LMFDB - Embedded newform 1323.4.a.bo.1.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9 $ x^8 - 54x^6 - 6x^5 + 555x^4 + 642x^3 - 218x^2 - 54x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$6.13087$ 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)

$f(q)$	$=$	$q+3.29273 q^{2} +2.84210 q^{4} +21.7165 q^{5} -16.9836 q^{8} +O(q^{10})$	$q+3.29273 q^{2} +2.84210 q^{4} +21.7165 q^{5} -16.9836 q^{8} +71.5066 q^{10} -41.9640 q^{11} +46.0329 q^{13} -78.6593 q^{16} +2.01763 q^{17} +73.2487 q^{19} +61.7204 q^{20} -138.176 q^{22} -24.1677 q^{23} +346.605 q^{25} +151.574 q^{26} +90.7240 q^{29} +53.2052 q^{31} -123.135 q^{32} +6.64353 q^{34} +67.8619 q^{37} +241.189 q^{38} -368.824 q^{40} +341.211 q^{41} +509.713 q^{43} -119.266 q^{44} -79.5778 q^{46} +38.4702 q^{47} +1141.28 q^{50} +130.830 q^{52} +389.388 q^{53} -911.311 q^{55} +298.730 q^{58} +450.621 q^{59} +225.903 q^{61} +175.191 q^{62} +223.822 q^{64} +999.672 q^{65} -772.012 q^{67} +5.73431 q^{68} -962.655 q^{71} -1053.79 q^{73} +223.451 q^{74} +208.180 q^{76} +33.5907 q^{79} -1708.20 q^{80} +1123.52 q^{82} +446.386 q^{83} +43.8159 q^{85} +1678.35 q^{86} +712.700 q^{88} +489.798 q^{89} -68.6870 q^{92} +126.672 q^{94} +1590.70 q^{95} +460.958 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 48 q^{4}+O(q^{10}) $ 8 * q + 48 * q^4	$ 8 q + 48 q^{4} + 44 q^{10} - 84 q^{13} + 156 q^{16} + 12 q^{19} + 224 q^{22} + 408 q^{25} + 800 q^{31} - 948 q^{34} + 692 q^{37} + 96 q^{40} + 1456 q^{43} + 1524 q^{46} + 1972 q^{52} - 1280 q^{55} + 2372 q^{58} + 216 q^{61} + 4964 q^{64} + 684 q^{67} - 4564 q^{73} - 380 q^{76} + 556 q^{79} + 3340 q^{82} + 1296 q^{85} + 6696 q^{88} - 492 q^{94} + 584 q^{97}+O(q^{100}) $ 8 * q + 48 * q^4 + 44 * q^10 - 84 * q^13 + 156 * q^16 + 12 * q^19 + 224 * q^22 + 408 * q^25 + 800 * q^31 - 948 * q^34 + 692 * q^37 + 96 * q^40 + 1456 * q^43 + 1524 * q^46 + 1972 * q^52 - 1280 * q^55 + 2372 * q^58 + 216 * q^61 + 4964 * q^64 + 684 * q^67 - 4564 * q^73 - 380 * q^76 + 556 * q^79 + 3340 * q^82 + 1296 * q^85 + 6696 * q^88 - 492 * q^94 + 584 * 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 $

$2$	3.29273	1.16416	0.582079	−	0.813132i	$-0.302240\pi$
$2$	3.29273	1.16416	0.582079	+	0.813132i	$0.302240\pi$
$3$	0	0
$3$	0	0
$4$	2.84210	0.355263
$5$	21.7165	1.94238	0.971190	−	0.238305i	$-0.0765918\pi$
$5$	21.7165	1.94238	0.971190	+	0.238305i	$0.0765918\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−16.9836	−0.750576
$9$	0	0
$10$	71.5066	2.26124
$11$	−41.9640	−1.15024	−0.575119	−	0.818069i	$-0.695044\pi$
$11$	−41.9640	−1.15024	−0.575119	+	0.818069i	$0.695044\pi$
$12$	0	0
$13$	46.0329	0.982095	0.491047	−	0.871133i	$-0.336614\pi$
$13$	46.0329	0.982095	0.491047	+	0.871133i	$0.336614\pi$
$14$	0	0
$15$	0	0
$16$	−78.6593	−1.22905
$17$	2.01763	0.0287852	0.0143926	−	0.999896i	$-0.495419\pi$
$17$	2.01763	0.0287852	0.0143926	+	0.999896i	$0.495419\pi$
$18$	0	0
$19$	73.2487	0.884442	0.442221	−	0.896906i	$-0.354191\pi$
$19$	73.2487	0.884442	0.442221	+	0.896906i	$0.354191\pi$
$20$	61.7204	0.690055
$21$	0	0
$22$	−138.176	−1.33906
$23$	−24.1677	−0.219100	−0.109550	−	0.993981i	$-0.534941\pi$
$23$	−24.1677	−0.219100	−0.109550	+	0.993981i	$0.534941\pi$
$24$	0	0
$25$	346.605	2.77284
$26$	151.574	1.14331
$27$	0	0
$28$	0	0
$29$	90.7240	0.580932	0.290466	−	0.956885i	$-0.406190\pi$
$29$	90.7240	0.580932	0.290466	+	0.956885i	$0.406190\pi$
$30$	0	0
$31$	53.2052	0.308256	0.154128	−	0.988051i	$-0.450743\pi$
$31$	53.2052	0.308256	0.154128	+	0.988051i	$0.450743\pi$
$32$	−123.135	−0.680233
$33$	0	0
$34$	6.64353	0.0335105
$35$	0	0
$36$	0	0
$37$	67.8619	0.301525	0.150762	−	0.988570i	$-0.451827\pi$
$37$	67.8619	0.301525	0.150762	+	0.988570i	$0.451827\pi$
$38$	241.189	1.02963
$39$	0	0
$40$	−368.824	−1.45790
$41$	341.211	1.29971	0.649856	−	0.760057i	$-0.274829\pi$
$41$	341.211	1.29971	0.649856	+	0.760057i	$0.274829\pi$
$42$	0	0
$43$	509.713	1.80769	0.903843	−	0.427863i	$-0.140734\pi$
$43$	509.713	1.80769	0.903843	+	0.427863i	$0.140734\pi$
$44$	−119.266	−0.408637
$45$	0	0
$46$	−79.5778	−0.255067
$47$	38.4702	0.119393	0.0596963	−	0.998217i	$-0.480987\pi$
$47$	38.4702	0.119393	0.0596963	+	0.998217i	$0.480987\pi$
$48$	0	0
$49$	0	0
$50$	1141.28	3.22803
$51$	0	0
$52$	130.830	0.348902
$53$	389.388	1.00918	0.504590	−	0.863359i	$-0.331644\pi$
$53$	389.388	1.00918	0.504590	+	0.863359i	$0.331644\pi$
$54$	0	0
$55$	−911.311	−2.23420
$56$	0	0
$57$	0	0
$58$	298.730	0.676296
$59$	450.621	0.994336	0.497168	−	0.867654i	$-0.334373\pi$
$59$	450.621	0.994336	0.497168	+	0.867654i	$0.334373\pi$
$60$	0	0
$61$	225.903	0.474162	0.237081	−	0.971490i	$-0.423809\pi$
$61$	225.903	0.474162	0.237081	+	0.971490i	$0.423809\pi$
$62$	175.191	0.358859
$63$	0	0
$64$	223.822	0.437152
$65$	999.672	1.90760
$66$	0	0
$67$	−772.012	−1.40771	−0.703853	−	0.710346i	$-0.748539\pi$
