LMFDB - Embedded newform 1323.4.a.ba.1.2

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:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 4 $ x^4 - 9*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.92081$ 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)

$f(q)$	$=$	$q-1.55133 q^{2} -5.59339 q^{4} +9.81086 q^{5} +21.0878 q^{8} +O(q^{10})$	$q-1.55133 q^{2} -5.59339 q^{4} +9.81086 q^{5} +21.0878 q^{8} -15.2198 q^{10} -70.6023 q^{11} -55.5603 q^{13} +12.0331 q^{16} +13.4164 q^{17} -91.3735 q^{19} -54.8759 q^{20} +109.527 q^{22} +113.784 q^{23} -28.7471 q^{25} +86.1922 q^{26} +12.4959 q^{29} -222.934 q^{31} -187.369 q^{32} -20.8132 q^{34} +257.440 q^{37} +141.750 q^{38} +206.889 q^{40} +286.765 q^{41} +4.81323 q^{43} +394.906 q^{44} -176.516 q^{46} -609.517 q^{47} +44.5961 q^{50} +310.770 q^{52} +691.107 q^{53} -692.669 q^{55} -19.3852 q^{58} +217.936 q^{59} +764.747 q^{61} +345.843 q^{62} +194.407 q^{64} -545.094 q^{65} -98.5487 q^{67} -75.0432 q^{68} -921.274 q^{71} +219.934 q^{73} -399.373 q^{74} +511.088 q^{76} -9.42805 q^{79} +118.055 q^{80} -444.866 q^{82} -800.364 q^{83} +131.626 q^{85} -7.46689 q^{86} -1488.85 q^{88} -253.574 q^{89} -636.436 q^{92} +945.560 q^{94} -896.453 q^{95} +59.2529 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 26 q^{4}+O(q^{10}) $ 4 * q + 26 * q^4	$ 4 q + 26 q^{4} - 206 q^{10} + 68 q^{13} + 290 q^{16} - 172 q^{19} - 94 q^{22} + 272 q^{25} - 408 q^{31} - 180 q^{34} + 1320 q^{37} - 1446 q^{40} + 116 q^{43} + 1374 q^{46} + 3952 q^{52} - 352 q^{55} - 1432 q^{58} + 2672 q^{61} + 826 q^{64} + 1444 q^{67} + 396 q^{73} + 1222 q^{76} + 1220 q^{79} + 3590 q^{82} + 720 q^{85} - 6294 q^{88} + 3492 q^{94} + 624 q^{97}+O(q^{100}) $ 4 * q + 26 * q^4 - 206 * q^10 + 68 * q^13 + 290 * q^16 - 172 * q^19 - 94 * q^22 + 272 * q^25 - 408 * q^31 - 180 * q^34 + 1320 * q^37 - 1446 * q^40 + 116 * q^43 + 1374 * q^46 + 3952 * q^52 - 352 * q^55 - 1432 * q^58 + 2672 * q^61 + 826 * q^64 + 1444 * q^67 + 396 * q^73 + 1222 * q^76 + 1220 * q^79 + 3590 * q^82 + 720 * q^85 - 6294 * q^88 + 3492 * q^94 + 624 * 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 $

$2$	−1.55133	−0.548477	−0.274238	−	0.961662i	$-0.588426\pi$
$2$	−1.55133	−0.548477	−0.274238	+	0.961662i	$0.588426\pi$
$3$	0	0
$3$	0	0
$4$	−5.59339	−0.699173
$5$	9.81086	0.877510	0.438755	−	0.898607i	$-0.355420\pi$
$5$	9.81086	0.877510	0.438755	+	0.898607i	$0.355420\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.0878	0.931957
$9$	0	0
$10$	−15.2198	−0.481294
$11$	−70.6023	−1.93522	−0.967609	−	0.252453i	$-0.918763\pi$
$11$	−70.6023	−1.93522	−0.967609	+	0.252453i	$0.918763\pi$
$12$	0	0
$13$	−55.5603	−1.18536	−0.592679	−	0.805439i	$-0.701930\pi$
$13$	−55.5603	−1.18536	−0.592679	+	0.805439i	$0.701930\pi$
$14$	0	0
$15$	0	0
$16$	12.0331	0.188017
$17$	13.4164	0.191409	0.0957046	−	0.995410i	$-0.469490\pi$
$17$	13.4164	0.191409	0.0957046	+	0.995410i	$0.469490\pi$
$18$	0	0
$19$	−91.3735	−1.10329	−0.551646	−	0.834079i	$-0.686000\pi$
$19$	−91.3735	−1.10329	−0.551646	+	0.834079i	$0.686000\pi$
$20$	−54.8759	−0.613531
$21$	0	0
$22$	109.527	1.06142
$23$	113.784	1.03155	0.515773	−	0.856726i	$-0.327505\pi$
$23$	113.784	1.03155	0.515773	+	0.856726i	$0.327505\pi$
$24$	0	0
$25$	−28.7471	−0.229977
$26$	86.1922	0.650141
$27$	0	0
$28$	0	0
$29$	12.4959	0.0800147	0.0400073	−	0.999199i	$-0.487262\pi$
$29$	12.4959	0.0800147	0.0400073	+	0.999199i	$0.487262\pi$
$30$	0	0
$31$	−222.934	−1.29162	−0.645808	−	0.763500i	$-0.723479\pi$
$31$	−222.934	−1.29162	−0.645808	+	0.763500i	$0.723479\pi$
$32$	−187.369	−1.03508
$33$	0	0
$34$	−20.8132	−0.104983
$35$	0	0
$36$	0	0
$37$	257.440	1.14386	0.571930	−	0.820302i	$-0.306195\pi$
$37$	257.440	1.14386	0.571930	+	0.820302i	$0.306195\pi$
$38$	141.750	0.605129
$39$	0	0
$40$	206.889	0.817801
$41$	286.765	1.09232	0.546160	−	0.837681i	$-0.316089\pi$
$41$	286.765	1.09232	0.546160	+	0.837681i	$0.316089\pi$
$42$	0	0
$43$	4.81323	0.0170700	0.00853500	−	0.999964i	$-0.497283\pi$
$43$	4.81323	0.0170700	0.00853500	+	0.999964i	$0.497283\pi$
$44$	394.906	1.35305
$45$	0	0
$46$	−176.516	−0.565778
$47$	−609.517	−1.89164	−0.945822	−	0.324686i	$-0.894741\pi$
$47$	−609.517	−1.89164	−0.945822	+	0.324686i	$0.894741\pi$
$48$	0	0
$49$	0	0
$50$	44.5961	0.126137
$51$	0	0
$52$	310.770	0.828771
$53$	691.107	1.79115	0.895574	−	0.444913i	$-0.146765\pi$
$53$	691.107	1.79115	0.895574	+	0.444913i	$0.146765\pi$
$54$	0	0
$55$	−692.669	−1.69817
$56$	0	0
$57$	0	0
$58$	−19.3852	−0.0438862
$59$	217.936	0.480895	0.240448	−	0.970662i	$-0.422706\pi$
$59$	217.936	0.480895	0.240448	+	0.970662i	$0.422706\pi$
$60$	0	0
$61$	764.747	1.60518	0.802589	−	0.596533i	$-0.203455\pi$
$61$	764.747	1.60518	0.802589	+	0.596533i	$0.203455\pi$
$62$	345.843	0.708421
$63$	0	0
$64$	194.407	0.379700
$65$	−545.094	−1.04016
$66$	0	0
