LMFDB - Embedded newform 1323.4.a.bk.1.5

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			1.5
Root			$1.44197$ 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)

$f(q)$	$=$	$q+1.44197 q^{2} -5.92074 q^{4} -6.60364 q^{5} -20.0732 q^{8} -9.52221 q^{10} +29.1645 q^{11} -16.7957 q^{13} +18.4210 q^{16} -47.1072 q^{17} -3.43125 q^{19} +39.0984 q^{20} +42.0542 q^{22} -29.5426 q^{23} -81.3920 q^{25} -24.2188 q^{26} -223.924 q^{29} -120.781 q^{31} +187.148 q^{32} -67.9269 q^{34} +34.9788 q^{37} -4.94775 q^{38} +132.556 q^{40} +192.991 q^{41} -5.61007 q^{43} -172.675 q^{44} -42.5994 q^{46} -359.246 q^{47} -117.364 q^{50} +99.4428 q^{52} -33.2997 q^{53} -192.592 q^{55} -322.890 q^{58} +742.285 q^{59} -658.795 q^{61} -174.162 q^{62} +122.493 q^{64} +110.913 q^{65} +941.546 q^{67} +278.909 q^{68} +871.189 q^{71} +732.553 q^{73} +50.4382 q^{74} +20.3156 q^{76} +1342.65 q^{79} -121.646 q^{80} +278.287 q^{82} -588.404 q^{83} +311.079 q^{85} -8.08952 q^{86} -585.426 q^{88} -1247.78 q^{89} +174.914 q^{92} -518.020 q^{94} +22.6588 q^{95} -691.412 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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 $

$2$	1.44197	0.509812	0.254906	−	0.966966i	$-0.417956\pi$
$2$	1.44197	0.509812	0.254906	+	0.966966i	$0.417956\pi$
$3$	0	0
$3$	0	0
$4$	−5.92074	−0.740092
$5$	−6.60364	−0.590647	−0.295324	−	0.955397i	$-0.595427\pi$
$5$	−6.60364	−0.590647	−0.295324	+	0.955397i	$0.595427\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−20.0732	−0.887119
$9$	0	0
$10$	−9.52221	−0.301119
$11$	29.1645	0.799403	0.399701	−	0.916645i	$-0.369114\pi$
$11$	29.1645	0.799403	0.399701	+	0.916645i	$0.369114\pi$
$12$	0	0
$13$	−16.7957	−0.358330	−0.179165	−	0.983819i	$-0.557340\pi$
$13$	−16.7957	−0.358330	−0.179165	+	0.983819i	$0.557340\pi$
$14$	0	0
$15$	0	0
$16$	18.4210	0.287828
$17$	−47.1072	−0.672069	−0.336034	−	0.941850i	$-0.609086\pi$
$17$	−47.1072	−0.672069	−0.336034	+	0.941850i	$0.609086\pi$
$18$	0	0
$19$	−3.43125	−0.0414307	−0.0207154	−	0.999785i	$-0.506594\pi$
$19$	−3.43125	−0.0414307	−0.0207154	+	0.999785i	$0.506594\pi$
$20$	39.0984	0.437133
$21$	0	0
$22$	42.0542	0.407545
$23$	−29.5426	−0.267828	−0.133914	−	0.990993i	$-0.542755\pi$
$23$	−29.5426	−0.267828	−0.133914	+	0.990993i	$0.542755\pi$
$24$	0	0
$25$	−81.3920	−0.651136
$26$	−24.2188	−0.182681
$27$	0	0
$28$	0	0
$29$	−223.924	−1.43385	−0.716924	−	0.697152i	$-0.754450\pi$
$29$	−223.924	−1.43385	−0.716924	+	0.697152i	$0.754450\pi$
$30$	0	0
$31$	−120.781	−0.699770	−0.349885	−	0.936793i	$-0.613779\pi$
$31$	−120.781	−0.699770	−0.349885	+	0.936793i	$0.613779\pi$
$32$	187.148	1.03386
$33$	0	0
$34$	−67.9269	−0.342628
$35$	0	0
$36$	0	0
$37$	34.9788	0.155418	0.0777092	−	0.996976i	$-0.475239\pi$
$37$	34.9788	0.155418	0.0777092	+	0.996976i	$0.475239\pi$
$38$	−4.94775	−0.0211219
$39$	0	0
$40$	132.556	0.523974
$41$	192.991	0.735127	0.367563	−	0.929998i	$-0.380192\pi$
$41$	192.991	0.735127	0.367563	+	0.929998i	$0.380192\pi$
$42$	0	0
$43$	−5.61007	−0.0198960	−0.00994799	−	0.999951i	$-0.503167\pi$
$43$	−5.61007	−0.0198960	−0.00994799	+	0.999951i	$0.503167\pi$
$44$	−172.675	−0.591632
$45$	0	0
$46$	−42.5994	−0.136542
$47$	−359.246	−1.11492	−0.557462	−	0.830202i	$-0.688225\pi$
$47$	−359.246	−1.11492	−0.557462	+	0.830202i	$0.688225\pi$
$48$	0	0
$49$	0	0
$50$	−117.364	−0.331957
$51$	0	0
$52$	99.4428	0.265197
$53$	−33.2997	−0.0863030	−0.0431515	−	0.999069i	$-0.513740\pi$
$53$	−33.2997	−0.0863030	−0.0431515	+	0.999069i	$0.513740\pi$
$54$	0	0
$55$	−192.592	−0.472165
$56$	0	0
$57$	0	0
$58$	−322.890	−0.730992
$59$	742.285	1.63792	0.818960	−	0.573851i	$-0.194551\pi$
$59$	742.285	1.63792	0.818960	+	0.573851i	$0.194551\pi$
$60$	0	0
$61$	−658.795	−1.38279	−0.691394	−	0.722478i	$-0.743003\pi$
$61$	−658.795	−1.38279	−0.691394	+	0.722478i	$0.743003\pi$
$62$	−174.162	−0.356751
$63$	0	0
$64$	122.493	0.239244
$65$	110.913	0.211646
$66$	0	0
$67$	941.546	1.71684	0.858419	−	0.512949i	$-0.171447\pi$
$67$	941.546	1.71684	0.858419	+	0.512949i	$0.171447\pi$
$68$	278.909	0.497393
$69$	0	0
$70$	0	0
$71$	871.189	1.45621	0.728107	−	0.685464i	$-0.240400\pi$
$71$	871.189	1.45621	0.728107	+	0.685464i	$0.240400\pi$
$72$	0	0
$73$	732.553	1.17451	0.587253	−	0.809404i	$-0.300210\pi$
$73$	732.553	1.17451	0.587253	+	0.809404i	$0.300210\pi$
$74$	50.4382	0.0792341
$75$	0	0
$76$	20.3156	0.0306626
$77$	0	0
$78$	0	0
$79$	1342.65	1.91216	0.956078	−	0.293112i	$-0.0946910\pi$
$79$	1342.65	1.91216	0.956078	+	0.293112i	$0.0946910\pi$
$80$	−121.646	−0.170005
$81$	0	0
$82$	278.287	0.374776
$83$	−588.404	−0.778142	−0.389071	−	0.921208i	$-0.627204\pi$
$83$	−588.404	−0.778142	−0.389071	+	0.921208i	$0.627204\pi$
$84$	0	0
