
$17$	43.7810	0.624615	0.312308	+	0.949981i	$0.398898\pi$
$18$	0	0
$19$	131.880	1.59239	0.796194	−	0.605041i	$-0.206843\pi$
$19$	131.880	1.59239	0.796194	+	0.605041i	$0.206843\pi$
$20$	−333.413	−3.72768
$21$	0	0
$22$	−5.37857	−0.0521234
$23$	161.410	1.46332	0.731659	−	0.681671i	$-0.238747\pi$
$23$	161.410	1.46332	0.731659	+	0.681671i	$0.238747\pi$
$24$	0	0
$25$	220.906	1.76725
$26$	−299.239	−2.25714
$27$	0	0
$28$	0	0
$29$	64.0250	0.409970	0.204985	−	0.978765i	$-0.434285\pi$
$29$	64.0250	0.409970	0.204985	+	0.978765i	$0.434285\pi$
$30$	0	0
$31$	55.9772	0.324316	0.162158	−	0.986765i	$-0.448155\pi$
$31$	55.9772	0.324316	0.162158	+	0.986765i	$0.448155\pi$
$32$	175.885	0.971634
$33$	0	0
$34$	222.926	1.12446
$35$	0	0
$36$	0	0
$37$	296.995	1.31961	0.659807	−	0.751435i	$-0.270638\pi$
$37$	296.995	1.31961	0.659807	+	0.751435i	$0.270638\pi$
$38$	671.513	2.86668
$39$	0	0
$40$	−940.082	−3.71600
$41$	80.6724	0.307290	0.153645	−	0.988126i	$-0.450899\pi$
$41$	80.6724	0.307290	0.153645	+	0.988126i	$0.450899\pi$
$42$	0	0
$43$	−134.280	−0.476222	−0.238111	−	0.971238i	$-0.576528\pi$
$43$	−134.280	−0.476222	−0.238111	+	0.971238i	$0.576528\pi$
$44$	−18.9363	−0.0648809
$45$	0	0
$46$	821.874	2.63432
$47$	233.366	0.724255	0.362128	−	0.932129i	$-0.382050\pi$
$47$	233.366	0.724255	0.362128	+	0.932129i	$0.382050\pi$
$48$	0	0
$49$	0	0
$50$	1124.82	3.18147
$51$	0	0
$52$	−1053.53	−2.80959
$53$	387.066	1.00316	0.501581	−	0.865111i	$-0.332752\pi$
$53$	387.066	1.00316	0.501581	+	0.865111i	$0.332752\pi$
$54$	0	0
$55$	19.6458	0.0481645
$56$	0	0
$57$	0	0
$58$	326.005	0.738044
$59$	722.871	1.59508	0.797541	−	0.603265i	$-0.206134\pi$
$59$	722.871	1.59508	0.797541	+	0.603265i	$0.206134\pi$
$60$	0	0
$61$	−388.144	−0.814701	−0.407351	−	0.913272i	$-0.633547\pi$
$61$	−388.144	−0.814701	−0.407351	+	0.913272i	$0.633547\pi$
$62$	285.027	0.583846
$63$	0	0
$64$	−16.0817	−0.0314095
$65$	1093.01	2.08570
$66$	0	0
$67$	−730.041	−1.33117	−0.665587	−	0.746320i	$-0.731819\pi$
$67$	−730.041	−1.33117	−0.665587	+	0.746320i	$0.731819\pi$
$68$	784.856	1.39967
$69$	0	0
$70$	0	0
$71$	−685.470	−1.14578	−0.572890	−	0.819632i	$-0.694178\pi$
$71$	−685.470	−1.14578	−0.572890	+	0.819632i	$0.694178\pi$
$72$	0	0
$73$	275.371	0.441503	0.220751	−	0.975330i	$-0.429149\pi$
$73$	275.371	0.441503	0.220751	+	0.975330i	$0.429149\pi$
$74$	1512.25	2.37562
$75$	0	0
$76$	2364.20	3.56831
$77$	0	0
$78$	0	0
$79$	854.544	1.21701	0.608504	−	0.793551i	$-0.291770\pi$
$79$	854.544	1.21701	0.608504	+	0.793551i	$0.291770\pi$
$80$	−2119.44	−2.96201
$81$	0	0
$82$	410.771	0.553196
$83$	922.479	1.21994	0.609971	−	0.792424i	$-0.291181\pi$
$83$	922.479	1.21994	0.609971	+	0.792424i	$0.291181\pi$
$84$	0	0
$85$	−814.263	−1.03905
$86$	−683.734	−0.857313
$87$	0	0
$88$	−53.3922	−0.0646776
$89$	−301.729	−0.359362	−0.179681	−	0.983725i	$-0.557507\pi$
$89$	−301.729	−0.359362	−0.179681	+	0.983725i	$0.557507\pi$
$90$	0	0
$91$	0	0
$92$	2893.57	3.27908
$93$	0	0
$94$	1188.27	1.30383
$95$	−2452.78	−2.64895
$96$	0	0
$97$	−913.706	−0.956421	−0.478210	−	0.878245i	$-0.658714\pi$
$97$	−913.706	−0.956421	−0.478210	+	0.878245i	$0.658714\pi$
$98$	0	0
$99$	0	0
$100$	3960.15	3.96015
$101$	−354.715	−0.349460	−0.174730	−	0.984616i	$-0.555905\pi$
$101$	−354.715	−0.349460	−0.174730	+	0.984616i	$0.555905\pi$
$102$	0	0
$103$	−142.105	−0.135942	−0.0679708	−	0.997687i	$-0.521652\pi$
$103$	−142.105	−0.135942	−0.0679708	+	0.997687i	$0.521652\pi$
$104$	−2970.50	−2.80079
$105$	0	0
$106$	1970.88	1.80593
$107$	559.789	0.505765	0.252882	−	0.967497i	$-0.418621\pi$
$107$	559.789	0.505765	0.252882	+	0.967497i	$0.418621\pi$
$108$	0	0
$109$	−457.761	−0.402252	−0.201126	−	0.979565i	$-0.564460\pi$
$109$	−457.761	−0.402252	−0.201126	+	0.979565i	$0.564460\pi$
$110$	100.034	0.0867075
$111$	0	0
$112$	0	0
$113$	−647.946	−0.539412	−0.269706	−	0.962943i	$-0.586927\pi$
$113$	−647.946	−0.539412	−0.269706	+	0.962943i	$0.586927\pi$
$114$	0	0
$115$	−3001.99	−2.43424
$116$	1147.77	0.918685
$117$	0	0
$118$	3680.75	2.87153
$119$	0	0
$120$	0	0
$121$	−1329.88	−0.999162
$122$	−1976.37	−1.46666
$123$	0	0
$124$	1003.50	0.726746
$125$	−1783.71	−1.27632
$126$	0	0
$127$	1108.02	0.774180	0.387090	−	0.922042i	$-0.373480\pi$
$127$	1108.02	0.774180	0.387090	+	0.922042i	$0.373480\pi$
$128$	−1488.96	−1.02818
$129$	0	0
$130$	5565.42	3.75476