$67$	−98.5487	−0.179696	−0.0898481	−	0.995955i	$-0.528638\pi$
$67$	−98.5487	−0.179696	−0.0898481	+	0.995955i	$0.528638\pi$
$68$	−75.0432	−0.133828
$69$	0	0
$70$	0	0
$71$	−921.274	−1.53993	−0.769966	−	0.638085i	$-0.779727\pi$
$71$	−921.274	−1.53993	−0.769966	+	0.638085i	$0.779727\pi$
$72$	0	0
$73$	219.934	0.352621	0.176310	−	0.984335i	$-0.443584\pi$
$73$	219.934	0.352621	0.176310	+	0.984335i	$0.443584\pi$
$74$	−399.373	−0.627381
$75$	0	0
$76$	511.088	0.771392
$77$	0	0
$78$	0	0
$79$	−9.42805	−0.0134271	−0.00671354	−	0.999977i	$-0.502137\pi$
$79$	−9.42805	−0.0134271	−0.00671354	+	0.999977i	$0.502137\pi$
$80$	118.055	0.164986
$81$	0	0
$82$	−444.866	−0.599112
$83$	−800.364	−1.05845	−0.529225	−	0.848481i	$-0.677517\pi$
$83$	−800.364	−1.05845	−0.529225	+	0.848481i	$0.677517\pi$
$84$	0	0
$85$	131.626	0.167963
$86$	−7.46689	−0.00936250
$87$	0	0
$88$	−1488.85	−1.80354
$89$	−253.574	−0.302008	−0.151004	−	0.988533i	$-0.548251\pi$
$89$	−253.574	−0.302008	−0.151004	+	0.988533i	$0.548251\pi$
$90$	0	0
$91$	0	0
$92$	−636.436	−0.721229
$93$	0	0
$94$	945.560	1.03752
$95$	−896.453	−0.968149
$96$	0	0
$97$	59.2529	0.0620229	0.0310114	−	0.999519i	$-0.490127\pi$
$97$	59.2529	0.0620229	0.0310114	+	0.999519i	$0.490127\pi$
$98$	0	0
$99$	0	0
$100$	160.794	0.160794
$101$	−683.043	−0.672924	−0.336462	−	0.941697i	$-0.609230\pi$
$101$	−683.043	−0.672924	−0.336462	+	0.941697i	$0.609230\pi$
$102$	0	0
$103$	520.537	0.497962	0.248981	−	0.968508i	$-0.419904\pi$
$103$	520.537	0.497962	0.248981	+	0.968508i	$0.419904\pi$
$104$	−1171.64	−1.10470
$105$	0	0
$106$	−1072.13	−0.982403
$107$	−714.428	−0.645480	−0.322740	−	0.946488i	$-0.604604\pi$
$107$	−714.428	−0.645480	−0.322740	+	0.946488i	$0.604604\pi$
$108$	0	0
$109$	−182.802	−0.160635	−0.0803175	−	0.996769i	$-0.525593\pi$
$109$	−182.802	−0.160635	−0.0803175	+	0.996769i	$0.525593\pi$
$110$	1074.56	0.931408
$111$	0	0
$112$	0	0
$113$	154.433	0.128565	0.0642825	−	0.997932i	$-0.479524\pi$
$113$	154.433	0.128565	0.0642825	+	0.997932i	$0.479524\pi$
$114$	0	0
$115$	1116.32	0.905191
$116$	−69.8942	−0.0559441
$117$	0	0
$118$	−338.089	−0.263760
$119$	0	0
$120$	0	0
$121$	3653.69	2.74507
$122$	−1186.37	−0.880402
$123$	0	0
$124$	1246.96	0.903064
$125$	−1508.39	−1.07932
$126$	0	0
$127$	2072.00	1.44772	0.723858	−	0.689949i	$-0.242367\pi$
$127$	2072.00	1.44772	0.723858	+	0.689949i	$0.242367\pi$
$128$	1197.37	0.826823
$129$	0	0
$130$	845.619	0.570505
$131$	25.5712	0.0170547	0.00852736	−	0.999964i	$-0.497286\pi$
$131$	25.5712	0.0170547	0.00852736	+	0.999964i	$0.497286\pi$
$132$	0	0
$133$	0	0
$134$	152.881	0.0985591
$135$	0	0
$136$	282.922	0.178385
$137$	581.202	0.362448	0.181224	−	0.983442i	$-0.441994\pi$
$137$	581.202	0.362448	0.181224	+	0.983442i	$0.441994\pi$
$138$	0	0
$139$	−1185.01	−0.723103	−0.361552	−	0.932352i	$-0.617753\pi$
$139$	−1185.01	−0.723103	−0.361552	+	0.932352i	$0.617753\pi$
$140$	0	0
$141$	0	0
$142$	1429.20	0.844616
$143$	3922.69	2.29393
$144$	0	0
$145$	122.595	0.0702136
$146$	−341.189	−0.193404
$147$	0	0
$148$	−1439.96	−0.799756
$149$	471.892	0.259456	0.129728	−	0.991550i	$-0.458590\pi$
$149$	471.892	0.259456	0.129728	+	0.991550i	$0.458590\pi$
$150$	0	0
$151$	2635.18	1.42019	0.710093	−	0.704108i	$-0.248653\pi$
$151$	2635.18	1.42019	0.710093	+	0.704108i	$0.248653\pi$
$152$	−1926.87	−1.02822
$153$	0	0
$154$	0	0
$155$	−2187.17	−1.13341
$156$	0	0
$157$	359.210	0.182599	0.0912996	−	0.995823i	$-0.470898\pi$
$157$	359.210	0.182599	0.0912996	+	0.995823i	$0.470898\pi$
$158$	14.6260	0.00736444
$159$	0	0
$160$	−1838.25	−0.908292
$161$	0	0
$162$	0	0
$163$	2659.91	1.27816	0.639081	−	0.769140i	$-0.279315\pi$
$163$	2659.91	1.27816	0.639081	+	0.769140i	$0.279315\pi$
$164$	−1603.99	−0.763722
$165$	0	0
$166$	1241.63	0.580536
$167$	−702.648	−0.325584	−0.162792	−	0.986660i	$-0.552050\pi$
$167$	−702.648	−0.325584	−0.162792	+	0.986660i	$0.552050\pi$
$168$	0	0
$169$	889.949	0.405075
$170$	−204.196	−0.0921240
$171$	0	0
$172$	−26.9222	−0.0119349
$173$	738.814	0.324688	0.162344	−	0.986734i	$-0.448095\pi$
$173$	738.814	0.324688	0.162344	+	0.986734i	$0.448095\pi$
$174$	0	0
$175$	0	0
$176$	−849.563	−0.363853
$177$	0	0
$178$	393.375	0.165645
$179$	3011.14	1.25734	0.628668	−	0.777674i	$-0.283600\pi$
$179$	3011.14	1.25734	0.628668	+	0.777674i	$0.283600\pi$
$180$	0	0
$181$	−204.288	−0.0838928	−0.0419464	−	0.999120i	$-0.513356\pi$
$181$	−204.288	−0.0838928	−0.0419464	+	0.999120i	$0.513356\pi$
$182$	0	0
$183$	0	0
$184$	2399.45	0.961356
$185$	2525.70	1.00375
$186$	0	0
$187$	−947.230	−0.370419
$188$	3409.27	1.32259
$189$	0	0
$190$	1390.69	0.531007
$191$	3802.34	1.44046	0.720229	−	0.693736i	$-0.244037\pi$
$191$	3802.34	1.44046	0.720229	+	0.693736i	$0.244037\pi$
$192$	0	0
$193$	−3085.69	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$193$	−3085.69	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$194$	−91.9206	−0.0340181