$85$	311.079	0.396956
$86$	−8.08952	−0.0101432
$87$	0	0
$88$	−585.426	−0.709166
$89$	−1247.78	−1.48612	−0.743061	−	0.669224i	$-0.766627\pi$
$89$	−1247.78	−1.48612	−0.743061	+	0.669224i	$0.766627\pi$
$90$	0	0
$91$	0	0
$92$	174.914	0.198218
$93$	0	0
$94$	−518.020	−0.568401
$95$	22.6588	0.0244709
$96$	0	0
$97$	−691.412	−0.723735	−0.361867	−	0.932230i	$-0.617861\pi$
$97$	−691.412	−0.723735	−0.361867	+	0.932230i	$0.617861\pi$
$98$	0	0
$99$	0	0
$100$	481.901	0.481901
$101$	1352.30	1.33227	0.666134	−	0.745832i	$-0.267948\pi$
$101$	1352.30	1.33227	0.666134	+	0.745832i	$0.267948\pi$
$102$	0	0
$103$	922.650	0.882635	0.441318	−	0.897351i	$-0.354511\pi$
$103$	922.650	0.882635	0.441318	+	0.897351i	$0.354511\pi$
$104$	337.143	0.317881
$105$	0	0
$106$	−48.0169	−0.0439983
$107$	1005.99	0.908908	0.454454	−	0.890770i	$-0.349834\pi$
$107$	1005.99	0.908908	0.454454	+	0.890770i	$0.349834\pi$
$108$	0	0
$109$	912.295	0.801670	0.400835	−	0.916150i	$-0.368720\pi$
$109$	912.295	0.801670	0.400835	+	0.916150i	$0.368720\pi$
$110$	−277.711	−0.240715
$111$	0	0
$112$	0	0
$113$	1287.35	1.07172	0.535858	−	0.844308i	$-0.319988\pi$
$113$	1287.35	1.07172	0.535858	+	0.844308i	$0.319988\pi$
$114$	0	0
$115$	195.088	0.158192
$116$	1325.79	1.06118
$117$	0	0
$118$	1070.35	0.835031
$119$	0	0
$120$	0	0
$121$	−480.431	−0.360955
$122$	−949.959	−0.704961
$123$	0	0
$124$	715.111	0.517894
$125$	1362.94	0.975239
$126$	0	0
$127$	−521.970	−0.364704	−0.182352	−	0.983233i	$-0.558371\pi$
$127$	−521.970	−0.364704	−0.182352	+	0.983233i	$0.558371\pi$
$128$	−1320.55	−0.911888
$129$	0	0
$130$	159.932	0.107900
$131$	−952.092	−0.634998	−0.317499	−	0.948259i	$-0.602843\pi$
$131$	−952.092	−0.634998	−0.317499	+	0.948259i	$0.602843\pi$
$132$	0	0
$133$	0	0
$134$	1357.68	0.875264
$135$	0	0
$136$	945.593	0.596205
$137$	2917.18	1.81921	0.909604	−	0.415476i	$-0.136385\pi$
$137$	2917.18	1.81921	0.909604	+	0.415476i	$0.136385\pi$
$138$	0	0
$139$	−1223.71	−0.746717	−0.373358	−	0.927687i	$-0.621794\pi$
$139$	−1223.71	−0.746717	−0.373358	+	0.927687i	$0.621794\pi$
$140$	0	0
$141$	0	0
$142$	1256.22	0.742395
$143$	−489.838	−0.286450
$144$	0	0
$145$	1478.71	0.846898
$146$	1056.32	0.598776
$147$	0	0
$148$	−207.100	−0.115024
$149$	2502.62	1.37599	0.687997	−	0.725714i	$-0.258490\pi$
$149$	2502.62	1.37599	0.687997	+	0.725714i	$0.258490\pi$
$150$	0	0
$151$	−264.177	−0.142374	−0.0711868	−	0.997463i	$-0.522679\pi$
$151$	−264.177	−0.142374	−0.0711868	+	0.997463i	$0.522679\pi$
$152$	68.8763	0.0367540
$153$	0	0
$154$	0	0
$155$	797.592	0.413317
$156$	0	0
$157$	−191.509	−0.0973511	−0.0486755	−	0.998815i	$-0.515500\pi$
$157$	−191.509	−0.0973511	−0.0486755	+	0.998815i	$0.515500\pi$
$158$	1936.06	0.974839
$159$	0	0
$160$	−1235.86	−0.610645
$161$	0	0
$162$	0	0
$163$	−1716.96	−0.825048	−0.412524	−	0.910947i	$-0.635353\pi$
$163$	−1716.96	−0.825048	−0.412524	+	0.910947i	$0.635353\pi$
$164$	−1142.65	−0.544062
$165$	0	0
$166$	−848.458	−0.396706
$167$	2927.93	1.35671	0.678354	−	0.734735i	$-0.262694\pi$
$167$	2927.93	1.35671	0.678354	+	0.734735i	$0.262694\pi$
$168$	0	0
$169$	−1914.90	−0.871600
$170$	448.565	0.202373
$171$	0	0
$172$	33.2157	0.0147249
$173$	−2768.31	−1.21660	−0.608298	−	0.793709i	$-0.708147\pi$
$173$	−2768.31	−1.21660	−0.608298	+	0.793709i	$0.708147\pi$
$174$	0	0
$175$	0	0
$176$	537.240	0.230091
$177$	0	0
$178$	−1799.26	−0.757642
$179$	3579.36	1.49460	0.747302	−	0.664485i	$-0.231349\pi$
$179$	3579.36	1.49460	0.747302	+	0.664485i	$0.231349\pi$
$180$	0	0
$181$	−721.949	−0.296475	−0.148238	−	0.988952i	$-0.547360\pi$
$181$	−721.949	−0.296475	−0.148238	+	0.988952i	$0.547360\pi$
$182$	0	0
$183$	0	0
$184$	593.015	0.237596
$185$	−230.987	−0.0917975
$186$	0	0
$187$	−1373.86	−0.537254
$188$	2127.00	0.825147
$189$	0	0
$190$	32.6731	0.0124756
$191$	815.525	0.308949	0.154475	−	0.987997i	$-0.450632\pi$
$191$	815.525	0.308949	0.154475	+	0.987997i	$0.450632\pi$
$192$	0	0
$193$	−2864.97	−1.06852	−0.534261	−	0.845319i	$-0.679410\pi$
$193$	−2864.97	−1.06852	−0.534261	+	0.845319i	$0.679410\pi$
$194$	−996.992	−0.368968
$195$	0	0
$196$	0	0
$197$	25.5763	0.00924992	0.00462496	−	0.999989i	$-0.498528\pi$
$197$	25.5763	0.00924992	0.00462496	+	0.999989i	$0.498528\pi$
$198$	0	0
$199$	2847.18	1.01423	0.507114	−	0.861879i	$-0.330712\pi$
$199$	2847.18	1.01423	0.507114	+	0.861879i	$0.330712\pi$
$200$	1633.80	0.577635
$201$	0	0
$202$	1949.97	0.679206
$203$	0	0
$204$	0	0
$205$	−1274.45	−0.434201
$206$	1330.43	0.449978
$207$	0	0
$208$	−309.394	−0.103137
$209$	−100.071	−0.0331198
$210$	0	0
$211$	2059.84	0.672063	0.336031	−	0.941851i	$-0.390915\pi$
$211$	2059.84	0.672063	0.336031	+	0.941851i	$0.390915\pi$
$212$	197.158	0.0638722
$213$	0	0
