LMFDB - Embedded newform 1323.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,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, 2, names="a")

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

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$10.5642081874$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69963$ 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)

$f(q)$	$=$	$q-2.69963 q^{2} +5.28799 q^{4} +1.58836 q^{5} -8.87636 q^{8} +O(q^{10})$	$q-2.69963 q^{2} +5.28799 q^{4} +1.58836 q^{5} -8.87636 q^{8} -4.28799 q^{10} -0.300372 q^{11} +2.81089 q^{13} +13.3869 q^{16} -5.87636 q^{17} -2.28799 q^{19} +8.39926 q^{20} +0.810892 q^{22} -1.88874 q^{23} -2.47710 q^{25} -7.58836 q^{26} -2.52290 q^{29} -4.81089 q^{31} -18.3869 q^{32} +15.8640 q^{34} -4.47710 q^{37} +6.17673 q^{38} -14.0989 q^{40} -8.90978 q^{41} -9.09888 q^{43} -1.58836 q^{44} +5.09888 q^{46} +3.21015 q^{47} +6.68725 q^{50} +14.8640 q^{52} -2.01238 q^{53} -0.477100 q^{55} +6.81089 q^{58} +4.88874 q^{59} +7.57598 q^{61} +12.9876 q^{62} +22.8640 q^{64} +4.46472 q^{65} +0.712008 q^{67} -31.0741 q^{68} +12.8640 q^{71} -11.6749 q^{73} +12.0865 q^{74} -12.0989 q^{76} +1.66621 q^{79} +21.2632 q^{80} +24.0531 q^{82} +5.43268 q^{83} -9.33379 q^{85} +24.5636 q^{86} +2.66621 q^{88} +9.35346 q^{89} -9.98762 q^{92} -8.66621 q^{94} -3.63416 q^{95} -12.5760 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8	$ 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8} - q^{10} - 7 q^{11} + 2 q^{13} + 10 q^{16} + 5 q^{19} + 13 q^{20} - 4 q^{22} - 6 q^{23} - 2 q^{25} - 17 q^{26} - 13 q^{29} - 8 q^{31} - 25 q^{32} + 12 q^{34} - 8 q^{37} + 7 q^{38} - 24 q^{40} - 2 q^{41} - 9 q^{43} + q^{44} - 3 q^{46} - 9 q^{47} - 4 q^{50} + 9 q^{52} - 24 q^{53} + 4 q^{55} + 14 q^{58} + 15 q^{59} - q^{61} + 21 q^{62} + 33 q^{64} - 10 q^{65} + 14 q^{67} - 39 q^{68} + 3 q^{71} + 7 q^{73} - 18 q^{76} + 6 q^{79} + 16 q^{80} + 43 q^{82} - 3 q^{83} - 27 q^{85} + 32 q^{86} + 9 q^{88} + 5 q^{89} - 12 q^{92} - 27 q^{94} - 16 q^{95} - 14 q^{97}+O(q^{100}) $ 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8 - q^10 - 7 * q^11 + 2 * q^13 + 10 * q^16 + 5 * q^19 + 13 * q^20 - 4 * q^22 - 6 * q^23 - 2 * q^25 - 17 * q^26 - 13 * q^29 - 8 * q^31 - 25 * q^32 + 12 * q^34 - 8 * q^37 + 7 * q^38 - 24 * q^40 - 2 * q^41 - 9 * q^43 + q^44 - 3 * q^46 - 9 * q^47 - 4 * q^50 + 9 * q^52 - 24 * q^53 + 4 * q^55 + 14 * q^58 + 15 * q^59 - q^61 + 21 * q^62 + 33 * q^64 - 10 * q^65 + 14 * q^67 - 39 * q^68 + 3 * q^71 + 7 * q^73 - 18 * q^76 + 6 * q^79 + 16 * q^80 + 43 * q^82 - 3 * q^83 - 27 * q^85 + 32 * q^86 + 9 * q^88 + 5 * q^89 - 12 * q^92 - 27 * q^94 - 16 * q^95 - 14 * 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 $

$2$	−2.69963	−1.90893	−0.954463	−	0.298330i	$-0.903570\pi$
$2$	−2.69963	−1.90893	−0.954463	+	0.298330i	$0.903570\pi$
$3$	0	0
$3$	0	0
$4$	5.28799	2.64400
$5$	1.58836	0.710338	0.355169	−	0.934802i	$-0.384423\pi$
$5$	1.58836	0.710338	0.355169	+	0.934802i	$0.384423\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.87636	−3.13827
$9$	0	0
$10$	−4.28799	−1.35598
$11$	−0.300372	−0.0905655	−0.0452828	−	0.998974i	$-0.514419\pi$
$11$	−0.300372	−0.0905655	−0.0452828	+	0.998974i	$0.514419\pi$
$12$	0	0
$13$	2.81089	0.779601	0.389801	−	0.920899i	$-0.372544\pi$
$13$	2.81089	0.779601	0.389801	+	0.920899i	$0.372544\pi$
$14$	0	0
$15$	0	0
$16$	13.3869	3.34672
$17$	−5.87636	−1.42523	−0.712613	−	0.701557i	$-0.752488\pi$
$17$	−5.87636	−1.42523	−0.712613	+	0.701557i	$0.752488\pi$
$18$	0	0
$19$	−2.28799	−0.524901	−0.262451	−	0.964945i	$-0.584531\pi$
$19$	−2.28799	−0.524901	−0.262451	+	0.964945i	$0.584531\pi$
$20$	8.39926	1.87813
$21$	0	0
$22$	0.810892	0.172883
$23$	−1.88874	−0.393829	−0.196914	−	0.980421i	$-0.563092\pi$
$23$	−1.88874	−0.393829	−0.196914	+	0.980421i	$0.563092\pi$
$24$	0	0
$25$	−2.47710	−0.495420
$26$	−7.58836	−1.48820
$27$	0	0
$28$	0	0
$29$	−2.52290	−0.468491	−0.234245	−	0.972178i	$-0.575262\pi$
$29$	−2.52290	−0.468491	−0.234245	+	0.972178i	$0.575262\pi$
$30$	0	0
$31$	−4.81089	−0.864062	−0.432031	−	0.901859i	$-0.642203\pi$
$31$	−4.81089	−0.864062	−0.432031	+	0.901859i	$0.642203\pi$
$32$	−18.3869	−3.25037
$33$	0	0
$34$	15.8640	2.72065
$35$	0	0
$36$	0	0
$37$	−4.47710	−0.736031	−0.368015	−	0.929820i	$-0.619963\pi$
$37$	−4.47710	−0.736031	−0.368015	+	0.929820i	$0.619963\pi$
$38$	6.17673	1.00200
$39$	0	0
$40$	−14.0989	−2.22923
$41$	−8.90978	−1.39147	−0.695737	−	0.718297i	$-0.744922\pi$
$41$	−8.90978	−1.39147	−0.695737	+	0.718297i	$0.744922\pi$
$42$	0	0
$43$	−9.09888	−1.38757	−0.693783	−	0.720184i	$-0.744058\pi$
$43$	−9.09888	−1.38757	−0.693783	+	0.720184i	$0.744058\pi$
$44$	−1.58836	−0.239455
$45$	0	0
$46$	5.09888	0.751789
$47$	3.21015	0.468248	0.234124	−	0.972207i	$-0.424778\pi$
$47$	3.21015	0.468248	0.234124	+	0.972207i	$0.424778\pi$
$48$	0	0
$49$	0	0
$50$	6.68725	0.945720
$51$	0	0
$52$	14.8640	2.06126
$53$	−2.01238	−0.276422	−0.138211	−	0.990403i	$-0.544135\pi$
$53$	−2.01238	−0.276422	−0.138211	+	0.990403i	$0.544135\pi$