$195$	0	0
$196$	0	0
$197$	2710.10	0.980134	0.490067	−	0.871685i	$-0.336972\pi$
$197$	2710.10	0.980134	0.490067	+	0.871685i	$0.336972\pi$
$198$	0	0
$199$	4807.21	1.71243	0.856217	−	0.516616i	$-0.172809\pi$
$199$	4807.21	1.71243	0.856217	+	0.516616i	$0.172809\pi$
$200$	−606.212	−0.214328
$201$	0	0
$202$	1059.62	0.369083
$203$	0	0
$204$	0	0
$205$	2813.41	0.958522
$206$	−807.523	−0.273120
$207$	0	0
$208$	−668.561	−0.222867
$209$	6451.19	2.13511
$210$	0	0
$211$	5648.81	1.84303	0.921517	−	0.388338i	$-0.126951\pi$
$211$	5648.81	1.84303	0.921517	+	0.388338i	$0.126951\pi$
$212$	−3865.63	−1.25232
$213$	0	0
$214$	1108.31	0.354031
$215$	47.2219	0.0149791
$216$	0	0
$217$	0	0
$218$	283.585	0.0881046
$219$	0	0
$220$	3874.37	1.18732
$221$	−745.420	−0.226889
$222$	0	0
$223$	−4595.61	−1.38002	−0.690011	−	0.723799i	$-0.742394\pi$
$223$	−4595.61	−1.38002	−0.690011	+	0.723799i	$0.742394\pi$
$224$	0	0
$225$	0	0
$226$	−239.576	−0.0705150
$227$	−1856.53	−0.542829	−0.271414	−	0.962463i	$-0.587491\pi$
$227$	−1856.53	−0.542829	−0.271414	+	0.962463i	$0.587491\pi$
$228$	0	0
$229$	3830.37	1.10532	0.552659	−	0.833408i	$-0.313613\pi$
$229$	3830.37	1.10532	0.552659	+	0.833408i	$0.313613\pi$
$230$	−1731.77	−0.496476
$231$	0	0
$232$	263.510	0.0745702
$233$	−2244.80	−0.631166	−0.315583	−	0.948898i	$-0.602200\pi$
$233$	−2244.80	−0.631166	−0.315583	+	0.948898i	$0.602200\pi$
$234$	0	0
$235$	−5979.89	−1.65994
$236$	−1219.00	−0.336229
$237$	0	0
$238$	0	0
$239$	3184.89	0.861980	0.430990	−	0.902357i	$-0.358164\pi$
$239$	3184.89	0.861980	0.430990	+	0.902357i	$0.358164\pi$
$240$	0	0
$241$	−1607.14	−0.429564	−0.214782	−	0.976662i	$-0.568904\pi$
$241$	−1607.14	−0.429564	−0.214782	+	0.976662i	$0.568904\pi$
$242$	−5668.06	−1.50561
$243$	0	0
$244$	−4277.53	−1.12230
$245$	0	0
$246$	0	0
$247$	5076.74	1.30780
$248$	−4701.18	−1.20373
$249$	0	0
$250$	2340.01	0.591980
$251$	−3398.32	−0.854582	−0.427291	−	0.904114i	$-0.640532\pi$
$251$	−3398.32	−0.854582	−0.427291	+	0.904114i	$0.640532\pi$
$252$	0	0
$253$	−8033.39	−1.99627
$254$	−3214.34	−0.794039
$255$	0	0
$256$	−3412.76	−0.833193
$257$	3127.70	0.759147	0.379573	−	0.925162i	$-0.376071\pi$
$257$	3127.70	0.759147	0.379573	+	0.925162i	$0.376071\pi$
$258$	0	0
$259$	0	0
$260$	3048.92	0.727255
$261$	0	0
$262$	−39.6693	−0.00935412
$263$	−4470.12	−1.04806	−0.524030	−	0.851700i	$-0.675572\pi$
$263$	−4470.12	−1.04806	−0.524030	+	0.851700i	$0.675572\pi$
$264$	0	0
$265$	6780.35	1.57175
$266$	0	0
$267$	0	0
$268$	551.221	0.125639
$269$	−3021.24	−0.684789	−0.342394	−	0.939556i	$-0.611238\pi$
$269$	−3021.24	−0.684789	−0.342394	+	0.939556i	$0.611238\pi$
$270$	0	0
$271$	6635.95	1.48747	0.743736	−	0.668473i	$-0.233052\pi$
$271$	6635.95	1.48747	0.743736	+	0.668473i	$0.233052\pi$
$272$	161.441	0.0359881
$273$	0	0
$274$	−901.634	−0.198794
$275$	2029.61	0.445055
$276$	0	0
$277$	−2223.61	−0.482324	−0.241162	−	0.970485i	$-0.577529\pi$
$277$	−2223.61	−0.482324	−0.241162	+	0.970485i	$0.577529\pi$
$278$	1838.34	0.396605
$279$	0	0
$280$	0	0
$281$	6027.19	1.27954	0.639772	−	0.768565i	$-0.279029\pi$
$281$	6027.19	1.27954	0.639772	+	0.768565i	$0.279029\pi$
$282$	0	0
$283$	769.183	0.161566	0.0807830	−	0.996732i	$-0.474258\pi$
$283$	769.183	0.161566	0.0807830	+	0.996732i	$0.474258\pi$
$284$	5153.04	1.07668
$285$	0	0
$286$	−6085.37	−1.25817
$287$	0	0
$288$	0	0
$289$	−4733.00	−0.963363
$290$	−190.185	−0.0385105
$291$	0	0
$292$	−1230.18	−0.246543
$293$	5232.52	1.04330	0.521650	−	0.853160i	$-0.325317\pi$
$293$	5232.52	1.04330	0.521650	+	0.853160i	$0.325317\pi$
$294$	0	0
$295$	2138.14	0.421990
$296$	5428.83	1.06603
$297$	0	0
$298$	−732.059	−0.142305
$299$	−6321.86	−1.22275
$300$	0	0
$301$	0	0
$302$	−4088.03	−0.778939
$303$	0	0
$304$	−1099.50	−0.207437
$305$	7502.82	1.40856
$306$	0	0
$307$	1705.68	0.317096	0.158548	−	0.987351i	$-0.449319\pi$
$307$	1705.68	0.317096	0.158548	+	0.987351i	$0.449319\pi$
$308$	0	0
$309$	0	0
$310$	3393.02	0.621647
$311$	7220.87	1.31659	0.658293	−	0.752762i	$-0.271279\pi$
$311$	7220.87	1.31659	0.658293	+	0.752762i	$0.271279\pi$
$312$	0	0
$313$	9173.62	1.65663	0.828313	−	0.560266i	$-0.189301\pi$
$313$	9173.62	1.65663	0.828313	+	0.560266i	$0.189301\pi$
$314$	−557.252	−0.100151
$315$	0	0
$316$	52.7347	0.00938785
$317$	−1082.48	−0.191793	−0.0958965	−	0.995391i	$-0.530572\pi$
$317$	−1082.48	−0.191793	−0.0958965	+	0.995391i	$0.530572\pi$
$318$	0	0
$319$	−882.238	−0.154846
$320$	1907.30	0.333191
$321$	0	0
$322$	0	0
$323$	−1225.90	−0.211180
$324$	0	0
$325$	1597.20	0.272605
$326$	−4126.39	−0.701042
$327$	0	0
$328$	6047.23	1.01800
$329$	0	0
$330$	0	0
$331$	−1753.27	−0.291143	−0.145572	−	0.989348i	$-0.546502\pi$
$331$	−1753.27	−0.291143	−0.145572	+	0.989348i	$0.546502\pi$
$332$	4476.75	0.740041
$333$	0	0
$334$	1090.04	0.178575
$335$	−966.847	−0.157685
$336$	0	0