$214$	1450.61	0.463372
$215$	37.0468	0.0117515
$216$	0	0
$217$	0	0
$218$	1315.50	0.408701
$219$	0	0
$220$	1140.29	0.349446
$221$	791.197	0.240822
$222$	0	0
$223$	826.484	0.248186	0.124093	−	0.992271i	$-0.460398\pi$
$223$	826.484	0.248186	0.124093	+	0.992271i	$0.460398\pi$
$224$	0	0
$225$	0	0
$226$	1856.32	0.546373
$227$	2311.44	0.675840	0.337920	−	0.941175i	$-0.390277\pi$
$227$	2311.44	0.675840	0.337920	+	0.941175i	$0.390277\pi$
$228$	0	0
$229$	−6638.39	−1.91562	−0.957811	−	0.287399i	$-0.907209\pi$
$229$	−6638.39	−1.91562	−0.957811	+	0.287399i	$0.907209\pi$
$230$	281.311	0.0806482
$231$	0	0
$232$	4494.87	1.27199
$233$	−66.4886	−0.0186945	−0.00934724	−	0.999956i	$-0.502975\pi$
$233$	−66.4886	−0.0186945	−0.00934724	+	0.999956i	$0.502975\pi$
$234$	0	0
$235$	2372.33	0.658527
$236$	−4394.87	−1.21221
$237$	0	0
$238$	0	0
$239$	6136.26	1.66076	0.830380	−	0.557198i	$-0.188124\pi$
$239$	6136.26	1.66076	0.830380	+	0.557198i	$0.188124\pi$
$240$	0	0
$241$	−6054.14	−1.61818	−0.809091	−	0.587684i	$-0.800040\pi$
$241$	−6054.14	−1.61818	−0.809091	+	0.587684i	$0.800040\pi$
$242$	−692.765	−0.184019
$243$	0	0
$244$	3900.55	1.02339
$245$	0	0
$246$	0	0
$247$	57.6303	0.0148459
$248$	2424.46	0.620779
$249$	0	0
$250$	1965.31	0.497188
$251$	−4244.07	−1.06727	−0.533633	−	0.845716i	$-0.679173\pi$
$251$	−4244.07	−1.06727	−0.533633	+	0.845716i	$0.679173\pi$
$252$	0	0
$253$	−861.595	−0.214103
$254$	−752.663	−0.185930
$255$	0	0
$256$	−2884.14	−0.704135
$257$	−3223.76	−0.782462	−0.391231	−	0.920293i	$-0.627951\pi$
$257$	−3223.76	−0.782462	−0.391231	+	0.920293i	$0.627951\pi$
$258$	0	0
$259$	0	0
$260$	−656.684	−0.156638
$261$	0	0
$262$	−1372.88	−0.323729
$263$	4249.87	0.996418	0.498209	−	0.867057i	$-0.333991\pi$
$263$	4249.87	0.996418	0.498209	+	0.867057i	$0.333991\pi$
$264$	0	0
$265$	219.899	0.0509746
$266$	0	0
$267$	0	0
$268$	−5574.64	−1.27062
$269$	4435.75	1.00540	0.502700	−	0.864461i	$-0.332340\pi$
$269$	4435.75	1.00540	0.502700	+	0.864461i	$0.332340\pi$
$270$	0	0
$271$	5819.59	1.30448	0.652242	−	0.758011i	$-0.273829\pi$
$271$	5819.59	1.30448	0.652242	+	0.758011i	$0.273829\pi$
$272$	−867.762	−0.193441
$273$	0	0
$274$	4206.47	0.927454
$275$	−2373.76	−0.520520
$276$	0	0
$277$	154.892	0.0335976	0.0167988	−	0.999859i	$-0.494653\pi$
$277$	154.892	0.0335976	0.0167988	+	0.999859i	$0.494653\pi$
$278$	−1764.55	−0.380685
$279$	0	0
$280$	0	0
$281$	−376.133	−0.0798514	−0.0399257	−	0.999203i	$-0.512712\pi$
$281$	−376.133	−0.0798514	−0.0399257	+	0.999203i	$0.512712\pi$
$282$	0	0
$283$	23.8430	0.00500820	0.00250410	−	0.999997i	$-0.499203\pi$
$283$	23.8430	0.00500820	0.00250410	+	0.999997i	$0.499203\pi$
$284$	−5158.08	−1.07773
$285$	0	0
$286$	−706.329	−0.146035
$287$	0	0
$288$	0	0
$289$	−2693.91	−0.548324
$290$	2132.25	0.431758
$291$	0	0
$292$	−4337.26	−0.869242
$293$	1801.80	0.359256	0.179628	−	0.983735i	$-0.442511\pi$
$293$	1801.80	0.359256	0.179628	+	0.983735i	$0.442511\pi$
$294$	0	0
$295$	−4901.78	−0.967433
$296$	−702.137	−0.137875
$297$	0	0
$298$	3608.70	0.701497
$299$	496.188	0.0959708
$300$	0	0
$301$	0	0
$302$	−380.934	−0.0725837
$303$	0	0
$304$	−63.2072	−0.0119249
$305$	4350.44	0.816739
$306$	0	0
$307$	−1120.29	−0.208269	−0.104134	−	0.994563i	$-0.533207\pi$
$307$	−1120.29	−0.208269	−0.104134	+	0.994563i	$0.533207\pi$
$308$	0	0
$309$	0	0
$310$	1150.10	0.210714
$311$	5287.18	0.964014	0.482007	−	0.876167i	$-0.339908\pi$
$311$	5287.18	0.964014	0.482007	+	0.876167i	$0.339908\pi$
$312$	0	0
$313$	−2453.40	−0.443050	−0.221525	−	0.975155i	$-0.571103\pi$
$313$	−2453.40	−0.443050	−0.221525	+	0.975155i	$0.571103\pi$
$314$	−276.150	−0.0496307
$315$	0	0
$316$	−7949.50	−1.41517
$317$	3808.40	0.674767	0.337384	−	0.941367i	$-0.390458\pi$
$317$	3808.40	0.674767	0.337384	+	0.941367i	$0.390458\pi$
$318$	0	0
$319$	−6530.62	−1.14622
$320$	−808.899	−0.141309
$321$	0	0
$322$	0	0
$323$	161.637	0.0278443
$324$	0	0
$325$	1367.03	0.233321
$326$	−2475.80	−0.420619
$327$	0	0
$328$	−3873.96	−0.652145
$329$	0	0
$330$	0	0
$331$	−3389.02	−0.562771	−0.281386	−	0.959595i	$-0.590794\pi$
$331$	−3389.02	−0.562771	−0.281386	+	0.959595i	$0.590794\pi$
$332$	3483.79	0.575896
$333$	0	0
$334$	4221.98	0.691665
$335$	−6217.63	−1.01405
$336$	0	0
$337$	75.8509	0.0122607	0.00613036	−	0.999981i	$-0.498049\pi$
$337$	75.8509	0.0122607	0.00613036	+	0.999981i	$0.498049\pi$
$338$	−2761.23	−0.444352
$339$	0	0
$340$	−1841.82	−0.293784
$341$	−3522.51	−0.559398
$342$	0	0
$343$	0	0
$344$	112.612	0.0176501
$345$	0	0
$346$	−3991.81	−0.620234
$347$	3320.56	0.513709	0.256854	−	0.966450i	$-0.417314\pi$
$347$	3320.56	0.513709	0.256854	+	0.966450i	$0.417314\pi$
$348$	0	0
$349$	8840.82	1.35598	0.677991	−	0.735070i	$-0.262851\pi$