$54$	0	0
$55$	−0.477100	−0.0643321
$56$	0	0
$57$	0	0
$58$	6.81089	0.894314
$59$	4.88874	0.636459	0.318230	−	0.948014i	$-0.396912\pi$
$59$	4.88874	0.636459	0.318230	+	0.948014i	$0.396912\pi$
$60$	0	0
$61$	7.57598	0.970005	0.485003	−	0.874513i	$-0.338819\pi$
$61$	7.57598	0.970005	0.485003	+	0.874513i	$0.338819\pi$
$62$	12.9876	1.64943
$63$	0	0
$64$	22.8640	2.85800
$65$	4.46472	0.553780
$66$	0	0
$67$	0.712008	0.0869856	0.0434928	−	0.999054i	$-0.486151\pi$
$67$	0.712008	0.0869856	0.0434928	+	0.999054i	$0.486151\pi$
$68$	−31.0741	−3.76829
$69$	0	0
$70$	0	0
$71$	12.8640	1.52667	0.763337	−	0.646001i	$-0.223560\pi$
$71$	12.8640	1.52667	0.763337	+	0.646001i	$0.223560\pi$
$72$	0	0
$73$	−11.6749	−1.36644	−0.683220	−	0.730213i	$-0.739421\pi$
$73$	−11.6749	−1.36644	−0.683220	+	0.730213i	$0.739421\pi$
$74$	12.0865	1.40503
$75$	0	0
$76$	−12.0989	−1.38784
$77$	0	0
$78$	0	0
$79$	1.66621	0.187463	0.0937315	−	0.995598i	$-0.470120\pi$
$79$	1.66621	0.187463	0.0937315	+	0.995598i	$0.470120\pi$
$80$	21.2632	2.37730
$81$	0	0
$82$	24.0531	2.65622
$83$	5.43268	0.596314	0.298157	−	0.954517i	$-0.403628\pi$
$83$	5.43268	0.596314	0.298157	+	0.954517i	$0.403628\pi$
$84$	0	0
$85$	−9.33379	−1.01239
$86$	24.5636	2.64876
$87$	0	0
$88$	2.66621	0.284219
$89$	9.35346	0.991464	0.495732	−	0.868475i	$-0.334900\pi$
$89$	9.35346	0.991464	0.495732	+	0.868475i	$0.334900\pi$
$90$	0	0
$91$	0	0
$92$	−9.98762	−1.04128
$93$	0	0
$94$	−8.66621	−0.893851
$95$	−3.63416	−0.372857
$96$	0	0
$97$	−12.5760	−1.27690	−0.638449	−	0.769664i	$-0.720424\pi$
$97$	−12.5760	−1.27690	−0.638449	+	0.769664i	$0.720424\pi$
$98$	0	0
$99$	0	0
$100$	−13.0989	−1.30989
$101$	14.6094	1.45369	0.726845	−	0.686801i	$-0.240986\pi$
$101$	14.6094	1.45369	0.726845	+	0.686801i	$0.240986\pi$
$102$	0	0
$103$	9.19777	0.906283	0.453142	−	0.891439i	$-0.350303\pi$
$103$	9.19777	0.906283	0.453142	+	0.891439i	$0.350303\pi$
$104$	−24.9505	−2.44660
$105$	0	0
$106$	5.43268	0.527668
$107$	−12.5105	−1.20944	−0.604719	−	0.796439i	$-0.706715\pi$
$107$	−12.5105	−1.20944	−0.604719	+	0.796439i	$0.706715\pi$
$108$	0	0
$109$	−3.04580	−0.291735	−0.145867	−	0.989304i	$-0.546597\pi$
$109$	−3.04580	−0.291735	−0.145867	+	0.989304i	$0.546597\pi$
$110$	1.28799	0.122805
$111$	0	0
$112$	0	0
$113$	−9.95420	−0.936412	−0.468206	−	0.883619i	$-0.655100\pi$
$113$	−9.95420	−0.936412	−0.468206	+	0.883619i	$0.655100\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−13.3411	−1.23869
$117$	0	0
$118$	−13.1978	−1.21495
$119$	0	0
$120$	0	0
$121$	−10.9098	−0.991798
$122$	−20.4523	−1.85167
$123$	0	0
$124$	−25.4400	−2.28458
$125$	−11.8764	−1.06225
$126$	0	0
$127$	−13.4400	−1.19260	−0.596302	−	0.802760i	$-0.703364\pi$
$127$	−13.4400	−1.19260	−0.596302	+	0.802760i	$0.703364\pi$
$128$	−24.9505	−2.20533
$129$	0	0
$130$	−12.0531	−1.05713
$131$	−11.2756	−0.985155	−0.492577	−	0.870269i	$-0.663945\pi$
$131$	−11.2756	−0.985155	−0.492577	+	0.870269i	$0.663945\pi$
$132$	0	0
$133$	0	0
$134$	−1.92216	−0.166049
$135$	0	0
$136$	52.1606	4.47274
$137$	−3.77747	−0.322731	−0.161366	−	0.986895i	$-0.551590\pi$
$137$	−3.77747	−0.322731	−0.161366	+	0.986895i	$0.551590\pi$
$138$	0	0
$139$	16.7193	1.41811	0.709056	−	0.705152i	$-0.249121\pi$
$139$	16.7193	1.41811	0.709056	+	0.705152i	$0.249121\pi$
$140$	0	0
$141$	0	0
$142$	−34.7280	−2.91431
$143$	−0.844313	−0.0706050
$144$	0	0
$145$	−4.00728	−0.332787
$146$	31.5178	2.60843
$147$	0	0
$148$	−23.6749	−1.94606
$149$	−10.8887	−0.892040	−0.446020	−	0.895023i	$-0.647159\pi$
$149$	−10.8887	−0.892040	−0.446020	+	0.895023i	$0.647159\pi$
$150$	0	0
$151$	−12.0989	−0.984593	−0.492297	−	0.870427i	$-0.663842\pi$
$151$	−12.0989	−0.984593	−0.492297	+	0.870427i	$0.663842\pi$
$152$	20.3090	1.64728
$153$	0	0
$154$	0	0
$155$	−7.64145	−0.613776
$156$	0	0
$157$	−0.189108	−0.0150924	−0.00754622	−	0.999972i	$-0.502402\pi$
$157$	−0.189108	−0.0150924	−0.00754622	+	0.999972i	$0.502402\pi$
$158$	−4.49814	−0.357853
$159$	0	0
$160$	−29.2051	−2.30886
$161$	0	0
$162$	0	0
$163$	−8.95558	−0.701455	−0.350727	−	0.936478i	$-0.614066\pi$
$163$	−8.95558	−0.701455	−0.350727	+	0.936478i	$0.614066\pi$
$164$	−47.1148	−3.67905
$165$	0	0
$166$	−14.6662	−1.13832
$167$	4.34108	0.335923	0.167961	−	0.985794i	$-0.446282\pi$
$167$	4.34108	0.335923	0.167961	+	0.985794i	$0.446282\pi$
$168$	0	0
$169$	−5.09888	−0.392222
$170$	25.1978	1.93258
$171$	0	0
$172$	−48.1148	−3.66872
$173$	6.30037	0.479008	0.239504	−	0.970895i	$-0.423015\pi$
$173$	6.30037	0.479008	0.239504	+	0.970895i	$0.423015\pi$
$174$	0	0
$175$	0	0
$176$	−4.02104	−0.303097
$177$	0	0
$178$	−25.2509	−1.89263
$179$	13.9418	1.04206	0.521030	−	0.853538i	$-0.325548\pi$
$179$	13.9418	1.04206	0.521030	+	0.853538i	$0.325548\pi$
$180$	0	0
$181$	18.3411	1.36328	0.681641	−	0.731687i	$-0.261267\pi$
$181$	18.3411	1.36328	0.681641	+	0.731687i	$0.261267\pi$
$182$	0	0
$183$	0	0
$184$	16.7651	1.23594
$185$	−7.11126	−0.522831
$186$	0	0
$187$	1.76509	0.129076
$188$	16.9752	1.23805
$189$	0	0
$190$	9.81089	0.711757
$191$	−7.60803	−0.550498	−0.275249	−	0.961373i	$-0.588760\pi$