$67$	−772.012	−1.40771	−0.703853	+	0.710346i	$0.748539\pi$
$68$	5.73431	0.0102263
$69$	0	0
$70$	0	0
$71$	−962.655	−1.60910	−0.804550	−	0.593885i	$-0.797593\pi$
$71$	−962.655	−1.60910	−0.804550	+	0.593885i	$0.797593\pi$
$72$	0	0
$73$	−1053.79	−1.68955	−0.844775	−	0.535121i	$-0.820266\pi$
$73$	−1053.79	−1.68955	−0.844775	+	0.535121i	$0.820266\pi$
$74$	223.451	0.351023
$75$	0	0
$76$	208.180	0.314209
$77$	0	0
$78$	0	0
$79$	33.5907	0.0478386	0.0239193	−	0.999714i	$-0.492386\pi$
$79$	33.5907	0.0478386	0.0239193	+	0.999714i	$0.492386\pi$
$80$	−1708.20	−2.38729
$81$	0	0
$82$	1123.52	1.51307
$83$	446.386	0.590328	0.295164	−	0.955447i	$-0.404626\pi$
$83$	446.386	0.590328	0.295164	+	0.955447i	$0.404626\pi$
$84$	0	0
$85$	43.8159	0.0559117
$86$	1678.35	2.10443
$87$	0	0
$88$	712.700	0.863342
$89$	489.798	0.583354	0.291677	−	0.956517i	$-0.405787\pi$
$89$	489.798	0.583354	0.291677	+	0.956517i	$0.405787\pi$
$90$	0	0
$91$	0	0
$92$	−68.6870	−0.0778382
$93$	0	0
$94$	126.672	0.138992
$95$	1590.70	1.71792
$96$	0	0
$97$	460.958	0.482507	0.241253	−	0.970462i	$-0.422441\pi$
$97$	460.958	0.482507	0.241253	+	0.970462i	$0.422441\pi$
$98$	0	0
$99$	0	0
$100$	985.088	0.985088
$101$	−1542.74	−1.51989	−0.759945	−	0.649988i	$-0.774774\pi$
$101$	−1542.74	−1.51989	−0.759945	+	0.649988i	$0.774774\pi$
$102$	0	0
$103$	1142.64	1.09308	0.546540	−	0.837433i	$-0.315945\pi$
$103$	1142.64	1.09308	0.546540	+	0.837433i	$0.315945\pi$
$104$	−781.804	−0.737136
$105$	0	0
$106$	1282.15	1.17485
$107$	−802.561	−0.725107	−0.362554	−	0.931963i	$-0.618095\pi$
$107$	−802.561	−0.725107	−0.362554	+	0.931963i	$0.618095\pi$
$108$	0	0
$109$	1498.96	1.31720	0.658598	−	0.752495i	$-0.271150\pi$
$109$	1498.96	1.31720	0.658598	+	0.752495i	$0.271150\pi$
$110$	−3000.70	−2.60096
$111$	0	0
$112$	0	0
$113$	−651.941	−0.542738	−0.271369	−	0.962475i	$-0.587476\pi$
$113$	−651.941	−0.542738	−0.271369	+	0.962475i	$0.587476\pi$
$114$	0	0
$115$	−524.837	−0.425576
$116$	257.847	0.206383
$117$	0	0
$118$	1483.77	1.15756
$119$	0	0
$120$	0	0
$121$	429.979	0.323050
$122$	743.838	0.551999
$123$	0	0
$124$	151.215	0.109512
$125$	4812.49	3.44354
$126$	0	0
$127$	−634.311	−0.443197	−0.221598	−	0.975138i	$-0.571127\pi$
$127$	−634.311	−0.443197	−0.221598	+	0.975138i	$0.571127\pi$
$128$	1722.07	1.18915
$129$	0	0
$130$	3291.66	2.22075
$131$	−2290.16	−1.52742	−0.763709	−	0.645560i	$-0.776624\pi$
$131$	−2290.16	−1.52742	−0.763709	+	0.645560i	$0.776624\pi$
$132$	0	0
$133$	0	0
$134$	−2542.03	−1.63879
$135$	0	0
$136$	−34.2666	−0.0216054
$137$	730.350	0.455460	0.227730	−	0.973724i	$-0.426870\pi$
$137$	730.350	0.455460	0.227730	+	0.973724i	$0.426870\pi$
$138$	0	0
$139$	−1195.39	−0.729436	−0.364718	−	0.931118i	$-0.618835\pi$
$139$	−1195.39	−0.729436	−0.364718	+	0.931118i	$0.618835\pi$
$140$	0	0
$141$	0	0
$142$	−3169.77	−1.87325
$143$	−1931.73	−1.12964
$144$	0	0
$145$	1970.21	1.12839
$146$	−3469.86	−1.96690
$147$	0	0
$148$	192.870	0.107121
$149$	1269.38	0.697932	0.348966	−	0.937135i	$-0.386533\pi$
$149$	1269.38	0.697932	0.348966	+	0.937135i	$0.386533\pi$
$150$	0	0
$151$	438.971	0.236576	0.118288	−	0.992979i	$-0.462259\pi$
$151$	438.971	0.236576	0.118288	+	0.992979i	$0.462259\pi$
$152$	−1244.03	−0.663841
$153$	0	0
$154$	0	0
$155$	1155.43	0.598751
$156$	0	0
$157$	−1786.27	−0.908027	−0.454013	−	0.890995i	$-0.650008\pi$
$157$	−1786.27	−0.908027	−0.454013	+	0.890995i	$0.650008\pi$
$158$	110.605	0.0556917
$159$	0	0
$160$	−2674.07	−1.32127
$161$	0	0
$162$	0	0
$163$	−847.231	−0.407118	−0.203559	−	0.979063i	$-0.565251\pi$
$163$	−847.231	−0.407118	−0.203559	+	0.979063i	$0.565251\pi$
$164$	969.756	0.461739
$165$	0	0
$166$	1469.83	0.687235
$167$	1454.66	0.674040	0.337020	−	0.941498i	$-0.390581\pi$
$167$	1454.66	0.674040	0.337020	+	0.941498i	$0.390581\pi$
$168$	0	0
$169$	−77.9721	−0.0354903
$170$	144.274	0.0650901
$171$	0	0
$172$	1448.66	0.642204
$173$	849.511	0.373336	0.186668	−	0.982423i	$-0.440231\pi$
$173$	849.511	0.373336	0.186668	+	0.982423i	$0.440231\pi$
$174$	0	0
$175$	0	0
$176$	3300.86	1.41370
$177$	0	0
$178$	1612.78	0.679116
$179$	1320.02	0.551189	0.275595	−	0.961274i	$-0.411125\pi$
$179$	1320.02	0.551189	0.275595	+	0.961274i	$0.411125\pi$
$180$	0	0
$181$	−4005.73	−1.64499	−0.822495	−	0.568772i	$-0.807419\pi$
$181$	−4005.73	−1.64499	−0.822495	+	0.568772i	$0.807419\pi$
$182$	0	0
$183$	0	0
$184$	410.454	0.164451
$185$	1473.72	0.585676
$186$	0	0
$187$	−84.6680	−0.0331098
$188$	109.336	0.0424157
$189$	0	0
$190$	5237.77	1.99993
$191$	−1711.51	−0.648380	−0.324190	−	0.945992i	$-0.605092\pi$
$191$	−1711.51	−0.648380	−0.324190	+	0.945992i	$0.605092\pi$
$192$	0	0
$193$	4383.94	1.63504	0.817521	−	0.575898i	$-0.195348\pi$
$193$	4383.94	1.63504	0.817521	+	0.575898i	$0.195348\pi$
$194$	1517.81	0.561714
$195$	0	0
$196$	0	0
$197$	−1269.36	−0.459076	−0.229538	−	0.973300i	$-0.573721\pi$
$197$	−1269.36	−0.459076	−0.229538	+	0.973300i	$0.573721\pi$