$131$	706.818	0.471412	0.235706	−	0.971824i	$-0.424260\pi$
$131$	706.818	0.471412	0.235706	+	0.971824i	$0.424260\pi$
$132$	0	0
$133$	0	0
$134$	−3717.25	−2.39643
$135$	0	0
$136$	2212.95	1.39529
$137$	−319.732	−0.199391	−0.0996955	−	0.995018i	$-0.531787\pi$
$137$	−319.732	−0.199391	−0.0996955	+	0.995018i	$0.531787\pi$
$138$	0	0
$139$	−439.298	−0.268063	−0.134032	−	0.990977i	$-0.542792\pi$
$139$	−439.298	−0.268063	−0.134032	+	0.990977i	$0.542792\pi$
$140$	0	0
$141$	0	0
$142$	−3490.31	−2.06268
$143$	62.0777	0.0363021
$144$	0	0
$145$	−1190.77	−0.681987
$146$	1402.14	0.794810
$147$	0	0
$148$	5324.19	2.95706
$149$	1362.30	0.749017	0.374509	−	0.927223i	$-0.377811\pi$
$149$	1362.30	0.749017	0.374509	+	0.927223i	$0.377811\pi$
$150$	0	0
$151$	2209.41	1.19072	0.595362	−	0.803458i	$-0.297009\pi$
$151$	2209.41	1.19072	0.595362	+	0.803458i	$0.297009\pi$
$152$	6666.01	3.55714
$153$	0	0
$154$	0	0
$155$	−1041.09	−0.539502
$156$	0	0
$157$	−1109.10	−0.563795	−0.281898	−	0.959444i	$-0.590964\pi$
$157$	−1109.10	−0.563795	−0.281898	+	0.959444i	$0.590964\pi$
$158$	4351.20	2.19091
$159$	0	0
$160$	−3271.20	−1.61632
$161$	0	0
$162$	0	0
$163$	1477.44	0.709953	0.354977	−	0.934875i	$-0.384489\pi$
$163$	1477.44	0.709953	0.354977	+	0.934875i	$0.384489\pi$
$164$	1446.20	0.688594
$165$	0	0
$166$	4697.12	2.19619
$167$	1848.67	0.856615	0.428307	−	0.903633i	$-0.359110\pi$
$167$	1848.67	0.856615	0.428307	+	0.903633i	$0.359110\pi$
$168$	0	0
$169$	1256.72	0.572017
$170$	−4146.10	−1.87054
$171$	0	0
$172$	−2407.22	−1.06715
$173$	2269.80	0.997513	0.498757	−	0.866742i	$-0.333790\pi$
$173$	2269.80	0.997513	0.498757	+	0.866742i	$0.333790\pi$
$174$	0	0
$175$	0	0
$176$	−120.374	−0.0515543
$177$	0	0
$178$	−1536.36	−0.646937
$179$	3666.49	1.53098	0.765492	−	0.643445i	$-0.222495\pi$
$179$	3666.49	1.53098	0.765492	+	0.643445i	$0.222495\pi$
$180$	0	0
$181$	3237.15	1.32937	0.664685	−	0.747124i	$-0.268566\pi$
$181$	3237.15	1.32937	0.664685	+	0.747124i	$0.268566\pi$
$182$	0	0
$183$	0	0
$184$	8158.62	3.26881
$185$	−5523.68	−2.19518
$186$	0	0
$187$	−46.2464	−0.0180849
$188$	4183.53	1.62295
$189$	0	0
$190$	−12489.2	−4.76873
$191$	−3353.99	−1.27061	−0.635304	−	0.772262i	$-0.719125\pi$
$191$	−3353.99	−1.27061	−0.635304	+	0.772262i	$0.719125\pi$
$192$	0	0
$193$	−5321.04	−1.98454	−0.992271	−	0.124086i	$-0.960400\pi$
$193$	−5321.04	−1.98454	−0.992271	+	0.124086i	$0.960400\pi$
$194$	−4652.45	−1.72178
$195$	0	0
$196$	0	0
$197$	1750.53	0.633098	0.316549	−	0.948576i	$-0.397476\pi$
$197$	1750.53	0.633098	0.316549	+	0.948576i	$0.397476\pi$
$198$	0	0
$199$	1572.10	0.560017	0.280009	−	0.959997i	$-0.409663\pi$
$199$	1572.10	0.560017	0.280009	+	0.959997i	$0.409663\pi$
$200$	11165.9	3.94774
$201$	0	0
$202$	−1806.15	−0.629110
$203$	0	0
$204$	0	0
$205$	−1500.39	−0.511179
$206$	−723.574	−0.244727
$207$	0	0
$208$	−6697.08	−2.23250
$209$	−139.306	−0.0461054
$210$	0	0
$211$	−2884.12	−0.941000	−0.470500	−	0.882400i	$-0.655926\pi$
$211$	−2884.12	−0.941000	−0.470500	+	0.882400i	$0.655926\pi$
$212$	6938.87	2.24794
$213$	0	0
$214$	2850.36	0.910497
$215$	2497.42	0.792198
$216$	0	0
$217$	0	0
$218$	−2330.84	−0.724150
$219$	0	0
$220$	352.188	0.107930
$221$	−2572.94	−0.783143
$222$	0	0
$223$	1267.27	0.380551	0.190276	−	0.981731i	$-0.439062\pi$
$223$	1267.27	0.380551	0.190276	+	0.981731i	$0.439062\pi$
$224$	0	0
$225$	0	0
$226$	−3299.24	−0.971071
$227$	910.785	0.266304	0.133152	−	0.991096i	$-0.457490\pi$
$227$	910.785	0.266304	0.133152	+	0.991096i	$0.457490\pi$
$228$	0	0
$229$	677.564	0.195523	0.0977613	−	0.995210i	$-0.468832\pi$
$229$	677.564	0.195523	0.0977613	+	0.995210i	$0.468832\pi$
$230$	−15285.7	−4.38220
$231$	0	0
$232$	3236.20	0.915807
$233$	−3312.78	−0.931449	−0.465724	−	0.884930i	$-0.654206\pi$
$233$	−3312.78	−0.931449	−0.465724	+	0.884930i	$0.654206\pi$
$234$	0	0
$235$	−4340.28	−1.20480
$236$	12958.8	3.57435
$237$	0	0
$238$	0	0
$239$	4933.62	1.33527	0.667634	−	0.744490i	$-0.267307\pi$
$239$	4933.62	1.33527	0.667634	+	0.744490i	$0.267307\pi$
$240$	0	0
$241$	−3931.98	−1.05096	−0.525479	−	0.850807i	$-0.676114\pi$
$241$	−3931.98	−1.05096	−0.525479	+	0.850807i	$0.676114\pi$
$242$	−6771.56	−1.79873
$243$	0	0
$244$	−6958.20	−1.82563
$245$	0	0
$246$	0	0
$247$	−7750.38	−1.99654
$248$	2829.42	0.724470
$249$	0	0
$250$	−9082.38	−2.29768
$251$	1637.68	0.411831	0.205915	−	0.978570i	$-0.433983\pi$
