LMFDB - Embedded newform 1323.4.a.bc.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:	$6$
Coefficient field:	6.6.346909504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 $ x^6 - x^5 - 12x^4 + 2x^3 + 39x^2 + 25x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.69129$ 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)

$f(q)$	$=$	$q-2.78484 q^{2} -0.244684 q^{4} +20.8311 q^{5} +22.9601 q^{8} +O(q^{10})$	$q-2.78484 q^{2} -0.244684 q^{4} +20.8311 q^{5} +22.9601 q^{8} -58.0112 q^{10} -65.7898 q^{11} +50.9140 q^{13} -61.9827 q^{16} -46.1303 q^{17} -62.1235 q^{19} -5.09704 q^{20} +183.214 q^{22} -125.329 q^{23} +308.935 q^{25} -141.787 q^{26} +168.840 q^{29} -187.101 q^{31} -11.0692 q^{32} +128.465 q^{34} -70.0464 q^{37} +173.004 q^{38} +478.284 q^{40} +188.366 q^{41} +239.590 q^{43} +16.0977 q^{44} +349.020 q^{46} +591.775 q^{47} -860.332 q^{50} -12.4579 q^{52} +464.090 q^{53} -1370.47 q^{55} -470.191 q^{58} -325.056 q^{59} +261.058 q^{61} +521.045 q^{62} +526.687 q^{64} +1060.59 q^{65} +340.985 q^{67} +11.2874 q^{68} +752.254 q^{71} -245.233 q^{73} +195.068 q^{74} +15.2006 q^{76} +546.822 q^{79} -1291.17 q^{80} -524.568 q^{82} -144.397 q^{83} -960.945 q^{85} -667.219 q^{86} -1510.54 q^{88} -603.758 q^{89} +30.6659 q^{92} -1648.00 q^{94} -1294.10 q^{95} -1355.27 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{4} + 24 q^{5}+O(q^{10}) $ 6 * q + 18 * q^4 + 24 * q^5	$ 6 q + 18 q^{4} + 24 q^{5} - 30 q^{16} + 42 q^{17} + 12 q^{20} + 132 q^{22} + 222 q^{25} + 366 q^{26} - 312 q^{37} + 336 q^{38} + 360 q^{41} + 654 q^{43} + 774 q^{46} + 1812 q^{47} - 378 q^{58} - 6 q^{59} + 2058 q^{62} + 66 q^{64} + 42 q^{67} + 2910 q^{68} + 1956 q^{79} - 2868 q^{80} + 2892 q^{83} - 1944 q^{85} - 2532 q^{88} + 1518 q^{89}+O(q^{100}) $ 6 * q + 18 * q^4 + 24 * q^5 - 30 * q^16 + 42 * q^17 + 12 * q^20 + 132 * q^22 + 222 * q^25 + 366 * q^26 - 312 * q^37 + 336 * q^38 + 360 * q^41 + 654 * q^43 + 774 * q^46 + 1812 * q^47 - 378 * q^58 - 6 * q^59 + 2058 * q^62 + 66 * q^64 + 42 * q^67 + 2910 * q^68 + 1956 * q^79 - 2868 * q^80 + 2892 * q^83 - 1944 * q^85 - 2532 * q^88 + 1518 * q^89

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 $

$2$	−2.78484	−0.984588	−0.492294	−	0.870429i	$-0.663842\pi$
$2$	−2.78484	−0.984588	−0.492294	+	0.870429i	$0.663842\pi$
$3$	0	0
$3$	0	0
$4$	−0.244684	−0.0305855
$5$	20.8311	1.86319	0.931595	−	0.363498i	$-0.118418\pi$
$5$	20.8311	1.86319	0.931595	+	0.363498i	$0.118418\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.9601	1.01470
$9$	0	0
$10$	−58.0112	−1.83448
$11$	−65.7898	−1.80331	−0.901654	−	0.432459i	$-0.857646\pi$
$11$	−65.7898	−1.80331	−0.901654	+	0.432459i	$0.857646\pi$
$12$	0	0
$13$	50.9140	1.08623	0.543115	−	0.839658i	$-0.317245\pi$
$13$	50.9140	1.08623	0.543115	+	0.839658i	$0.317245\pi$
$14$	0	0
$15$	0	0
$16$	−61.9827	−0.968479
$17$	−46.1303	−0.658132	−0.329066	−	0.944307i	$-0.606734\pi$
$17$	−46.1303	−0.658132	−0.329066	+	0.944307i	$0.606734\pi$
$18$	0	0
$19$	−62.1235	−0.750110	−0.375055	−	0.927002i	$-0.622376\pi$
$19$	−62.1235	−0.750110	−0.375055	+	0.927002i	$0.622376\pi$
$20$	−5.09704	−0.0569867
$21$	0	0
$22$	183.214	1.77552
$23$	−125.329	−1.13621	−0.568105	−	0.822956i	$-0.692323\pi$
$23$	−125.329	−1.13621	−0.568105	+	0.822956i	$0.692323\pi$
$24$	0	0
$25$	308.935	2.47148
$26$	−141.787	−1.06949
$27$	0	0
$28$	0	0
$29$	168.840	1.08113	0.540565	−	0.841303i	$-0.318211\pi$
$29$	168.840	1.08113	0.540565	+	0.841303i	$0.318211\pi$
$30$	0	0
$31$	−187.101	−1.08401	−0.542005	−	0.840376i	$-0.682334\pi$
$31$	−187.101	−1.08401	−0.542005	+	0.840376i	$0.682334\pi$
$32$	−11.0692	−0.0611494
$33$	0	0
$34$	128.465	0.647989
$35$	0	0
$36$	0	0
$37$	−70.0464	−0.311231	−0.155616	−	0.987818i	$-0.549736\pi$
$37$	−70.0464	−0.311231	−0.155616	+	0.987818i	$0.549736\pi$
$38$	173.004	0.738550
$39$	0	0
$40$	478.284	1.89058
$41$	188.366	0.717507	0.358753	−	0.933432i	$-0.383202\pi$
$41$	188.366	0.717507	0.358753	+	0.933432i	$0.383202\pi$
$42$	0	0
$43$	239.590	0.849701	0.424851	−	0.905264i	$-0.360327\pi$
$43$	239.590	0.849701	0.424851	+	0.905264i	$0.360327\pi$
$44$	16.0977	0.0551551
$45$	0	0
$46$	349.020	1.11870
$47$	591.775	1.83658	0.918289	−	0.395910i	$-0.129571\pi$
$47$	591.775	1.83658	0.918289	+	0.395910i	$0.129571\pi$
$48$	0	0
$49$	0	0
$50$	−860.332	−2.43339
$51$	0	0
$52$	−12.4579	−0.0332230
$53$	464.090	1.20279	0.601393	−	0.798953i	$-0.294612\pi$
$53$	464.090	1.20279	0.601393	+	0.798953i	$0.294612\pi$
$54$	0	0
$55$	−1370.47	−3.35990
$56$	0	0
$57$	0	0
$58$	−470.191	−1.06447
$59$	−325.056	−0.717265	−0.358633	−	0.933479i	$-0.616757\pi$
$59$	−325.056	−0.717265	−0.358633	+	0.933479i	$0.616757\pi$
$60$	0	0
$61$	261.058	0.547951	0.273976	−	0.961737i	$-0.411661\pi$
$61$	261.058	0.547951	0.273976	+	0.961737i	$0.411661\pi$
$62$	521.045	1.06730
$63$	0	0
$64$	526.687	1.02869