$198$	0	0
$199$	586.566	0.208948	0.104474	−	0.994528i	$-0.466684\pi$
$199$	586.566	0.208948	0.104474	+	0.994528i	$0.466684\pi$
$200$	−5886.60	−2.08123
$201$	0	0
$202$	−5079.85	−1.76939
$203$	0	0
$204$	0	0
$205$	7409.90	2.52454
$206$	3762.40	1.27252
$207$	0	0
$208$	−3620.91	−1.20704
$209$	−3073.81	−1.01732
$210$	0	0
$211$	1037.46	0.338491	0.169246	−	0.985574i	$-0.445867\pi$
$211$	1037.46	0.338491	0.169246	+	0.985574i	$0.445867\pi$
$212$	1106.68	0.358524
$213$	0	0
$214$	−2642.62	−0.844139
$215$	11069.2	3.51122
$216$	0	0
$217$	0	0
$218$	4935.68	1.53342
$219$	0	0
$220$	−2590.04	−0.793729
$221$	92.8775	0.0282697
$222$	0	0
$223$	5421.68	1.62808	0.814041	−	0.580807i	$-0.197263\pi$
$223$	5421.68	1.62808	0.814041	+	0.580807i	$0.197263\pi$
$224$	0	0
$225$	0	0
$226$	−2146.67	−0.631833
$227$	−1830.60	−0.535248	−0.267624	−	0.963523i	$-0.586238\pi$
$227$	−1830.60	−0.535248	−0.267624	+	0.963523i	$0.586238\pi$
$228$	0	0
$229$	68.7870	0.0198497	0.00992483	−	0.999951i	$-0.496841\pi$
$229$	68.7870	0.0198497	0.00992483	+	0.999951i	$0.496841\pi$
$230$	−1728.15	−0.495438
$231$	0	0
$232$	−1540.82	−0.436033
$233$	−4871.93	−1.36983	−0.684916	−	0.728622i	$-0.740161\pi$
$233$	−4871.93	−1.36983	−0.684916	+	0.728622i	$0.740161\pi$
$234$	0	0
$235$	835.437	0.231906
$236$	1280.71	0.353251
$237$	0	0
$238$	0	0
$239$	−3786.86	−1.02490	−0.512451	−	0.858716i	$-0.671263\pi$
$239$	−3786.86	−1.02490	−0.512451	+	0.858716i	$0.671263\pi$
$240$	0	0
$241$	1806.60	0.482877	0.241438	−	0.970416i	$-0.422381\pi$
$241$	1806.60	0.482877	0.241438	+	0.970416i	$0.422381\pi$
$242$	1415.81	0.376081
$243$	0	0
$244$	642.039	0.168452
$245$	0	0
$246$	0	0
$247$	3371.85	0.868606
$248$	−903.616	−0.231370
$249$	0	0
$250$	15846.2	4.00882
$251$	235.616	0.0592508	0.0296254	−	0.999561i	$-0.490569\pi$
$251$	235.616	0.0592508	0.0296254	+	0.999561i	$0.490569\pi$
$252$	0	0
$253$	1014.17	0.252018
$254$	−2088.62	−0.515951
$255$	0	0
$256$	3879.74	0.947203
$257$	2084.35	0.505908	0.252954	−	0.967478i	$-0.418598\pi$
$257$	2084.35	0.505908	0.252954	+	0.967478i	$0.418598\pi$
$258$	0	0
$259$	0	0
$260$	2841.17	0.677700
$261$	0	0
$262$	−7540.87	−1.77816
$263$	−5884.72	−1.37972	−0.689862	−	0.723941i	$-0.742329\pi$
$263$	−5884.72	−1.37972	−0.689862	+	0.723941i	$0.742329\pi$
$264$	0	0
$265$	8456.14	1.96021
$266$	0	0
$267$	0	0
$268$	−2194.14	−0.500105
$269$	7076.62	1.60397	0.801987	−	0.597342i	$-0.203776\pi$
$269$	7076.62	1.60397	0.801987	+	0.597342i	$0.203776\pi$
$270$	0	0
$271$	2818.66	0.631814	0.315907	−	0.948790i	$-0.397691\pi$
$271$	2818.66	0.631814	0.315907	+	0.948790i	$0.397691\pi$
$272$	−158.705	−0.0353784
$273$	0	0
$274$	2404.85	0.530227
$275$	−14545.0	−3.18943
$276$	0	0
$277$	4368.87	0.947652	0.473826	−	0.880618i	$-0.342872\pi$
$277$	4368.87	0.947652	0.473826	+	0.880618i	$0.342872\pi$
$278$	−3936.10	−0.849178
$279$	0	0
$280$	0	0
$281$	520.957	0.110597	0.0552984	−	0.998470i	$-0.482389\pi$
$281$	520.957	0.110597	0.0552984	+	0.998470i	$0.482389\pi$
$282$	0	0
$283$	−2832.71	−0.595008	−0.297504	−	0.954720i	$-0.596154\pi$
$283$	−2832.71	−0.595008	−0.297504	+	0.954720i	$0.596154\pi$
$284$	−2735.96	−0.571653
$285$	0	0
$286$	−6360.66	−1.31508
$287$	0	0
$288$	0	0
$289$	−4908.93	−0.999171
$290$	6487.36	1.31362
$291$	0	0
$292$	−2994.99	−0.600234
$293$	3591.98	0.716196	0.358098	−	0.933684i	$-0.383425\pi$
$293$	3591.98	0.716196	0.358098	+	0.933684i	$0.383425\pi$
$294$	0	0
$295$	9785.90	1.93138
$296$	−1152.54	−0.226317
$297$	0	0
$298$	4179.74	0.812503
$299$	−1112.51	−0.215177
$300$	0	0
$301$	0	0
$302$	1445.42	0.275412
$303$	0	0
$304$	−5761.69	−1.08702
$305$	4905.81	0.921003
$306$	0	0
$307$	2168.14	0.403069	0.201535	−	0.979481i	$-0.435407\pi$
$307$	2168.14	0.403069	0.201535	+	0.979481i	$0.435407\pi$
$308$	0	0
$309$	0	0
$310$	3804.52	0.697040
$311$	−7218.02	−1.31606	−0.658032	−	0.752990i	$-0.728611\pi$
$311$	−7218.02	−1.31606	−0.658032	+	0.752990i	$0.728611\pi$
$312$	0	0
$313$	3044.54	0.549800	0.274900	−	0.961473i	$-0.411355\pi$
$313$	3044.54	0.549800	0.274900	+	0.961473i	$0.411355\pi$
$314$	−5881.73	−1.05709
$315$	0	0
$316$	95.4682	0.0169953
$317$	6110.77	1.08270	0.541349	−	0.840798i	$-0.317914\pi$
$317$	6110.77	1.08270	0.541349	+	0.840798i	$0.317914\pi$
$318$	0	0
$319$	−3807.14	−0.668210
$320$	4860.63	0.849116
$321$	0	0
$322$	0	0
$323$	147.789	0.0254588
$324$	0	0
$325$	15955.3	2.72319
$326$	−2789.71	−0.473950
$327$	0	0
$328$	−5794.99	−0.975533
$329$	0	0
$330$	0	0
$331$	1497.42	0.248657	0.124329	−	0.992241i	$-0.460322\pi$
$331$	1497.42	0.248657	0.124329	+	0.992241i	$0.460322\pi$
$332$	1268.67	0.209722
$333$	0	0
$334$	4789.80	0.784689
$335$	−16765.4	−2.73430
$336$	0	0
$337$	−8157.61	−1.31861	−0.659307	−	0.751873i	$-0.729150\pi$
$337$	−8157.61	−1.31861	−0.659307	+	0.751873i	$0.729150\pi$
$338$	−256.742	−0.0413163
$339$	0	0
$340$	124.529	0.0198634
$341$	−2232.70	−0.354568