$251$	1637.68	0.411831	0.205915	+	0.978570i	$0.433983\pi$
$252$	0	0
$253$	−170.499	−0.0423683
$254$	5641.86	1.39371
$255$	0	0
$256$	−7452.90	−1.81956
$257$	−7868.11	−1.90973	−0.954863	−	0.297047i	$-0.903998\pi$
$257$	−7868.11	−1.90973	−0.954863	+	0.297047i	$0.903998\pi$
$258$	0	0
$259$	0	0
$260$	19594.2	4.67376
$261$	0	0
$262$	3599.01	0.848654
$263$	3047.36	0.714480	0.357240	−	0.934013i	$-0.383718\pi$
$263$	3047.36	0.714480	0.357240	+	0.934013i	$0.383718\pi$
$264$	0	0
$265$	−7198.86	−1.66876
$266$	0	0
$267$	0	0
$268$	−13087.3	−2.98297
$269$	2891.25	0.655326	0.327663	−	0.944795i	$-0.393739\pi$
$269$	2891.25	0.655326	0.327663	+	0.944795i	$0.393739\pi$
$270$	0	0
$271$	−1804.39	−0.404461	−0.202230	−	0.979338i	$-0.564819\pi$
$271$	−1804.39	−0.404461	−0.202230	+	0.979338i	$0.564819\pi$
$272$	4989.17	1.11218
$273$	0	0
$274$	−1628.03	−0.358951
$275$	−233.345	−0.0511682
$276$	0	0
$277$	−309.700	−0.0671772	−0.0335886	−	0.999436i	$-0.510694\pi$
$277$	−309.700	−0.0671772	−0.0335886	+	0.999436i	$0.510694\pi$
$278$	−2236.84	−0.482577
$279$	0	0
$280$	0	0
$281$	−1192.16	−0.253090	−0.126545	−	0.991961i	$-0.540389\pi$
$281$	−1192.16	−0.253090	−0.126545	+	0.991961i	$0.540389\pi$
$282$	0	0
$283$	2248.09	0.472210	0.236105	−	0.971728i	$-0.424129\pi$
$283$	2248.09	0.472210	0.236105	+	0.971728i	$0.424129\pi$
$284$	−12288.3	−2.56753
$285$	0	0
$286$	316.090	0.0653523
$287$	0	0
$288$	0	0
$289$	−2996.22	−0.609856
$290$	−6063.22	−1.22774
$291$	0	0
$292$	4936.53	0.989344
$293$	−9472.83	−1.88877	−0.944383	−	0.328847i	$-0.893340\pi$
$293$	−9472.83	−1.88877	−0.944383	+	0.328847i	$0.893340\pi$
$294$	0	0
$295$	−13444.4	−2.65343
$296$	15011.9	2.94780
$297$	0	0
$298$	6936.59	1.34841
$299$	−9485.80	−1.83471
$300$	0	0
$301$	0	0
$302$	11250.0	2.14359
$303$	0	0
$304$	15028.7	2.83538
$305$	7218.92	1.35526
$306$	0	0
$307$	−2379.45	−0.442352	−0.221176	−	0.975234i	$-0.570990\pi$
$307$	−2379.45	−0.442352	−0.221176	+	0.975234i	$0.570990\pi$
$308$	0	0
$309$	0	0
$310$	−5301.09	−0.971231
$311$	−1756.21	−0.320210	−0.160105	−	0.987100i	$-0.551183\pi$
$311$	−1756.21	−0.320210	−0.160105	+	0.987100i	$0.551183\pi$
$312$	0	0
$313$	8379.01	1.51313	0.756565	−	0.653918i	$-0.226876\pi$
$313$	8379.01	1.51313	0.756565	+	0.653918i	$0.226876\pi$
$314$	−5647.37	−1.01497
$315$	0	0
$316$	15319.3	2.72714
$317$	6933.28	1.22843	0.614214	−	0.789139i	$-0.289473\pi$
$317$	6933.28	1.22843	0.614214	+	0.789139i	$0.289473\pi$
$318$	0	0
$319$	−67.6303	−0.0118701
$320$	299.095	0.0522498
$321$	0	0
$322$	0	0
$323$	5773.85	0.994630
$324$	0	0
$325$	−12982.3	−2.21578
$326$	7522.91	1.27808
$327$	0	0
$328$	4077.66	0.686437
$329$	0	0
$330$	0	0
$331$	11171.1	1.85504	0.927520	−	0.373773i	$-0.121936\pi$
$331$	11171.1	1.85504	0.927520	+	0.373773i	$0.121936\pi$
$332$	16537.1	2.73372
$333$	0	0
$334$	9413.16	1.54211
$335$	13577.7	2.21442
$336$	0	0
$337$	−6360.51	−1.02813	−0.514064	−	0.857752i	$-0.671861\pi$
$337$	−6360.51	−1.02813	−0.514064	+	0.857752i	$0.671861\pi$
$338$	6399.03	1.02977
$339$	0	0
$340$	−14597.2	−2.32836
$341$	−59.1293	−0.00939012
$342$	0	0
$343$	0	0
$344$	−6787.33	−1.06380
$345$	0	0
$346$	11557.5	1.79576
$347$	1727.97	0.267326	0.133663	−	0.991027i	$-0.457326\pi$
$347$	1727.97	0.267326	0.133663	+	0.991027i	$0.457326\pi$
$348$	0	0
$349$	−2514.58	−0.385680	−0.192840	−	0.981230i	$-0.561770\pi$
$349$	−2514.58	−0.385680	−0.192840	+	0.981230i	$0.561770\pi$
$350$	0	0
$351$	0	0
$352$	−185.789	−0.0281323
$353$	5642.22	0.850722	0.425361	−	0.905024i	$-0.360147\pi$
$353$	5642.22	0.850722	0.425361	+	0.905024i	$0.360147\pi$
$354$	0	0
$355$	12748.7	1.90601
$356$	−5409.05	−0.805278
$357$	0	0
$358$	18669.2	2.75614
$359$	1725.87	0.253727	0.126863	−	0.991920i	$-0.459509\pi$
$359$	1725.87	0.253727	0.126863	+	0.991920i	$0.459509\pi$
$360$	0	0
$361$	10533.4	1.53570
$362$	16483.1	2.39318
$363$	0	0
$364$	0	0
$365$	−5121.50	−0.734442
$366$	0	0
$367$	8800.56	1.25173	0.625866	−	0.779931i	$-0.284746\pi$
$367$	8800.56	1.25173	0.625866	+	0.779931i	$0.284746\pi$
$368$	18393.8	2.60556
$369$	0	0
$370$	−28125.7	−3.95185
$371$	0	0
$372$	0	0
$373$	−10805.8	−1.50001	−0.750005	−	0.661432i	$-0.769949\pi$
$373$	−10805.8	−1.50001	−0.750005	+	0.661432i	$0.769949\pi$
$374$	−235.479	−0.0325570
$375$	0	0
$376$	11795.7	1.61787
$377$	−3762.64	−0.514021
$378$	0	0