$65$	1060.59	2.02385
$66$	0	0
$67$	340.985	0.621760	0.310880	−	0.950449i	$-0.399376\pi$
$67$	340.985	0.621760	0.310880	+	0.950449i	$0.399376\pi$
$68$	11.2874	0.0201293
$69$	0	0
$70$	0	0
$71$	752.254	1.25741	0.628705	−	0.777644i	$-0.283585\pi$
$71$	752.254	1.25741	0.628705	+	0.777644i	$0.283585\pi$
$72$	0	0
$73$	−245.233	−0.393183	−0.196591	−	0.980485i	$-0.562987\pi$
$73$	−245.233	−0.393183	−0.196591	+	0.980485i	$0.562987\pi$
$74$	195.068	0.306435
$75$	0	0
$76$	15.2006	0.0229425
$77$	0	0
$78$	0	0
$79$	546.822	0.778763	0.389382	−	0.921076i	$-0.372689\pi$
$79$	546.822	0.778763	0.389382	+	0.921076i	$0.372689\pi$
$80$	−1291.17	−1.80446
$81$	0	0
$82$	−524.568	−0.706449
$83$	−144.397	−0.190959	−0.0954794	−	0.995431i	$-0.530438\pi$
$83$	−144.397	−0.190959	−0.0954794	+	0.995431i	$0.530438\pi$
$84$	0	0
$85$	−960.945	−1.22622
$86$	−667.219	−0.836606
$87$	0	0
$88$	−1510.54	−1.82982
$89$	−603.758	−0.719081	−0.359540	−	0.933129i	$-0.617067\pi$
$89$	−603.758	−0.719081	−0.359540	+	0.933129i	$0.617067\pi$
$90$	0	0
$91$	0	0
$92$	30.6659	0.0347516
$93$	0	0
$94$	−1648.00	−1.80827
$95$	−1294.10	−1.39760
$96$	0	0
$97$	−1355.27	−1.41863	−0.709315	−	0.704891i	$-0.750996\pi$
$97$	−1355.27	−1.41863	−0.709315	+	0.704891i	$0.750996\pi$
$98$	0	0
$99$	0	0
$100$	−75.5914	−0.0755914
$101$	1690.26	1.66522	0.832608	−	0.553863i	$-0.186847\pi$
$101$	1690.26	1.66522	0.832608	+	0.553863i	$0.186847\pi$
$102$	0	0
$103$	−189.737	−0.181508	−0.0907542	−	0.995873i	$-0.528928\pi$
$103$	−189.737	−0.181508	−0.0907542	+	0.995873i	$0.528928\pi$
$104$	1168.99	1.10220
$105$	0	0
$106$	−1292.42	−1.18425
$107$	−741.325	−0.669781	−0.334891	−	0.942257i	$-0.608699\pi$
$107$	−741.325	−0.669781	−0.334891	+	0.942257i	$0.608699\pi$
$108$	0	0
$109$	−607.599	−0.533921	−0.266961	−	0.963707i	$-0.586019\pi$
$109$	−607.599	−0.533921	−0.266961	+	0.963707i	$0.586019\pi$
$110$	3816.55	3.30812
$111$	0	0
$112$	0	0
$113$	1475.76	1.22856	0.614281	−	0.789088i	$-0.289446\pi$
$113$	1475.76	1.22856	0.614281	+	0.789088i	$0.289446\pi$
$114$	0	0
$115$	−2610.73	−2.11697
$116$	−41.3124	−0.0330669
$117$	0	0
$118$	905.227	0.706211
$119$	0	0
$120$	0	0
$121$	2997.30	2.25192
$122$	−727.004	−0.539507
$123$	0	0
$124$	45.7806	0.0331550
$125$	3831.56	2.74164
$126$	0	0
$127$	−1560.29	−1.09018	−0.545090	−	0.838377i	$-0.683505\pi$
$127$	−1560.29	−1.09018	−0.545090	+	0.838377i	$0.683505\pi$
$128$	−1378.18	−0.951683
$129$	0	0
$130$	−2953.58	−1.99266
$131$	736.880	0.491462	0.245731	−	0.969338i	$-0.420972\pi$
$131$	736.880	0.491462	0.245731	+	0.969338i	$0.420972\pi$
$132$	0	0
$133$	0	0
$134$	−949.587	−0.612178
$135$	0	0
$136$	−1059.16	−0.667808
$137$	373.279	0.232784	0.116392	−	0.993203i	$-0.462867\pi$
$137$	373.279	0.232784	0.116392	+	0.993203i	$0.462867\pi$
$138$	0	0
$139$	−1292.10	−0.788450	−0.394225	−	0.919014i	$-0.628987\pi$
$139$	−1292.10	−0.788450	−0.394225	+	0.919014i	$0.628987\pi$
$140$	0	0
$141$	0	0
$142$	−2094.90	−1.23803
$143$	−3349.62	−1.95881
$144$	0	0
$145$	3517.12	2.01435
$146$	682.934	0.387123
$147$	0	0
$148$	17.1393	0.00951918
$149$	−593.676	−0.326415	−0.163207	−	0.986592i	$-0.552184\pi$
$149$	−593.676	−0.326415	−0.163207	+	0.986592i	$0.552184\pi$
$150$	0	0
$151$	1172.62	0.631965	0.315982	−	0.948765i	$-0.397666\pi$
$151$	1172.62	0.631965	0.315982	+	0.948765i	$0.397666\pi$
$152$	−1426.36	−0.761139
$153$	0	0
$154$	0	0
$155$	−3897.51	−2.01971
$156$	0	0
$157$	2308.19	1.17333	0.586667	−	0.809828i	$-0.300440\pi$
$157$	2308.19	1.17333	0.586667	+	0.809828i	$0.300440\pi$
$158$	−1522.81	−0.766762
$159$	0	0
$160$	−230.584	−0.113933
$161$	0	0
$162$	0	0
$163$	550.709	0.264631	0.132316	−	0.991208i	$-0.457759\pi$
$163$	550.709	0.264631	0.132316	+	0.991208i	$0.457759\pi$
$164$	−46.0901	−0.0219453
$165$	0	0
$166$	402.121	0.188016
$167$	1875.76	0.869167	0.434584	−	0.900631i	$-0.356896\pi$
$167$	1875.76	0.869167	0.434584	+	0.900631i	$0.356896\pi$
$168$	0	0
$169$	395.234	0.179897
$170$	2676.07	1.20733
$171$	0	0
$172$	−58.6239	−0.0259886
$173$	3576.61	1.57182	0.785910	−	0.618341i	$-0.212195\pi$
$173$	3576.61	1.57182	0.785910	+	0.618341i	$0.212195\pi$
$174$	0	0
$175$	0	0
$176$	4077.83	1.74646
$177$	0	0
$178$	1681.37	0.707999
$179$	1847.79	0.771567	0.385783	−	0.922589i	$-0.373931\pi$
$179$	1847.79	0.771567	0.385783	+	0.922589i	$0.373931\pi$
$180$	0	0
$181$	3789.93	1.55637	0.778187	−	0.628033i	$-0.216140\pi$
$181$	3789.93	1.55637	0.778187	+	0.628033i	$0.216140\pi$
$182$	0	0
$183$	0	0
$184$	−2877.56	−1.15291
$185$	−1459.14	−0.579883
$186$	0	0
$187$	3034.90	1.18681
$188$	−144.798	−0.0561728
$189$	0	0
$190$	3603.86	1.37606
$191$	3446.71	1.30573	0.652867	−	0.757472i	$-0.273566\pi$
$191$	3446.71	1.30573	0.652867	+	0.757472i	$0.273566\pi$
$192$	0	0
$193$	1264.46	0.471594	0.235797	−	0.971802i	$-0.424230\pi$
$193$	1264.46	0.471594	0.235797	+	0.971802i	$0.424230\pi$
$194$	3774.22	1.39677