$342$	0	0
$343$	0	0
$344$	−8656.76	−1.35681
$345$	0	0
$346$	2797.21	0.434622
$347$	−9133.12	−1.41294	−0.706472	−	0.707741i	$-0.749714\pi$
$347$	−9133.12	−1.41294	−0.706472	+	0.707741i	$0.749714\pi$
$348$	0	0
$349$	−9528.48	−1.46145	−0.730727	−	0.682670i	$-0.760819\pi$
$349$	−9528.48	−1.46145	−0.730727	+	0.682670i	$0.760819\pi$
$350$	0	0
$351$	0	0
$352$	5167.26	0.782431
$353$	−3357.34	−0.506214	−0.253107	−	0.967438i	$-0.581452\pi$
$353$	−3357.34	−0.506214	−0.253107	+	0.967438i	$0.581452\pi$
$354$	0	0
$355$	−20905.5	−3.12548
$356$	1392.06	0.207244
$357$	0	0
$358$	4346.47	0.641671
$359$	92.9735	0.0136684	0.00683420	−	0.999977i	$-0.497825\pi$
$359$	92.9735	0.0136684	0.00683420	+	0.999977i	$0.497825\pi$
$360$	0	0
$361$	−1493.63	−0.217762
$362$	−13189.8	−1.91503
$363$	0	0
$364$	0	0
$365$	−22884.7	−3.28175
$366$	0	0
$367$	−6785.90	−0.965181	−0.482590	−	0.875846i	$-0.660304\pi$
$367$	−6785.90	−0.965181	−0.482590	+	0.875846i	$0.660304\pi$
$368$	1901.01	0.269286
$369$	0	0
$370$	4852.57	0.681820
$371$	0	0
$372$	0	0
$373$	−7451.05	−1.03432	−0.517159	−	0.855889i	$-0.673010\pi$
$373$	−7451.05	−1.03432	−0.517159	+	0.855889i	$0.673010\pi$
$374$	−278.789	−0.0385450
$375$	0	0
$376$	−653.362	−0.0896132
$377$	4176.29	0.570530
$378$	0	0
$379$	4014.67	0.544115	0.272058	−	0.962281i	$-0.412296\pi$
$379$	4014.67	0.544115	0.272058	+	0.962281i	$0.412296\pi$
$380$	4520.94	0.610314
$381$	0	0
$382$	−5635.55	−0.754816
$383$	−6464.87	−0.862505	−0.431253	−	0.902231i	$-0.641928\pi$
$383$	−6464.87	−0.862505	−0.431253	+	0.902231i	$0.641928\pi$
$384$	0	0
$385$	0	0
$386$	14435.2	1.90345
$387$	0	0
$388$	1310.09	0.171417
$389$	−6944.06	−0.905085	−0.452542	−	0.891743i	$-0.649483\pi$
$389$	−6944.06	−0.905085	−0.452542	+	0.891743i	$0.649483\pi$
$390$	0	0
$391$	−48.7615	−0.00630684
$392$	0	0
$393$	0	0
$394$	−4179.65	−0.534436
$395$	729.472	0.0929208
$396$	0	0
$397$	1363.07	0.172319	0.0861594	−	0.996281i	$-0.472541\pi$
$397$	1363.07	0.172319	0.0861594	+	0.996281i	$0.472541\pi$
$398$	1931.41	0.243248
$399$	0	0
$400$	−27263.7	−3.40797
$401$	3889.07	0.484317	0.242158	−	0.970237i	$-0.422145\pi$
$401$	3889.07	0.484317	0.242158	+	0.970237i	$0.422145\pi$
$402$	0	0
$403$	2449.19	0.302737
$404$	−4384.64	−0.539960
$405$	0	0
$406$	0	0
$407$	−2847.76	−0.346826
$408$	0	0
$409$	−8373.76	−1.01236	−0.506180	−	0.862428i	$-0.668943\pi$
$409$	−8373.76	−1.01236	−0.506180	+	0.862428i	$0.668943\pi$
$410$	24398.8	2.93896
$411$	0	0
$412$	3247.49	0.388331
$413$	0	0
$414$	0	0
$415$	9693.93	1.14664
$416$	−5668.28	−0.668053
$417$	0	0
$418$	−10121.2	−1.18432
$419$	−11115.8	−1.29604	−0.648022	−	0.761621i	$-0.724404\pi$
$419$	−11115.8	−1.29604	−0.648022	+	0.761621i	$0.724404\pi$
$420$	0	0
$421$	9201.47	1.06521	0.532604	−	0.846365i	$-0.321214\pi$
$421$	9201.47	1.06521	0.532604	+	0.846365i	$0.321214\pi$
$422$	3416.08	0.394057
$423$	0	0
$424$	−6613.21	−0.757467
$425$	699.322	0.0798167
$426$	0	0
$427$	0	0
$428$	−2280.96	−0.257603
$429$	0	0
$430$	36447.9	4.08761
$431$	2997.12	0.334956	0.167478	−	0.985876i	$-0.446438\pi$
$431$	2997.12	0.334956	0.167478	+	0.985876i	$0.446438\pi$
$432$	0	0
$433$	−4285.85	−0.475669	−0.237835	−	0.971306i	$-0.576438\pi$
$433$	−4285.85	−0.475669	−0.237835	+	0.971306i	$0.576438\pi$
$434$	0	0
$435$	0	0
$436$	4260.20	0.467951
$437$	−1770.25	−0.193782
$438$	0	0
$439$	3051.15	0.331716	0.165858	−	0.986150i	$-0.446961\pi$
$439$	3051.15	0.331716	0.165858	+	0.986150i	$0.446961\pi$
$440$	15477.3	1.67694
$441$	0	0
$442$	305.821	0.0329104
$443$	−11571.0	−1.24098	−0.620491	−	0.784214i	$-0.713066\pi$
$443$	−11571.0	−1.24098	−0.620491	+	0.784214i	$0.713066\pi$
$444$	0	0
$445$	10636.7	1.13310
$446$	17852.1	1.89534
$447$	0	0
$448$	0	0
$449$	−3588.18	−0.377142	−0.188571	−	0.982060i	$-0.560386\pi$
$449$	−3588.18	−0.377142	−0.188571	+	0.982060i	$0.560386\pi$
$450$	0	0
$451$	−14318.6	−1.49498
$452$	−1852.88	−0.192815
$453$	0	0
$454$	−6027.68	−0.623113
$455$	0	0
$456$	0	0
$457$	−9367.40	−0.958837	−0.479419	−	0.877586i	$-0.659152\pi$
$457$	−9367.40	−0.958837	−0.479419	+	0.877586i	$0.659152\pi$
$458$	226.497	0.0231081
$459$	0	0
$460$	−1491.64	−0.151191
$461$	1091.46	0.110269	0.0551346	−	0.998479i	$-0.482441\pi$
$461$	1091.46	0.110269	0.0551346	+	0.998479i	$0.482441\pi$
$462$	0	0
$463$	13226.1	1.32758	0.663790	−	0.747919i	$-0.268947\pi$
$463$	13226.1	1.32758	0.663790	+	0.747919i	$0.268947\pi$
$464$	−7136.28	−0.713995
$465$	0	0
$466$	−16042.0	−1.59470
$467$	−11341.4	−1.12381	−0.561903	−	0.827203i	$-0.689931\pi$
$467$	−11341.4	−1.12381	−0.561903	+	0.827203i	$0.689931\pi$
$468$	0	0
$469$	0	0
$470$	2750.87	0.269975
$471$	0	0
$472$	−7653.16	−0.746325
$473$	−21389.6	−2.07927
$474$	0	0
$475$	25388.4	2.45242
$476$	0	0
$477$	0	0
$478$	−12469.1	−1.19315
$479$	−17024.4	−1.62394	−0.811968	−	0.583702i	$-0.801604\pi$
$479$	−17024.4	−1.62394	−0.811968	+	0.583702i	$0.801604\pi$