$337$	8918.33	1.44158	0.720790	−	0.693153i	$-0.243779\pi$
$337$	8918.33	1.44158	0.720790	+	0.693153i	$0.243779\pi$
$338$	−1380.60	−0.222174
$339$	0	0
$340$	−736.238	−0.117436
$341$	15739.7	2.49956
$342$	0	0
$343$	0	0
$344$	101.500	0.0159085
$345$	0	0
$346$	−1146.14	−0.178084
$347$	6811.49	1.05377	0.526887	−	0.849935i	$-0.323359\pi$
$347$	6811.49	1.05377	0.526887	+	0.849935i	$0.323359\pi$
$348$	0	0
$349$	−9254.60	−1.41945	−0.709724	−	0.704480i	$-0.751180\pi$
$349$	−9254.60	−1.41945	−0.709724	+	0.704480i	$0.751180\pi$
$350$	0	0
$351$	0	0
$352$	13228.7	2.00311
$353$	−8005.75	−1.20709	−0.603546	−	0.797329i	$-0.706246\pi$
$353$	−8005.75	−1.20709	−0.603546	+	0.797329i	$0.706246\pi$
$354$	0	0
$355$	−9038.49	−1.35130
$356$	1418.34	0.211156
$357$	0	0
$358$	−4671.26	−0.689619
$359$	−6611.34	−0.971958	−0.485979	−	0.873970i	$-0.661537\pi$
$359$	−6611.34	−0.971958	−0.485979	+	0.873970i	$0.661537\pi$
$360$	0	0
$361$	1490.12	0.217251
$362$	316.917	0.0460132
$363$	0	0
$364$	0	0
$365$	2157.74	0.309428
$366$	0	0
$367$	−1836.21	−0.261170	−0.130585	−	0.991437i	$-0.541686\pi$
$367$	−1836.21	−0.261170	−0.130585	+	0.991437i	$0.541686\pi$
$368$	1369.17	0.193948
$369$	0	0
$370$	−3918.19	−0.550533
$371$	0	0
$372$	0	0
$373$	−4819.81	−0.669062	−0.334531	−	0.942385i	$-0.608578\pi$
$373$	−4819.81	−0.669062	−0.334531	+	0.942385i	$0.608578\pi$
$374$	1469.46	0.203166
$375$	0	0
$376$	−12853.4	−1.76293
$377$	−694.275	−0.0948461
$378$	0	0
$379$	6710.00	0.909418	0.454709	−	0.890640i	$-0.349743\pi$
$379$	6710.00	0.909418	0.454709	+	0.890640i	$0.349743\pi$
$380$	5014.21	0.676904
$381$	0	0
$382$	−5898.67	−0.790058
$383$	−11584.9	−1.54559	−0.772797	−	0.634653i	$-0.781143\pi$
$383$	−11584.9	−1.54559	−0.772797	+	0.634653i	$0.781143\pi$
$384$	0	0
$385$	0	0
$386$	4786.91	0.631211
$387$	0	0
$388$	−331.424	−0.0433648
$389$	7523.59	0.980620	0.490310	−	0.871548i	$-0.336884\pi$
$389$	7523.59	0.980620	0.490310	+	0.871548i	$0.336884\pi$
$390$	0	0
$391$	1526.57	0.197447
$392$	0	0
$393$	0	0
$394$	−4204.24	−0.537581
$395$	−92.4973	−0.0117824
$396$	0	0
$397$	3302.26	0.417471	0.208735	−	0.977972i	$-0.433065\pi$
$397$	3302.26	0.417471	0.208735	+	0.977972i	$0.433065\pi$
$398$	−7457.56	−0.939230
$399$	0	0
$400$	−345.916	−0.0432395
$401$	6996.80	0.871331	0.435665	−	0.900109i	$-0.356513\pi$
$401$	6996.80	0.871331	0.435665	+	0.900109i	$0.356513\pi$
$402$	0	0
$403$	12386.3	1.53103
$404$	3820.52	0.470491
$405$	0	0
$406$	0	0
$407$	−18175.8	−2.21362
$408$	0	0
$409$	4534.39	0.548193	0.274097	−	0.961702i	$-0.411621\pi$
$409$	4534.39	0.548193	0.274097	+	0.961702i	$0.411621\pi$
$410$	−4364.52	−0.525727
$411$	0	0
$412$	−2911.57	−0.348161
$413$	0	0
$414$	0	0
$415$	−7852.26	−0.928801
$416$	10410.3	1.22694
$417$	0	0
$418$	−10007.9	−1.17106
$419$	1497.50	0.174601	0.0873004	−	0.996182i	$-0.472176\pi$
$419$	1497.50	0.174601	0.0873004	+	0.996182i	$0.472176\pi$
$420$	0	0
$421$	−4669.50	−0.540564	−0.270282	−	0.962781i	$-0.587117\pi$
$421$	−4669.50	−0.540564	−0.270282	+	0.962781i	$0.587117\pi$
$422$	−8763.15	−1.01086
$423$	0	0
$424$	14573.9	1.66927
$425$	−385.683	−0.0440197
$426$	0	0
$427$	0	0
$428$	3996.07	0.451302
$429$	0	0
$430$	−73.2565	−0.00821568
$431$	−11985.2	−1.33946	−0.669732	−	0.742603i	$-0.733591\pi$
$431$	−11985.2	−1.33946	−0.669732	+	0.742603i	$0.733591\pi$
$432$	0	0
$433$	8226.59	0.913036	0.456518	−	0.889714i	$-0.349096\pi$
$433$	8226.59	0.913036	0.456518	+	0.889714i	$0.349096\pi$
$434$	0	0
$435$	0	0
$436$	1022.48	0.112312
$437$	−10396.8	−1.13809
$438$	0	0
$439$	8154.12	0.886503	0.443251	−	0.896397i	$-0.353825\pi$
$439$	8154.12	0.886503	0.443251	+	0.896397i	$0.353825\pi$
$440$	−14606.9	−1.58262
$441$	0	0
$442$	1156.39	0.124443
$443$	−11406.0	−1.22329	−0.611644	−	0.791133i	$-0.709492\pi$
$443$	−11406.0	−1.22329	−0.611644	+	0.791133i	$0.709492\pi$
$444$	0	0
$445$	−2487.77	−0.265015
$446$	7129.29	0.756910
$447$	0	0
$448$	0	0
$449$	2463.59	0.258940	0.129470	−	0.991583i	$-0.458672\pi$
$449$	2463.59	0.258940	0.129470	+	0.991583i	$0.458672\pi$
$450$	0	0
$451$	−20246.3	−2.11388
$452$	−863.805	−0.0898893
$453$	0	0
$454$	2880.08	0.297729
$455$	0	0
$456$	0	0
$457$	−16349.9	−1.67356	−0.836778	−	0.547542i	$-0.815564\pi$
$457$	−16349.9	−1.67356	−0.836778	+	0.547542i	$0.815564\pi$
$458$	−5942.15	−0.606241
$459$	0	0
$460$	−6243.98	−0.632885
$461$	2872.86	0.290244	0.145122	−	0.989414i	$-0.453643\pi$
$461$	2872.86	0.290244	0.145122	+	0.989414i	$0.453643\pi$
$462$	0	0
$463$	2984.20	0.299541	0.149771	−	0.988721i	$-0.452146\pi$
$463$	2984.20	0.299541	0.149771	+	0.988721i	$0.452146\pi$
$464$	150.364	0.0150441
$465$	0	0
$466$	3482.42	0.346180
$467$	−19419.5	−1.92426	−0.962129	−	0.272595i	$-0.912118\pi$
$467$	−19419.5	−1.92426	−0.962129	+	0.272595i	$0.912118\pi$
$468$	0	0
$469$	0	0
$470$	9276.76	0.910436
$471$	0	0
$472$	4595.78	0.448174
$473$	−339.825	−0.0330342
$474$	0	0
$475$	2626.72	0.253731