$195$	0	0
$196$	0	0
$197$	−2591.45	−0.937226	−0.468613	−	0.883404i	$-0.655246\pi$
$197$	−2591.45	−0.937226	−0.468613	+	0.883404i	$0.655246\pi$
$198$	0	0
$199$	86.8060	0.0309222	0.0154611	−	0.999880i	$-0.495078\pi$
$199$	86.8060	0.0309222	0.0154611	+	0.999880i	$0.495078\pi$
$200$	7093.17	2.50781
$201$	0	0
$202$	−4707.09	−1.63955
$203$	0	0
$204$	0	0
$205$	3923.86	1.33685
$206$	528.388	0.178711
$207$	0	0
$208$	−3155.78	−1.05199
$209$	4087.09	1.35268
$210$	0	0
$211$	−1586.52	−0.517633	−0.258817	−	0.965927i	$-0.583333\pi$
$211$	−1586.52	−0.517633	−0.258817	+	0.965927i	$0.583333\pi$
$212$	−113.556	−0.0367879
$213$	0	0
$214$	2064.47	0.659459
$215$	4990.92	1.58315
$216$	0	0
$217$	0	0
$218$	1692.06	0.525693
$219$	0	0
$220$	335.334	0.102764
$221$	−2348.68	−0.714883
$222$	0	0
$223$	−519.207	−0.155913	−0.0779567	−	0.996957i	$-0.524840\pi$
$223$	−519.207	−0.155913	−0.0779567	+	0.996957i	$0.524840\pi$
$224$	0	0
$225$	0	0
$226$	−4109.74	−1.20963
$227$	−931.422	−0.272338	−0.136169	−	0.990686i	$-0.543479\pi$
$227$	−931.422	−0.272338	−0.136169	+	0.990686i	$0.543479\pi$
$228$	0	0
$229$	−5040.17	−1.45443	−0.727213	−	0.686412i	$-0.759185\pi$
$229$	−5040.17	−1.45443	−0.727213	+	0.686412i	$0.759185\pi$
$230$	7270.46	2.08435
$231$	0	0
$232$	3876.58	1.09702
$233$	191.642	0.0538835	0.0269418	−	0.999637i	$-0.491423\pi$
$233$	191.642	0.0538835	0.0269418	+	0.999637i	$0.491423\pi$
$234$	0	0
$235$	12327.3	3.42190
$236$	79.5361	0.0219380
$237$	0	0
$238$	0	0
$239$	2806.82	0.759658	0.379829	−	0.925057i	$-0.375983\pi$
$239$	2806.82	0.759658	0.379829	+	0.925057i	$0.375983\pi$
$240$	0	0
$241$	3827.25	1.02297	0.511484	−	0.859293i	$-0.329096\pi$
$241$	3827.25	1.02297	0.511484	+	0.859293i	$0.329096\pi$
$242$	−8346.99	−2.21721
$243$	0	0
$244$	−63.8768	−0.0167594
$245$	0	0
$246$	0	0
$247$	−3162.95	−0.814793
$248$	−4295.85	−1.09995
$249$	0	0
$250$	−10670.3	−2.69939
$251$	5242.63	1.31837	0.659187	−	0.751979i	$-0.270900\pi$
$251$	5242.63	1.31837	0.659187	+	0.751979i	$0.270900\pi$
$252$	0	0
$253$	8245.34	2.04893
$254$	4345.14	1.07338
$255$	0	0
$256$	−375.480	−0.0916699
$257$	−1722.90	−0.418176	−0.209088	−	0.977897i	$-0.567050\pi$
$257$	−1722.90	−0.418176	−0.209088	+	0.977897i	$0.567050\pi$
$258$	0	0
$259$	0	0
$260$	−259.511	−0.0619007
$261$	0	0
$262$	−2052.09	−0.483888
$263$	−1478.24	−0.346586	−0.173293	−	0.984870i	$-0.555441\pi$
$263$	−1478.24	−0.346586	−0.173293	+	0.984870i	$0.555441\pi$
$264$	0	0
$265$	9667.51	2.24102
$266$	0	0
$267$	0	0
$268$	−83.4337	−0.0190169
$269$	2540.01	0.575714	0.287857	−	0.957673i	$-0.407057\pi$
$269$	2540.01	0.575714	0.287857	+	0.957673i	$0.407057\pi$
$270$	0	0
$271$	2439.03	0.546718	0.273359	−	0.961912i	$-0.411865\pi$
$271$	2439.03	0.546718	0.273359	+	0.961912i	$0.411865\pi$
$272$	2859.28	0.637387
$273$	0	0
$274$	−1039.52	−0.229196
$275$	−20324.7	−4.45683
$276$	0	0
$277$	−4322.03	−0.937493	−0.468747	−	0.883333i	$-0.655294\pi$
$277$	−4322.03	−0.937493	−0.468747	+	0.883333i	$0.655294\pi$
$278$	3598.29	0.776299
$279$	0	0
$280$	0	0
$281$	−2793.78	−0.593107	−0.296553	−	0.955016i	$-0.595837\pi$
$281$	−2793.78	−0.593107	−0.296553	+	0.955016i	$0.595837\pi$
$282$	0	0
$283$	4897.86	1.02879	0.514395	−	0.857553i	$-0.328016\pi$
$283$	4897.86	1.02879	0.514395	+	0.857553i	$0.328016\pi$
$284$	−184.065	−0.0384586
$285$	0	0
$286$	9328.15	1.92862
$287$	0	0
$288$	0	0
$289$	−2785.00	−0.566863
$290$	−9794.59	−1.98330
$291$	0	0
$292$	60.0047	0.0120257
$293$	−6039.14	−1.20413	−0.602065	−	0.798447i	$-0.705655\pi$
$293$	−6039.14	−1.20413	−0.602065	+	0.798447i	$0.705655\pi$
$294$	0	0
$295$	−6771.27	−1.33640
$296$	−1608.27	−0.315807
$297$	0	0
$298$	1653.29	0.321384
$299$	−6380.98	−1.23419
$300$	0	0
$301$	0	0
$302$	−3265.56	−0.622225
$303$	0	0
$304$	3850.58	0.726466
$305$	5438.12	1.02094
$306$	0	0
$307$	−7116.12	−1.32293	−0.661463	−	0.749978i	$-0.730064\pi$
$307$	−7116.12	−1.32293	−0.661463	+	0.749978i	$0.730064\pi$
$308$	0	0
$309$	0	0
$310$	10853.9	1.98859
$311$	10565.3	1.92638	0.963191	−	0.268819i	$-0.0866335\pi$
$311$	10565.3	1.92638	0.963191	+	0.268819i	$0.0866335\pi$
$312$	0	0
$313$	3419.87	0.617580	0.308790	−	0.951130i	$-0.400076\pi$
$313$	3419.87	0.617580	0.308790	+	0.951130i	$0.400076\pi$
$314$	−6427.93	−1.15525
$315$	0	0
$316$	−133.799	−0.0238189
$317$	10098.2	1.78918	0.894589	−	0.446889i	$-0.147468\pi$
$317$	10098.2	1.78918	0.894589	+	0.446889i	$0.147468\pi$
$318$	0	0
$319$	−11107.9	−1.94961
$320$	10971.5	1.91664
$321$	0	0
$322$	0	0
$323$	2865.77	0.493672
$324$	0	0
$325$	15729.1	2.68459
$326$	−1533.64	−0.260553
$327$	0	0
$328$	4324.90	0.728056
$329$	0	0
$330$	0	0
$331$	−6978.18	−1.15878	−0.579389	−	0.815051i	$-0.696709\pi$
$331$	−6978.18	−1.15878	−0.579389	+	0.815051i	$0.696709\pi$
$332$	35.3316	0.00584058
$333$	0	0
$334$	−5223.69	−0.855772
$335$	7103.09	1.15846
$336$	0	0