$480$	0	0
$481$	3123.88	0.296126
$482$	5948.66	0.562145
$483$	0	0
$484$	1222.04	0.114767
$485$	10010.4	0.937212
$486$	0	0
$487$	9724.70	0.904863	0.452431	−	0.891799i	$-0.350557\pi$
$487$	9724.70	0.904863	0.452431	+	0.891799i	$0.350557\pi$
$488$	−3836.64	−0.355895
$489$	0	0
$490$	0	0
$491$	16363.4	1.50402	0.752008	−	0.659154i	$-0.229085\pi$
$491$	16363.4	1.50402	0.752008	+	0.659154i	$0.229085\pi$
$492$	0	0
$493$	183.048	0.0167222
$494$	11102.6	1.01119
$495$	0	0
$496$	−4185.08	−0.378862
$497$	0	0
$498$	0	0
$499$	−5452.46	−0.489149	−0.244575	−	0.969630i	$-0.578648\pi$
$499$	−5452.46	−0.489149	−0.244575	+	0.969630i	$0.578648\pi$
$500$	13677.6	1.22336
$501$	0	0
$502$	775.820	0.0689772
$503$	3703.99	0.328336	0.164168	−	0.986432i	$-0.447506\pi$
$503$	3703.99	0.328336	0.164168	+	0.986432i	$0.447506\pi$
$504$	0	0
$505$	−33503.0	−2.95220
$506$	3339.40	0.293388
$507$	0	0
$508$	−1802.78	−0.157451
$509$	−1653.69	−0.144005	−0.0720023	−	0.997404i	$-0.522939\pi$
$509$	−1653.69	−0.144005	−0.0720023	+	0.997404i	$0.522939\pi$
$510$	0	0
$511$	0	0
$512$	−1001.60	−0.0864546
$513$	0	0
$514$	6863.22	0.588956
$515$	24814.0	2.12318
$516$	0	0
$517$	−1614.36	−0.137330
$518$	0	0
$519$	0	0
$520$	−16978.0	−1.43180
$521$	3100.30	0.260704	0.130352	−	0.991468i	$-0.458389\pi$
$521$	3100.30	0.260704	0.130352	+	0.991468i	$0.458389\pi$
$522$	0	0
$523$	−3125.43	−0.261311	−0.130655	−	0.991428i	$-0.541708\pi$
$523$	−3125.43	−0.261311	−0.130655	+	0.991428i	$0.541708\pi$
$524$	−6508.85	−0.542635
$525$	0	0
$526$	−19376.8	−1.60622
$527$	107.349	0.00887320
$528$	0	0
$529$	−11582.9	−0.951995
$530$	27843.8	2.28200
$531$	0	0
$532$	0	0
$533$	15706.9	1.27644
$534$	0	0
$535$	−17428.8	−1.40843
$536$	13111.5	1.05659
$537$	0	0
$538$	23301.4	1.86728
$539$	0	0
$540$	0	0
$541$	21514.3	1.70975	0.854874	−	0.518836i	$-0.173634\pi$
$541$	21514.3	1.70975	0.854874	+	0.518836i	$0.173634\pi$
$542$	9281.10	0.735531
$543$	0	0
$544$	−248.442	−0.0195806
$545$	32552.2	2.55850
$546$	0	0
$547$	−13104.4	−1.02432	−0.512161	−	0.858889i	$-0.671155\pi$
$547$	−13104.4	−1.02432	−0.512161	+	0.858889i	$0.671155\pi$
$548$	2075.73	0.161808
$549$	0	0
$550$	−47892.7	−3.71300
$551$	6645.42	0.513801
$552$	0	0
$553$	0	0
$554$	14385.5	1.10322
$555$	0	0
$556$	−3397.42	−0.259141
$557$	−18794.9	−1.42974	−0.714870	−	0.699257i	$-0.753514\pi$
$557$	−18794.9	−1.42974	−0.714870	+	0.699257i	$0.753514\pi$
$558$	0	0
$559$	23463.6	1.77532
$560$	0	0
$561$	0	0
$562$	1715.37	0.128752
$563$	−17449.7	−1.30625	−0.653124	−	0.757251i	$-0.726542\pi$
$563$	−17449.7	−1.30625	−0.653124	+	0.757251i	$0.726542\pi$
$564$	0	0
$565$	−14157.9	−1.05420
$566$	−9327.38	−0.692684
$567$	0	0
$568$	16349.3	1.20775
$569$	22840.9	1.68285	0.841423	−	0.540376i	$-0.181718\pi$
$569$	22840.9	1.68285	0.841423	+	0.540376i	$0.181718\pi$
$570$	0	0
$571$	896.042	0.0656711	0.0328355	−	0.999461i	$-0.489546\pi$
$571$	896.042	0.0656711	0.0328355	+	0.999461i	$0.489546\pi$
$572$	−5490.16	−0.401320
$573$	0	0
$574$	0	0
$575$	−8376.65	−0.607531
$576$	0	0
$577$	−2966.19	−0.214011	−0.107005	−	0.994258i	$-0.534126\pi$
$577$	−2966.19	−0.214011	−0.107005	+	0.994258i	$0.534126\pi$
$578$	−16163.8	−1.16319
$579$	0	0
$580$	5599.52	0.400875
$581$	0	0
$582$	0	0
$583$	−16340.3	−1.16080
$584$	17897.2	1.26814
$585$	0	0
$586$	11827.4	0.833765
$587$	1872.80	0.131684	0.0658420	−	0.997830i	$-0.479027\pi$
$587$	1872.80	0.131684	0.0658420	+	0.997830i	$0.479027\pi$
$588$	0	0
$589$	3897.21	0.272635
$590$	32222.4	2.24843
$591$	0	0
$592$	−5337.97	−0.370590
$593$	17099.0	1.18410	0.592051	−	0.805901i	$-0.298318\pi$
$593$	17099.0	1.18410	0.592051	+	0.805901i	$0.298318\pi$
$594$	0	0
$595$	0	0
$596$	3607.71	0.247949
$597$	0	0
$598$	−3663.20	−0.250500
$599$	−11418.2	−0.778856	−0.389428	−	0.921057i	$-0.627327\pi$
$599$	−11418.2	−0.778856	−0.389428	+	0.921057i	$0.627327\pi$
$600$	0	0
$601$	−10231.6	−0.694432	−0.347216	−	0.937785i	$-0.612873\pi$
$601$	−10231.6	−0.694432	−0.347216	+	0.937785i	$0.612873\pi$
$602$	0	0
$603$	0	0
$604$	1247.60	0.0840466
$605$	9337.63	0.627485
$606$	0	0
$607$	−12191.4	−0.815212	−0.407606	−	0.913158i	$-0.633636\pi$
$607$	−12191.4	−0.815212	−0.407606	+	0.913158i	$0.633636\pi$
$608$	−9019.51	−0.601627
$609$	0	0
$610$	16153.5	1.07219
$611$	1770.89	0.117255
$612$	0	0
$613$	−7649.75	−0.504030	−0.252015	−	0.967723i	$-0.581093\pi$
$613$	−7649.75	−0.504030	−0.252015	+	0.967723i	$0.581093\pi$
$614$	7139.11	0.469236
$615$	0	0
$616$	0	0
$617$	19107.4	1.24673	0.623367	−	0.781929i	$-0.285764\pi$
$617$	19107.4	1.24673	0.623367	+	0.781929i	$0.285764\pi$
$618$	0	0
$619$	17663.3	1.14693	0.573464	−	0.819231i	$-0.305599\pi$
$619$	17663.3	1.14693	0.573464	+	0.819231i	$0.305599\pi$
$620$	3283.85	0.212714
$621$	0	0
$622$	−23767.0	−1.53211
$623$	0	0
$624$	0	0
$625$	61184.6	3.91582
$626$	10024.8	0.640053
$627$	0	0
$628$	−5076.77	−0.322588
$629$	136.920	0.00867944