$349$	8840.82	1.35598	0.677991	+	0.735070i	$0.262851\pi$
$350$	0	0
$351$	0	0
$352$	5458.09	0.826469
$353$	−10966.0	−1.65343	−0.826717	−	0.562618i	$-0.809794\pi$
$353$	−10966.0	−1.65343	−0.826717	+	0.562618i	$0.809794\pi$
$354$	0	0
$355$	−5753.02	−0.860108
$356$	7387.80	1.09987
$357$	0	0
$358$	5161.31	0.761966
$359$	−4597.22	−0.675854	−0.337927	−	0.941172i	$-0.609726\pi$
$359$	−4597.22	−0.675854	−0.337927	+	0.941172i	$0.609726\pi$
$360$	0	0
$361$	−6847.23	−0.998283
$362$	−1041.03	−0.151147
$363$	0	0
$364$	0	0
$365$	−4837.52	−0.693718
$366$	0	0
$367$	9073.27	1.29052	0.645260	−	0.763963i	$-0.276749\pi$
$367$	9073.27	1.29052	0.645260	+	0.763963i	$0.276749\pi$
$368$	−544.204	−0.0770886
$369$	0	0
$370$	−333.076	−0.0467994
$371$	0	0
$372$	0	0
$373$	−5313.30	−0.737566	−0.368783	−	0.929516i	$-0.620225\pi$
$373$	−5313.30	−0.737566	−0.368783	+	0.929516i	$0.620225\pi$
$374$	−1981.06	−0.273898
$375$	0	0
$376$	7211.23	0.989071
$377$	3760.95	0.513790
$378$	0	0
$379$	−10987.4	−1.48914	−0.744571	−	0.667543i	$-0.767346\pi$
$379$	−10987.4	−1.48914	−0.744571	+	0.667543i	$0.767346\pi$
$380$	−134.157	−0.0181108
$381$	0	0
$382$	1175.96	0.157506
$383$	14557.0	1.94211	0.971053	−	0.238864i	$-0.0767749\pi$
$383$	14557.0	1.94211	0.971053	+	0.238864i	$0.0767749\pi$
$384$	0	0
$385$	0	0
$386$	−4131.18	−0.544745
$387$	0	0
$388$	4093.67	0.535631
$389$	8451.48	1.10156	0.550780	−	0.834650i	$-0.314330\pi$
$389$	8451.48	1.10156	0.550780	+	0.834650i	$0.314330\pi$
$390$	0	0
$391$	1391.67	0.179999
$392$	0	0
$393$	0	0
$394$	36.8801	0.00471572
$395$	−8866.40	−1.12941
$396$	0	0
$397$	11550.7	1.46023	0.730117	−	0.683322i	$-0.239465\pi$
$397$	11550.7	1.46023	0.730117	+	0.683322i	$0.239465\pi$
$398$	4105.54	0.517065
$399$	0	0
$400$	−1499.32	−0.187415
$401$	11935.6	1.48638	0.743188	−	0.669083i	$-0.233313\pi$
$401$	11935.6	1.48638	0.743188	+	0.669083i	$0.233313\pi$
$402$	0	0
$403$	2028.59	0.250748
$404$	−8006.63	−0.986001
$405$	0	0
$406$	0	0
$407$	1020.14	0.124242
$408$	0	0
$409$	−7300.90	−0.882656	−0.441328	−	0.897346i	$-0.645492\pi$
$409$	−7300.90	−0.882656	−0.441328	+	0.897346i	$0.645492\pi$
$410$	−1837.71	−0.221361
$411$	0	0
$412$	−5462.77	−0.653232
$413$	0	0
$414$	0	0
$415$	3885.61	0.459607
$416$	−3143.28	−0.370462
$417$	0	0
$418$	−144.299	−0.0168849
$419$	1840.07	0.214542	0.107271	−	0.994230i	$-0.465789\pi$
$419$	1840.07	0.214542	0.107271	+	0.994230i	$0.465789\pi$
$420$	0	0
$421$	1332.32	0.154236	0.0771181	−	0.997022i	$-0.475428\pi$
$421$	1332.32	0.154236	0.0771181	+	0.997022i	$0.475428\pi$
$422$	2970.22	0.342625
$423$	0	0
$424$	668.431	0.0765610
$425$	3834.15	0.437608
$426$	0	0
$427$	0	0
$428$	−5956.23	−0.672675
$429$	0	0
$430$	53.4203	0.00599105
$431$	−9329.41	−1.04265	−0.521325	−	0.853358i	$-0.674562\pi$
$431$	−9329.41	−1.04265	−0.521325	+	0.853358i	$0.674562\pi$
$432$	0	0
$433$	1295.25	0.143755	0.0718775	−	0.997413i	$-0.477101\pi$
$433$	1295.25	0.143755	0.0718775	+	0.997413i	$0.477101\pi$
$434$	0	0
$435$	0	0
$436$	−5401.46	−0.593310
$437$	101.368	0.0110963
$438$	0	0
$439$	−11862.2	−1.28964	−0.644821	−	0.764334i	$-0.723068\pi$
$439$	−11862.2	−1.28964	−0.644821	+	0.764334i	$0.723068\pi$
$440$	3865.94	0.418867
$441$	0	0
$442$	1140.88	0.122774
$443$	−8321.32	−0.892455	−0.446228	−	0.894920i	$-0.647233\pi$
$443$	−8321.32	−0.892455	−0.446228	+	0.894920i	$0.647233\pi$
$444$	0	0
$445$	8239.91	0.877774
$446$	1191.76	0.126528
$447$	0	0
$448$	0	0
$449$	−2710.45	−0.284886	−0.142443	−	0.989803i	$-0.545496\pi$
$449$	−2710.45	−0.284886	−0.142443	+	0.989803i	$0.545496\pi$
$450$	0	0
$451$	5628.50	0.587663
$452$	−7622.07	−0.793168
$453$	0	0
$454$	3333.02	0.344551
$455$	0	0
$456$	0	0
$457$	8893.10	0.910288	0.455144	−	0.890418i	$-0.349588\pi$
$457$	8893.10	0.910288	0.455144	+	0.890418i	$0.349588\pi$
$458$	−9572.33	−0.976606
$459$	0	0
$460$	−1155.07	−0.117077
$461$	−3569.04	−0.360578	−0.180289	−	0.983614i	$-0.557703\pi$
$461$	−3569.04	−0.360578	−0.180289	+	0.983614i	$0.557703\pi$
$462$	0	0
$463$	18634.5	1.87045	0.935227	−	0.354048i	$-0.115195\pi$
$463$	18634.5	1.87045	0.935227	+	0.354048i	$0.115195\pi$
$464$	−4124.90	−0.412702
$465$	0	0
$466$	−95.8743	−0.00953067
$467$	11292.2	1.11893	0.559467	−	0.828853i	$-0.311006\pi$
$467$	11292.2	1.11893	0.559467	+	0.828853i	$0.311006\pi$
$468$	0	0
$469$	0	0
$470$	3420.82	0.335725
$471$	0	0
$472$	−14900.0	−1.45303
$473$	−163.615	−0.0159049
$474$	0	0
$475$	279.277	0.0269770
$476$	0	0
$477$	0	0
$478$	8848.27	0.846674
$479$	3271.94	0.312106	0.156053	−	0.987749i	$-0.450123\pi$
$479$	3271.94	0.312106	0.156053	+	0.987749i	$0.450123\pi$
$480$	0	0
$481$	−587.493	−0.0556910
$482$	−8729.86	−0.824968
$483$	0	0
$484$	2844.51	0.267140
$485$	4565.84	0.427472
$486$	0	0