$337$	−3376.48	−0.545782	−0.272891	−	0.962045i	$-0.587980\pi$
$337$	−3376.48	−0.545782	−0.272891	+	0.962045i	$0.587980\pi$
$338$	−1100.66	−0.177125
$339$	0	0
$340$	235.128	0.0375047
$341$	12309.3	1.95480
$342$	0	0
$343$	0	0
$344$	5501.01	0.862194
$345$	0	0
$346$	−9960.28	−1.54760
$347$	−3781.51	−0.585020	−0.292510	−	0.956262i	$-0.594490\pi$
$347$	−3781.51	−0.585020	−0.292510	+	0.956262i	$0.594490\pi$
$348$	0	0
$349$	985.586	0.151167	0.0755834	−	0.997139i	$-0.475918\pi$
$349$	985.586	0.151167	0.0755834	+	0.997139i	$0.475918\pi$
$350$	0	0
$351$	0	0
$352$	728.242	0.110271
$353$	−5674.48	−0.855587	−0.427793	−	0.903877i	$-0.640709\pi$
$353$	−5674.48	−0.855587	−0.427793	+	0.903877i	$0.640709\pi$
$354$	0	0
$355$	15670.3	2.34279
$356$	147.730	0.0219935
$357$	0	0
$358$	−5145.80	−0.759676
$359$	8038.50	1.18177	0.590886	−	0.806755i	$-0.298778\pi$
$359$	8038.50	1.18177	0.590886	+	0.806755i	$0.298778\pi$
$360$	0	0
$361$	−2999.68	−0.437334
$362$	−10554.3	−1.53239
$363$	0	0
$364$	0	0
$365$	−5108.47	−0.732574
$366$	0	0
$367$	3466.45	0.493044	0.246522	−	0.969137i	$-0.420712\pi$
$367$	3466.45	0.493044	0.246522	+	0.969137i	$0.420712\pi$
$368$	7768.20	1.10039
$369$	0	0
$370$	4063.48	0.570946
$371$	0	0
$372$	0	0
$373$	10339.9	1.43534	0.717670	−	0.696384i	$-0.245209\pi$
$373$	10339.9	1.43534	0.717670	+	0.696384i	$0.245209\pi$
$374$	−8451.71	−1.16852
$375$	0	0
$376$	13587.2	1.86358
$377$	8596.30	1.17436
$378$	0	0
$379$	−1299.74	−0.176156	−0.0880780	−	0.996114i	$-0.528072\pi$
$379$	−1299.74	−0.176156	−0.0880780	+	0.996114i	$0.528072\pi$
$380$	316.646	0.0427463
$381$	0	0
$382$	−9598.53	−1.28561
$383$	−10646.4	−1.42038	−0.710190	−	0.704010i	$-0.751391\pi$
$383$	−10646.4	−1.42038	−0.710190	+	0.704010i	$0.751391\pi$
$384$	0	0
$385$	0	0
$386$	−3521.31	−0.464326
$387$	0	0
$388$	331.614	0.0433896
$389$	11895.3	1.55042	0.775212	−	0.631701i	$-0.217643\pi$
$389$	11895.3	1.55042	0.775212	+	0.631701i	$0.217643\pi$
$390$	0	0
$391$	5781.44	0.747775
$392$	0	0
$393$	0	0
$394$	7216.78	0.922781
$395$	11390.9	1.45098
$396$	0	0
$397$	−12111.6	−1.53114	−0.765570	−	0.643353i	$-0.777543\pi$
$397$	−12111.6	−1.53114	−0.765570	+	0.643353i	$0.777543\pi$
$398$	−241.741	−0.0304456
$399$	0	0
$400$	−19148.6	−2.39357
$401$	7394.86	0.920902	0.460451	−	0.887685i	$-0.347688\pi$
$401$	7394.86	0.920902	0.460451	+	0.887685i	$0.347688\pi$
$402$	0	0
$403$	−9526.04	−1.17748
$404$	−413.579	−0.0509315
$405$	0	0
$406$	0	0
$407$	4608.34	0.561246
$408$	0	0
$409$	15003.4	1.81386	0.906931	−	0.421280i	$-0.138419\pi$
$409$	15003.4	1.81386	0.906931	+	0.421280i	$0.138419\pi$
$410$	−10927.3	−1.31625
$411$	0	0
$412$	46.4258	0.00555154
$413$	0	0
$414$	0	0
$415$	−3007.94	−0.355793
$416$	−563.578	−0.0664224
$417$	0	0
$418$	−11381.9	−1.33183
$419$	1102.65	0.128563	0.0642815	−	0.997932i	$-0.479524\pi$
$419$	1102.65	0.128563	0.0642815	+	0.997932i	$0.479524\pi$
$420$	0	0
$421$	7596.04	0.879355	0.439678	−	0.898156i	$-0.355093\pi$
$421$	7596.04	0.879355	0.439678	+	0.898156i	$0.355093\pi$
$422$	4418.20	0.509656
$423$	0	0
$424$	10655.6	1.22047
$425$	−14251.2	−1.62656
$426$	0	0
$427$	0	0
$428$	181.391	0.0204856
$429$	0	0
$430$	−13898.9	−1.55876
$431$	2127.52	0.237770	0.118885	−	0.992908i	$-0.462068\pi$
$431$	2127.52	0.237770	0.118885	+	0.992908i	$0.462068\pi$
$432$	0	0
$433$	−6315.20	−0.700899	−0.350449	−	0.936582i	$-0.613971\pi$
$433$	−6315.20	−0.700899	−0.350449	+	0.936582i	$0.613971\pi$
$434$	0	0
$435$	0	0
$436$	148.670	0.0163303
$437$	7785.84	0.852282
$438$	0	0
$439$	−10983.8	−1.19414	−0.597068	−	0.802190i	$-0.703668\pi$
$439$	−10983.8	−1.19414	−0.597068	+	0.802190i	$0.703668\pi$
$440$	−31466.2	−3.40930
$441$	0	0
$442$	6540.68	0.703865
$443$	11095.3	1.18997	0.594983	−	0.803738i	$-0.297159\pi$
$443$	11095.3	1.18997	0.594983	+	0.803738i	$0.297159\pi$
$444$	0	0
$445$	−12576.9	−1.33978
$446$	1445.91	0.153510
$447$	0	0
$448$	0	0
$449$	9190.22	0.965954	0.482977	−	0.875633i	$-0.339555\pi$
$449$	9190.22	0.965954	0.482977	+	0.875633i	$0.339555\pi$
$450$	0	0
$451$	−12392.5	−1.29389
$452$	−361.094	−0.0375762
$453$	0	0
$454$	2593.86	0.268140
$455$	0	0
$456$	0	0
$457$	−295.522	−0.0302493	−0.0151247	−	0.999886i	$-0.504815\pi$
$457$	−295.522	−0.0302493	−0.0151247	+	0.999886i	$0.504815\pi$
$458$	14036.0	1.43201
$459$	0	0
$460$	638.805	0.0647488
$461$	−10319.9	−1.04262	−0.521309	−	0.853368i	$-0.674556\pi$
$461$	−10319.9	−1.04262	−0.521309	+	0.853368i	$0.674556\pi$
$462$	0	0
$463$	−10886.8	−1.09277	−0.546385	−	0.837534i	$-0.683996\pi$
$463$	−10886.8	−1.09277	−0.546385	+	0.837534i	$0.683996\pi$
$464$	−10465.1	−1.04705
$465$	0	0
$466$	−533.691	−0.0530531
$467$	5056.59	0.501052	0.250526	−	0.968110i	$-0.419396\pi$
$467$	5056.59	0.501052	0.250526	+	0.968110i	$0.419396\pi$
$468$	0	0
$469$	0	0
$470$	−34329.6	−3.36916
$471$	0	0
$472$	−7463.31	−0.727811
$473$	−15762.6	−1.53227
$474$	0	0
$475$	−19192.1	−1.85388