$630$	0	0
$631$	5523.07	0.348447	0.174223	−	0.984706i	$-0.444259\pi$
$631$	5523.07	0.348447	0.174223	+	0.984706i	$0.444259\pi$
$632$	−570.491	−0.0359065
$633$	0	0
$634$	20121.2	1.26043
$635$	−13775.0	−0.860857
$636$	0	0
$637$	0	0
$638$	−12535.9	−0.777902
$639$	0	0
$640$	37397.3	2.30978
$641$	17454.0	1.07549	0.537747	−	0.843106i	$-0.319276\pi$
$641$	17454.0	1.07549	0.537747	+	0.843106i	$0.319276\pi$
$642$	0	0
$643$	−8175.22	−0.501398	−0.250699	−	0.968065i	$-0.580660\pi$
$643$	−8175.22	−0.501398	−0.250699	+	0.968065i	$0.580660\pi$
$644$	0	0
$645$	0	0
$646$	486.630	0.0296381
$647$	−25000.6	−1.51913	−0.759564	−	0.650432i	$-0.774588\pi$
$647$	−25000.6	−1.51913	−0.759564	+	0.650432i	$0.774588\pi$
$648$	0	0
$649$	−18909.9	−1.14372
$650$	52536.4	3.17023
$651$	0	0
$652$	−2407.92	−0.144634
$653$	−769.091	−0.0460901	−0.0230451	−	0.999734i	$-0.507336\pi$
$653$	−769.091	−0.0460901	−0.0230451	+	0.999734i	$0.507336\pi$
$654$	0	0
$655$	−49734.1	−2.96683
$656$	−26839.4	−1.59741
$657$	0	0
$658$	0	0
$659$	7037.98	0.416025	0.208013	−	0.978126i	$-0.433300\pi$
$659$	7037.98	0.416025	0.208013	+	0.978126i	$0.433300\pi$
$660$	0	0
$661$	4089.75	0.240655	0.120327	−	0.992734i	$-0.461606\pi$
$661$	4089.75	0.240655	0.120327	+	0.992734i	$0.461606\pi$
$662$	4930.60	0.289476
$663$	0	0
$664$	−7581.24	−0.443086
$665$	0	0
$666$	0	0
$667$	−2192.59	−0.127282
$668$	4134.28	0.239461
$669$	0	0
$670$	−55204.0	−3.18316
$671$	−9479.79	−0.545400
$672$	0	0
$673$	2151.39	0.123224	0.0616121	−	0.998100i	$-0.480376\pi$
$673$	2151.39	0.123224	0.0616121	+	0.998100i	$0.480376\pi$
$674$	−26860.8	−1.53508
$675$	0	0
$676$	−221.605	−0.0126084
$677$	−7675.61	−0.435742	−0.217871	−	0.975978i	$-0.569911\pi$
$677$	−7675.61	−0.435742	−0.217871	+	0.975978i	$0.569911\pi$
$678$	0	0
$679$	0	0
$680$	−744.151	−0.0419660
$681$	0	0
$682$	−7351.70	−0.412773
$683$	16098.6	0.901896	0.450948	−	0.892550i	$-0.351086\pi$
$683$	16098.6	0.901896	0.450948	+	0.892550i	$0.351086\pi$
$684$	0	0
$685$	15860.6	0.884676
$686$	0	0
$687$	0	0
$688$	−40093.7	−2.22174
$689$	17924.7	0.991111
$690$	0	0
$691$	19083.8	1.05063	0.525313	−	0.850909i	$-0.323948\pi$
$691$	19083.8	1.05063	0.525313	+	0.850909i	$0.323948\pi$
$692$	2414.40	0.132632
$693$	0	0
$694$	−30072.9	−1.64489
$695$	−25959.6	−1.41684
$696$	0	0
$697$	688.438	0.0374124
$698$	−31374.7	−1.70136
$699$	0	0
$700$	0	0
$701$	−29898.7	−1.61093	−0.805463	−	0.592646i	$-0.798083\pi$
$701$	−29898.7	−1.61093	−0.805463	+	0.592646i	$0.798083\pi$
$702$	0	0
$703$	4970.79	0.266681
$704$	−9392.47	−0.502830
$705$	0	0
$706$	−11054.8	−0.589312
$707$	0	0
$708$	0	0
$709$	19879.1	1.05300	0.526500	−	0.850175i	$-0.323504\pi$
$709$	19879.1	1.05300	0.526500	+	0.850175i	$0.323504\pi$
$710$	−68836.2	−3.63856
$711$	0	0
$712$	−8318.53	−0.437851
$713$	−1285.85	−0.0675390
$714$	0	0
$715$	−41950.3	−2.19420
$716$	3751.63	0.195817
$717$	0	0
$718$	306.137	0.0159122
$719$	−13640.5	−0.707517	−0.353758	−	0.935337i	$-0.615097\pi$
$719$	−13640.5	−0.707517	−0.353758	+	0.935337i	$0.615097\pi$
$720$	0	0
$721$	0	0
$722$	−4918.12	−0.253509
$723$	0	0
$724$	−11384.7	−0.584404
$725$	31445.4	1.61083
$726$	0	0
$727$	−10021.8	−0.511262	−0.255631	−	0.966774i	$-0.582283\pi$
$727$	−10021.8	−0.511262	−0.255631	+	0.966774i	$0.582283\pi$
$728$	0	0
$729$	0	0
$730$	−75353.2	−3.82047
$731$	1028.41	0.0520345
$732$	0	0
$733$	−24088.0	−1.21380	−0.606898	−	0.794780i	$-0.707586\pi$
$733$	−24088.0	−1.21380	−0.606898	+	0.794780i	$0.707586\pi$
$734$	−22344.2	−1.12362
$735$	0	0
$736$	2975.90	0.149039
$737$	32396.7	1.61920
$738$	0	0
$739$	24657.9	1.22741	0.613705	−	0.789536i	$-0.289679\pi$
$739$	24657.9	1.22741	0.613705	+	0.789536i	$0.289679\pi$
$740$	4188.46	0.208069
$741$	0	0
$742$	0	0
$743$	−13467.0	−0.664948	−0.332474	−	0.943112i	$-0.607883\pi$
$743$	−13467.0	−0.664948	−0.332474	+	0.943112i	$0.607883\pi$
$744$	0	0
$745$	27566.5	1.35565
$746$	−24534.3	−1.20411
$747$	0	0
$748$	−240.635	−0.0117627
$749$	0	0
$750$	0	0
$751$	−6880.15	−0.334301	−0.167151	−	0.985931i	$-0.553457\pi$
$751$	−6880.15	−0.334301	−0.167151	+	0.985931i	$0.553457\pi$
$752$	−3026.04	−0.146740
$753$	0	0
$754$	13751.4	0.664187
$755$	9532.91	0.459521
$756$	0	0
$757$	33528.2	1.60978	0.804889	−	0.593425i	$-0.202225\pi$
$757$	33528.2	1.60978	0.804889	+	0.593425i	$0.202225\pi$
$758$	13219.2	0.633436
$759$	0	0
$760$	−27015.9	−1.28943
$761$	1090.91	0.0519653	0.0259826	−	0.999662i	$-0.491729\pi$
$761$	1090.91	0.0519653	0.0259826	+	0.999662i	$0.491729\pi$
$762$	0	0
$763$	0	0
$764$	−4864.29	−0.230345
$765$	0	0
$766$	−21287.1	−1.00409
$767$	20743.4	0.976532
$768$	0	0
$769$	30520.8	1.43122	0.715609	−	0.698501i	$-0.246149\pi$
$769$	30520.8	1.43122	0.715609	+	0.698501i	$0.246149\pi$
$770$	0	0
$771$	0	0
$772$	12459.6	0.580870
$773$	7848.30	0.365179	0.182590	−	0.983189i	$-0.441552\pi$
$773$	7848.30	0.365179	0.182590	+	0.983189i	$0.441552\pi$