$191$	−7.60803	−0.550498	−0.275249	+	0.961373i	$0.588760\pi$
$192$	0	0
$193$	7.66621	0.551826	0.275913	−	0.961183i	$-0.411020\pi$
$193$	7.66621	0.551826	0.275913	+	0.961183i	$0.411020\pi$
$194$	33.9505	2.43750
$195$	0	0
$196$	0	0
$197$	−18.3869	−1.31001	−0.655005	−	0.755624i	$-0.727334\pi$
$197$	−18.3869	−1.31001	−0.655005	+	0.755624i	$0.727334\pi$
$198$	0	0
$199$	5.95420	0.422082	0.211041	−	0.977477i	$-0.432315\pi$
$199$	5.95420	0.422082	0.211041	+	0.977477i	$0.432315\pi$
$200$	21.9876	1.55476
$201$	0	0
$202$	−39.4400	−2.77499
$203$	0	0
$204$	0	0
$205$	−14.1520	−0.988416
$206$	−24.8306	−1.73003
$207$	0	0
$208$	37.6291	2.60911
$209$	0.687248	0.0475380
$210$	0	0
$211$	6.67487	0.459517	0.229758	−	0.973248i	$-0.426206\pi$
$211$	6.67487	0.459517	0.229758	+	0.973248i	$0.426206\pi$
$212$	−10.6414	−0.730858
$213$	0	0
$214$	33.7738	2.30873
$215$	−14.4523	−0.985641
$216$	0	0
$217$	0	0
$218$	8.22253	0.556900
$219$	0	0
$220$	−2.52290	−0.170094
$221$	−16.5178	−1.11111
$222$	0	0
$223$	6.34108	0.424630	0.212315	−	0.977201i	$-0.431900\pi$
$223$	6.34108	0.424630	0.212315	+	0.977201i	$0.431900\pi$
$224$	0	0
$225$	0	0
$226$	26.8726	1.78754
$227$	27.0865	1.79779	0.898897	−	0.438160i	$-0.144370\pi$
$227$	27.0865	1.79779	0.898897	+	0.438160i	$0.144370\pi$
$228$	0	0
$229$	8.19639	0.541633	0.270816	−	0.962631i	$-0.412706\pi$
$229$	8.19639	0.541633	0.270816	+	0.962631i	$0.412706\pi$
$230$	8.09888	0.534025
$231$	0	0
$232$	22.3942	1.47025
$233$	−27.4203	−1.79636	−0.898182	−	0.439624i	$-0.855112\pi$
$233$	−27.4203	−1.79636	−0.898182	+	0.439624i	$0.855112\pi$
$234$	0	0
$235$	5.09888	0.332615
$236$	25.8516	1.68280
$237$	0	0
$238$	0	0
$239$	27.8640	1.80237	0.901185	−	0.433434i	$-0.142698\pi$
$239$	27.8640	1.80237	0.901185	+	0.433434i	$0.142698\pi$
$240$	0	0
$241$	−22.5302	−1.45130	−0.725648	−	0.688066i	$-0.758460\pi$
$241$	−22.5302	−1.45130	−0.725648	+	0.688066i	$0.758460\pi$
$242$	29.4523	1.89327
$243$	0	0
$244$	40.0617	2.56469
$245$	0	0
$246$	0	0
$247$	−6.43130	−0.409214
$248$	42.7032	2.71166
$249$	0	0
$250$	32.0617	2.02776
$251$	−31.2509	−1.97254	−0.986268	−	0.165152i	$-0.947189\pi$
$251$	−31.2509	−1.97254	−0.986268	+	0.165152i	$0.947189\pi$
$252$	0	0
$253$	0.567323	0.0356673
$254$	36.2829	2.27659
$255$	0	0
$256$	21.6291	1.35182
$257$	10.2015	0.636351	0.318176	−	0.948032i	$-0.396930\pi$
$257$	10.2015	0.636351	0.318176	+	0.948032i	$0.396930\pi$
$258$	0	0
$259$	0	0
$260$	23.6094	1.46419
$261$	0	0
$262$	30.4400	1.88059
$263$	−28.4610	−1.75498	−0.877490	−	0.479594i	$-0.840784\pi$
$263$	−28.4610	−1.75498	−0.877490	+	0.479594i	$0.840784\pi$
$264$	0	0
$265$	−3.19639	−0.196353
$266$	0	0
$267$	0	0
$268$	3.76509	0.229990
$269$	−14.8764	−0.907027	−0.453514	−	0.891249i	$-0.649830\pi$
$269$	−14.8764	−0.907027	−0.453514	+	0.891249i	$0.649830\pi$
$270$	0	0
$271$	0.0444229	0.00269850	0.00134925	−	0.999999i	$-0.499571\pi$
$271$	0.0444229	0.00269850	0.00134925	+	0.999999i	$0.499571\pi$
$272$	−78.6661	−4.76983
$273$	0	0
$274$	10.1978	0.616070
$275$	0.744051	0.0448680
$276$	0	0
$277$	15.6749	0.941812	0.470906	−	0.882184i	$-0.343927\pi$
$277$	15.6749	0.941812	0.470906	+	0.882184i	$0.343927\pi$
$278$	−45.1359	−2.70707
$279$	0	0
$280$	0	0
$281$	−11.9098	−0.710478	−0.355239	−	0.934776i	$-0.615600\pi$
$281$	−11.9098	−0.710478	−0.355239	+	0.934776i	$0.615600\pi$
$282$	0	0
$283$	6.00728	0.357096	0.178548	−	0.983931i	$-0.442860\pi$
$283$	6.00728	0.357096	0.178548	+	0.983931i	$0.442860\pi$
$284$	68.0246	4.03652
$285$	0	0
$286$	2.27933	0.134780
$287$	0	0
$288$	0	0
$289$	17.5316	1.03127
$290$	10.8182	0.635265
$291$	0	0
$292$	−61.7366	−3.61286
$293$	4.95558	0.289508	0.144754	−	0.989468i	$-0.453761\pi$
$293$	4.95558	0.289508	0.144754	+	0.989468i	$0.453761\pi$
$294$	0	0
$295$	7.76509	0.452101
$296$	39.7403	2.30986
$297$	0	0
$298$	29.3955	1.70284
$299$	−5.30903	−0.307029
$300$	0	0
$301$	0	0
$302$	32.6625	1.87952
$303$	0	0
$304$	−30.6291	−1.75670
$305$	12.0334	0.689032
$306$	0	0
$307$	−22.0531	−1.25864	−0.629318	−	0.777148i	$-0.716666\pi$
$307$	−22.0531	−1.25864	−0.629318	+	0.777148i	$0.716666\pi$
$308$	0	0
$309$	0	0
$310$	20.6291	1.17165
$311$	−7.97524	−0.452234	−0.226117	−	0.974100i	$-0.572603\pi$
$311$	−7.97524	−0.452234	−0.226117	+	0.974100i	$0.572603\pi$
$312$	0	0
$313$	−22.5302	−1.27348	−0.636741	−	0.771078i	$-0.719718\pi$
$313$	−22.5302	−1.27348	−0.636741	+	0.771078i	$0.719718\pi$
$314$	0.510520	0.0288103
$315$	0	0
$316$	8.81089	0.495651
$317$	19.9381	1.11984	0.559918	−	0.828548i	$-0.310833\pi$
$317$	19.9381	1.11984	0.559918	+	0.828548i	$0.310833\pi$
$318$	0	0
$319$	0.757808	0.0424291
$320$	36.3163	2.03014
$321$	0	0
$322$	0	0
$323$	13.4451	0.748103
$324$	0	0
$325$	−6.96286	−0.386230
$326$	24.1767	1.33903
$327$	0	0
$328$	79.0864	4.36681
$329$	0	0
$330$	0	0
$331$	23.6304	1.29885	0.649423	−	0.760427i	$-0.275010\pi$
$331$	23.6304	1.29885	0.649423	+	0.760427i	$0.275010\pi$
$332$	28.7280	1.57665
$333$	0	0
$334$	−11.7193	−0.641251
$335$	1.13093	0.0617892
$336$	0	0
$337$	−12.3855	−0.674681	−0.337341	−	0.941383i	$-0.609527\pi$