$379$	−7808.27	−1.05827	−0.529134	−	0.848538i	$-0.677483\pi$
$379$	−7808.27	−1.05827	−0.529134	+	0.848538i	$0.677483\pi$
$380$	−43970.6	−5.93591
$381$	0	0
$382$	−17078.0	−2.28740
$383$	−6071.57	−0.810033	−0.405016	−	0.914309i	$-0.632734\pi$
$383$	−6071.57	−0.810033	−0.405016	+	0.914309i	$0.632734\pi$
$384$	0	0
$385$	0	0
$386$	−27093.9	−3.57265
$387$	0	0
$388$	−16379.9	−2.14320
$389$	−2769.36	−0.360956	−0.180478	−	0.983579i	$-0.557764\pi$
$389$	−2769.36	−0.360956	−0.180478	+	0.983579i	$0.557764\pi$
$390$	0	0
$391$	7066.69	0.914010
$392$	0	0
$393$	0	0
$394$	8913.44	1.13973
$395$	−15893.3	−2.02450
$396$	0	0
$397$	8064.82	1.01955	0.509775	−	0.860308i	$-0.329729\pi$
$397$	8064.82	1.01955	0.509775	+	0.860308i	$0.329729\pi$
$398$	8004.90	1.00816
$399$	0	0
$400$	25173.8	3.14673
$401$	−4.73201	−0.000589290 0	−0.000294645	−	1.00000i	$-0.500094\pi$
$401$	−4.73201	−0.000589290 0	−0.000294645		1.00000i	$0.500094\pi$
$402$	0	0
$403$	−3289.69	−0.406628
$404$	−6358.92	−0.783089
$405$	0	0
$406$	0	0
$407$	−313.719	−0.0382076
$408$	0	0
$409$	−12101.6	−1.46304	−0.731522	−	0.681818i	$-0.761190\pi$
$409$	−12101.6	−1.46304	−0.731522	+	0.681818i	$0.761190\pi$
$410$	−7639.74	−0.920244
$411$	0	0
$412$	−2547.49	−0.304626
$413$	0	0
$414$	0	0
$415$	−17156.8	−2.02938
$416$	−10336.4	−1.21824
$417$	0	0
$418$	−709.326	−0.0830007
$419$	13439.5	1.56698	0.783488	−	0.621407i	$-0.213438\pi$
$419$	13439.5	1.56698	0.783488	+	0.621407i	$0.213438\pi$
$420$	0	0
$421$	−9748.84	−1.12857	−0.564287	−	0.825579i	$-0.690849\pi$
$421$	−9748.84	−1.12857	−0.564287	+	0.825579i	$0.690849\pi$
$422$	−14685.5	−1.69402
$423$	0	0
$424$	19564.6	2.24090
$425$	9671.49	1.10385
$426$	0	0
$427$	0	0
$428$	10035.3	1.13335
$429$	0	0
$430$	12716.5	1.42614
$431$	−90.9615	−0.0101658	−0.00508290	−	0.999987i	$-0.501618\pi$
$431$	−90.9615	−0.0101658	−0.00508290	+	0.999987i	$0.501618\pi$
$432$	0	0
$433$	13636.5	1.51346	0.756728	−	0.653730i	$-0.226797\pi$
$433$	13636.5	1.51346	0.756728	+	0.653730i	$0.226797\pi$
$434$	0	0
$435$	0	0
$436$	−8206.21	−0.901390
$437$	21286.8	2.33017
$438$	0	0
$439$	−2295.45	−0.249558	−0.124779	−	0.992185i	$-0.539822\pi$
$439$	−2295.45	−0.249558	−0.124779	+	0.992185i	$0.539822\pi$
$440$	993.018	0.107592
$441$	0	0
$442$	−13101.0	−1.40984
$443$	2112.15	0.226526	0.113263	−	0.993565i	$-0.463870\pi$
$443$	2112.15	0.226526	0.113263	+	0.993565i	$0.463870\pi$
$444$	0	0
$445$	5611.72	0.597800
$446$	6452.76	0.685083
$447$	0	0
$448$	0	0
$449$	−6924.57	−0.727818	−0.363909	−	0.931434i	$-0.618558\pi$
$449$	−6924.57	−0.727818	−0.363909	+	0.931434i	$0.618558\pi$
$450$	0	0
$451$	−85.2151	−0.00889716
$452$	−11615.6	−1.20875
$453$	0	0
$454$	4637.57	0.479410
$455$	0	0
$456$	0	0
$457$	−11733.6	−1.20104	−0.600519	−	0.799610i	$-0.705039\pi$
$457$	−11733.6	−1.20104	−0.600519	+	0.799610i	$0.705039\pi$
$458$	3450.05	0.351987
$459$	0	0
$460$	−53816.2	−5.45477
$461$	4623.07	0.467067	0.233533	−	0.972349i	$-0.424971\pi$
$461$	4623.07	0.467067	0.233533	+	0.972349i	$0.424971\pi$
$462$	0	0
$463$	932.150	0.0935652	0.0467826	−	0.998905i	$-0.485103\pi$
$463$	932.150	0.0935652	0.0467826	+	0.998905i	$0.485103\pi$
$464$	7296.11	0.729986
$465$	0	0
$466$	−16868.2	−1.67683
$467$	−2223.55	−0.220329	−0.110164	−	0.993913i	$-0.535138\pi$
$467$	−2223.55	−0.220329	−0.110164	+	0.993913i	$0.535138\pi$
$468$	0	0
$469$	0	0
$470$	−22100.0	−2.16893
$471$	0	0
$472$	36538.2	3.56315
$473$	141.842	0.0137883
$474$	0	0
$475$	29133.1	2.81414
$476$	0	0
$477$	0	0
$478$	25121.2	2.40380
$479$	1738.85	0.165867	0.0829333	−	0.996555i	$-0.473571\pi$
$479$	1738.85	0.165867	0.0829333	+	0.996555i	$0.473571\pi$
$480$	0	0
$481$	−17453.9	−1.65453
$482$	−20021.0	−1.89197
$483$	0	0
$484$	−23840.6	−2.23898
$485$	16993.6	1.59101
$486$	0	0
$487$	−13284.4	−1.23608	−0.618041	−	0.786146i	$-0.712074\pi$
$487$	−13284.4	−1.23608	−0.618041	+	0.786146i	$0.712074\pi$
$488$	−19619.1	−1.81991
$489$	0	0
$490$	0	0
$491$	−7762.78	−0.713501	−0.356751	−	0.934200i	$-0.616115\pi$
$491$	−7762.78	−0.713501	−0.356751	+	0.934200i	$0.616115\pi$
$492$	0	0
$493$	2803.08	0.256074
$494$	−39463.7	−3.59424
$495$	0	0
$496$	6379.01	0.577472
$497$	0	0
$498$	0	0
$499$	18195.4	1.63234	0.816170	−	0.577811i	$-0.196093\pi$
$499$	18195.4	1.63234	0.816170	+	0.577811i	$0.196093\pi$
$500$	−31976.3	−2.86005
$501$	0	0
$502$	8338.81	0.741393