$774$	0	0
$775$	18441.2	0.854746
$776$	−7828.72	−0.362158
$777$	0	0
$778$	−22865.0	−1.05366
$779$	24993.3	1.14952
$780$	0	0
$781$	40396.9	1.85085
$782$	−160.559	−0.00734216
$783$	0	0
$784$	0	0
$785$	−38791.6	−1.76373
$786$	0	0
$787$	18778.4	0.850545	0.425272	−	0.905065i	$-0.360178\pi$
$787$	18778.4	0.850545	0.425272	+	0.905065i	$0.360178\pi$
$788$	−3607.64	−0.163092
$789$	0	0
$790$	2401.96	0.108174
$791$	0	0
$792$	0	0
$793$	10399.0	0.465672
$794$	4488.23	0.200606
$795$	0	0
$796$	1667.08	0.0742313
$797$	−34662.3	−1.54053	−0.770264	−	0.637725i	$-0.779876\pi$
$797$	−34662.3	−1.54053	−0.770264	+	0.637725i	$0.779876\pi$
$798$	0	0
$799$	77.6187	0.00343674
$800$	−42679.4	−1.88618
$801$	0	0
$802$	12805.7	0.563821
$803$	44221.4	1.94339
$804$	0	0
$805$	0	0
$806$	8064.53	0.352433
$807$	0	0
$808$	26201.3	1.14079
$809$	−27491.8	−1.19476	−0.597380	−	0.801958i	$-0.703792\pi$
$809$	−27491.8	−1.19476	−0.597380	+	0.801958i	$0.703792\pi$
$810$	0	0
$811$	19150.6	0.829183	0.414592	−	0.910008i	$-0.363924\pi$
$811$	19150.6	0.829183	0.414592	+	0.910008i	$0.363924\pi$
$812$	0	0
$813$	0	0
$814$	−9376.91	−0.403760
$815$	−18398.9	−0.790779
$816$	0	0
$817$	37335.8	1.59879
$818$	−27572.6	−1.17855
$819$	0	0
$820$	21059.7	0.896874
$821$	−2547.05	−0.108274	−0.0541368	−	0.998534i	$-0.517241\pi$
$821$	−2547.05	−0.108274	−0.0541368	+	0.998534i	$0.517241\pi$
$822$	0	0
$823$	21290.5	0.901748	0.450874	−	0.892588i	$-0.351112\pi$
$823$	21290.5	0.901748	0.450874	+	0.892588i	$0.351112\pi$
$824$	−19406.1	−0.820439
$825$	0	0
$826$	0	0
$827$	23131.7	0.972632	0.486316	−	0.873783i	$-0.338340\pi$
$827$	23131.7	0.972632	0.486316	+	0.873783i	$0.338340\pi$
$828$	0	0
$829$	−11897.4	−0.498450	−0.249225	−	0.968446i	$-0.580176\pi$
$829$	−11897.4	−0.498450	−0.249225	+	0.968446i	$0.580176\pi$
$830$	31919.6	1.33487
$831$	0	0
$832$	10303.2	0.429325
$833$	0	0
$834$	0	0
$835$	31590.0	1.30924
$836$	−8736.08	−0.361416
$837$	0	0
$838$	−36601.4	−1.50880
$839$	21937.3	0.902693	0.451347	−	0.892349i	$-0.350944\pi$
$839$	21937.3	0.902693	0.451347	+	0.892349i	$0.350944\pi$
$840$	0	0
$841$	−16158.2	−0.662518
$842$	30298.0	1.24007
$843$	0	0
$844$	2948.56	0.120253
$845$	−1693.28	−0.0689356
$846$	0	0
$847$	0	0
$848$	−30629.0	−1.24033
$849$	0	0
$850$	2302.68	0.0929192
$851$	−1640.06	−0.0660643
$852$	0	0
$853$	−7087.36	−0.284486	−0.142243	−	0.989832i	$-0.545431\pi$
$853$	−7087.36	−0.284486	−0.142243	+	0.989832i	$0.545431\pi$
$854$	0	0
$855$	0	0
$856$	13630.4	0.544248
$857$	29202.6	1.16399	0.581996	−	0.813192i	$-0.302272\pi$
$857$	29202.6	1.16399	0.581996	+	0.813192i	$0.302272\pi$
$858$	0	0
$859$	−6946.42	−0.275912	−0.137956	−	0.990438i	$-0.544053\pi$
$859$	−6946.42	−0.275912	−0.137956	+	0.990438i	$0.544053\pi$
$860$	31459.7	1.24740
$861$	0	0
$862$	9868.72	0.389942
$863$	−18125.7	−0.714956	−0.357478	−	0.933922i	$-0.616363\pi$
$863$	−18125.7	−0.714956	−0.357478	+	0.933922i	$0.616363\pi$
$864$	0	0
$865$	18448.4	0.725161
$866$	−14112.2	−0.553754
$867$	0	0
$868$	0	0
$869$	−1409.60	−0.0550258
$870$	0	0
$871$	−35538.0	−1.38250
$872$	−25457.7	−0.988656
$873$	0	0
$874$	−5828.97	−0.225592
$875$	0	0
$876$	0	0
$877$	16741.7	0.644614	0.322307	−	0.946635i	$-0.395542\pi$
$877$	16741.7	0.644614	0.322307	+	0.946635i	$0.395542\pi$
$878$	10046.6	0.386170
$879$	0	0
$880$	71683.0	2.74595
$881$	−38779.3	−1.48298	−0.741491	−	0.670963i	$-0.765881\pi$
$881$	−38779.3	−1.48298	−0.741491	+	0.670963i	$0.765881\pi$
$882$	0	0
$883$	32874.4	1.25290	0.626451	−	0.779461i	$-0.284507\pi$
$883$	32874.4	1.25290	0.626451	+	0.779461i	$0.284507\pi$
$884$	263.967	0.0100432
$885$	0	0
$886$	−38100.2	−1.44470
$887$	1849.20	0.0700001	0.0350000	−	0.999387i	$-0.488857\pi$
$887$	1849.20	0.0700001	0.0350000	+	0.999387i	$0.488857\pi$
$888$	0	0
$889$	0	0
$890$	35023.8	1.31910
$891$	0	0
$892$	15409.0	0.578397
$893$	2817.89	0.105596
$894$	0	0
$895$	28666.2	1.07062
$896$	0	0
$897$	0	0
$898$	−11814.9	−0.439053
$899$	4826.99	0.179076
$900$	0	0
$901$	785.642	0.0290494
$902$	−47147.3	−1.74039
$903$	0	0
$904$	11072.3	0.407366
$905$	−86990.3	−3.19520
$906$	0	0
$907$	−33145.2	−1.21342	−0.606708	−	0.794925i	$-0.707510\pi$
$907$	−33145.2	−1.21342	−0.606708	+	0.794925i	$0.707510\pi$
$908$	−5202.75	−0.190154
$909$	0	0
$910$	0	0
$911$	25056.8	0.911272	0.455636	−	0.890166i	$-0.349412\pi$
$911$	25056.8	0.911272	0.455636	+	0.890166i	$0.349412\pi$
$912$	0	0
$913$	−18732.2	−0.679018
$914$	−30844.4	−1.11624
$915$	0	0
$916$	195.500	0.00705184
$917$	0	0
$918$	0	0
$919$	19673.4	0.706164	0.353082	−	0.935592i	$-0.385134\pi$
$919$	19673.4	0.706164	0.353082	+	0.935592i	$0.385134\pi$
$920$	8913.62	0.319427
$921$	0	0
$922$	3593.87	0.128371
$923$	−44313.8	−1.58029
$924$	0	0
$925$	23521.3	0.836081
$926$	43550.1	1.54551
$927$	0	0
$928$	−11171.3	−0.395169
$929$	23323.5	0.823702	0.411851	−	0.911251i	$-0.364882\pi$
$929$	23323.5	0.823702	0.411851	+	0.911251i	$0.364882\pi$