$476$	0	0
$477$	0	0
$478$	−7816.54	−0.747950
$479$	389.667	0.0371698	0.0185849	−	0.999827i	$-0.494084\pi$
$479$	389.667	0.0371698	0.0185849	+	0.999827i	$0.494084\pi$
$480$	0	0
$481$	−3566.34	−0.338069
$482$	−10658.3	−1.00720
$483$	0	0
$484$	−733.392	−0.0688761
$485$	−28231.8	−2.64318
$486$	0	0
$487$	15067.5	1.40200	0.700998	−	0.713163i	$-0.252738\pi$
$487$	15067.5	1.40200	0.700998	+	0.713163i	$0.252738\pi$
$488$	5993.91	0.556008
$489$	0	0
$490$	0	0
$491$	17552.7	1.61333	0.806663	−	0.591012i	$-0.201271\pi$
$491$	17552.7	1.61333	0.806663	+	0.591012i	$0.201271\pi$
$492$	0	0
$493$	−7788.63	−0.711525
$494$	8808.31	0.802236
$495$	0	0
$496$	11597.0	1.04984
$497$	0	0
$498$	0	0
$499$	−19298.9	−1.73133	−0.865667	−	0.500620i	$-0.833105\pi$
$499$	−19298.9	−1.73133	−0.865667	+	0.500620i	$0.833105\pi$
$500$	−937.522	−0.0838545
$501$	0	0
$502$	−14599.9	−1.29806
$503$	10038.8	0.889881	0.444940	−	0.895560i	$-0.353225\pi$
$503$	10038.8	0.889881	0.444940	+	0.895560i	$0.353225\pi$
$504$	0	0
$505$	35209.9	3.10261
$506$	−22961.9	−2.01736
$507$	0	0
$508$	381.777	0.0333438
$509$	−7594.91	−0.661372	−0.330686	−	0.943741i	$-0.607280\pi$
$509$	−7594.91	−0.661372	−0.330686	+	0.943741i	$0.607280\pi$
$510$	0	0
$511$	0	0
$512$	12071.1	1.04194
$513$	0	0
$514$	4797.98	0.411732
$515$	−3952.44	−0.338185
$516$	0	0
$517$	−38932.7	−3.31192
$518$	0	0
$519$	0	0
$520$	24351.3	2.05361
$521$	−13207.5	−1.11062	−0.555309	−	0.831644i	$-0.687400\pi$
$521$	−13207.5	−1.11062	−0.555309	+	0.831644i	$0.687400\pi$
$522$	0	0
$523$	21135.4	1.76709	0.883544	−	0.468349i	$-0.155151\pi$
$523$	21135.4	1.76709	0.883544	+	0.468349i	$0.155151\pi$
$524$	−180.303	−0.0150316
$525$	0	0
$526$	4116.66	0.341245
$527$	8631.01	0.713421
$528$	0	0
$529$	3540.25	0.290971
$530$	−26922.4	−2.20648
$531$	0	0
$532$	0	0
$533$	9590.45	0.779378
$534$	0	0
$535$	−15442.6	−1.24793
$536$	7829.05	0.630902
$537$	0	0
$538$	−7073.51	−0.566841
$539$	0	0
$540$	0	0
$541$	−16839.0	−1.33820	−0.669099	−	0.743173i	$-0.733320\pi$
$541$	−16839.0	−1.33820	−0.669099	+	0.743173i	$0.733320\pi$
$542$	−6792.30	−0.538292
$543$	0	0
$544$	510.627	0.0402444
$545$	−12656.9	−0.994797
$546$	0	0
$547$	−6413.01	−0.501281	−0.250640	−	0.968080i	$-0.580641\pi$
$547$	−6413.01	−0.501281	−0.250640	+	0.968080i	$0.580641\pi$
$548$	−91.3355	−0.00711982
$549$	0	0
$550$	56601.1	4.38814
$551$	−10488.9	−0.810966
$552$	0	0
$553$	0	0
$554$	12036.2	0.923045
$555$	0	0
$556$	316.157	0.0241152
$557$	−18970.8	−1.44312	−0.721562	−	0.692350i	$-0.756575\pi$
$557$	−18970.8	−1.44312	−0.721562	+	0.692350i	$0.756575\pi$
$558$	0	0
$559$	12198.5	0.922971
$560$	0	0
$561$	0	0
$562$	7780.23	0.583966
$563$	11466.8	0.858378	0.429189	−	0.903215i	$-0.358799\pi$
$563$	11466.8	0.858378	0.429189	+	0.903215i	$0.358799\pi$
$564$	0	0
$565$	30741.6	2.28904
$566$	−13639.7	−1.01294
$567$	0	0
$568$	17271.8	1.27590
$569$	−1329.17	−0.0979293	−0.0489647	−	0.998801i	$-0.515592\pi$
$569$	−1329.17	−0.0979293	−0.0489647	+	0.998801i	$0.515592\pi$
$570$	0	0
$571$	−5171.50	−0.379020	−0.189510	−	0.981879i	$-0.560690\pi$
$571$	−5171.50	−0.379020	−0.189510	+	0.981879i	$0.560690\pi$
$572$	819.600	0.0599112
$573$	0	0
$574$	0	0
$575$	−38718.3	−2.80811
$576$	0	0
$577$	−18629.2	−1.34410	−0.672048	−	0.740507i	$-0.734585\pi$
$577$	−18629.2	−1.34410	−0.672048	+	0.740507i	$0.734585\pi$
$578$	7755.76	0.558126
$579$	0	0
$580$	−860.583	−0.0616099
$581$	0	0
$582$	0	0
$583$	−30532.4	−2.16899
$584$	−5630.57	−0.398964
$585$	0	0
$586$	16818.0	1.18557
$587$	−21184.8	−1.48959	−0.744795	−	0.667293i	$-0.767453\pi$
$587$	−21184.8	−1.48959	−0.744795	+	0.667293i	$0.767453\pi$
$588$	0	0
$589$	11623.3	0.813127
$590$	18856.9	1.31581
$591$	0	0
$592$	4341.66	0.301421
$593$	−25230.1	−1.74718	−0.873588	−	0.486666i	$-0.838213\pi$
$593$	−25230.1	−1.74718	−0.873588	+	0.486666i	$0.838213\pi$
$594$	0	0
$595$	0	0
$596$	145.263	0.00998357
$597$	0	0
$598$	17770.0	1.21516
$599$	1180.40	0.0805173	0.0402587	−	0.999189i	$-0.487182\pi$
$599$	1180.40	0.0805173	0.0402587	+	0.999189i	$0.487182\pi$
$600$	0	0
$601$	14156.0	0.960793	0.480396	−	0.877052i	$-0.340493\pi$
$601$	14156.0	0.960793	0.480396	+	0.877052i	$0.340493\pi$
$602$	0	0
$603$	0	0
$604$	−286.922	−0.0193290
$605$	62437.1	4.19575
$606$	0	0
$607$	5158.89	0.344964	0.172482	−	0.985013i	$-0.444821\pi$
$607$	5158.89	0.344964	0.172482	+	0.985013i	$0.444821\pi$
$608$	687.658	0.0458688
$609$	0	0
$610$	−15144.3	−1.00520
$611$	30129.6	1.99495
$612$	0	0
$613$	−5700.96	−0.375628	−0.187814	−	0.982205i	$-0.560140\pi$
$613$	−5700.96	−0.375628	−0.187814	+	0.982205i	$0.560140\pi$
$614$	19817.2	1.30254
$615$	0	0
$616$	0	0
$617$	−7424.49	−0.484439	−0.242219	−	0.970221i	$-0.577875\pi$
$617$	−7424.49	−0.484439	−0.242219	+	0.970221i	$0.577875\pi$
$618$	0	0
$619$	25672.5	1.66699	0.833494	−	0.552529i	$-0.186337\pi$
$619$	25672.5	1.66699	0.833494	+	0.552529i	$0.186337\pi$