$476$	0	0
$477$	0	0
$478$	−4940.80	−0.472776
$479$	−18229.3	−1.73887	−0.869435	−	0.494047i	$-0.835517\pi$
$479$	−18229.3	−1.73887	−0.869435	+	0.494047i	$0.835517\pi$
$480$	0	0
$481$	−14303.4	−1.35588
$482$	2493.20	0.235606
$483$	0	0
$484$	−20436.5	−1.91928
$485$	581.322	0.0544257
$486$	0	0
$487$	17079.6	1.58922	0.794609	−	0.607122i	$-0.207676\pi$
$487$	17079.6	1.58922	0.794609	+	0.607122i	$0.207676\pi$
$488$	16126.8	1.49596
$489$	0	0
$490$	0	0
$491$	1087.13	0.0999214	0.0499607	−	0.998751i	$-0.484090\pi$
$491$	1087.13	0.0999214	0.0499607	+	0.998751i	$0.484090\pi$
$492$	0	0
$493$	167.650	0.0153155
$494$	−7875.69	−0.717295
$495$	0	0
$496$	−2682.58	−0.242845
$497$	0	0
$498$	0	0
$499$	18785.8	1.68530	0.842652	−	0.538459i	$-0.180993\pi$
$499$	18785.8	1.68530	0.842652	+	0.538459i	$0.180993\pi$
$500$	8437.01	0.754629
$501$	0	0
$502$	5271.91	0.468719
$503$	12278.7	1.08843	0.544214	−	0.838947i	$-0.316828\pi$
$503$	12278.7	1.08843	0.544214	+	0.838947i	$0.316828\pi$
$504$	0	0
$505$	−6701.24	−0.590497
$506$	12462.4	1.09490
$507$	0	0
$508$	−11589.5	−1.01220
$509$	−1226.83	−0.106834	−0.0534168	−	0.998572i	$-0.517011\pi$
$509$	−1226.83	−0.106834	−0.0534168	+	0.998572i	$0.517011\pi$
$510$	0	0
$511$	0	0
$512$	−4284.63	−0.369836
$513$	0	0
$514$	−4852.09	−0.416374
$515$	5106.91	0.436966
$516$	0	0
$517$	43033.3	3.66074
$518$	0	0
$519$	0	0
$520$	−11494.8	−0.969388
$521$	12571.3	1.05712	0.528559	−	0.848897i	$-0.322733\pi$
$521$	12571.3	1.05712	0.528559	+	0.848897i	$0.322733\pi$
$522$	0	0
$523$	7828.18	0.654498	0.327249	−	0.944938i	$-0.393878\pi$
$523$	7828.18	0.654498	0.327249	+	0.944938i	$0.393878\pi$
$524$	−143.030	−0.0119242
$525$	0	0
$526$	6934.62	0.574836
$527$	−2990.97	−0.247227
$528$	0	0
$529$	779.727	0.0640854
$530$	−10518.5	−0.862068
$531$	0	0
$532$	0	0
$533$	−15932.7	−1.29479
$534$	0	0
$535$	−7009.15	−0.566415
$536$	−2078.17	−0.167469
$537$	0	0
$538$	4686.93	0.375591
$539$	0	0
$540$	0	0
$541$	−5726.90	−0.455118	−0.227559	−	0.973764i	$-0.573074\pi$
$541$	−5726.90	−0.455118	−0.227559	+	0.973764i	$0.573074\pi$
$542$	−10294.5	−0.815844
$543$	0	0
$544$	−2513.82	−0.198124
$545$	−1793.44	−0.140959
$546$	0	0
$547$	−7738.41	−0.604882	−0.302441	−	0.953168i	$-0.597801\pi$
$547$	−7738.41	−0.604882	−0.302441	+	0.953168i	$0.597801\pi$
$548$	−3250.89	−0.253414
$549$	0	0
$550$	−3148.59	−0.244102
$551$	−1141.79	−0.0882795
$552$	0	0
$553$	0	0
$554$	3449.55	0.264544
$555$	0	0
$556$	6628.23	0.505575
$557$	6714.94	0.510810	0.255405	−	0.966834i	$-0.417791\pi$
$557$	6714.94	0.510810	0.255405	+	0.966834i	$0.417791\pi$
$558$	0	0
$559$	−267.424	−0.0202341
$560$	0	0
$561$	0	0
$562$	−9350.13	−0.701800
$563$	25016.6	1.87269	0.936343	−	0.351087i	$-0.114188\pi$
$563$	25016.6	1.87269	0.936343	+	0.351087i	$0.114188\pi$
$564$	0	0
$565$	1515.12	0.112817
$566$	−1193.25	−0.0886152
$567$	0	0
$568$	−19427.6	−1.43515
$569$	11748.3	0.865578	0.432789	−	0.901495i	$-0.357530\pi$
$569$	11748.3	0.865578	0.432789	+	0.901495i	$0.357530\pi$
$570$	0	0
$571$	2168.65	0.158941	0.0794705	−	0.996837i	$-0.474677\pi$
$571$	2168.65	0.158941	0.0794705	+	0.996837i	$0.474677\pi$
$572$	−21941.1	−1.60385
$573$	0	0
$574$	0	0
$575$	−3270.95	−0.237231
$576$	0	0
$577$	−3941.26	−0.284362	−0.142181	−	0.989841i	$-0.545411\pi$
$577$	−3941.26	−0.284362	−0.142181	+	0.989841i	$0.545411\pi$
$578$	7342.43	0.528382
$579$	0	0
$580$	−685.722	−0.0490915
$581$	0	0
$582$	0	0
$583$	−48793.8	−3.46626
$584$	4637.92	0.328627
$585$	0	0
$586$	−8117.34	−0.572226
$587$	13873.7	0.975521	0.487760	−	0.872978i	$-0.337814\pi$
$587$	13873.7	0.975521	0.487760	+	0.872978i	$0.337814\pi$
$588$	0	0
$589$	20370.3	1.42503
$590$	−3316.95	−0.231452
$591$	0	0
$592$	3097.79	0.215065
$593$	−9754.07	−0.675467	−0.337733	−	0.941242i	$-0.609660\pi$
$593$	−9754.07	−0.675467	−0.337733	+	0.941242i	$0.609660\pi$
$594$	0	0
$595$	0	0
$596$	−2639.48	−0.181405
$597$	0	0
$598$	9807.26	0.670650
$599$	−15994.0	−1.09098	−0.545491	−	0.838116i	$-0.683657\pi$
$599$	−15994.0	−1.09098	−0.545491	+	0.838116i	$0.683657\pi$
$600$	0	0
$601$	−15183.7	−1.03054	−0.515272	−	0.857027i	$-0.672309\pi$
$601$	−15183.7	−1.03054	−0.515272	+	0.857027i	$0.672309\pi$
$602$	0	0
$603$	0	0
$604$	−14739.6	−0.992957
$605$	35845.8	2.40883
$606$	0	0
$607$	−18747.2	−1.25358	−0.626792	−	0.779187i	$-0.715632\pi$
$607$	−18747.2	−1.25358	−0.626792	+	0.779187i	$0.715632\pi$
$608$	17120.6	1.14199
$609$	0	0
$610$	−11639.3	−0.772562
$611$	33865.0	2.24228
$612$	0	0
$613$	−460.802	−0.0303615	−0.0151808	−	0.999885i	$-0.504832\pi$
$613$	−460.802	−0.0303615	−0.0151808	+	0.999885i	$0.504832\pi$
$614$	−2646.07	−0.173920
$615$	0	0
$616$	0	0
$617$	−27174.7	−1.77311	−0.886557	−	0.462620i	$-0.846910\pi$
$617$	−27174.7	−1.77311	−0.886557	+	0.462620i	$0.846910\pi$
$618$	0	0
$619$	−29663.1	−1.92611	−0.963055	−	0.269305i	$-0.913206\pi$