$487$	19777.7	1.84028	0.920139	−	0.391592i	$-0.128076\pi$
$487$	19777.7	1.84028	0.920139	+	0.391592i	$0.128076\pi$
$488$	13224.1	1.22670
$489$	0	0
$490$	0	0
$491$	4477.06	0.411501	0.205751	−	0.978604i	$-0.434036\pi$
$491$	4477.06	0.411501	0.205751	+	0.978604i	$0.434036\pi$
$492$	0	0
$493$	10548.4	0.963644
$494$	83.1008	0.00756859
$495$	0	0
$496$	−2224.90	−0.201414
$497$	0	0
$498$	0	0
$499$	−9095.77	−0.815997	−0.407999	−	0.912983i	$-0.633773\pi$
$499$	−9095.77	−0.815997	−0.407999	+	0.912983i	$0.633773\pi$
$500$	−8069.60	−0.721767
$501$	0	0
$502$	−6119.81	−0.544104
$503$	−20102.5	−1.78196	−0.890981	−	0.454041i	$-0.849982\pi$
$503$	−20102.5	−1.78196	−0.890981	+	0.454041i	$0.849982\pi$
$504$	0	0
$505$	−8930.11	−0.786901
$506$	−1242.39	−0.109152
$507$	0	0
$508$	3090.45	0.269914
$509$	1431.90	0.124692	0.0623458	−	0.998055i	$-0.480142\pi$
$509$	1431.90	0.124692	0.0623458	+	0.998055i	$0.480142\pi$
$510$	0	0
$511$	0	0
$512$	6405.61	0.552912
$513$	0	0
$514$	−4648.55	−0.398908
$515$	−6092.85	−0.521326
$516$	0	0
$517$	−10477.2	−0.891274
$518$	0	0
$519$	0	0
$520$	−2226.37	−0.187756
$521$	11682.9	0.982415	0.491207	−	0.871043i	$-0.336556\pi$
$521$	11682.9	0.982415	0.491207	+	0.871043i	$0.336556\pi$
$522$	0	0
$523$	14402.3	1.20414	0.602071	−	0.798442i	$-0.294342\pi$
$523$	14402.3	1.20414	0.602071	+	0.798442i	$0.294342\pi$
$524$	5637.09	0.469957
$525$	0	0
$526$	6128.16	0.507986
$527$	5689.64	0.470293
$528$	0	0
$529$	−11294.2	−0.928268
$530$	317.086	0.0259875
$531$	0	0
$532$	0	0
$533$	−3241.42	−0.263418
$534$	0	0
$535$	−6643.22	−0.536844
$536$	−18899.9	−1.52304
$537$	0	0
$538$	6396.20	0.512565
$539$	0	0
$540$	0	0
$541$	13866.3	1.10196	0.550978	−	0.834519i	$-0.314255\pi$
$541$	13866.3	1.10196	0.550978	+	0.834519i	$0.314255\pi$
$542$	8391.65	0.665041
$543$	0	0
$544$	−8816.02	−0.694823
$545$	−6024.46	−0.473504
$546$	0	0
$547$	8793.92	0.687387	0.343694	−	0.939082i	$-0.388322\pi$
$547$	8793.92	0.687387	0.343694	+	0.939082i	$0.388322\pi$
$548$	−17271.9	−1.34638
$549$	0	0
$550$	−3422.88	−0.265367
$551$	768.339	0.0594053
$552$	0	0
$553$	0	0
$554$	223.348	0.0171284
$555$	0	0
$556$	7245.26	0.552639
$557$	−20085.0	−1.52788	−0.763941	−	0.645286i	$-0.776738\pi$
$557$	−20085.0	−1.52788	−0.763941	+	0.645286i	$0.776738\pi$
$558$	0	0
$559$	94.2249	0.00712932
$560$	0	0
$561$	0	0
$562$	−542.371	−0.0407092
$563$	−3172.18	−0.237463	−0.118731	−	0.992926i	$-0.537883\pi$
$563$	−3172.18	−0.237463	−0.118731	+	0.992926i	$0.537883\pi$
$564$	0	0
$565$	−8501.20	−0.633006
$566$	34.3808	0.00255324
$567$	0	0
$568$	−17487.6	−1.29183
$569$	18767.7	1.38274	0.691372	−	0.722499i	$-0.257007\pi$
$569$	18767.7	1.38274	0.691372	+	0.722499i	$0.257007\pi$
$570$	0	0
$571$	−18650.3	−1.36688	−0.683442	−	0.730005i	$-0.739518\pi$
$571$	−18650.3	−1.36688	−0.683442	+	0.730005i	$0.739518\pi$
$572$	2900.20	0.211999
$573$	0	0
$574$	0	0
$575$	2404.53	0.174393
$576$	0	0
$577$	21527.6	1.55322	0.776610	−	0.629982i	$-0.216938\pi$
$577$	21527.6	1.55322	0.776610	+	0.629982i	$0.216938\pi$
$578$	−3884.53	−0.279542
$579$	0	0
$580$	−8755.05	−0.626782
$581$	0	0
$582$	0	0
$583$	−971.168	−0.0689909
$584$	−14704.7	−1.04193
$585$	0	0
$586$	2598.13	0.183153
$587$	−17142.7	−1.20537	−0.602686	−	0.797978i	$-0.705903\pi$
$587$	−17142.7	−1.20537	−0.602686	+	0.797978i	$0.705903\pi$
$588$	0	0
$589$	414.429	0.0289920
$590$	−7068.20	−0.493209
$591$	0	0
$592$	644.346	0.0447338
$593$	23471.7	1.62541	0.812704	−	0.582677i	$-0.197995\pi$
$593$	23471.7	1.62541	0.812704	+	0.582677i	$0.197995\pi$
$594$	0	0
$595$	0	0
$596$	−14817.4	−1.01836
$597$	0	0
$598$	715.485	0.0489270
$599$	−9698.12	−0.661527	−0.330763	−	0.943714i	$-0.607306\pi$
$599$	−9698.12	−0.661527	−0.330763	+	0.943714i	$0.607306\pi$
$600$	0	0
$601$	−18513.2	−1.25652	−0.628261	−	0.778002i	$-0.716233\pi$
$601$	−18513.2	−1.25652	−0.628261	+	0.778002i	$0.716233\pi$
$602$	0	0
$603$	0	0
$604$	1564.12	0.105370
$605$	3172.59	0.213197
$606$	0	0
$607$	−5168.65	−0.345616	−0.172808	−	0.984956i	$-0.555284\pi$
$607$	−5168.65	−0.345616	−0.172808	+	0.984956i	$0.555284\pi$
$608$	−642.153	−0.0428335
$609$	0	0
$610$	6273.18	0.416383
$611$	6033.79	0.399510
$612$	0	0
$613$	3359.81	0.221372	0.110686	−	0.993855i	$-0.464695\pi$
$613$	3359.81	0.221372	0.110686	+	0.993855i	$0.464695\pi$
$614$	−1615.42	−0.106178
$615$	0	0
$616$	0	0
$617$	−10966.7	−0.715561	−0.357780	−	0.933806i	$-0.616466\pi$
$617$	−10966.7	−0.715561	−0.357780	+	0.933806i	$0.616466\pi$
$618$	0	0
$619$	−8466.21	−0.549734	−0.274867	−	0.961482i	$-0.588634\pi$
$619$	−8466.21	−0.549734	−0.274867	+	0.961482i	$0.588634\pi$
$620$	−4722.33	−0.305893
$621$	0	0
$622$	7623.93	0.491466
$623$	0	0
$624$	0	0
$625$	1173.65	0.0751139