$337$	−12.3855	−0.674681	−0.337341	+	0.941383i	$0.609527\pi$
$338$	13.7651	0.748722
$339$	0	0
$340$	−49.3570	−2.67676
$341$	1.44506	0.0782542
$342$	0	0
$343$	0	0
$344$	80.7649	4.35455
$345$	0	0
$346$	−17.0087	−0.914391
$347$	3.84431	0.206374	0.103187	−	0.994662i	$-0.467096\pi$
$347$	3.84431	0.206374	0.103187	+	0.994662i	$0.467096\pi$
$348$	0	0
$349$	−12.1891	−0.652468	−0.326234	−	0.945289i	$-0.605780\pi$
$349$	−12.1891	−0.652468	−0.326234	+	0.945289i	$0.605780\pi$
$350$	0	0
$351$	0	0
$352$	5.52290	0.294372
$353$	3.56360	0.189672	0.0948358	−	0.995493i	$-0.469767\pi$
$353$	3.56360	0.189672	0.0948358	+	0.995493i	$0.469767\pi$
$354$	0	0
$355$	20.4327	1.08445
$356$	49.4610	2.62143
$357$	0	0
$358$	−37.6377	−1.98922
$359$	0.967957	0.0510868	0.0255434	−	0.999674i	$-0.491868\pi$
$359$	0.967957	0.0510868	0.0255434	+	0.999674i	$0.491868\pi$
$360$	0	0
$361$	−13.7651	−0.724479
$362$	−49.5141	−2.60240
$363$	0	0
$364$	0	0
$365$	−18.5439	−0.970634
$366$	0	0
$367$	20.9542	1.09380	0.546900	−	0.837198i	$-0.315808\pi$
$367$	20.9542	1.09380	0.546900	+	0.837198i	$0.315808\pi$
$368$	−25.2843	−1.31803
$369$	0	0
$370$	19.1978	0.998044
$371$	0	0
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−4.76509	−0.246397
$375$	0	0
$376$	−28.4944	−1.46949
$377$	−7.09160	−0.365236
$378$	0	0
$379$	−7.14331	−0.366927	−0.183464	−	0.983027i	$-0.558731\pi$
$379$	−7.14331	−0.366927	−0.183464	+	0.983027i	$0.558731\pi$
$380$	−19.2174	−0.985833
$381$	0	0
$382$	20.5388	1.05086
$383$	11.9766	0.611977	0.305988	−	0.952035i	$-0.401013\pi$
$383$	11.9766	0.611977	0.305988	+	0.952035i	$0.401013\pi$
$384$	0	0
$385$	0	0
$386$	−20.6959	−1.05339
$387$	0	0
$388$	−66.5017	−3.37611
$389$	−13.7788	−0.698615	−0.349308	−	0.937008i	$-0.613583\pi$
$389$	−13.7788	−0.698615	−0.349308	+	0.937008i	$0.613583\pi$
$390$	0	0
$391$	11.0989	0.561295
$392$	0	0
$393$	0	0
$394$	49.6377	2.50071
$395$	2.64654	0.133162
$396$	0	0
$397$	8.38688	0.420925	0.210463	−	0.977602i	$-0.432503\pi$
$397$	8.38688	0.420925	0.210463	+	0.977602i	$0.432503\pi$
$398$	−16.0741	−0.805723
$399$	0	0
$400$	−33.1606	−1.65803
$401$	−27.6167	−1.37911	−0.689556	−	0.724233i	$-0.742194\pi$
$401$	−27.6167	−1.37911	−0.689556	+	0.724233i	$0.742194\pi$
$402$	0	0
$403$	−13.5229	−0.673624
$404$	77.2544	3.84355
$405$	0	0
$406$	0	0
$407$	1.34479	0.0666590
$408$	0	0
$409$	30.5316	1.50969	0.754844	−	0.655904i	$-0.227712\pi$
$409$	30.5316	1.50969	0.754844	+	0.655904i	$0.227712\pi$
$410$	38.2051	1.88681
$411$	0	0
$412$	48.6377	2.39621
$413$	0	0
$414$	0	0
$415$	8.62907	0.423584
$416$	−51.6835	−2.53399
$417$	0	0
$418$	−1.85532	−0.0907464
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	34.7293	1.69260	0.846302	−	0.532703i	$-0.178824\pi$
$421$	34.7293	1.69260	0.846302	+	0.532703i	$0.178824\pi$
$422$	−18.0197	−0.877183
$423$	0	0
$424$	17.8626	0.867484
$425$	14.5563	0.706085
$426$	0	0
$427$	0	0
$428$	−66.1555	−3.19775
$429$	0	0
$430$	39.0159	1.88152
$431$	−31.6043	−1.52233	−0.761163	−	0.648561i	$-0.775371\pi$
$431$	−31.6043	−1.52233	−0.761163	+	0.648561i	$0.775371\pi$
$432$	0	0
$433$	−6.48576	−0.311686	−0.155843	−	0.987782i	$-0.549809\pi$
$433$	−6.48576	−0.311686	−0.155843	+	0.987782i	$0.549809\pi$
$434$	0	0
$435$	0	0
$436$	−16.1062	−0.771346
$437$	4.32141	0.206721
$438$	0	0
$439$	−15.5760	−0.743401	−0.371701	−	0.928353i	$-0.621225\pi$
$439$	−15.5760	−0.743401	−0.371701	+	0.928353i	$0.621225\pi$
$440$	4.23491	0.201891
$441$	0	0
$442$	44.5919	2.12102
$443$	25.0741	1.19131	0.595654	−	0.803241i	$-0.296893\pi$
$443$	25.0741	1.19131	0.595654	+	0.803241i	$0.296893\pi$
$444$	0	0
$445$	14.8567	0.704275
$446$	−17.1185	−0.810587
$447$	0	0
$448$	0	0
$449$	15.2967	0.721894	0.360947	−	0.932586i	$-0.382454\pi$
$449$	15.2967	0.721894	0.360947	+	0.932586i	$0.382454\pi$
$450$	0	0
$451$	2.67625	0.126020
$452$	−52.6377	−2.47587
$453$	0	0
$454$	−73.1235	−3.43186
$455$	0	0
$456$	0	0
$457$	20.4400	0.956141	0.478071	−	0.878321i	$-0.341336\pi$
$457$	20.4400	0.956141	0.478071	+	0.878321i	$0.341336\pi$
$458$	−22.1272	−1.03394
$459$	0	0
$460$	−15.8640	−0.739662
$461$	40.8182	1.90109	0.950546	−	0.310584i	$-0.100525\pi$
$461$	40.8182	1.90109	0.950546	+	0.310584i	$0.100525\pi$
$462$	0	0
$463$	9.90840	0.460482	0.230241	−	0.973134i	$-0.426048\pi$
$463$	9.90840	0.460482	0.230241	+	0.973134i	$0.426048\pi$
$464$	−33.7738	−1.56791
$465$	0	0
$466$	74.0246	3.42912
$467$	17.7390	0.820861	0.410430	−	0.911892i	$-0.365378\pi$
$467$	17.7390	0.820861	0.410430	+	0.911892i	$0.365378\pi$
$468$	0	0
$469$	0	0
$470$	−13.7651	−0.634936
$471$	0	0
$472$	−43.3942	−1.99738
$473$	2.73305	0.125666
$474$	0	0
$475$	5.66758	0.260047
$476$	0	0
$477$	0	0
$478$	−75.2224	−3.44059
$479$	23.6094	1.07874	0.539371	−	0.842068i	$-0.318662\pi$
$479$	23.6094	1.07874	0.539371	+	0.842068i	$0.318662\pi$
$480$	0	0
$481$	−12.5846	−0.573810
$482$	60.8231	2.77042
$483$	0	0
$484$	−57.6908	−2.62231
$485$	−19.9752	−0.907029
$486$	0	0
$487$	34.8726	1.58023	0.790115	−	0.612959i	$-0.210021\pi$
$487$	34.8726	1.58023	0.790115	+	0.612959i	$0.210021\pi$