$930$	0	0
$931$	0	0
$932$	−13846.5	−0.486650
$933$	0	0
$934$	−37344.2	−1.30829
$935$	−1838.69	−0.0643119
$936$	0	0
$937$	22330.3	0.778547	0.389274	−	0.921122i	$-0.372726\pi$
$937$	22330.3	0.778547	0.389274	+	0.921122i	$0.372726\pi$
$938$	0	0
$939$	0	0
$940$	2374.40	0.0823875
$941$	42050.1	1.45674	0.728371	−	0.685183i	$-0.240278\pi$
$941$	42050.1	1.45674	0.728371	+	0.685183i	$0.240278\pi$
$942$	0	0
$943$	−8246.28	−0.284768
$944$	−35445.5	−1.22209
$945$	0	0
$946$	−70430.3	−2.42060
$947$	−27731.1	−0.951572	−0.475786	−	0.879561i	$-0.657836\pi$
$947$	−27731.1	−0.951572	−0.475786	+	0.879561i	$0.657836\pi$
$948$	0	0
$949$	−48509.2	−1.65930
$950$	83597.2	2.85500
$951$	0	0
$952$	0	0
$953$	−1378.80	−0.0468666	−0.0234333	−	0.999725i	$-0.507460\pi$
$953$	−1378.80	−0.0468666	−0.0234333	+	0.999725i	$0.507460\pi$
$954$	0	0
$955$	−37168.0	−1.25940
$956$	−10762.6	−0.364110
$957$	0	0
$958$	−56056.9	−1.89052
$959$	0	0
$960$	0	0
$961$	−26960.2	−0.904978
$962$	10286.1	0.344737
$963$	0	0
$964$	5134.54	0.171548
$965$	95203.8	3.17588
$966$	0	0
$967$	−35157.0	−1.16916	−0.584578	−	0.811338i	$-0.698740\pi$
$967$	−35157.0	−1.16916	−0.584578	+	0.811338i	$0.698740\pi$
$968$	−7302.59	−0.242473
$969$	0	0
$970$	32961.5	1.09106
$971$	−24254.7	−0.801617	−0.400808	−	0.916162i	$-0.631271\pi$
$971$	−24254.7	−0.801617	−0.400808	+	0.916162i	$0.631271\pi$
$972$	0	0
$973$	0	0
$974$	32020.9	1.05340
$975$	0	0
$976$	−17769.3	−0.582769
$977$	−12769.3	−0.418142	−0.209071	−	0.977900i	$-0.567044\pi$
$977$	−12769.3	−0.418142	−0.209071	+	0.977900i	$0.567044\pi$
$978$	0	0
$979$	−20553.9	−0.670997
$980$	0	0
$981$	0	0
$982$	53880.5	1.75091
$983$	−4753.76	−0.154244	−0.0771218	−	0.997022i	$-0.524573\pi$
$983$	−4753.76	−0.154244	−0.0771218	+	0.997022i	$0.524573\pi$
$984$	0	0
$985$	−27565.9	−0.891700
$986$	602.727	0.0194673
$987$	0	0
$988$	9583.14	0.308583
$989$	−12318.6	−0.396065
$990$	0	0
$991$	−23723.1	−0.760433	−0.380216	−	0.924898i	$-0.624150\pi$
$991$	−23723.1	−0.760433	−0.380216	+	0.924898i	$0.624150\pi$
$992$	−6551.44	−0.209686
$993$	0	0
$994$	0	0
$995$	12738.2	0.405856
$996$	0	0
$997$	−14203.6	−0.451185	−0.225593	−	0.974222i	$-0.572432\pi$
$997$	−14203.6	−0.451185	−0.225593	+	0.974222i	$0.572432\pi$
$998$	−17953.5	−0.569447
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bo.1.6		8
3.2	odd	2	inner	1323.4.a.bo.1.3		8
7.2	even	3		189.4.e.h.109.3	✓	16
7.4	even	3		189.4.e.h.163.3	yes	16
7.6	odd	2		1323.4.a.bn.1.6		8
21.2	odd	6		189.4.e.h.109.6	yes	16
21.11	odd	6		189.4.e.h.163.6	yes	16
21.20	even	2		1323.4.a.bn.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.h.109.3	✓	16	7.2	even	3
189.4.e.h.109.6	yes	16	21.2	odd	6
189.4.e.h.163.3	yes	16	7.4	even	3
189.4.e.h.163.6	yes	16	21.11	odd	6
1323.4.a.bn.1.3		8	21.20	even	2
1323.4.a.bn.1.6		8	7.6	odd	2
1323.4.a.bo.1.3		8	3.2	odd	2	inner
1323.4.a.bo.1.6		8	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)

\(f(q)\)	\(=\)	\(q+3.29273 q^{2} +2.84210 q^{4} +21.7165 q^{5} -16.9836 q^{8} +O(q^{10})\)	\(q+3.29273 q^{2} +2.84210 q^{4} +21.7165 q^{5} -16.9836 q^{8} +71.5066 q^{10} -41.9640 q^{11} +46.0329 q^{13} -78.6593 q^{16} +2.01763 q^{17} +73.2487 q^{19} +61.7204 q^{20} -138.176 q^{22} -24.1677 q^{23} +346.605 q^{25} +151.574 q^{26} +90.7240 q^{29} +53.2052 q^{31} -123.135 q^{32} +6.64353 q^{34} +67.8619 q^{37} +241.189 q^{38} -368.824 q^{40} +341.211 q^{41} +509.713 q^{43} -119.266 q^{44} -79.5778 q^{46} +38.4702 q^{47} +1141.28 q^{50} +130.830 q^{52} +389.388 q^{53} -911.311 q^{55} +298.730 q^{58} +450.621 q^{59} +225.903 q^{61} +175.191 q^{62} +223.822 q^{64} +999.672 q^{65} -772.012 q^{67} +5.73431 q^{68} -962.655 q^{71} -1053.79 q^{73} +223.451 q^{74} +208.180 q^{76} +33.5907 q^{79} -1708.20 q^{80} +1123.52 q^{82} +446.386 q^{83} +43.8159 q^{85} +1678.35 q^{86} +712.700 q^{88} +489.798 q^{89} -68.6870 q^{92} +126.672 q^{94} +1590.70 q^{95} +460.958 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 48 q^{4}+O(q^{10}) \) 8 * q + 48 * q^4	\( 8 q + 48 q^{4} + 44 q^{10} - 84 q^{13} + 156 q^{16} + 12 q^{19} + 224 q^{22} + 408 q^{25} + 800 q^{31} - 948 q^{34} + 692 q^{37} + 96 q^{40} + 1456 q^{43} + 1524 q^{46} + 1972 q^{52} - 1280 q^{55} + 2372 q^{58} + 216 q^{61} + 4964 q^{64} + 684 q^{67} - 4564 q^{73} - 380 q^{76} + 556 q^{79} + 3340 q^{82} + 1296 q^{85} + 6696 q^{88} - 492 q^{94} + 584 q^{97}+O(q^{100}) \) 8 * q + 48 * q^4 + 44 * q^10 - 84 * q^13 + 156 * q^16 + 12 * q^19 + 224 * q^22 + 408 * q^25 + 800 * q^31 - 948 * q^34 + 692 * q^37 + 96 * q^40 + 1456 * q^43 + 1524 * q^46 + 1972 * q^52 - 1280 * q^55 + 2372 * q^58 + 216 * q^61 + 4964 * q^64 + 684 * q^67 - 4564 * q^73 - 380 * q^76 + 556 * q^79 + 3340 * q^82 + 1296 * q^85 + 6696 * q^88 - 492 * q^94 + 584 * q^97

Introduction

L-functions

Modular forms

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