$626$	−3537.72	−0.225872
$627$	0	0
$628$	1133.88	0.0720488
$629$	−1647.75	−0.104452
$630$	0	0
$631$	−6200.95	−0.391214	−0.195607	−	0.980682i	$-0.562668\pi$
$631$	−6200.95	−0.391214	−0.195607	+	0.980682i	$0.562668\pi$
$632$	−26951.4	−1.69631
$633$	0	0
$634$	5491.58	0.344004
$635$	3446.90	0.215411
$636$	0	0
$637$	0	0
$638$	−9416.93	−0.584357
$639$	0	0
$640$	8720.46	0.538604
$641$	−27467.7	−1.69252	−0.846262	−	0.532766i	$-0.821153\pi$
$641$	−27467.7	−1.69252	−0.846262	+	0.532766i	$0.821153\pi$
$642$	0	0
$643$	927.437	0.0568811	0.0284405	−	0.999595i	$-0.490946\pi$
$643$	927.437	0.0568811	0.0284405	+	0.999595i	$0.490946\pi$
$644$	0	0
$645$	0	0
$646$	233.075	0.0141953
$647$	16172.3	0.982689	0.491344	−	0.870965i	$-0.336506\pi$
$647$	16172.3	0.982689	0.491344	+	0.870965i	$0.336506\pi$
$648$	0	0
$649$	21648.4	1.30936
$650$	1971.22	0.118950
$651$	0	0
$652$	10165.7	0.610611
$653$	−4251.10	−0.254760	−0.127380	−	0.991854i	$-0.540657\pi$
$653$	−4251.10	−0.254760	−0.127380	+	0.991854i	$0.540657\pi$
$654$	0	0
$655$	6287.27	0.375060
$656$	3555.10	0.211590
$657$	0	0
$658$	0	0
$659$	−16537.0	−0.977524	−0.488762	−	0.872417i	$-0.662551\pi$
$659$	−16537.0	−0.977524	−0.488762	+	0.872417i	$0.662551\pi$
$660$	0	0
$661$	−22520.3	−1.32517	−0.662585	−	0.748987i	$-0.730541\pi$
$661$	−22520.3	−1.32517	−0.662585	+	0.748987i	$0.730541\pi$
$662$	−4886.84	−0.286907
$663$	0	0
$664$	11811.2	0.690304
$665$	0	0
$666$	0	0
$667$	6615.28	0.384025
$668$	−17335.5	−1.00409
$669$	0	0
$670$	−8965.60	−0.516972
$671$	−19213.4	−1.10540
$672$	0	0
$673$	−14730.1	−0.843690	−0.421845	−	0.906668i	$-0.638617\pi$
$673$	−14730.1	−0.843690	−0.421845	+	0.906668i	$0.638617\pi$
$674$	109.374	0.00625066
$675$	0	0
$676$	11337.6	0.645064
$677$	−2930.78	−0.166380	−0.0831898	−	0.996534i	$-0.526511\pi$
$677$	−2930.78	−0.166380	−0.0831898	+	0.996534i	$0.526511\pi$
$678$	0	0
$679$	0	0
$680$	−6244.35	−0.352147
$681$	0	0
$682$	−5079.34	−0.285188
$683$	17706.6	0.991983	0.495992	−	0.868327i	$-0.334805\pi$
$683$	17706.6	0.991983	0.495992	+	0.868327i	$0.334805\pi$
$684$	0	0
$685$	−19264.0	−1.07451
$686$	0	0
$687$	0	0
$688$	−103.343	−0.00572663
$689$	559.290	0.0309249
$690$	0	0
$691$	14313.6	0.788010	0.394005	−	0.919108i	$-0.371089\pi$
$691$	14313.6	0.788010	0.394005	+	0.919108i	$0.371089\pi$
$692$	16390.5	0.900393
$693$	0	0
$694$	4788.13	0.261895
$695$	8080.93	0.441046
$696$	0	0
$697$	−9091.28	−0.494056
$698$	12748.1	0.691296
$699$	0	0
$700$	0	0
$701$	−18506.7	−0.997131	−0.498565	−	0.866852i	$-0.666140\pi$
$701$	−18506.7	−0.997131	−0.498565	+	0.866852i	$0.666140\pi$
$702$	0	0
$703$	−120.021	−0.00643910
$704$	3572.45	0.191252
$705$	0	0
$706$	−15812.6	−0.842940
$707$	0	0
$708$	0	0
$709$	−6306.66	−0.334064	−0.167032	−	0.985951i	$-0.553418\pi$
$709$	−6306.66	−0.334064	−0.167032	+	0.985951i	$0.553418\pi$
$710$	−8295.65	−0.438493
$711$	0	0
$712$	25047.0	1.31837
$713$	3568.17	0.187418
$714$	0	0
$715$	3234.71	0.169191
$716$	−21192.5	−1.10614
$717$	0	0
$718$	−6629.02	−0.344558
$719$	4391.44	0.227779	0.113889	−	0.993493i	$-0.463669\pi$
$719$	4391.44	0.227779	0.113889	+	0.993493i	$0.463669\pi$
$720$	0	0
$721$	0	0
$722$	−9873.46	−0.508937
$723$	0	0
$724$	4274.47	0.219419
$725$	18225.6	0.933630
$726$	0	0
$727$	3740.47	0.190820	0.0954102	−	0.995438i	$-0.469584\pi$
$727$	3740.47	0.190820	0.0954102	+	0.995438i	$0.469584\pi$
$728$	0	0
$729$	0	0
$730$	−6975.53	−0.353666
$731$	264.274	0.0133715
$732$	0	0
$733$	−26104.4	−1.31540	−0.657699	−	0.753281i	$-0.728470\pi$
$733$	−26104.4	−1.31540	−0.657699	+	0.753281i	$0.728470\pi$
$734$	13083.3	0.657922
$735$	0	0
$736$	−5528.84	−0.276896
$737$	27459.7	1.37245
$738$	0	0
$739$	4656.46	0.231787	0.115894	−	0.993262i	$-0.463027\pi$
$739$	4656.46	0.231787	0.115894	+	0.993262i	$0.463027\pi$
$740$	1367.62	0.0679386
$741$	0	0
$742$	0	0
$743$	−14101.5	−0.696276	−0.348138	−	0.937443i	$-0.613186\pi$
$743$	−14101.5	−0.696276	−0.348138	+	0.937443i	$0.613186\pi$
$744$	0	0
$745$	−16526.4	−0.812727
$746$	−7661.59	−0.376020
$747$	0	0
$748$	8134.25	0.397617
$749$	0	0
$750$	0	0
$751$	5235.30	0.254379	0.127190	−	0.991878i	$-0.459404\pi$
$751$	5235.30	0.254379	0.127190	+	0.991878i	$0.459404\pi$
$752$	−6617.68	−0.320907
$753$	0	0
$754$	5423.16	0.261936
$755$	1744.53	0.0840925
$756$	0	0
$757$	−17983.1	−0.863417	−0.431709	−	0.902013i	$-0.642089\pi$
$757$	−17983.1	−0.863417	−0.431709	+	0.902013i	$0.642089\pi$
$758$	−15843.5	−0.759182
$759$	0	0
$760$	−454.834	−0.0217086
$761$	−19710.8	−0.938919	−0.469459	−	0.882954i	$-0.655551\pi$
$761$	−19710.8	−0.938919	−0.469459	+	0.882954i	$0.655551\pi$
$762$	0	0
$763$	0	0
$764$	−4828.51	−0.228651
$765$	0	0
$766$	20990.6	0.990108
$767$	−12467.2	−0.586915
$768$	0	0