$488$	−67.2471	−3.04413
$489$	0	0
$490$	0	0
$491$	−23.2051	−1.04723	−0.523615	−	0.851955i	$-0.675417\pi$
$491$	−23.2051	−1.04723	−0.523615	+	0.851955i	$0.675417\pi$
$492$	0	0
$493$	14.8255	0.667705
$494$	17.3621	0.781159
$495$	0	0
$496$	−64.4028	−2.89177
$497$	0	0
$498$	0	0
$499$	−5.32513	−0.238386	−0.119193	−	0.992871i	$-0.538031\pi$
$499$	−5.32513	−0.238386	−0.119193	+	0.992871i	$0.538031\pi$
$500$	−62.8021	−2.80859
$501$	0	0
$502$	84.3657	3.76542
$503$	10.4313	0.465109	0.232554	−	0.972583i	$-0.425292\pi$
$503$	10.4313	0.465109	0.232554	+	0.972583i	$0.425292\pi$
$504$	0	0
$505$	23.2051	1.03261
$506$	−1.53156	−0.0680862
$507$	0	0
$508$	−71.0704	−3.15324
$509$	−1.50186	−0.0665687	−0.0332844	−	0.999446i	$-0.510597\pi$
$509$	−1.50186	−0.0665687	−0.0332844	+	0.999446i	$0.510597\pi$
$510$	0	0
$511$	0	0
$512$	−8.48948	−0.375186
$513$	0	0
$514$	−27.5402	−1.21475
$515$	14.6094	0.643767
$516$	0	0
$517$	−0.964238	−0.0424072
$518$	0	0
$519$	0	0
$520$	−39.6304	−1.73791
$521$	−37.7417	−1.65349	−0.826747	−	0.562574i	$-0.809811\pi$
$521$	−37.7417	−1.65349	−0.826747	+	0.562574i	$0.809811\pi$
$522$	0	0
$523$	−17.9556	−0.785143	−0.392571	−	0.919722i	$-0.628414\pi$
$523$	−17.9556	−0.785143	−0.392571	+	0.919722i	$0.628414\pi$
$524$	−59.6253	−2.60475
$525$	0	0
$526$	76.8341	3.35013
$527$	28.2705	1.23148
$528$	0	0
$529$	−19.4327	−0.844899
$530$	8.62907	0.374823
$531$	0	0
$532$	0	0
$533$	−25.0444	−1.08479
$534$	0	0
$535$	−19.8713	−0.859109
$536$	−6.32004	−0.272984
$537$	0	0
$538$	40.1606	1.73145
$539$	0	0
$540$	0	0
$541$	8.37093	0.359894	0.179947	−	0.983676i	$-0.442407\pi$
$541$	8.37093	0.359894	0.179947	+	0.983676i	$0.442407\pi$
$542$	−0.119925	−0.00515123
$543$	0	0
$544$	108.048	4.63251
$545$	−4.83784	−0.207230
$546$	0	0
$547$	30.1075	1.28731	0.643653	−	0.765318i	$-0.277418\pi$
$547$	30.1075	1.28731	0.643653	+	0.765318i	$0.277418\pi$
$548$	−19.9752	−0.853300
$549$	0	0
$550$	−2.00866	−0.0856496
$551$	5.77238	0.245911
$552$	0	0
$553$	0	0
$554$	−42.3163	−1.79785
$555$	0	0
$556$	88.4115	3.74948
$557$	−16.5353	−0.700622	−0.350311	−	0.936633i	$-0.613924\pi$
$557$	−16.5353	−0.700622	−0.350311	+	0.936633i	$0.613924\pi$
$558$	0	0
$559$	−25.5760	−1.08175
$560$	0	0
$561$	0	0
$562$	32.1520	1.35625
$563$	22.2312	0.936933	0.468466	−	0.883481i	$-0.344807\pi$
$563$	22.2312	0.936933	0.468466	+	0.883481i	$0.344807\pi$
$564$	0	0
$565$	−15.8109	−0.665169
$566$	−16.2174	−0.681670
$567$	0	0
$568$	−114.185	−4.79111
$569$	−2.31275	−0.0969556	−0.0484778	−	0.998824i	$-0.515437\pi$
$569$	−2.31275	−0.0969556	−0.0484778	+	0.998824i	$0.515437\pi$
$570$	0	0
$571$	−6.09888	−0.255230	−0.127615	−	0.991824i	$-0.540732\pi$
$571$	−6.09888	−0.255230	−0.127615	+	0.991824i	$0.540732\pi$
$572$	−4.46472	−0.186679
$573$	0	0
$574$	0	0
$575$	4.67859	0.195111
$576$	0	0
$577$	20.2064	0.841205	0.420602	−	0.907245i	$-0.361819\pi$
$577$	20.2064	0.841205	0.420602	+	0.907245i	$0.361819\pi$
$578$	−47.3287	−1.96861
$579$	0	0
$580$	−21.1905	−0.879887
$581$	0	0
$582$	0	0
$583$	0.604462	0.0250343
$584$	103.630	4.28825
$585$	0	0
$586$	−13.3782	−0.552649
$587$	−38.6822	−1.59658	−0.798292	−	0.602271i	$-0.794263\pi$
$587$	−38.6822	−1.59658	−0.798292	+	0.602271i	$0.794263\pi$
$588$	0	0
$589$	11.0073	0.453547
$590$	−20.9629	−0.863027
$591$	0	0
$592$	−59.9344	−2.46329
$593$	−23.7156	−0.973882	−0.486941	−	0.873435i	$-0.661887\pi$
$593$	−23.7156	−0.973882	−0.486941	+	0.873435i	$0.661887\pi$
$594$	0	0
$595$	0	0
$596$	−57.5795	−2.35855
$597$	0	0
$598$	14.3324	0.586096
$599$	34.4807	1.40884	0.704421	−	0.709783i	$-0.251207\pi$
$599$	34.4807	1.40884	0.704421	+	0.709783i	$0.251207\pi$
$600$	0	0
$601$	−7.28071	−0.296986	−0.148493	−	0.988913i	$-0.547442\pi$
$601$	−7.28071	−0.296986	−0.148493	+	0.988913i	$0.547442\pi$
$602$	0	0
$603$	0	0
$604$	−63.9788	−2.60326
$605$	−17.3287	−0.704512
$606$	0	0
$607$	−25.1593	−1.02118	−0.510591	−	0.859824i	$-0.670573\pi$
$607$	−25.1593	−1.02118	−0.510591	+	0.859824i	$0.670573\pi$
$608$	42.0690	1.70612
$609$	0	0
$610$	−32.4858	−1.31531
$611$	9.02338	0.365047
$612$	0	0
$613$	−37.5461	−1.51647	−0.758237	−	0.651979i	$-0.773939\pi$
$613$	−37.5461	−1.51647	−0.758237	+	0.651979i	$0.773939\pi$
$614$	59.5351	2.40264
$615$	0	0
$616$	0	0
$617$	13.7280	0.552667	0.276333	−	0.961062i	$-0.410881\pi$
$617$	13.7280	0.552667	0.276333	+	0.961062i	$0.410881\pi$
$618$	0	0
$619$	11.8109	0.474720	0.237360	−	0.971422i	$-0.423718\pi$
$619$	11.8109	0.474720	0.237360	+	0.971422i	$0.423718\pi$
$620$	−40.4079	−1.62282
$621$	0	0
$622$	21.5302	0.863282
$623$	0	0
$624$	0	0
$625$	−6.47848	−0.259139
$626$	60.8231	2.43098
$627$	0	0
$628$	−1.00000	−0.0399043
$629$	26.3090	1.04901
$630$	0	0
$631$	29.6304	1.17957	0.589785	−	0.807561i	$-0.299213\pi$
$631$	29.6304	1.17957	0.589785	+	0.807561i	$0.299213\pi$
$632$	−14.7899	−0.588309
$633$	0	0
$634$	−53.8255	−2.13768
$635$	−21.3475	−0.847152
$636$	0	0
$637$	0	0
$638$	−2.04580	−0.0809940
$639$	0	0
$640$	−39.6304	−1.56653
$641$	14.6525	0.578737	0.289368	−	0.957218i	$-0.406555\pi$