$620$	953.660	0.0617741
$621$	0	0
$622$	−29422.7	−1.89669
$623$	0	0
$624$	0	0
$625$	41198.7	2.63672
$626$	−9523.78	−0.608062
$627$	0	0
$628$	−564.778	−0.0358871
$629$	3231.26	0.204831
$630$	0	0
$631$	−7093.10	−0.447499	−0.223749	−	0.974647i	$-0.571830\pi$
$631$	−7093.10	−0.447499	−0.223749	+	0.974647i	$0.571830\pi$
$632$	12555.1	0.790213
$633$	0	0
$634$	−28121.8	−1.76160
$635$	−32502.5	−2.03121
$636$	0	0
$637$	0	0
$638$	30933.8	1.91956
$639$	0	0
$640$	−28709.1	−1.77317
$641$	3465.01	0.213510	0.106755	−	0.994285i	$-0.465954\pi$
$641$	3465.01	0.213510	0.106755	+	0.994285i	$0.465954\pi$
$642$	0	0
$643$	−5460.79	−0.334919	−0.167459	−	0.985879i	$-0.553556\pi$
$643$	−5460.79	−0.334919	−0.167459	+	0.985879i	$0.553556\pi$
$644$	0	0
$645$	0	0
$646$	−7980.71	−0.486063
$647$	4535.16	0.275573	0.137786	−	0.990462i	$-0.456001\pi$
$647$	4535.16	0.275573	0.137786	+	0.990462i	$0.456001\pi$
$648$	0	0
$649$	21385.4	1.29345
$650$	−43802.9	−2.64322
$651$	0	0
$652$	−134.750	−0.00809389
$653$	961.151	0.0575999	0.0288000	−	0.999585i	$-0.490831\pi$
$653$	961.151	0.0575999	0.0288000	+	0.999585i	$0.490831\pi$
$654$	0	0
$655$	15350.0	0.915687
$656$	−11675.4	−0.694890
$657$	0	0
$658$	0	0
$659$	17654.3	1.04357	0.521787	−	0.853076i	$-0.325266\pi$
$659$	17654.3	1.04357	0.521787	+	0.853076i	$0.325266\pi$
$660$	0	0
$661$	−27374.3	−1.61080	−0.805400	−	0.592732i	$-0.798049\pi$
$661$	−27374.3	−1.61080	−0.805400	+	0.592732i	$0.798049\pi$
$662$	19433.1	1.14092
$663$	0	0
$664$	−3315.36	−0.193766
$665$	0	0
$666$	0	0
$667$	−21160.4	−1.22839
$668$	−458.970	−0.0265839
$669$	0	0
$670$	−19780.9	−1.14060
$671$	−17174.9	−0.988125
$672$	0	0
$673$	23933.1	1.37080	0.685402	−	0.728165i	$-0.259626\pi$
$673$	23933.1	1.37080	0.685402	+	0.728165i	$0.259626\pi$
$674$	9402.94	0.537370
$675$	0	0
$676$	−96.7075	−0.00550225
$677$	17537.7	0.995610	0.497805	−	0.867289i	$-0.334139\pi$
$677$	17537.7	0.995610	0.497805	+	0.867289i	$0.334139\pi$
$678$	0	0
$679$	0	0
$680$	−22063.4	−1.24425
$681$	0	0
$682$	−34279.5	−1.92467
$683$	15170.6	0.849908	0.424954	−	0.905215i	$-0.360290\pi$
$683$	15170.6	0.849908	0.424954	+	0.905215i	$0.360290\pi$
$684$	0	0
$685$	7775.81	0.433720
$686$	0	0
$687$	0	0
$688$	−14850.4	−0.822918
$689$	23628.7	1.30650
$690$	0	0
$691$	−8107.90	−0.446366	−0.223183	−	0.974777i	$-0.571645\pi$
$691$	−8107.90	−0.446366	−0.223183	+	0.974777i	$0.571645\pi$
$692$	−875.141	−0.0480750
$693$	0	0
$694$	10530.9	0.576004
$695$	−26915.9	−1.46903
$696$	0	0
$697$	−8689.37	−0.472214
$698$	−2744.70	−0.148837
$699$	0	0
$700$	0	0
$701$	−24388.3	−1.31403	−0.657015	−	0.753877i	$-0.728181\pi$
$701$	−24388.3	−1.31403	−0.657015	+	0.753877i	$0.728181\pi$
$702$	0	0
$703$	4351.53	0.233458
$704$	−34650.7	−1.85504
$705$	0	0
$706$	15802.5	0.842401
$707$	0	0
$708$	0	0
$709$	16879.7	0.894119	0.447060	−	0.894504i	$-0.352471\pi$
$709$	16879.7	0.894119	0.447060	+	0.894504i	$0.352471\pi$
$710$	−43639.1	−2.30669
$711$	0	0
$712$	−13862.3	−0.729653
$713$	23449.1	1.23166
$714$	0	0
$715$	−69776.3	−3.64963
$716$	−452.126	−0.0235988
$717$	0	0
$718$	−22385.9	−1.16356
$719$	−18111.5	−0.939421	−0.469710	−	0.882821i	$-0.655642\pi$
$719$	−18111.5	−0.939421	−0.469710	+	0.882821i	$0.655642\pi$
$720$	0	0
$721$	0	0
$722$	8353.61	0.430594
$723$	0	0
$724$	−927.338	−0.0476025
$725$	52160.4	2.67198
$726$	0	0
$727$	18001.3	0.918339	0.459170	−	0.888349i	$-0.348147\pi$
$727$	18001.3	0.918339	0.459170	+	0.888349i	$0.348147\pi$
$728$	0	0
$729$	0	0
$730$	14226.3	0.721284
$731$	−11052.4	−0.559215
$732$	0	0
$733$	27617.2	1.39163	0.695816	−	0.718220i	$-0.255043\pi$
$733$	27617.2	1.39163	0.695816	+	0.718220i	$0.255043\pi$
$734$	−9653.49	−0.485445
$735$	0	0
$736$	1387.29	0.0694785
$737$	−22433.3	−1.12122
$738$	0	0
$739$	5128.96	0.255307	0.127653	−	0.991819i	$-0.459255\pi$
$739$	5128.96	0.255307	0.127653	+	0.991819i	$0.459255\pi$
$740$	357.030	0.0177360
$741$	0	0
$742$	0	0
$743$	−30347.6	−1.49845	−0.749224	−	0.662317i	$-0.769573\pi$
$743$	−30347.6	−1.49845	−0.749224	+	0.662317i	$0.769573\pi$
$744$	0	0
$745$	−12366.9	−0.608173
$746$	−28795.0	−1.41322
$747$	0	0
$748$	−742.593	−0.0362993
$749$	0	0
$750$	0	0
$751$	23545.1	1.14404	0.572020	−	0.820239i	$-0.306160\pi$
$751$	23545.1	1.14404	0.572020	+	0.820239i	$0.306160\pi$
$752$	−36679.8	−1.77869
$753$	0	0
$754$	−23939.3	−1.15626
$755$	24427.0	1.17747
$756$	0	0
$757$	10693.8	0.513439	0.256719	−	0.966486i	$-0.417358\pi$
$757$	10693.8	0.513439	0.256719	+	0.966486i	$0.417358\pi$
$758$	3619.56	0.173441
$759$	0	0
$760$	−29712.7	−1.41815
$761$	−6456.89	−0.307572	−0.153786	−	0.988104i	$-0.549147\pi$
$761$	−6456.89	−0.307572	−0.153786	+	0.988104i	$0.549147\pi$
$762$	0	0
$763$	0	0
$764$	−843.356	−0.0399366
$765$	0	0
$766$	29648.5	1.39849
$767$	−16549.9	−0.779116
$768$	0	0
$769$	−2661.03	−0.124784	−0.0623922	−	0.998052i	$-0.519873\pi$