$619$	−29663.1	−1.92611	−0.963055	+	0.269305i	$0.913206\pi$
$620$	12233.7	0.792447
$621$	0	0
$622$	−11201.9	−0.722117
$623$	0	0
$624$	0	0
$625$	−11205.2	−0.717134
$626$	−14231.3	−0.908621
$627$	0	0
$628$	−2009.20	−0.127669
$629$	3453.92	0.218945
$630$	0	0
$631$	8710.14	0.549517	0.274758	−	0.961513i	$-0.411402\pi$
$631$	8710.14	0.549517	0.274758	+	0.961513i	$0.411402\pi$
$632$	−198.817	−0.0125135
$633$	0	0
$634$	1679.29	0.105194
$635$	20328.1	1.27038
$636$	0	0
$637$	0	0
$638$	1368.64	0.0849293
$639$	0	0
$640$	11747.2	0.725545
$641$	−14429.7	−0.889138	−0.444569	−	0.895745i	$-0.646643\pi$
$641$	−14429.7	−0.889138	−0.444569	+	0.895745i	$0.646643\pi$
$642$	0	0
$643$	25865.5	1.58637	0.793184	−	0.608982i	$-0.208422\pi$
$643$	25865.5	1.58637	0.793184	+	0.608982i	$0.208422\pi$
$644$	0	0
$645$	0	0
$646$	1901.78	0.115827
$647$	−5059.78	−0.307451	−0.153725	−	0.988114i	$-0.549127\pi$
$647$	−5059.78	−0.307451	−0.153725	+	0.988114i	$0.549127\pi$
$648$	0	0
$649$	−15386.8	−0.930637
$650$	−2477.77	−0.149517
$651$	0	0
$652$	−14877.9	−0.893656
$653$	15389.9	0.922287	0.461143	−	0.887326i	$-0.347439\pi$
$653$	15389.9	0.922287	0.461143	+	0.887326i	$0.347439\pi$
$654$	0	0
$655$	250.876	0.0149657
$656$	3450.66	0.205375
$657$	0	0
$658$	0	0
$659$	30375.3	1.79553	0.897765	−	0.440474i	$-0.145190\pi$
$659$	30375.3	1.79553	0.897765	+	0.440474i	$0.145190\pi$
$660$	0	0
$661$	−12410.4	−0.730270	−0.365135	−	0.930955i	$-0.618977\pi$
$661$	−12410.4	−0.730270	−0.365135	+	0.930955i	$0.618977\pi$
$662$	2719.89	0.159685
$663$	0	0
$664$	−16877.9	−0.986431
$665$	0	0
$666$	0	0
$667$	1421.83	0.0825387
$668$	3930.18	0.227640
$669$	0	0
$670$	1499.90	0.0864866
$671$	−53992.9	−3.10637
$672$	0	0
$673$	23632.8	1.35361	0.676804	−	0.736163i	$-0.263364\pi$
$673$	23632.8	1.35361	0.676804	+	0.736163i	$0.263364\pi$
$674$	−13835.2	−0.790673
$675$	0	0
$676$	−4977.83	−0.283217
$677$	−7460.16	−0.423511	−0.211756	−	0.977323i	$-0.567918\pi$
$677$	−7460.16	−0.423511	−0.211756	+	0.977323i	$0.567918\pi$
$678$	0	0
$679$	0	0
$680$	2775.71	0.156535
$681$	0	0
$682$	−24417.3	−1.37095
$683$	7798.73	0.436911	0.218455	−	0.975847i	$-0.429898\pi$
$683$	7798.73	0.436911	0.218455	+	0.975847i	$0.429898\pi$
$684$	0	0
$685$	5702.09	0.318052
$686$	0	0
$687$	0	0
$688$	57.9179	0.00320945
$689$	−38398.1	−2.12315
$690$	0	0
$691$	−4748.63	−0.261427	−0.130714	−	0.991420i	$-0.541727\pi$
$691$	−4748.63	−0.261427	−0.130714	+	0.991420i	$0.541727\pi$
$692$	−4132.47	−0.227013
$693$	0	0
$694$	−10566.8	−0.577971
$695$	−11626.0	−0.634530
$696$	0	0
$697$	3847.35	0.209080
$698$	14356.9	0.778534
$699$	0	0
$700$	0	0
$701$	−13784.1	−0.742681	−0.371341	−	0.928497i	$-0.621102\pi$
$701$	−13784.1	−0.742681	−0.371341	+	0.928497i	$0.621102\pi$
$702$	0	0
$703$	−23523.2	−1.26201
$704$	−13725.6	−0.734803
$705$	0	0
$706$	12419.5	0.662061
$707$	0	0
$708$	0	0
$709$	577.799	0.0306060	0.0153030	−	0.999883i	$-0.495129\pi$
$709$	577.799	0.0306060	0.0153030	+	0.999883i	$0.495129\pi$
$710$	14021.6	0.741159
$711$	0	0
$712$	−5347.30	−0.281459
$713$	−25366.2	−1.33236
$714$	0	0
$715$	38484.9	2.01294
$716$	−16842.5	−0.879096
$717$	0	0
$718$	10256.3	0.533097
$719$	20274.7	1.05162	0.525812	−	0.850601i	$-0.323761\pi$
$719$	20274.7	1.05162	0.525812	+	0.850601i	$0.323761\pi$
$720$	0	0
$721$	0	0
$722$	−2311.67	−0.119157
$723$	0	0
$724$	1142.66	0.0586556
$725$	−359.220	−0.0184015
$726$	0	0
$727$	−28861.8	−1.47239	−0.736193	−	0.676772i	$-0.763378\pi$
$727$	−28861.8	−1.47239	−0.736193	+	0.676772i	$0.763378\pi$
$728$	0	0
$729$	0	0
$730$	−3347.36	−0.169714
$731$	64.5762	0.00326736
$732$	0	0
$733$	26169.5	1.31868	0.659340	−	0.751844i	$-0.270836\pi$
$733$	26169.5	1.31868	0.659340	+	0.751844i	$0.270836\pi$
$734$	2848.56	0.143246
$735$	0	0
$736$	−21319.6	−1.06773
$737$	6957.77	0.347751
$738$	0	0
$739$	−8351.16	−0.415700	−0.207850	−	0.978161i	$-0.566647\pi$
$739$	−8351.16	−0.415700	−0.207850	+	0.978161i	$0.566647\pi$
$740$	−14127.2	−0.701794
$741$	0	0
$742$	0	0
$743$	20732.7	1.02370	0.511849	−	0.859075i	$-0.328961\pi$
$743$	20732.7	1.02370	0.511849	+	0.859075i	$0.328961\pi$
$744$	0	0
$745$	4629.67	0.227675
$746$	7477.09	0.366965
$747$	0	0
$748$	5298.22	0.258987
$749$	0	0
$750$	0	0
$751$	13055.4	0.634351	0.317176	−	0.948367i	$-0.397265\pi$
$751$	13055.4	0.634351	0.317176	+	0.948367i	$0.397265\pi$
$752$	−7334.36	−0.355661
$753$	0	0
$754$	1077.05	0.0520208
$755$	25853.4	1.24623
$756$	0	0
$757$	14425.9	0.692628	0.346314	−	0.938119i	$-0.387433\pi$
$757$	14425.9	0.692628	0.346314	+	0.938119i	$0.387433\pi$
$758$	−10409.4	−0.498794
$759$	0	0
$760$	−18904.2	−0.902273
$761$	5232.01	0.249225	0.124613	−	0.992205i	$-0.460231\pi$
$761$	5232.01	0.249225	0.124613	+	0.992205i	$0.460231\pi$
$762$	0	0
$763$	0	0
$764$	−21268.0	−1.00713
$765$	0	0
$766$	17972.0	0.847723
$767$	−12108.6	−0.570033
$768$	0	0
$769$	18570.2	0.870818	0.435409	−	0.900233i	$-0.356604\pi$