$641$	14.6525	0.578737	0.289368	+	0.957218i	$0.406555\pi$
$642$	0	0
$643$	2.03714	0.0803369	0.0401685	−	0.999193i	$-0.487211\pi$
$643$	2.03714	0.0803369	0.0401685	+	0.999193i	$0.487211\pi$
$644$	0	0
$645$	0	0
$646$	−36.2967	−1.42807
$647$	−11.7861	−0.463361	−0.231680	−	0.972792i	$-0.574422\pi$
$647$	−11.7861	−0.463361	−0.231680	+	0.972792i	$0.574422\pi$
$648$	0	0
$649$	−1.46844	−0.0576413
$650$	18.7971	0.737284
$651$	0	0
$652$	−47.3570	−1.85464
$653$	−34.1840	−1.33772	−0.668862	−	0.743387i	$-0.733218\pi$
$653$	−34.1840	−1.33772	−0.668862	+	0.743387i	$0.733218\pi$
$654$	0	0
$655$	−17.9098	−0.699793
$656$	−119.274	−4.65687
$657$	0	0
$658$	0	0
$659$	24.8640	0.968563	0.484282	−	0.874912i	$-0.339081\pi$
$659$	24.8640	0.968563	0.484282	+	0.874912i	$0.339081\pi$
$660$	0	0
$661$	25.5388	0.993346	0.496673	−	0.867938i	$-0.334555\pi$
$661$	25.5388	0.993346	0.496673	+	0.867938i	$0.334555\pi$
$662$	−63.7934	−2.47940
$663$	0	0
$664$	−48.2224	−1.87139
$665$	0	0
$666$	0	0
$667$	4.76509	0.184505
$668$	22.9556	0.888178
$669$	0	0
$670$	−3.05308	−0.117951
$671$	−2.27561	−0.0878490
$672$	0	0
$673$	−3.81955	−0.147233	−0.0736165	−	0.997287i	$-0.523454\pi$
$673$	−3.81955	−0.147233	−0.0736165	+	0.997287i	$0.523454\pi$
$674$	33.4362	1.28792
$675$	0	0
$676$	−26.9629	−1.03703
$677$	−17.2225	−0.661916	−0.330958	−	0.943646i	$-0.607372\pi$
$677$	−17.2225	−0.661916	−0.330958	+	0.943646i	$0.607372\pi$
$678$	0	0
$679$	0	0
$680$	82.8501	3.17715
$681$	0	0
$682$	−3.90112	−0.149381
$683$	33.7170	1.29014	0.645072	−	0.764122i	$-0.276828\pi$
$683$	33.7170	1.29014	0.645072	+	0.764122i	$0.276828\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	−121.806	−4.64380
$689$	−5.65658	−0.215499
$690$	0	0
$691$	−25.4930	−0.969801	−0.484901	−	0.874569i	$-0.661144\pi$
$691$	−25.4930	−0.969801	−0.484901	+	0.874569i	$0.661144\pi$
$692$	33.3163	1.26650
$693$	0	0
$694$	−10.3782	−0.393952
$695$	26.5563	1.00734
$696$	0	0
$697$	52.3570	1.98316
$698$	32.9061	1.24551
$699$	0	0
$700$	0	0
$701$	10.1606	0.383762	0.191881	−	0.981418i	$-0.438541\pi$
$701$	10.1606	0.383762	0.191881	+	0.981418i	$0.438541\pi$
$702$	0	0
$703$	10.2436	0.386344
$704$	−6.86769	−0.258836
$705$	0	0
$706$	−9.62041	−0.362069
$707$	0	0
$708$	0	0
$709$	−15.9469	−0.598899	−0.299449	−	0.954112i	$-0.596803\pi$
$709$	−15.9469	−0.598899	−0.299449	+	0.954112i	$0.596803\pi$
$710$	−55.1606	−2.07014
$711$	0	0
$712$	−83.0246	−3.11148
$713$	9.08650	0.340292
$714$	0	0
$715$	−1.34108	−0.0501534
$716$	73.7242	2.75520
$717$	0	0
$718$	−2.61312	−0.0975209
$719$	−16.4175	−0.612271	−0.306136	−	0.951988i	$-0.599036\pi$
$719$	−16.4175	−0.612271	−0.306136	+	0.951988i	$0.599036\pi$
$720$	0	0
$721$	0	0
$722$	37.1606	1.38298
$723$	0	0
$724$	96.9875	3.60451
$725$	6.24948	0.232100
$726$	0	0
$727$	−35.8282	−1.32879	−0.664397	−	0.747379i	$-0.731312\pi$
$727$	−35.8282	−1.32879	−0.664397	+	0.747379i	$0.731312\pi$
$728$	0	0
$729$	0	0
$730$	50.0617	1.85287
$731$	53.4683	1.97760
$732$	0	0
$733$	39.2880	1.45114	0.725568	−	0.688151i	$-0.241577\pi$
$733$	39.2880	1.45114	0.725568	+	0.688151i	$0.241577\pi$
$734$	−56.5685	−2.08798
$735$	0	0
$736$	34.7280	1.28009
$737$	−0.213867	−0.00787790
$738$	0	0
$739$	2.09022	0.0768901	0.0384451	−	0.999261i	$-0.487760\pi$
$739$	2.09022	0.0768901	0.0384451	+	0.999261i	$0.487760\pi$
$740$	−37.6043	−1.38236
$741$	0	0
$742$	0	0
$743$	30.7266	1.12725	0.563624	−	0.826031i	$-0.309407\pi$
$743$	30.7266	1.12725	0.563624	+	0.826031i	$0.309407\pi$
$744$	0	0
$745$	−17.2953	−0.633650
$746$	−5.39926	−0.197681
$747$	0	0
$748$	9.33379	0.341277
$749$	0	0
$750$	0	0
$751$	20.7738	0.758045	0.379023	−	0.925387i	$-0.376260\pi$
$751$	20.7738	0.758045	0.379023	+	0.925387i	$0.376260\pi$
$752$	42.9739	1.56710
$753$	0	0
$754$	19.1447	0.697208
$755$	−19.2174	−0.699394
$756$	0	0
$757$	16.9257	0.615176	0.307588	−	0.951520i	$-0.400478\pi$
$757$	16.9257	0.615176	0.307588	+	0.951520i	$0.400478\pi$
$758$	19.2843	0.700436
$759$	0	0
$760$	32.2581	1.17013
$761$	32.8392	1.19042	0.595210	−	0.803570i	$-0.297069\pi$
$761$	32.8392	1.19042	0.595210	+	0.803570i	$0.297069\pi$
$762$	0	0
$763$	0	0
$764$	−40.2312	−1.45551
$765$	0	0
$766$	−32.3324	−1.16822
$767$	13.7417	0.496184
$768$	0	0
$769$	31.7293	1.14419	0.572094	−	0.820188i	$-0.306131\pi$
$769$	31.7293	1.14419	0.572094	+	0.820188i	$0.306131\pi$
$770$	0	0
$771$	0	0
$772$	40.5388	1.45902
$773$	6.37312	0.229225	0.114613	−	0.993410i	$-0.463437\pi$
$773$	6.37312	0.229225	0.114613	+	0.993410i	$0.463437\pi$
$774$	0	0
$775$	11.9171	0.428073
$776$	111.629	4.00724
$777$	0	0
$778$	37.1978	1.33360
$779$	20.3855	0.730386
$780$	0	0
$781$	−3.86398	−0.138264
$782$	−29.9629	−1.07147
$783$	0	0
$784$	0	0
$785$	−0.300372	−0.0107207
$786$	0	0
$787$	−19.8799	−0.708643	−0.354321	−	0.935124i	$-0.615288\pi$
$787$	−19.8799	−0.708643	−0.354321	+	0.935124i	$0.615288\pi$
$788$	−97.2297	−3.46366
$789$	0	0
$790$	−7.14468	−0.254196
$791$	0	0
$792$	0	0
$793$	21.2953	0.756217
$794$	−22.6414	−0.803515
$795$	0	0
$796$	31.4858	1.11598
$797$	−5.81955	−0.206139	−0.103070	−	0.994674i	$-0.532866\pi$