$503$	3509.89	0.311130	0.155565	−	0.987826i	$-0.450280\pi$
$503$	3509.89	0.311130	0.155565	+	0.987826i	$0.450280\pi$
$504$	0	0
$505$	6597.17	0.581328
$506$	−868.154	−0.0762730
$507$	0	0
$508$	19863.3	1.73483
$509$	618.568	0.0538655	0.0269328	−	0.999637i	$-0.491426\pi$
$509$	618.568	0.0538655	0.0269328	+	0.999637i	$0.491426\pi$
$510$	0	0
$511$	0	0
$512$	−26037.3	−2.24746
$513$	0	0
$514$	−40063.2	−3.43796
$515$	2642.94	0.226139
$516$	0	0
$517$	−246.507	−0.0209698
$518$	0	0
$519$	0	0
$520$	55247.1	4.65912
$521$	−19934.5	−1.67629	−0.838143	−	0.545450i	$-0.816359\pi$
$521$	−19934.5	−1.67629	−0.838143	+	0.545450i	$0.816359\pi$
$522$	0	0
$523$	4547.48	0.380206	0.190103	−	0.981764i	$-0.439118\pi$
$523$	4547.48	0.380206	0.190103	+	0.981764i	$0.439118\pi$
$524$	12671.0	1.05637
$525$	0	0
$526$	15516.7	1.28623
$527$	2450.74	0.202573
$528$	0	0
$529$	13886.2	1.14130
$530$	−36655.5	−3.00417
$531$	0	0
$532$	0	0
$533$	−4740.98	−0.385281
$534$	0	0
$535$	−10411.3	−0.841342
$536$	−36900.6	−2.97363
$537$	0	0
$538$	14721.8	1.17974
$539$	0	0
$540$	0	0
$541$	−15500.0	−1.23179	−0.615895	−	0.787828i	$-0.711206\pi$
$541$	−15500.0	−1.23179	−0.615895	+	0.787828i	$0.711206\pi$
$542$	−9187.66	−0.728125
$543$	0	0
$544$	7700.41	0.606897
$545$	8513.68	0.669149
$546$	0	0
$547$	−4515.69	−0.352974	−0.176487	−	0.984303i	$-0.556473\pi$
$547$	−4515.69	−0.352974	−0.176487	+	0.984303i	$0.556473\pi$
$548$	−5731.79	−0.446807
$549$	0	0
$550$	−1188.16	−0.0921149
$551$	8443.62	0.652832
$552$	0	0
$553$	0	0
$554$	−1576.95	−0.120935
$555$	0	0
$556$	−7875.23	−0.600691
$557$	11635.4	0.885112	0.442556	−	0.896741i	$-0.354072\pi$
$557$	11635.4	0.885112	0.442556	+	0.896741i	$0.354072\pi$
$558$	0	0
$559$	7891.43	0.597088
$560$	0	0
$561$	0	0
$562$	−6070.29	−0.455623
$563$	−2088.05	−0.156307	−0.0781536	−	0.996941i	$-0.524902\pi$
$563$	−2088.05	−0.156307	−0.0781536	+	0.996941i	$0.524902\pi$
$564$	0	0
$565$	12050.8	0.897315
$566$	11446.9	0.850090
$567$	0	0
$568$	−34647.8	−2.55948
$569$	22610.1	1.66585	0.832923	−	0.553388i	$-0.186665\pi$
$569$	22610.1	1.66585	0.832923	+	0.553388i	$0.186665\pi$
$570$	0	0
$571$	7300.32	0.535042	0.267521	−	0.963552i	$-0.413796\pi$
$571$	7300.32	0.535042	0.267521	+	0.963552i	$0.413796\pi$
$572$	1112.86	0.0813477
$573$	0	0
$574$	0	0
$575$	35656.4	2.58604
$576$	0	0
$577$	−23134.9	−1.66918	−0.834592	−	0.550869i	$-0.814296\pi$
$577$	−23134.9	−1.66918	−0.834592	+	0.550869i	$0.814296\pi$
$578$	−15256.3	−1.09789
$579$	0	0
$580$	−21346.8	−1.52824
$581$	0	0
$582$	0	0
$583$	−408.862	−0.0290452
$584$	13918.9	0.986245
$585$	0	0
$586$	−48234.1	−3.40023
$587$	263.501	0.0185278	0.00926392	−	0.999957i	$-0.497051\pi$
$587$	263.501	0.0185278	0.00926392	+	0.999957i	$0.497051\pi$
$588$	0	0
$589$	7382.28	0.516437
$590$	−68456.5	−4.77680
$591$	0	0
$592$	33844.7	2.34968
$593$	26652.2	1.84566	0.922829	−	0.385211i	$-0.125871\pi$
$593$	26652.2	1.84566	0.922829	+	0.385211i	$0.125871\pi$
$594$	0	0
$595$	0	0
$596$	24421.7	1.67844
$597$	0	0
$598$	−48300.2	−3.30291
$599$	−14447.5	−0.985491	−0.492746	−	0.870173i	$-0.664007\pi$
$599$	−14447.5	−0.985491	−0.492746	+	0.870173i	$0.664007\pi$
$600$	0	0
$601$	−26357.9	−1.78896	−0.894478	−	0.447112i	$-0.852452\pi$
$601$	−26357.9	−1.78896	−0.894478	+	0.447112i	$0.852452\pi$
$602$	0	0
$603$	0	0
$604$	39607.8	2.66824
$605$	24733.9	1.66211
$606$	0	0
$607$	7693.50	0.514447	0.257224	−	0.966352i	$-0.417192\pi$
$607$	7693.50	0.514447	0.257224	+	0.966352i	$0.417192\pi$
$608$	23195.7	1.54722
$609$	0	0
$610$	36757.6	2.43979
$611$	−13714.6	−0.908072
$612$	0	0
$613$	766.271	0.0504884	0.0252442	−	0.999681i	$-0.491964\pi$
$613$	766.271	0.0504884	0.0252442	+	0.999681i	$0.491964\pi$
$614$	−12115.8	−0.796340
$615$	0	0
$616$	0	0
$617$	−17976.0	−1.17291	−0.586455	−	0.809982i	$-0.699477\pi$
$617$	−17976.0	−1.17291	−0.586455	+	0.809982i	$0.699477\pi$
$618$	0	0
$619$	−7552.67	−0.490416	−0.245208	−	0.969471i	$-0.578856\pi$
$619$	−7552.67	−0.490416	−0.245208	+	0.969471i	$0.578856\pi$
$620$	−18663.6	−1.20895
$621$	0	0
$622$	−8942.32	−0.576454
$623$	0	0
$624$	0	0
$625$	5561.19	0.355916
$626$	42664.6	2.72399
$627$	0	0
$628$	−19882.7	−1.26338
$629$	13002.8	0.824251
$630$	0	0
$631$	−13073.2	−0.824779	−0.412390	−	0.911008i	$-0.635306\pi$
$631$	−13073.2	−0.824779	−0.412390	+	0.911008i	$0.635306\pi$
$632$	43193.7	2.71860
$633$	0	0