$769$	18570.2	0.870818	0.435409	+	0.900233i	$0.356604\pi$
$770$	0	0
$771$	0	0
$772$	17259.5	0.804639
$773$	−23462.7	−1.09171	−0.545857	−	0.837878i	$-0.683796\pi$
$773$	−23462.7	−1.09171	−0.545857	+	0.837878i	$0.683796\pi$
$774$	0	0
$775$	6408.70	0.297042
$776$	1249.51	0.0578027
$777$	0	0
$778$	−11671.5	−0.537847
$779$	−26202.7	−1.20515
$780$	0	0
$781$	65044.1	2.98010
$782$	−2368.21	−0.108295
$783$	0	0
$784$	0	0
$785$	3524.16	0.160233
$786$	0	0
$787$	−25626.0	−1.16070	−0.580348	−	0.814369i	$-0.697083\pi$
$787$	−25626.0	−1.16070	−0.580348	+	0.814369i	$0.697083\pi$
$788$	−15158.6	−0.685284
$789$	0	0
$790$	143.493	0.00646236
$791$	0	0
$792$	0	0
$793$	−42489.6	−1.90271
$794$	−5122.89	−0.228973
$795$	0	0
$796$	−26888.6	−1.19729
$797$	−12209.4	−0.542633	−0.271316	−	0.962490i	$-0.587459\pi$
$797$	−12209.4	−0.542633	−0.271316	+	0.962490i	$0.587459\pi$
$798$	0	0
$799$	−8177.53	−0.362078
$800$	5386.33	0.238044
$801$	0	0
$802$	−10854.3	−0.477905
$803$	−15527.8	−0.682398
$804$	0	0
$805$	0	0
$806$	−19215.2	−0.839733
$807$	0	0
$808$	−14403.9	−0.627136
$809$	7844.50	0.340912	0.170456	−	0.985365i	$-0.445476\pi$
$809$	7844.50	0.340912	0.170456	+	0.985365i	$0.445476\pi$
$810$	0	0
$811$	−7602.91	−0.329192	−0.164596	−	0.986361i	$-0.552632\pi$
$811$	−7602.91	−0.329192	−0.164596	+	0.986361i	$0.552632\pi$
$812$	0	0
$813$	0	0
$814$	28196.7	1.21412
$815$	26096.0	1.12160
$816$	0	0
$817$	−439.802	−0.0188332
$818$	−7034.31	−0.300671
$819$	0	0
$820$	−15736.5	−0.670173
$821$	−29146.5	−1.23900	−0.619501	−	0.784996i	$-0.712665\pi$
$821$	−29146.5	−1.23900	−0.619501	+	0.784996i	$0.712665\pi$
$822$	0	0
$823$	−22062.1	−0.934432	−0.467216	−	0.884143i	$-0.654743\pi$
$823$	−22062.1	−0.934432	−0.467216	+	0.884143i	$0.654743\pi$
$824$	10977.0	0.464079
$825$	0	0
$826$	0	0
$827$	−4571.98	−0.192241	−0.0961206	−	0.995370i	$-0.530643\pi$
$827$	−4571.98	−0.192241	−0.0961206	+	0.995370i	$0.530643\pi$
$828$	0	0
$829$	−10311.6	−0.432012	−0.216006	−	0.976392i	$-0.569303\pi$
$829$	−10311.6	−0.432012	−0.216006	+	0.976392i	$0.569303\pi$
$830$	12181.4	0.509426
$831$	0	0
$832$	−10801.3	−0.450081
$833$	0	0
$834$	0	0
$835$	−6893.57	−0.285703
$836$	−36084.0	−1.49281
$837$	0	0
$838$	−2323.11	−0.0957645
$839$	−3414.34	−0.140496	−0.0702480	−	0.997530i	$-0.522379\pi$
$839$	−3414.34	−0.140496	−0.0702480	+	0.997530i	$0.522379\pi$
$840$	0	0
$841$	−24232.9	−0.993598
$842$	7243.92	0.296487
$843$	0	0
$844$	−31596.0	−1.28860
$845$	8731.16	0.355457
$846$	0	0
$847$	0	0
$848$	8316.14	0.336766
$849$	0	0
$850$	598.320	0.0241438
$851$	29292.4	1.17994
$852$	0	0
$853$	−12093.2	−0.485419	−0.242709	−	0.970099i	$-0.578036\pi$
$853$	−12093.2	−0.485419	−0.242709	+	0.970099i	$0.578036\pi$
$854$	0	0
$855$	0	0
$856$	−15065.7	−0.601560
$857$	35003.7	1.39522	0.697610	−	0.716478i	$-0.254247\pi$
$857$	35003.7	1.39522	0.697610	+	0.716478i	$0.254247\pi$
$858$	0	0
$859$	1669.49	0.0663123	0.0331562	−	0.999450i	$-0.489444\pi$
$859$	1669.49	0.0663123	0.0331562	+	0.999450i	$0.489444\pi$
$860$	−264.130	−0.0104730
$861$	0	0
$862$	18593.0	0.734664
$863$	−1620.80	−0.0639312	−0.0319656	−	0.999489i	$-0.510177\pi$
$863$	−1620.80	−0.0639312	−0.0319656	+	0.999489i	$0.510177\pi$
$864$	0	0
$865$	7248.40	0.284917
$866$	−12762.1	−0.500779
$867$	0	0
$868$	0	0
$869$	665.642	0.0259843
$870$	0	0
$871$	5475.40	0.213004
$872$	−3854.88	−0.149705
$873$	0	0
$874$	16128.9	0.624218
$875$	0	0
$876$	0	0
$877$	−14449.2	−0.556346	−0.278173	−	0.960531i	$-0.589729\pi$
$877$	−14449.2	−0.556346	−0.278173	+	0.960531i	$0.589729\pi$
$878$	−12649.7	−0.486226
$879$	0	0
$880$	−8334.94	−0.319285
$881$	40111.6	1.53393	0.766966	−	0.641688i	$-0.221766\pi$
$881$	40111.6	1.53393	0.766966	+	0.641688i	$0.221766\pi$
$882$	0	0
$883$	10152.0	0.386911	0.193456	−	0.981109i	$-0.438030\pi$
$883$	10152.0	0.386911	0.193456	+	0.981109i	$0.438030\pi$
$884$	4169.42	0.158634
$885$	0	0
$886$	17694.5	0.670945
$887$	−18907.0	−0.715711	−0.357856	−	0.933777i	$-0.616492\pi$
$887$	−18907.0	−0.715711	−0.357856	+	0.933777i	$0.616492\pi$
$888$	0	0
$889$	0	0
$890$	3859.35	0.145355
$891$	0	0
$892$	25705.0	0.964875
$893$	55693.8	2.08703
$894$	0	0
$895$	29541.9	1.10332
$896$	0	0
$897$	0	0
$898$	−3821.83	−0.142022
$899$	−2785.75	−0.103348
$900$	0	0
$901$	9272.17	0.342842
$902$	31408.6	1.15941
$903$	0	0
$904$	3256.65	0.119817
$905$	−2004.24	−0.0736167
$906$	0	0
$907$	−6789.41	−0.248554	−0.124277	−	0.992248i	$-0.539661\pi$
$907$	−6789.41	−0.248554	−0.124277	+	0.992248i	$0.539661\pi$
$908$	10384.3	0.379531
$909$	0	0
$910$	0	0
$911$	45716.8	1.66264	0.831320	−	0.555794i	$-0.187586\pi$
$911$	45716.8	1.66264	0.831320	+	0.555794i	$0.187586\pi$
$912$	0	0
$913$	56507.6	2.04833
$914$	25364.0	0.917907
$915$	0	0
$916$	−21424.7	−0.772809
$917$	0	0
$918$	0	0
$919$	19232.7	0.690346	0.345173	−	0.938539i	$-0.387820\pi$
$919$	19232.7	0.690346	0.345173	+	0.938539i	$0.387820\pi$