$797$	−5.81955	−0.206139	−0.103070	+	0.994674i	$0.532866\pi$
$798$	0	0
$799$	−18.8640	−0.667360
$800$	45.5461	1.61030
$801$	0	0
$802$	74.5548	2.63262
$803$	3.50680	0.123752
$804$	0	0
$805$	0	0
$806$	36.5068	1.28590
$807$	0	0
$808$	−129.678	−4.56207
$809$	2.12227	0.0746149	0.0373075	−	0.999304i	$-0.488122\pi$
$809$	2.12227	0.0746149	0.0373075	+	0.999304i	$0.488122\pi$
$810$	0	0
$811$	−18.4327	−0.647259	−0.323629	−	0.946184i	$-0.604903\pi$
$811$	−18.4327	−0.647259	−0.323629	+	0.946184i	$0.604903\pi$
$812$	0	0
$813$	0	0
$814$	−3.63045	−0.127247
$815$	−14.2247	−0.498270
$816$	0	0
$817$	20.8182	0.728336
$818$	−82.4239	−2.88188
$819$	0	0
$820$	−74.8355	−2.61337
$821$	−45.1555	−1.57594	−0.787969	−	0.615714i	$-0.788868\pi$
$821$	−45.1555	−1.57594	−0.787969	+	0.615714i	$0.788868\pi$
$822$	0	0
$823$	25.0517	0.873248	0.436624	−	0.899644i	$-0.356174\pi$
$823$	25.0517	0.873248	0.436624	+	0.899644i	$0.356174\pi$
$824$	−81.6427	−2.84416
$825$	0	0
$826$	0	0
$827$	−14.2953	−0.497095	−0.248548	−	0.968620i	$-0.579953\pi$
$827$	−14.2953	−0.497095	−0.248548	+	0.968620i	$0.579953\pi$
$828$	0	0
$829$	31.8255	1.10534	0.552672	−	0.833399i	$-0.313608\pi$
$829$	31.8255	1.10534	0.552672	+	0.833399i	$0.313608\pi$
$830$	−23.2953	−0.808591
$831$	0	0
$832$	64.2682	2.22810
$833$	0	0
$834$	0	0
$835$	6.89521	0.238619
$836$	3.63416	0.125690
$837$	0	0
$838$	56.6922	1.95840
$839$	16.4785	0.568900	0.284450	−	0.958691i	$-0.408189\pi$
$839$	16.4785	0.568900	0.284450	+	0.958691i	$0.408189\pi$
$840$	0	0
$841$	−22.6350	−0.780516
$842$	−93.7563	−3.23105
$843$	0	0
$844$	35.2967	1.21496
$845$	−8.09888	−0.278610
$846$	0	0
$847$	0	0
$848$	−26.9395	−0.925106
$849$	0	0
$850$	−39.2967	−1.34786
$851$	8.45606	0.289870
$852$	0	0
$853$	−26.9986	−0.924415	−0.462208	−	0.886772i	$-0.652943\pi$
$853$	−26.9986	−0.924415	−0.462208	+	0.886772i	$0.652943\pi$
$854$	0	0
$855$	0	0
$856$	111.048	3.79554
$857$	−39.1630	−1.33778	−0.668891	−	0.743361i	$-0.733231\pi$
$857$	−39.1630	−1.33778	−0.668891	+	0.743361i	$0.733231\pi$
$858$	0	0
$859$	9.46844	0.323059	0.161529	−	0.986868i	$-0.448357\pi$
$859$	9.46844	0.323059	0.161529	+	0.986868i	$0.448357\pi$
$860$	−76.4239	−2.60603
$861$	0	0
$862$	85.3199	2.90601
$863$	2.01238	0.0685022	0.0342511	−	0.999413i	$-0.489095\pi$
$863$	2.01238	0.0685022	0.0342511	+	0.999413i	$0.489095\pi$
$864$	0	0
$865$	10.0073	0.340258
$866$	17.5091	0.594985
$867$	0	0
$868$	0	0
$869$	−0.500482	−0.0169777
$870$	0	0
$871$	2.00138	0.0678141
$872$	27.0356	0.915541
$873$	0	0
$874$	−11.6662	−0.394615
$875$	0	0
$876$	0	0
$877$	44.6995	1.50939	0.754697	−	0.656073i	$-0.227784\pi$
$877$	44.6995	1.50939	0.754697	+	0.656073i	$0.227784\pi$
$878$	42.0494	1.41910
$879$	0	0
$880$	−6.38688	−0.215302
$881$	12.7047	0.428033	0.214017	−	0.976830i	$-0.431345\pi$
$881$	12.7047	0.428033	0.214017	+	0.976830i	$0.431345\pi$
$882$	0	0
$883$	−17.0014	−0.572142	−0.286071	−	0.958208i	$-0.592349\pi$
$883$	−17.0014	−0.572142	−0.286071	+	0.958208i	$0.592349\pi$
$884$	−87.3460	−2.93776
$885$	0	0
$886$	−67.6908	−2.27412
$887$	1.20149	0.0403420	0.0201710	−	0.999797i	$-0.493579\pi$
$887$	1.20149	0.0403420	0.0201710	+	0.999797i	$0.493579\pi$
$888$	0	0
$889$	0	0
$890$	−40.1075	−1.34441
$891$	0	0
$892$	33.5316	1.12272
$893$	−7.34479	−0.245784
$894$	0	0
$895$	22.1447	0.740215
$896$	0	0
$897$	0	0
$898$	−41.2953	−1.37804
$899$	12.1374	0.404805
$900$	0	0
$901$	11.8255	0.393963
$902$	−7.22487	−0.240562
$903$	0	0
$904$	88.3570	2.93871
$905$	29.1323	0.968391
$906$	0	0
$907$	17.0631	0.566572	0.283286	−	0.959036i	$-0.408575\pi$
$907$	17.0631	0.566572	0.283286	+	0.959036i	$0.408575\pi$
$908$	143.233	4.75336
$909$	0	0
$910$	0	0
$911$	22.0014	0.728938	0.364469	−	0.931215i	$-0.381250\pi$
$911$	22.0014	0.728938	0.364469	+	0.931215i	$0.381250\pi$
$912$	0	0
$913$	−1.63182	−0.0540055
$914$	−55.1803	−1.82520
$915$	0	0
$916$	43.3425	1.43207
$917$	0	0
$918$	0	0
$919$	−29.5906	−0.976102	−0.488051	−	0.872815i	$-0.662292\pi$
$919$	−29.5906	−0.976102	−0.488051	+	0.872815i	$0.662292\pi$
$920$	26.6291	0.877934
$921$	0	0
$922$	−110.194	−3.62904
$923$	36.1593	1.19020
$924$	0	0
$925$	11.0902	0.364644
$926$	−26.7490	−0.879026
$927$	0	0
$928$	46.3883	1.52277
$929$	−2.69963	−0.0885719	−0.0442860	−	0.999019i	$-0.514101\pi$
$929$	−2.69963	−0.0885719	−0.0442860	+	0.999019i	$0.514101\pi$
$930$	0	0
$931$	0	0
$932$	−144.998	−4.74958
$933$	0	0
$934$	−47.8886	−1.56696
$935$	2.80361	0.0916878
$936$	0	0
$937$	39.4472	1.28869	0.644343	−	0.764737i	$-0.277131\pi$
$937$	39.4472	1.28869	0.644343	+	0.764737i	$0.277131\pi$
$938$	0	0
$939$	0	0
$940$	26.9629	0.879432
$941$	−39.7244	−1.29498	−0.647489	−	0.762075i	$-0.724181\pi$
$941$	−39.7244	−1.29498	−0.647489	+	0.762075i	$0.724181\pi$
$942$	0	0
$943$	16.8282	0.548002
$944$	65.4449	2.13005
$945$	0	0
$946$	−7.37822	−0.239886
$947$	15.7303	0.511166	0.255583	−	0.966787i	$-0.417733\pi$
$947$	15.7303	0.511166	0.255583	+	0.966787i	$0.417733\pi$
$948$	0	0
$949$	−32.8168	−1.06528
$950$	−15.3004	−0.496410
$951$	0	0
$952$	0	0