$634$	35303.2	2.21146
$635$	−20607.6	−1.28785
$636$	0	0
$637$	0	0
$638$	−344.363	−0.0213690
$639$	0	0
$640$	27692.5	1.71038
$641$	11900.9	0.733318	0.366659	−	0.930355i	$-0.380502\pi$
$641$	11900.9	0.733318	0.366659	+	0.930355i	$0.380502\pi$
$642$	0	0
$643$	27777.9	1.70366	0.851829	−	0.523820i	$-0.175494\pi$
$643$	27777.9	1.70366	0.851829	+	0.523820i	$0.175494\pi$
$644$	0	0
$645$	0	0
$646$	29399.5	1.79057
$647$	−14688.0	−0.892496	−0.446248	−	0.894909i	$-0.647240\pi$
$647$	−14688.0	−0.892496	−0.446248	+	0.894909i	$0.647240\pi$
$648$	0	0
$649$	−763.577	−0.0461834
$650$	−66103.7	−3.98893
$651$	0	0
$652$	26485.9	1.59090
$653$	17485.7	1.04788	0.523941	−	0.851754i	$-0.324461\pi$
$653$	17485.7	1.04788	0.523941	+	0.851754i	$0.324461\pi$
$654$	0	0
$655$	−13145.8	−0.784196
$656$	9193.20	0.547156
$657$	0	0
$658$	0	0
$659$	−18436.4	−1.08980	−0.544902	−	0.838500i	$-0.683433\pi$
$659$	−18436.4	−1.08980	−0.544902	+	0.838500i	$0.683433\pi$
$660$	0	0
$661$	9836.24	0.578798	0.289399	−	0.957209i	$-0.406545\pi$
$661$	9836.24	0.578798	0.289399	+	0.957209i	$0.406545\pi$
$662$	56881.4	3.33951
$663$	0	0
$664$	46627.6	2.72515
$665$	0	0
$666$	0	0
$667$	10334.3	0.599917
$668$	33140.9	1.91955
$669$	0	0
$670$	69135.5	3.98647
$671$	410.001	0.0235885
$672$	0	0
$673$	−9955.62	−0.570224	−0.285112	−	0.958494i	$-0.592031\pi$
$673$	−9955.62	−0.570224	−0.285112	+	0.958494i	$0.592031\pi$
$674$	−32386.7	−1.85088
$675$	0	0
$676$	22529.1	1.28181
$677$	−12851.6	−0.729581	−0.364791	−	0.931090i	$-0.618859\pi$
$677$	−12851.6	−0.729581	−0.364791	+	0.931090i	$0.618859\pi$
$678$	0	0
$679$	0	0
$680$	−41157.7	−2.32107
$681$	0	0
$682$	−301.077	−0.0169045
$683$	17472.2	0.978853	0.489426	−	0.872045i	$-0.337206\pi$
$683$	17472.2	0.978853	0.489426	+	0.872045i	$0.337206\pi$
$684$	0	0
$685$	5946.56	0.331688
$686$	0	0
$687$	0	0
$688$	−15302.2	−0.847952
$689$	−22747.2	−1.25777
$690$	0	0
$691$	8275.06	0.455569	0.227784	−	0.973712i	$-0.426852\pi$
$691$	8275.06	0.455569	0.227784	+	0.973712i	$0.426852\pi$
$692$	40690.4	2.23528
$693$	0	0
$694$	8798.55	0.481251
$695$	8170.31	0.445924
$696$	0	0
$697$	3531.92	0.191938
$698$	−12803.8	−0.694315
$699$	0	0
$700$	0	0
$701$	30709.9	1.65463	0.827315	−	0.561738i	$-0.189867\pi$
$701$	30709.9	1.65463	0.827315	+	0.561738i	$0.189867\pi$
$702$	0	0
$703$	39167.8	2.10134
$704$	16.9872	0.000909418 0
$705$	0	0
$706$	28729.3	1.53150
$707$	0	0
$708$	0	0
$709$	−18261.0	−0.967287	−0.483643	−	0.875265i	$-0.660687\pi$
$709$	−18261.0	−0.967287	−0.483643	+	0.875265i	$0.660687\pi$
$710$	64914.6	3.43127
$711$	0	0
$712$	−15251.2	−0.802756
$713$	9035.28	0.474578
$714$	0	0
$715$	−1154.55	−0.0603886
$716$	65728.6	3.43072
$717$	0	0
$718$	8787.86	0.456769
$719$	77.4789	0.00401874	0.00200937	−	0.999998i	$-0.499360\pi$
$719$	77.4789	0.00401874	0.00200937	+	0.999998i	$0.499360\pi$
$720$	0	0
$721$	0	0
$722$	53634.3	2.76463
$723$	0	0
$724$	58032.0	2.97893
$725$	14143.5	0.724519
$726$	0	0
$727$	−7339.56	−0.374428	−0.187214	−	0.982319i	$-0.559946\pi$
$727$	−7339.56	−0.374428	−0.187214	+	0.982319i	$0.559946\pi$
$728$	0	0
$729$	0	0
$730$	−26077.8	−1.32217
$731$	−5878.93	−0.297455
$732$	0	0
$733$	−21479.9	−1.08237	−0.541185	−	0.840903i	$-0.682024\pi$
$733$	−21479.9	−1.08237	−0.541185	+	0.840903i	$0.682024\pi$
$734$	44811.0	2.25341
$735$	0	0
$736$	28389.5	1.42181
$737$	771.150	0.0385423
$738$	0	0
$739$	19042.0	0.947863	0.473932	−	0.880562i	$-0.342834\pi$
$739$	19042.0	0.947863	0.473932	+	0.880562i	$0.342834\pi$
$740$	−99022.2	−4.91909
$741$	0	0
$742$	0	0
$743$	31563.4	1.55848	0.779238	−	0.626728i	$-0.215606\pi$
$743$	31563.4	1.55848	0.779238	+	0.626728i	$0.215606\pi$
$744$	0	0
$745$	−25336.7	−1.24599
$746$	−55021.5	−2.70038
$747$	0	0
$748$	−829.052	−0.0405256
$749$	0	0
$750$	0	0
$751$	30254.1	1.47003	0.735013	−	0.678053i	$-0.237176\pi$
$751$	30254.1	1.47003	0.735013	+	0.678053i	$0.237176\pi$
$752$	26593.8	1.28960
$753$	0	0
$754$	−19158.8	−0.925360
$755$	−41091.8	−1.98077
$756$	0	0
$757$	3384.65	0.162506	0.0812531	−	0.996693i	$-0.474108\pi$
$757$	3384.65	0.162506	0.0812531	+	0.996693i	$0.474108\pi$
$758$	−39758.5	−1.90514
$759$	0	0
$760$	−123978.	−5.91731
$761$	−552.737	−0.0263294	−0.0131647	−	0.999913i	$-0.504191\pi$
$761$	−552.737	−0.0263294	−0.0131647	+	0.999913i	$0.504191\pi$
$762$	0	0
$763$	0	0
$764$	−60126.4	−2.84725
$765$	0	0