$769$	−2661.03	−0.124784	−0.0623922	+	0.998052i	$0.519873\pi$
$770$	0	0
$771$	0	0
$772$	−309.393	−0.0144239
$773$	28404.4	1.32165	0.660825	−	0.750540i	$-0.270206\pi$
$773$	28404.4	1.32165	0.660825	+	0.750540i	$0.270206\pi$
$774$	0	0
$775$	−57801.9	−2.67910
$776$	−31117.2	−1.43949
$777$	0	0
$778$	−33126.4	−1.52653
$779$	−11701.9	−0.538209
$780$	0	0
$781$	−49490.6	−2.26750
$782$	−16100.4	−0.736251
$783$	0	0
$784$	0	0
$785$	48082.1	2.18615
$786$	0	0
$787$	−708.483	−0.0320898	−0.0160449	−	0.999871i	$-0.505107\pi$
$787$	−708.483	−0.0320898	−0.0160449	+	0.999871i	$0.505107\pi$
$788$	634.088	0.0286656
$789$	0	0
$790$	−31721.8	−1.42862
$791$	0	0
$792$	0	0
$793$	13291.5	0.595202
$794$	33728.8	1.50754
$795$	0	0
$796$	−21.2401	−0.000945772 0
$797$	10538.1	0.468355	0.234178	−	0.972194i	$-0.424760\pi$
$797$	10538.1	0.468355	0.234178	+	0.972194i	$0.424760\pi$
$798$	0	0
$799$	−27298.7	−1.20871
$800$	−3419.67	−0.151129
$801$	0	0
$802$	−20593.5	−0.906710
$803$	16133.8	0.709030
$804$	0	0
$805$	0	0
$806$	26528.5	1.15934
$807$	0	0
$808$	38808.4	1.68970
$809$	−38211.3	−1.66061	−0.830307	−	0.557306i	$-0.811835\pi$
$809$	−38211.3	−1.66061	−0.830307	+	0.557306i	$0.811835\pi$
$810$	0	0
$811$	−12627.8	−0.546758	−0.273379	−	0.961906i	$-0.588141\pi$
$811$	−12627.8	−0.546758	−0.273379	+	0.961906i	$0.588141\pi$
$812$	0	0
$813$	0	0
$814$	−12833.5	−0.552596
$815$	11471.9	0.493058
$816$	0	0
$817$	−14884.2	−0.637370
$818$	−41782.0	−1.78591
$819$	0	0
$820$	−960.108	−0.0408883
$821$	−15408.0	−0.654985	−0.327492	−	0.944854i	$-0.606204\pi$
$821$	−15408.0	−0.654985	−0.327492	+	0.944854i	$0.606204\pi$
$822$	0	0
$823$	12225.4	0.517803	0.258901	−	0.965904i	$-0.416640\pi$
$823$	12225.4	0.517803	0.258901	+	0.965904i	$0.416640\pi$
$824$	−4356.39	−0.184177
$825$	0	0
$826$	0	0
$827$	−2156.41	−0.0906721	−0.0453360	−	0.998972i	$-0.514436\pi$
$827$	−2156.41	−0.0906721	−0.0453360	+	0.998972i	$0.514436\pi$
$828$	0	0
$829$	20682.7	0.866515	0.433257	−	0.901270i	$-0.357364\pi$
$829$	20682.7	0.866515	0.433257	+	0.901270i	$0.357364\pi$
$830$	8376.62	0.350309
$831$	0	0
$832$	26815.7	1.11739
$833$	0	0
$834$	0	0
$835$	39074.2	1.61942
$836$	−1000.05	−0.0413724
$837$	0	0
$838$	−3070.70	−0.126582
$839$	−1930.29	−0.0794291	−0.0397145	−	0.999211i	$-0.512645\pi$
$839$	−1930.29	−0.0794291	−0.0397145	+	0.999211i	$0.512645\pi$
$840$	0	0
$841$	4117.84	0.168840
$842$	−21153.7	−0.865803
$843$	0	0
$844$	388.197	0.0158321
$845$	8233.15	0.335182
$846$	0	0
$847$	0	0
$848$	−28765.5	−1.16487
$849$	0	0
$850$	39687.4	1.60149
$851$	8778.82	0.353624
$852$	0	0
$853$	11991.2	0.481327	0.240663	−	0.970609i	$-0.422635\pi$
$853$	11991.2	0.481327	0.240663	+	0.970609i	$0.422635\pi$
$854$	0	0
$855$	0	0
$856$	−17020.9	−0.679629
$857$	32354.2	1.28961	0.644807	−	0.764346i	$-0.276938\pi$
$857$	32354.2	1.28961	0.644807	+	0.764346i	$0.276938\pi$
$858$	0	0
$859$	7053.85	0.280180	0.140090	−	0.990139i	$-0.455261\pi$
$859$	7053.85	0.280180	0.140090	+	0.990139i	$0.455261\pi$
$860$	−1221.20	−0.0484216
$861$	0	0
$862$	−5924.78	−0.234105
$863$	−19806.4	−0.781247	−0.390623	−	0.920551i	$-0.627741\pi$
$863$	−19806.4	−0.781247	−0.390623	+	0.920551i	$0.627741\pi$
$864$	0	0
$865$	74504.8	2.92860
$866$	17586.8	0.690097
$867$	0	0
$868$	0	0
$869$	−35975.3	−1.40435
$870$	0	0
$871$	17360.9	0.675375
$872$	−13950.5	−0.541771
$873$	0	0
$874$	−21682.3	−0.839147
$875$	0	0
$876$	0	0
$877$	7276.80	0.280183	0.140091	−	0.990139i	$-0.455260\pi$
$877$	7276.80	0.280183	0.140091	+	0.990139i	$0.455260\pi$
$878$	30588.0	1.17573
$879$	0	0
$880$	84945.6	3.25400
$881$	25903.2	0.990580	0.495290	−	0.868728i	$-0.335062\pi$
$881$	25903.2	0.990580	0.495290	+	0.868728i	$0.335062\pi$
$882$	0	0
$883$	14502.1	0.552701	0.276351	−	0.961057i	$-0.410875\pi$
$883$	14502.1	0.552701	0.276351	+	0.961057i	$0.410875\pi$
$884$	574.684	0.0218651
$885$	0	0
$886$	−30898.7	−1.17163
$887$	4327.73	0.163823	0.0819115	−	0.996640i	$-0.473898\pi$
$887$	4327.73	0.163823	0.0819115	+	0.996640i	$0.473898\pi$
$888$	0	0
$889$	0	0
$890$	35024.7	1.31914
$891$	0	0
$892$	127.042	0.00476869
$893$	−36763.1	−1.37764
$894$	0	0
$895$	38491.5	1.43758
$896$	0	0
$897$	0	0
$898$	−25593.3	−0.951067
$899$	−31590.0	−1.17195
$900$	0	0
$901$	−21408.6	−0.791592
$902$	34511.2	1.27394
$903$	0	0
$904$	33883.5	1.24662
$905$	78948.5	2.89982
$906$	0	0
$907$	−7456.79	−0.272986	−0.136493	−	0.990641i	$-0.543583\pi$
$907$	−7456.79	−0.272986	−0.136493	+	0.990641i	$0.543583\pi$
$908$	227.904	0.00832959
$909$	0	0
$910$	0	0
$911$	27823.5	1.01189	0.505946	−	0.862565i	$-0.331143\pi$
$911$	27823.5	1.01189	0.505946	+	0.862565i	$0.331143\pi$
$912$	0	0
$913$	9499.82	0.344357
$914$	822.981	0.0297831
$915$	0	0
$916$	1233.25	0.0444844
$917$	0	0
$918$	0	0
$919$	−27876.2	−1.00060	−0.500299	−	0.865853i	$-0.666777\pi$
$919$	−27876.2	−1.00060	−0.500299	+	0.865853i	$0.666777\pi$