$953$	−30.2064	−0.978482	−0.489241	−	0.872149i	$-0.662726\pi$
$953$	−30.2064	−0.978482	−0.489241	+	0.872149i	$0.662726\pi$
$954$	0	0
$955$	−12.0843	−0.391039
$956$	147.344	4.76546
$957$	0	0
$958$	−63.7366	−2.05924
$959$	0	0
$960$	0	0
$961$	−7.85532	−0.253397
$962$	33.9739	1.09536
$963$	0	0
$964$	−119.139	−3.83722
$965$	12.1767	0.391983
$966$	0	0
$967$	−6.18911	−0.199028	−0.0995141	−	0.995036i	$-0.531729\pi$
$967$	−6.18911	−0.199028	−0.0995141	+	0.995036i	$0.531729\pi$
$968$	96.8391	3.11253
$969$	0	0
$970$	53.9257	1.73145
$971$	2.17163	0.0696910	0.0348455	−	0.999393i	$-0.488906\pi$
$971$	2.17163	0.0696910	0.0348455	+	0.999393i	$0.488906\pi$
$972$	0	0
$973$	0	0
$974$	−94.1432	−3.01654
$975$	0	0
$976$	101.419	3.24634
$977$	11.4561	0.366512	0.183256	−	0.983065i	$-0.441336\pi$
$977$	11.4561	0.366512	0.183256	+	0.983065i	$0.441336\pi$
$978$	0	0
$979$	−2.80951	−0.0897925
$980$	0	0
$981$	0	0
$982$	62.6450	1.99908
$983$	−44.8021	−1.42896	−0.714482	−	0.699654i	$-0.753338\pi$
$983$	−44.8021	−1.42896	−0.714482	+	0.699654i	$0.753338\pi$
$984$	0	0
$985$	−29.2051	−0.930550
$986$	−40.0232	−1.27460
$987$	0	0
$988$	−34.0087	−1.08196
$989$	17.1854	0.546464
$990$	0	0
$991$	−24.2953	−0.771765	−0.385882	−	0.922548i	$-0.626103\pi$
$991$	−24.2953	−0.771765	−0.385882	+	0.922548i	$0.626103\pi$
$992$	88.4573	2.80852
$993$	0	0
$994$	0	0
$995$	9.45744	0.299821
$996$	0	0
$997$	−11.3251	−0.358671	−0.179335	−	0.983788i	$-0.557395\pi$
$997$	−11.3251	−0.358671	−0.179335	+	0.983788i	$0.557395\pi$
$998$	14.3759	0.455060
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.x.1.1		3
3.2	odd	2		1323.2.a.ba.1.3		3
7.2	even	3		189.2.e.f.109.3	yes	6
7.4	even	3		189.2.e.f.163.3	yes	6
7.6	odd	2		1323.2.a.y.1.1		3
21.2	odd	6		189.2.e.e.109.1	✓	6
21.11	odd	6		189.2.e.e.163.1	yes	6
21.20	even	2		1323.2.a.z.1.3		3
63.2	odd	6		567.2.g.h.109.1		6
63.4	even	3		567.2.g.i.541.3		6
63.11	odd	6		567.2.h.i.352.3		6
63.16	even	3		567.2.g.i.109.3		6
63.23	odd	6		567.2.h.i.298.3		6
63.25	even	3		567.2.h.h.352.1		6
63.32	odd	6		567.2.g.h.541.1		6
63.58	even	3		567.2.h.h.298.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.e.109.1	✓	6	21.2	odd	6
189.2.e.e.163.1	yes	6	21.11	odd	6
189.2.e.f.109.3	yes	6	7.2	even	3
189.2.e.f.163.3	yes	6	7.4	even	3
567.2.g.h.109.1		6	63.2	odd	6
567.2.g.h.541.1		6	63.32	odd	6
567.2.g.i.109.3		6	63.16	even	3
567.2.g.i.541.3		6	63.4	even	3
567.2.h.h.298.1		6	63.58	even	3
567.2.h.h.352.1		6	63.25	even	3
567.2.h.i.298.3		6	63.23	odd	6
567.2.h.i.352.3		6	63.11	odd	6
1323.2.a.x.1.1		3	1.1	even	1	trivial
1323.2.a.y.1.1		3	7.6	odd	2
1323.2.a.z.1.3		3	21.20	even	2
1323.2.a.ba.1.3		3	3.2	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69963 q^{2} +5.28799 q^{4} +1.58836 q^{5} -8.87636 q^{8} +O(q^{10})\)	\(q-2.69963 q^{2} +5.28799 q^{4} +1.58836 q^{5} -8.87636 q^{8} -4.28799 q^{10} -0.300372 q^{11} +2.81089 q^{13} +13.3869 q^{16} -5.87636 q^{17} -2.28799 q^{19} +8.39926 q^{20} +0.810892 q^{22} -1.88874 q^{23} -2.47710 q^{25} -7.58836 q^{26} -2.52290 q^{29} -4.81089 q^{31} -18.3869 q^{32} +15.8640 q^{34} -4.47710 q^{37} +6.17673 q^{38} -14.0989 q^{40} -8.90978 q^{41} -9.09888 q^{43} -1.58836 q^{44} +5.09888 q^{46} +3.21015 q^{47} +6.68725 q^{50} +14.8640 q^{52} -2.01238 q^{53} -0.477100 q^{55} +6.81089 q^{58} +4.88874 q^{59} +7.57598 q^{61} +12.9876 q^{62} +22.8640 q^{64} +4.46472 q^{65} +0.712008 q^{67} -31.0741 q^{68} +12.8640 q^{71} -11.6749 q^{73} +12.0865 q^{74} -12.0989 q^{76} +1.66621 q^{79} +21.2632 q^{80} +24.0531 q^{82} +5.43268 q^{83} -9.33379 q^{85} +24.5636 q^{86} +2.66621 q^{88} +9.35346 q^{89} -9.98762 q^{92} -8.66621 q^{94} -3.63416 q^{95} -12.5760 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8	\( 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8} - q^{10} - 7 q^{11} + 2 q^{13} + 10 q^{16} + 5 q^{19} + 13 q^{20} - 4 q^{22} - 6 q^{23} - 2 q^{25} - 17 q^{26} - 13 q^{29} - 8 q^{31} - 25 q^{32} + 12 q^{34} - 8 q^{37} + 7 q^{38} - 24 q^{40} - 2 q^{41} - 9 q^{43} + q^{44} - 3 q^{46} - 9 q^{47} - 4 q^{50} + 9 q^{52} - 24 q^{53} + 4 q^{55} + 14 q^{58} + 15 q^{59} - q^{61} + 21 q^{62} + 33 q^{64} - 10 q^{65} + 14 q^{67} - 39 q^{68} + 3 q^{71} + 7 q^{73} - 18 q^{76} + 6 q^{79} + 16 q^{80} + 43 q^{82} - 3 q^{83} - 27 q^{85} + 32 q^{86} + 9 q^{88} + 5 q^{89} - 12 q^{92} - 27 q^{94} - 16 q^{95} - 14 q^{97}+O(q^{100}) \) 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8 - q^10 - 7 * q^11 + 2 * q^13 + 10 * q^16 + 5 * q^19 + 13 * q^20 - 4 * q^22 - 6 * q^23 - 2 * q^25 - 17 * q^26 - 13 * q^29 - 8 * q^31 - 25 * q^32 + 12 * q^34 - 8 * q^37 + 7 * q^38 - 24 * q^40 - 2 * q^41 - 9 * q^43 + q^44 - 3 * q^46 - 9 * q^47 - 4 * q^50 + 9 * q^52 - 24 * q^53 + 4 * q^55 + 14 * q^58 + 15 * q^59 - q^61 + 21 * q^62 + 33 * q^64 - 10 * q^65 + 14 * q^67 - 39 * q^68 + 3 * q^71 + 7 * q^73 - 18 * q^76 + 6 * q^79 + 16 * q^80 + 43 * q^82 - 3 * q^83 - 27 * q^85 + 32 * q^86 + 9 * q^88 + 5 * q^89 - 12 * q^92 - 27 * q^94 - 16 * q^95 - 14 * q^97

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