$769$	32441.6	1.52129	0.760647	−	0.649165i	$-0.224882\pi$
$769$	32441.6	1.52129	0.760647	+	0.649165i	$0.224882\pi$
$770$	0	0
$771$	0	0
$772$	16962.7	0.790805
$773$	−16807.1	−0.782029	−0.391014	−	0.920385i	$-0.627876\pi$
$773$	−16807.1	−0.782029	−0.391014	+	0.920385i	$0.627876\pi$
$774$	0	0
$775$	9830.58	0.455645
$776$	13878.9	0.642039
$777$	0	0
$778$	12186.7	0.561588
$779$	−662.203	−0.0304568
$780$	0	0
$781$	25407.8	1.16410
$782$	2006.74	0.0917656
$783$	0	0
$784$	0	0
$785$	1264.66	0.0575001
$786$	0	0
$787$	38870.7	1.76060	0.880299	−	0.474420i	$-0.157342\pi$
$787$	38870.7	1.76060	0.880299	+	0.474420i	$0.157342\pi$
$788$	−151.430	−0.00684580
$789$	0	0
$790$	−12785.0	−0.575786
$791$	0	0
$792$	0	0
$793$	11064.9	0.495494
$794$	16655.7	0.744445
$795$	0	0
$796$	−16857.4	−0.750622
$797$	−15248.8	−0.677717	−0.338858	−	0.940837i	$-0.610041\pi$
$797$	−15248.8	−0.677717	−0.338858	+	0.940837i	$0.610041\pi$
$798$	0	0
$799$	16923.1	0.749306
$800$	−15232.4	−0.673182
$801$	0	0
$802$	17210.8	0.757772
$803$	21364.6	0.938903
$804$	0	0
$805$	0	0
$806$	2925.16	0.127834
$807$	0	0
$808$	−27145.1	−1.18188
$809$	3463.54	0.150521	0.0752605	−	0.997164i	$-0.476021\pi$
$809$	3463.54	0.150521	0.0752605	+	0.997164i	$0.476021\pi$
$810$	0	0
$811$	45402.7	1.96585	0.982925	−	0.184006i	$-0.0589066\pi$
$811$	45402.7	1.96585	0.982925	+	0.184006i	$0.0589066\pi$
$812$	0	0
$813$	0	0
$814$	1471.01	0.0633400
$815$	11338.2	0.487312
$816$	0	0
$817$	19.2496	0.000824305 0
$818$	−10527.6	−0.449988
$819$	0	0
$820$	7545.66	0.321349
$821$	12650.8	0.537778	0.268889	−	0.963171i	$-0.413344\pi$
$821$	12650.8	0.537778	0.268889	+	0.963171i	$0.413344\pi$
$822$	0	0
$823$	−26294.4	−1.11369	−0.556845	−	0.830617i	$-0.687988\pi$
$823$	−26294.4	−1.11369	−0.556845	+	0.830617i	$0.687988\pi$
$824$	−18520.6	−0.783003
$825$	0	0
$826$	0	0
$827$	−3948.55	−0.166027	−0.0830136	−	0.996548i	$-0.526454\pi$
$827$	−3948.55	−0.166027	−0.0830136	+	0.996548i	$0.526454\pi$
$828$	0	0
$829$	18967.4	0.794652	0.397326	−	0.917678i	$-0.369938\pi$
$829$	18967.4	0.794652	0.397326	+	0.917678i	$0.369938\pi$
$830$	5602.91	0.234313
$831$	0	0
$832$	−2057.35	−0.0857282
$833$	0	0
$834$	0	0
$835$	−19335.0	−0.801336
$836$	592.493	0.0245117
$837$	0	0
$838$	2653.32	0.109376
$839$	−21494.7	−0.884481	−0.442241	−	0.896896i	$-0.645816\pi$
$839$	−21494.7	−0.884481	−0.442241	+	0.896896i	$0.645816\pi$
$840$	0	0
$841$	25752.8	1.05592
$842$	1921.16	0.0786314
$843$	0	0
$844$	−12195.8	−0.497388
$845$	12645.3	0.514808
$846$	0	0
$847$	0	0
$848$	−613.414	−0.0248405
$849$	0	0
$850$	5528.71	0.223098
$851$	−1033.36	−0.0416255
$852$	0	0
$853$	−25163.6	−1.01006	−0.505032	−	0.863100i	$-0.668519\pi$
$853$	−25163.6	−1.01006	−0.505032	+	0.863100i	$0.668519\pi$
$854$	0	0
$855$	0	0
$856$	−20193.5	−0.806309
$857$	−2324.30	−0.0926449	−0.0463224	−	0.998927i	$-0.514750\pi$
$857$	−2324.30	−0.0926449	−0.0463224	+	0.998927i	$0.514750\pi$
$858$	0	0
$859$	−28144.9	−1.11792	−0.558959	−	0.829195i	$-0.688799\pi$
$859$	−28144.9	−1.11792	−0.558959	+	0.829195i	$0.688799\pi$
$860$	−219.345	−0.00869720
$861$	0	0
$862$	−13452.7	−0.531555
$863$	−11512.8	−0.454115	−0.227058	−	0.973881i	$-0.572911\pi$
$863$	−11512.8	−0.454115	−0.227058	+	0.973881i	$0.572911\pi$
$864$	0	0
$865$	18280.9	0.718579
$866$	1867.71	0.0732880
$867$	0	0
$868$	0	0
$869$	39157.8	1.52858
$870$	0	0
$871$	−15813.9	−0.615194
$872$	−18312.7	−0.711177
$873$	0	0
$874$	146.169	0.00565704
$875$	0	0
$876$	0	0
$877$	14143.0	0.544557	0.272278	−	0.962219i	$-0.412223\pi$
$877$	14143.0	0.544557	0.272278	+	0.962219i	$0.412223\pi$
$878$	−17104.9	−0.657474
$879$	0	0
$880$	−3547.74	−0.135903
$881$	1038.97	0.0397318	0.0198659	−	0.999803i	$-0.493676\pi$
$881$	1038.97	0.0397318	0.0198659	+	0.999803i	$0.493676\pi$
$882$	0	0
$883$	−48786.6	−1.85934	−0.929671	−	0.368391i	$-0.879909\pi$
$883$	−48786.6	−1.85934	−0.929671	+	0.368391i	$0.879909\pi$
$884$	−4684.47	−0.178231
$885$	0	0
$886$	−11999.0	−0.454984
$887$	−13831.7	−0.523589	−0.261794	−	0.965124i	$-0.584314\pi$
$887$	−13831.7	−0.523589	−0.261794	+	0.965124i	$0.584314\pi$
$888$	0	0
$889$	0	0
$890$	11881.7	0.447499
$891$	0	0
$892$	−4893.40	−0.183681
$893$	1232.67	0.0461921
$894$	0	0
$895$	−23636.8	−0.882783
$896$	0	0
$897$	0	0
$898$	−3908.37	−0.145238
$899$	27045.7	1.00336
$900$	0	0
$901$	1568.65	0.0580016
$902$	8116.11	0.299597
$903$	0	0
$904$	−25841.3	−0.950739
$905$	4767.49	0.175112
$906$	0	0
$907$	−2677.25	−0.0980118	−0.0490059	−	0.998798i	$-0.515605\pi$
$907$	−2677.25	−0.0980118	−0.0490059	+	0.998798i	$0.515605\pi$
$908$	−13685.4	−0.500184
$909$	0	0
$910$	0	0
$911$	35842.4	1.30352	0.651762	−	0.758424i	$-0.274030\pi$
$911$	35842.4	1.30352	0.651762	+	0.758424i	$0.274030\pi$