$766$	−30915.5	−1.45825
$767$	−42482.0	−1.99992
$768$	0	0
$769$	6199.24	0.290703	0.145351	−	0.989380i	$-0.453569\pi$
$769$	6199.24	0.290703	0.145351	+	0.989380i	$0.453569\pi$
$770$	0	0
$771$	0	0
$772$	−95389.5	−4.44708
$773$	−30857.1	−1.43577	−0.717887	−	0.696160i	$-0.754891\pi$
$773$	−30857.1	−1.43577	−0.717887	+	0.696160i	$0.754891\pi$
$774$	0	0
$775$	12365.7	0.573147
$776$	−46184.2	−2.13649
$777$	0	0
$778$	−14101.1	−0.649807
$779$	10639.1	0.489326
$780$	0	0
$781$	724.069	0.0331744
$782$	35982.5	1.64544
$783$	0	0
$784$	0	0
$785$	20627.7	0.937876
$786$	0	0
$787$	3181.28	0.144092	0.0720459	−	0.997401i	$-0.477047\pi$
$787$	3181.28	0.144092	0.0720459	+	0.997401i	$0.477047\pi$
$788$	31381.6	1.41868
$789$	0	0
$790$	−80926.0	−3.64458
$791$	0	0
$792$	0	0
$793$	22810.6	1.02147
$794$	41064.8	1.83543
$795$	0	0
$796$	28182.9	1.25492
$797$	−10742.3	−0.477429	−0.238714	−	0.971090i	$-0.576726\pi$
$797$	−10742.3	−0.477429	−0.238714	+	0.971090i	$0.576726\pi$
$798$	0	0
$799$	10217.0	0.452381
$800$	38853.9	1.71712
$801$	0	0
$802$	−24.0947	−0.00106086
$803$	−290.877	−0.0127831
$804$	0	0
$805$	0	0
$806$	−16750.6	−0.732027
$807$	0	0
$808$	−17929.4	−0.780636
$809$	17671.9	0.767998	0.383999	−	0.923334i	$-0.374547\pi$
$809$	17671.9	0.767998	0.383999	+	0.923334i	$0.374547\pi$
$810$	0	0
$811$	−34620.2	−1.49899	−0.749494	−	0.662012i	$-0.769703\pi$
$811$	−34620.2	−1.49899	−0.749494	+	0.662012i	$0.769703\pi$
$812$	0	0
$813$	0	0
$814$	−1597.41	−0.0687827
$815$	−27478.3	−1.18101
$816$	0	0
$817$	−17708.9	−0.758330
$818$	−61619.4	−2.63383
$819$	0	0
$820$	−26897.2	−1.14548
$821$	38889.0	1.65315	0.826574	−	0.562829i	$-0.190287\pi$
$821$	38889.0	1.65315	0.826574	+	0.562829i	$0.190287\pi$
$822$	0	0
$823$	−18404.8	−0.779529	−0.389764	−	0.920915i	$-0.627444\pi$
$823$	−18404.8	−0.779529	−0.389764	+	0.920915i	$0.627444\pi$
$824$	−7182.81	−0.303671
$825$	0	0
$826$	0	0
$827$	−20341.4	−0.855307	−0.427654	−	0.903943i	$-0.640660\pi$
$827$	−20341.4	−0.855307	−0.427654	+	0.903943i	$0.640660\pi$
$828$	0	0
$829$	−19988.8	−0.837442	−0.418721	−	0.908115i	$-0.637522\pi$
$829$	−19988.8	−0.837442	−0.418721	+	0.908115i	$0.637522\pi$
$830$	−87359.6	−3.65337
$831$	0	0
$832$	945.093	0.0393812
$833$	0	0
$834$	0	0
$835$	−34382.7	−1.42498
$836$	−2497.33	−0.103316
$837$	0	0
$838$	68431.9	2.82093
$839$	556.650	0.0229055	0.0114527	−	0.999934i	$-0.496354\pi$
$839$	556.650	0.0229055	0.0114527	+	0.999934i	$0.496354\pi$
$840$	0	0
$841$	−20289.8	−0.831924
$842$	−49639.5	−2.03170
$843$	0	0
$844$	−51703.2	−2.10865
$845$	−23373.2	−0.951553
$846$	0	0
$847$	0	0
$848$	44109.0	1.78621
$849$	0	0
$850$	49245.7	1.98719
$851$	47938.0	1.93101
$852$	0	0
$853$	−16702.9	−0.670455	−0.335227	−	0.942137i	$-0.608813\pi$
$853$	−16702.9	−0.670455	−0.335227	+	0.942137i	$0.608813\pi$
$854$	0	0
$855$	0	0
$856$	28295.1	1.12980
$857$	−8216.56	−0.327506	−0.163753	−	0.986501i	$-0.552360\pi$
$857$	−8216.56	−0.327506	−0.163753	+	0.986501i	$0.552360\pi$
$858$	0	0
$859$	−19311.8	−0.767068	−0.383534	−	0.923527i	$-0.625293\pi$
$859$	−19311.8	−0.767068	−0.383534	+	0.923527i	$0.625293\pi$
$860$	44770.8	1.77520
$861$	0	0
$862$	−463.161	−0.0183009
$863$	−4145.65	−0.163522	−0.0817611	−	0.996652i	$-0.526054\pi$
$863$	−4145.65	−0.163522	−0.0817611	+	0.996652i	$0.526054\pi$
$864$	0	0
$865$	−42215.0	−1.65937
$866$	69434.7	2.72458
$867$	0	0
$868$	0	0
$869$	−902.664	−0.0352368
$870$	0	0
$871$	42903.3	1.66903
$872$	−23137.9	−0.898566
$873$	0	0
$874$	108389.	4.19486
$875$	0	0
$876$	0	0
$877$	6801.41	0.261878	0.130939	−	0.991390i	$-0.458201\pi$
$877$	6801.41	0.261878	0.130939	+	0.991390i	$0.458201\pi$
$878$	−11688.1	−0.449263
$879$	0	0
$880$	2238.79	0.0857608
$881$	−36057.6	−1.37890	−0.689451	−	0.724333i	$-0.742148\pi$
$881$	−36057.6	−1.37890	−0.689451	+	0.724333i	$0.742148\pi$
$882$	0	0
$883$	−23743.3	−0.904898	−0.452449	−	0.891790i	$-0.649450\pi$
$883$	−23743.3	−0.904898	−0.452449	+	0.891790i	$0.649450\pi$
$884$	−46124.7	−1.75491
$885$	0	0
$886$	10754.7	0.407801
$887$	34610.0	1.31014	0.655068	−	0.755570i	$-0.272640\pi$
$887$	34610.0	1.31014	0.655068	+	0.755570i	$0.272640\pi$
$888$	0	0
$889$	0	0
$890$	28574.0	1.07618
$891$	0	0
$892$	22718.2	0.852761
$893$	30776.4	1.15330
$894$	0	0
$895$	−68191.4	−2.54680
$896$	0	0
$897$	0	0
$898$	−35258.8	−1.31025
$899$	3583.94	0.132960
$900$	0	0
$901$	16946.1	0.626590