$920$	−59942.7	−2.14810
$921$	0	0
$922$	28739.3	1.02655
$923$	38300.2	1.36584
$924$	0	0
$925$	−21639.8	−0.769201
$926$	30317.9	1.07593
$927$	0	0
$928$	−1868.92	−0.0661104
$929$	−26442.1	−0.933842	−0.466921	−	0.884299i	$-0.654637\pi$
$929$	−26442.1	−0.933842	−0.466921	+	0.884299i	$0.654637\pi$
$930$	0	0
$931$	0	0
$932$	−46.8917	−0.00164806
$933$	0	0
$934$	−14081.8	−0.493330
$935$	63220.4	2.21126
$936$	0	0
$937$	−1548.38	−0.0539843	−0.0269922	−	0.999636i	$-0.508593\pi$
$937$	−1548.38	−0.0539843	−0.0269922	+	0.999636i	$0.508593\pi$
$938$	0	0
$939$	0	0
$940$	−3016.30	−0.104661
$941$	−57386.7	−1.98805	−0.994024	−	0.109163i	$-0.965183\pi$
$941$	−57386.7	−1.98805	−0.994024	+	0.109163i	$0.965183\pi$
$942$	0	0
$943$	−23607.6	−0.815238
$944$	20147.8	0.694657
$945$	0	0
$946$	43896.2	1.50866
$947$	7852.31	0.269446	0.134723	−	0.990883i	$-0.456985\pi$
$947$	7852.31	0.269446	0.134723	+	0.990883i	$0.456985\pi$
$948$	0	0
$949$	−12485.8	−0.427087
$950$	53446.8	1.82531
$951$	0	0
$952$	0	0
$953$	17617.8	0.598842	0.299421	−	0.954121i	$-0.403206\pi$
$953$	17617.8	0.598842	0.299421	+	0.954121i	$0.403206\pi$
$954$	0	0
$955$	71798.8	2.43283
$956$	−686.785	−0.0232345
$957$	0	0
$958$	−1085.16	−0.0365970
$959$	0	0
$960$	0	0
$961$	5215.69	0.175076
$962$	9931.68	0.332859
$963$	0	0
$964$	−936.469	−0.0312880
$965$	26340.0	0.878669
$966$	0	0
$967$	40773.5	1.35593	0.677966	−	0.735093i	$-0.262862\pi$
$967$	40773.5	1.35593	0.677966	+	0.735093i	$0.262862\pi$
$968$	68818.3	2.28503
$969$	0	0
$970$	78621.0	2.60244
$971$	−4951.31	−0.163641	−0.0818203	−	0.996647i	$-0.526073\pi$
$971$	−4951.31	−0.163641	−0.0818203	+	0.996647i	$0.526073\pi$
$972$	0	0
$973$	0	0
$974$	−41960.4	−1.38039
$975$	0	0
$976$	−16181.1	−0.530679
$977$	54810.5	1.79483	0.897413	−	0.441192i	$-0.145444\pi$
$977$	54810.5	1.79483	0.897413	+	0.441192i	$0.145444\pi$
$978$	0	0
$979$	39721.1	1.29672
$980$	0	0
$981$	0	0
$982$	−48881.4	−1.58846
$983$	−27663.4	−0.897583	−0.448792	−	0.893636i	$-0.648145\pi$
$983$	−27663.4	−0.897583	−0.448792	+	0.893636i	$0.648145\pi$
$984$	0	0
$985$	−53982.8	−1.74623
$986$	21690.0	0.700560
$987$	0	0
$988$	773.925	0.0249209
$989$	−30027.5	−0.965438
$990$	0	0
$991$	45685.2	1.46442	0.732210	−	0.681079i	$-0.238489\pi$
$991$	45685.2	1.46442	0.732210	+	0.681079i	$0.238489\pi$
$992$	2071.06	0.0662865
$993$	0	0
$994$	0	0
$995$	1808.26	0.0576139
$996$	0	0
$997$	−17641.7	−0.560399	−0.280199	−	0.959942i	$-0.590401\pi$
$997$	−17641.7	−0.560399	−0.280199	+	0.959942i	$0.590401\pi$
$998$	53744.2	1.70465
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bc.1.2	yes	6
3.2	odd	2		1323.4.a.bb.1.5	yes	6
7.6	odd	2		1323.4.a.bb.1.2	✓	6
21.20	even	2	inner	1323.4.a.bc.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bb.1.2	✓	6	7.6	odd	2
1323.4.a.bb.1.5	yes	6	3.2	odd	2
1323.4.a.bc.1.2	yes	6	1.1	even	1	trivial
1323.4.a.bc.1.5	yes	6	21.20	even	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)

\(f(q)\)	\(=\)	\(q-2.78484 q^{2} -0.244684 q^{4} +20.8311 q^{5} +22.9601 q^{8} +O(q^{10})\)	\(q-2.78484 q^{2} -0.244684 q^{4} +20.8311 q^{5} +22.9601 q^{8} -58.0112 q^{10} -65.7898 q^{11} +50.9140 q^{13} -61.9827 q^{16} -46.1303 q^{17} -62.1235 q^{19} -5.09704 q^{20} +183.214 q^{22} -125.329 q^{23} +308.935 q^{25} -141.787 q^{26} +168.840 q^{29} -187.101 q^{31} -11.0692 q^{32} +128.465 q^{34} -70.0464 q^{37} +173.004 q^{38} +478.284 q^{40} +188.366 q^{41} +239.590 q^{43} +16.0977 q^{44} +349.020 q^{46} +591.775 q^{47} -860.332 q^{50} -12.4579 q^{52} +464.090 q^{53} -1370.47 q^{55} -470.191 q^{58} -325.056 q^{59} +261.058 q^{61} +521.045 q^{62} +526.687 q^{64} +1060.59 q^{65} +340.985 q^{67} +11.2874 q^{68} +752.254 q^{71} -245.233 q^{73} +195.068 q^{74} +15.2006 q^{76} +546.822 q^{79} -1291.17 q^{80} -524.568 q^{82} -144.397 q^{83} -960.945 q^{85} -667.219 q^{86} -1510.54 q^{88} -603.758 q^{89} +30.6659 q^{92} -1648.00 q^{94} -1294.10 q^{95} -1355.27 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{4} + 24 q^{5}+O(q^{10}) \) 6 * q + 18 * q^4 + 24 * q^5	\( 6 q + 18 q^{4} + 24 q^{5} - 30 q^{16} + 42 q^{17} + 12 q^{20} + 132 q^{22} + 222 q^{25} + 366 q^{26} - 312 q^{37} + 336 q^{38} + 360 q^{41} + 654 q^{43} + 774 q^{46} + 1812 q^{47} - 378 q^{58} - 6 q^{59} + 2058 q^{62} + 66 q^{64} + 42 q^{67} + 2910 q^{68} + 1956 q^{79} - 2868 q^{80} + 2892 q^{83} - 1944 q^{85} - 2532 q^{88} + 1518 q^{89}+O(q^{100}) \) 6 * q + 18 * q^4 + 24 * q^5 - 30 * q^16 + 42 * q^17 + 12 * q^20 + 132 * q^22 + 222 * q^25 + 366 * q^26 - 312 * q^37 + 336 * q^38 + 360 * q^41 + 654 * q^43 + 774 * q^46 + 1812 * q^47 - 378 * q^58 - 6 * q^59 + 2058 * q^62 + 66 * q^64 + 42 * q^67 + 2910 * q^68 + 1956 * q^79 - 2868 * q^80 + 2892 * q^83 - 1944 * q^85 - 2532 * q^88 + 1518 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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