$912$	0	0
$913$	−17160.5	−0.622049
$914$	12823.5	0.464075
$915$	0	0
$916$	39304.2	1.41774
$917$	0	0
$918$	0	0
$919$	−33473.5	−1.20151	−0.600756	−	0.799433i	$-0.705133\pi$
$919$	−33473.5	−1.20151	−0.600756	+	0.799433i	$0.705133\pi$
$920$	−3916.05	−0.140335
$921$	0	0
$922$	−5146.43	−0.183827
$923$	−14632.2	−0.521804
$924$	0	0
$925$	−2847.00	−0.101199
$926$	26870.4	0.953579
$927$	0	0
$928$	−41906.9	−1.48239
$929$	8801.66	0.310843	0.155421	−	0.987848i	$-0.450326\pi$
$929$	8801.66	0.310843	0.155421	+	0.987848i	$0.450326\pi$
$930$	0	0
$931$	0	0
$932$	393.662	0.0138356
$933$	0	0
$934$	16283.0	0.570446
$935$	9072.46	0.317327
$936$	0	0
$937$	−1144.38	−0.0398991	−0.0199495	−	0.999801i	$-0.506351\pi$
$937$	−1144.38	−0.0398991	−0.0199495	+	0.999801i	$0.506351\pi$
$938$	0	0
$939$	0	0
$940$	−14045.9	−0.487371
$941$	11014.1	0.381563	0.190782	−	0.981633i	$-0.438898\pi$
$941$	11014.1	0.381563	0.190782	+	0.981633i	$0.438898\pi$
$942$	0	0
$943$	−5701.47	−0.196888
$944$	13673.6	0.471440
$945$	0	0
$946$	−235.927	−0.00810851
$947$	47646.5	1.63496	0.817478	−	0.575960i	$-0.195371\pi$
$947$	47646.5	1.63496	0.817478	+	0.575960i	$0.195371\pi$
$948$	0	0
$949$	−12303.7	−0.420860
$950$	402.707	0.0137532
$951$	0	0
$952$	0	0
$953$	46771.5	1.58980	0.794898	−	0.606743i	$-0.207524\pi$
$953$	46771.5	1.58980	0.794898	+	0.606743i	$0.207524\pi$
$954$	0	0
$955$	−5385.43	−0.182480
$956$	−36331.2	−1.22911
$957$	0	0
$958$	4718.02	0.159115
$959$	0	0
$960$	0	0
$961$	−15203.0	−0.510323
$962$	−847.145	−0.0283919
$963$	0	0
$964$	35845.0	1.19760
$965$	18919.2	0.631120
$966$	0	0
$967$	22111.6	0.735326	0.367663	−	0.929959i	$-0.380158\pi$
$967$	22111.6	0.735326	0.367663	+	0.929959i	$0.380158\pi$
$968$	9643.79	0.320210
$969$	0	0
$970$	6583.78	0.217930
$971$	−24420.8	−0.807108	−0.403554	−	0.914956i	$-0.632225\pi$
$971$	−24420.8	−0.807108	−0.403554	+	0.914956i	$0.632225\pi$
$972$	0	0
$973$	0	0
$974$	28518.8	0.938195
$975$	0	0
$976$	−12135.7	−0.398005
$977$	41638.6	1.36350	0.681748	−	0.731587i	$-0.261220\pi$
$977$	41638.6	1.36350	0.681748	+	0.731587i	$0.261220\pi$
$978$	0	0
$979$	−36391.0	−1.18801
$980$	0	0
$981$	0	0
$982$	6455.77	0.209788
$983$	2225.94	0.0722243	0.0361122	−	0.999348i	$-0.488503\pi$
$983$	2225.94	0.0722243	0.0361122	+	0.999348i	$0.488503\pi$
$984$	0	0
$985$	−168.896	−0.00546344
$986$	15210.4	0.491277
$987$	0	0
$988$	−341.214	−0.0109873
$989$	165.736	0.00532871
$990$	0	0
$991$	10975.3	0.351808	0.175904	−	0.984407i	$-0.443715\pi$
$991$	10975.3	0.351808	0.175904	+	0.984407i	$0.443715\pi$
$992$	−22603.9	−0.723462
$993$	0	0
$994$	0	0
$995$	−18801.7	−0.599051
$996$	0	0
$997$	−38123.0	−1.21100	−0.605500	−	0.795846i	$-0.707027\pi$
$997$	−38123.0	−1.21100	−0.605500	+	0.795846i	$0.707027\pi$
$998$	−13115.8	−0.416005
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bk.1.5		7
3.2	odd	2		1323.4.a.bh.1.3		7
7.3	odd	6		189.4.e.f.163.3	yes	14
7.5	odd	6		189.4.e.f.109.3	✓	14
7.6	odd	2		1323.4.a.bj.1.5		7
21.5	even	6		189.4.e.g.109.5	yes	14
21.17	even	6		189.4.e.g.163.5	yes	14
21.20	even	2		1323.4.a.bi.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.f.109.3	✓	14	7.5	odd	6
189.4.e.f.163.3	yes	14	7.3	odd	6
189.4.e.g.109.5	yes	14	21.5	even	6
189.4.e.g.163.5	yes	14	21.17	even	6
1323.4.a.bh.1.3		7	3.2	odd	2
1323.4.a.bi.1.3		7	21.20	even	2
1323.4.a.bj.1.5		7	7.6	odd	2
1323.4.a.bk.1.5		7	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.44197 q^{2} -5.92074 q^{4} -6.60364 q^{5} -20.0732 q^{8} -9.52221 q^{10} +29.1645 q^{11} -16.7957 q^{13} +18.4210 q^{16} -47.1072 q^{17} -3.43125 q^{19} +39.0984 q^{20} +42.0542 q^{22} -29.5426 q^{23} -81.3920 q^{25} -24.2188 q^{26} -223.924 q^{29} -120.781 q^{31} +187.148 q^{32} -67.9269 q^{34} +34.9788 q^{37} -4.94775 q^{38} +132.556 q^{40} +192.991 q^{41} -5.61007 q^{43} -172.675 q^{44} -42.5994 q^{46} -359.246 q^{47} -117.364 q^{50} +99.4428 q^{52} -33.2997 q^{53} -192.592 q^{55} -322.890 q^{58} +742.285 q^{59} -658.795 q^{61} -174.162 q^{62} +122.493 q^{64} +110.913 q^{65} +941.546 q^{67} +278.909 q^{68} +871.189 q^{71} +732.553 q^{73} +50.4382 q^{74} +20.3156 q^{76} +1342.65 q^{79} -121.646 q^{80} +278.287 q^{82} -588.404 q^{83} +311.079 q^{85} -8.08952 q^{86} -585.426 q^{88} -1247.78 q^{89} +174.914 q^{92} -518.020 q^{94} +22.6588 q^{95} -691.412 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 31 q^{4} + q^{5} + 84 q^{8} - 12 q^{10} + 98 q^{11} + 124 q^{13} + 139 q^{16} - 30 q^{17} - 182 q^{19} + 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} + 245 q^{26} + 323 q^{29} - 26 q^{31}+ \cdots + 3730 q^{97}+O(q^{100}) \) 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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