$902$	−433.902	−0.0160170
$903$	0	0
$904$	−32751.0	−1.20496
$905$	−60206.4	−2.21141
$906$	0	0
$907$	6863.70	0.251274	0.125637	−	0.992076i	$-0.459903\pi$
$907$	6863.70	0.251274	0.125637	+	0.992076i	$0.459903\pi$
$908$	16327.5	0.596748
$909$	0	0
$910$	0	0
$911$	−13817.7	−0.502524	−0.251262	−	0.967919i	$-0.580846\pi$
$911$	−13817.7	−0.502524	−0.251262	+	0.967919i	$0.580846\pi$
$912$	0	0
$913$	−974.424	−0.0353217
$914$	−59745.6	−2.16216
$915$	0	0
$916$	12146.6	0.438138
$917$	0	0
$918$	0	0
$919$	30221.0	1.08476	0.542382	−	0.840132i	$-0.317523\pi$
$919$	30221.0	1.08476	0.542382	+	0.840132i	$0.317523\pi$
$920$	−151739.	−5.43769
$921$	0	0
$922$	23539.9	0.840831
$923$	40284.0	1.43658
$924$	0	0
$925$	65608.0	2.33208
$926$	4746.36	0.168440
$927$	0	0
$928$	11261.0	0.398341
$929$	−52669.0	−1.86008	−0.930040	−	0.367459i	$-0.880228\pi$
$929$	−52669.0	−1.86008	−0.930040	+	0.367459i	$0.880228\pi$
$930$	0	0
$931$	0	0
$932$	−59387.8	−2.08724
$933$	0	0
$934$	−11322.0	−0.396644
$935$	860.115	0.0300842
$936$	0	0
$937$	−54528.9	−1.90116	−0.950578	−	0.310486i	$-0.899508\pi$
$937$	−54528.9	−1.90116	−0.950578	+	0.310486i	$0.899508\pi$
$938$	0	0
$939$	0	0
$940$	−77807.5	−2.69979
$941$	1414.91	0.0490169	0.0245085	−	0.999700i	$-0.492198\pi$
$941$	1414.91	0.0490169	0.0245085	+	0.999700i	$0.492198\pi$
$942$	0	0
$943$	13021.3	0.449663
$944$	82376.4	2.84017
$945$	0	0
$946$	722.235	0.0248223
$947$	45287.6	1.55401	0.777005	−	0.629494i	$-0.216738\pi$
$947$	45287.6	1.55401	0.777005	+	0.629494i	$0.216738\pi$
$948$	0	0
$949$	−16183.1	−0.553557
$950$	148341.	5.06613
$951$	0	0
$952$	0	0
$953$	−21718.7	−0.738235	−0.369117	−	0.929383i	$-0.620340\pi$
$953$	−21718.7	−0.738235	−0.369117	+	0.929383i	$0.620340\pi$
$954$	0	0
$955$	62379.3	2.11366
$956$	88444.2	2.99214
$957$	0	0
$958$	8853.96	0.298599
$959$	0	0
$960$	0	0
$961$	−26657.6	−0.894819
$962$	−88872.6	−2.97855
$963$	0	0
$964$	−70487.9	−2.35505
$965$	98963.6	3.30130
$966$	0	0
$967$	38754.4	1.28879	0.644393	−	0.764694i	$-0.277110\pi$
$967$	38754.4	1.28879	0.644393	+	0.764694i	$0.277110\pi$
$968$	−67220.3	−2.23196
$969$	0	0
$970$	86528.8	2.86420
$971$	−53591.6	−1.77120	−0.885601	−	0.464446i	$-0.846253\pi$
$971$	−53591.6	−1.77120	−0.885601	+	0.464446i	$0.846253\pi$
$972$	0	0
$973$	0	0
$974$	−67641.9	−2.22524
$975$	0	0
$976$	−44231.8	−1.45064
$977$	−18556.3	−0.607645	−0.303822	−	0.952729i	$-0.598263\pi$
$977$	−18556.3	−0.607645	−0.303822	+	0.952729i	$0.598263\pi$
$978$	0	0
$979$	318.719	0.0104048
$980$	0	0
$981$	0	0
$982$	−39526.8	−1.28447
$983$	3557.17	0.115418	0.0577091	−	0.998333i	$-0.481620\pi$
$983$	3557.17	0.115418	0.0577091	+	0.998333i	$0.481620\pi$
$984$	0	0
$985$	−32557.4	−1.05316
$986$	14272.8	0.460994
$987$	0	0
$988$	−138940.	−4.47396
$989$	−21674.2	−0.696864
$990$	0	0
$991$	44666.7	1.43177	0.715886	−	0.698218i	$-0.246023\pi$
$991$	44666.7	1.43177	0.715886	+	0.698218i	$0.246023\pi$
$992$	9845.53	0.315117
$993$	0	0
$994$	0	0
$995$	−29238.8	−0.931592
$996$	0	0
$997$	−1142.64	−0.0362967	−0.0181484	−	0.999835i	$-0.505777\pi$
$997$	−1142.64	−0.0362967	−0.0181484	+	0.999835i	$0.505777\pi$
$998$	92648.1	2.93860
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+5.09184 q^{2} +17.9269 q^{4} -18.5985 q^{5} +50.5460 q^{8} +O(q^{10})\)	\(q+5.09184 q^{2} +17.9269 q^{4} -18.5985 q^{5} +50.5460 q^{8} -94.7009 q^{10} -1.05631 q^{11} -58.7684 q^{13} +113.957 q^{16} +43.7810 q^{17} +131.880 q^{19} -333.413 q^{20} -5.37857 q^{22} +161.410 q^{23} +220.906 q^{25} -299.239 q^{26} +64.0250 q^{29} +55.9772 q^{31} +175.885 q^{32} +222.926 q^{34} +296.995 q^{37} +671.513 q^{38} -940.082 q^{40} +80.6724 q^{41} -134.280 q^{43} -18.9363 q^{44} +821.874 q^{46} +233.366 q^{47} +1124.82 q^{50} -1053.53 q^{52} +387.066 q^{53} +19.6458 q^{55} +326.005 q^{58} +722.871 q^{59} -388.144 q^{61} +285.027 q^{62} -16.0817 q^{64} +1093.01 q^{65} -730.041 q^{67} +784.856 q^{68} -685.470 q^{71} +275.371 q^{73} +1512.25 q^{74} +2364.20 q^{76} +854.544 q^{79} -2119.44 q^{80} +410.771 q^{82} +922.479 q^{83} -814.263 q^{85} -683.734 q^{86} -53.3922 q^{88} -301.729 q^{89} +2893.57 q^{92} +1188.27 q^{94} -2452.78 q^{95} -913.706 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8}+O(q^{10}) \) 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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