$920$	23540.6	0.843599
$921$	0	0
$922$	−4456.74	−0.159192
$923$	51186.3	1.82537
$924$	0	0
$925$	−7400.64	−0.263061
$926$	−4629.47	−0.164291
$927$	0	0
$928$	−2341.34	−0.0828215
$929$	45139.4	1.59416	0.797081	−	0.603873i	$-0.206377\pi$
$929$	45139.4	1.59416	0.797081	+	0.603873i	$0.206377\pi$
$930$	0	0
$931$	0	0
$932$	12556.0	0.441295
$933$	0	0
$934$	30126.0	1.05541
$935$	−9293.13	−0.325046
$936$	0	0
$937$	3684.00	0.128443	0.0642215	−	0.997936i	$-0.479544\pi$
$937$	3684.00	0.128443	0.0642215	+	0.997936i	$0.479544\pi$
$938$	0	0
$939$	0	0
$940$	33447.8	1.16058
$941$	−3082.72	−0.106795	−0.0533973	−	0.998573i	$-0.517005\pi$
$941$	−3082.72	−0.106795	−0.0533973	+	0.998573i	$0.517005\pi$
$942$	0	0
$943$	32629.2	1.12678
$944$	2622.43	0.0904163
$945$	0	0
$946$	527.180	0.0181185
$947$	6475.69	0.222209	0.111104	−	0.993809i	$-0.464561\pi$
$947$	6475.69	0.222209	0.111104	+	0.993809i	$0.464561\pi$
$948$	0	0
$949$	−12219.6	−0.417982
$950$	−4074.91	−0.139166
$951$	0	0
$952$	0	0
$953$	−7643.58	−0.259811	−0.129906	−	0.991526i	$-0.541467\pi$
$953$	−7643.58	−0.259811	−0.129906	+	0.991526i	$0.541467\pi$
$954$	0	0
$955$	37304.2	1.26402
$956$	−17814.3	−0.602674
$957$	0	0
$958$	28279.6	0.953730
$959$	0	0
$960$	0	0
$961$	19908.5	0.668273
$962$	22189.3	0.743671
$963$	0	0
$964$	8989.36	0.300340
$965$	−30273.3	−1.00988
$966$	0	0
$967$	5636.41	0.187440	0.0937202	−	0.995599i	$-0.470124\pi$
$967$	5636.41	0.187440	0.0937202	+	0.995599i	$0.470124\pi$
$968$	77048.2	2.55829
$969$	0	0
$970$	−901.820	−0.0298512
$971$	−12541.0	−0.414479	−0.207240	−	0.978290i	$-0.566448\pi$
$971$	−12541.0	−0.414479	−0.207240	+	0.978290i	$0.566448\pi$
$972$	0	0
$973$	0	0
$974$	−26496.0	−0.871649
$975$	0	0
$976$	9202.25	0.301800
$977$	−45250.1	−1.48176	−0.740879	−	0.671638i	$-0.765591\pi$
$977$	−45250.1	−1.48176	−0.740879	+	0.671638i	$0.765591\pi$
$978$	0	0
$979$	17902.9	0.584452
$980$	0	0
$981$	0	0
$982$	−1686.49	−0.0548045
$983$	−48847.0	−1.58492	−0.792460	−	0.609923i	$-0.791200\pi$
$983$	−48847.0	−1.58492	−0.792460	+	0.609923i	$0.791200\pi$
$984$	0	0
$985$	26588.4	0.860077
$986$	−260.079	−0.00840022
$987$	0	0
$988$	−28396.2	−0.914376
$989$	547.667	0.0176085
$990$	0	0
$991$	−31801.6	−1.01939	−0.509693	−	0.860357i	$-0.670241\pi$
$991$	−31801.6	−1.01939	−0.509693	+	0.860357i	$0.670241\pi$
$992$	41771.0	1.33693
$993$	0	0
$994$	0	0
$995$	47162.9	1.50268
$996$	0	0
$997$	18382.5	0.583930	0.291965	−	0.956429i	$-0.405691\pi$
$997$	18382.5	0.583930	0.291965	+	0.956429i	$0.405691\pi$
$998$	−29142.8	−0.924349
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.ba.1.2		4
3.2	odd	2	inner	1323.4.a.ba.1.3		4
7.6	odd	2		189.4.a.l.1.2	✓	4
21.20	even	2		189.4.a.l.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.l.1.2	✓	4	7.6	odd	2
189.4.a.l.1.3	yes	4	21.20	even	2
1323.4.a.ba.1.2		4	1.1	even	1	trivial
1323.4.a.ba.1.3		4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.55133 q^{2} -5.59339 q^{4} +9.81086 q^{5} +21.0878 q^{8} +O(q^{10})\)	\(q-1.55133 q^{2} -5.59339 q^{4} +9.81086 q^{5} +21.0878 q^{8} -15.2198 q^{10} -70.6023 q^{11} -55.5603 q^{13} +12.0331 q^{16} +13.4164 q^{17} -91.3735 q^{19} -54.8759 q^{20} +109.527 q^{22} +113.784 q^{23} -28.7471 q^{25} +86.1922 q^{26} +12.4959 q^{29} -222.934 q^{31} -187.369 q^{32} -20.8132 q^{34} +257.440 q^{37} +141.750 q^{38} +206.889 q^{40} +286.765 q^{41} +4.81323 q^{43} +394.906 q^{44} -176.516 q^{46} -609.517 q^{47} +44.5961 q^{50} +310.770 q^{52} +691.107 q^{53} -692.669 q^{55} -19.3852 q^{58} +217.936 q^{59} +764.747 q^{61} +345.843 q^{62} +194.407 q^{64} -545.094 q^{65} -98.5487 q^{67} -75.0432 q^{68} -921.274 q^{71} +219.934 q^{73} -399.373 q^{74} +511.088 q^{76} -9.42805 q^{79} +118.055 q^{80} -444.866 q^{82} -800.364 q^{83} +131.626 q^{85} -7.46689 q^{86} -1488.85 q^{88} -253.574 q^{89} -636.436 q^{92} +945.560 q^{94} -896.453 q^{95} +59.2529 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 26 q^{4}+O(q^{10}) \) 4 * q + 26 * q^4	\( 4 q + 26 q^{4} - 206 q^{10} + 68 q^{13} + 290 q^{16} - 172 q^{19} - 94 q^{22} + 272 q^{25} - 408 q^{31} - 180 q^{34} + 1320 q^{37} - 1446 q^{40} + 116 q^{43} + 1374 q^{46} + 3952 q^{52} - 352 q^{55} - 1432 q^{58} + 2672 q^{61} + 826 q^{64} + 1444 q^{67} + 396 q^{73} + 1222 q^{76} + 1220 q^{79} + 3590 q^{82} + 720 q^{85} - 6294 q^{88} + 3492 q^{94} + 624 q^{97}+O(q^{100}) \) 4 * q + 26 * q^4 - 206 * q^10 + 68 * q^13 + 290 * q^16 - 172 * q^19 - 94 * q^22 + 272 * q^25 - 408 * q^31 - 180 * q^34 + 1320 * q^37 - 1446 * q^40 + 116 * q^43 + 1374 * q^46 + 3952 * q^52 - 352 * q^55 - 1432 * q^58 + 2672 * q^61 + 826 * q^64 + 1444 * q^67 + 396 * q^73 + 1222 * q^76 + 1220 * q^79 + 3590 * q^82 + 720 * q^85 - 6294 * q^88 + 3492 * q^94 + 624 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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