LMFDB - Embedded newform 1521.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +O(q^{10})$	$q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +3.00000 q^{10} +13.8564 q^{11} +24.0000 q^{14} +1.00000 q^{16} +117.000 q^{17} +114.315 q^{19} +8.66025 q^{20} -24.0000 q^{22} -78.0000 q^{23} -122.000 q^{25} +69.2820 q^{28} +141.000 q^{29} -155.885 q^{31} -181.865 q^{32} -202.650 q^{34} +24.0000 q^{35} +143.760 q^{37} -198.000 q^{38} -39.0000 q^{40} -271.932 q^{41} -104.000 q^{43} -69.2820 q^{44} +135.100 q^{46} -301.377 q^{47} -151.000 q^{49} +211.310 q^{50} -93.0000 q^{53} -24.0000 q^{55} -312.000 q^{56} -244.219 q^{58} +284.056 q^{59} +145.000 q^{61} +270.000 q^{62} +307.000 q^{64} -786.351 q^{67} -585.000 q^{68} -41.5692 q^{70} +1056.55 q^{71} +458.993 q^{73} -249.000 q^{74} -571.577 q^{76} -192.000 q^{77} +1276.00 q^{79} -1.73205 q^{80} +471.000 q^{82} -789.815 q^{83} -202.650 q^{85} +180.133 q^{86} +312.000 q^{88} -976.877 q^{89} +390.000 q^{92} +522.000 q^{94} -198.000 q^{95} -200.918 q^{97} +261.540 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{4}+O(q^{10}) $ 2 * q - 10 * q^4	$ 2 q - 10 q^{4} + 6 q^{10} + 48 q^{14} + 2 q^{16} + 234 q^{17} - 48 q^{22} - 156 q^{23} - 244 q^{25} + 282 q^{29} + 48 q^{35} - 396 q^{38} - 78 q^{40} - 208 q^{43} - 302 q^{49} - 186 q^{53} - 48 q^{55} - 624 q^{56} + 290 q^{61} + 540 q^{62} + 614 q^{64} - 1170 q^{68} - 498 q^{74} - 384 q^{77} + 2552 q^{79} + 942 q^{82} + 624 q^{88} + 780 q^{92} + 1044 q^{94} - 396 q^{95}+O(q^{100}) $ 2 * q - 10 * q^4 + 6 * q^10 + 48 * q^14 + 2 * q^16 + 234 * q^17 - 48 * q^22 - 156 * q^23 - 244 * q^25 + 282 * q^29 + 48 * q^35 - 396 * q^38 - 78 * q^40 - 208 * q^43 - 302 * q^49 - 186 * q^53 - 48 * q^55 - 624 * q^56 + 290 * q^61 + 540 * q^62 + 614 * q^64 - 1170 * q^68 - 498 * q^74 - 384 * q^77 + 2552 * q^79 + 942 * q^82 + 624 * q^88 + 780 * q^92 + 1044 * q^94 - 396 * q^95

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 $

$2$	−1.73205	−0.612372	−0.306186	−	0.951972i	$-0.599053\pi$
$2$	−1.73205	−0.612372	−0.306186	+	0.951972i	$0.599053\pi$
$3$	0	0
$3$	0	0
$4$	−5.00000	−0.625000
$5$	−1.73205	−0.154919	−0.0774597	−	0.996995i	$-0.524681\pi$
$5$	−1.73205	−0.154919	−0.0774597	+	0.996995i	$0.524681\pi$
$6$	0	0
$7$	−13.8564	−0.748176	−0.374088	−	0.927393i	$-0.622044\pi$
$7$	−13.8564	−0.748176	−0.374088	+	0.927393i	$0.622044\pi$
$8$	22.5167	0.995105
$9$	0	0
$10$	3.00000	0.0948683
$11$	13.8564	0.379806	0.189903	−	0.981803i	$-0.439183\pi$
$11$	13.8564	0.379806	0.189903	+	0.981803i	$0.439183\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	24.0000	0.458162
$15$	0	0
$16$	1.00000	0.0156250
$17$	117.000	1.66922	0.834608	−	0.550845i	$-0.185694\pi$
$17$	117.000	1.66922	0.834608	+	0.550845i	$0.185694\pi$
$18$	0	0
$19$	114.315	1.38030	0.690151	−	0.723665i	$-0.257544\pi$
$19$	114.315	1.38030	0.690151	+	0.723665i	$0.257544\pi$
$20$	8.66025	0.0968246
$21$	0	0
$22$	−24.0000	−0.232583
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	0	0
$25$	−122.000	−0.976000
$26$	0	0
$27$	0	0
$28$	69.2820	0.467610
$29$	141.000	0.902864	0.451432	−	0.892306i	$-0.350913\pi$
$29$	141.000	0.902864	0.451432	+	0.892306i	$0.350913\pi$
$30$	0	0
$31$	−155.885	−0.903151	−0.451576	−	0.892233i	$-0.649138\pi$
$31$	−155.885	−0.903151	−0.451576	+	0.892233i	$0.649138\pi$
$32$	−181.865	−1.00467
$33$	0	0
$34$	−202.650	−1.02218
$35$	24.0000	0.115907
$36$	0	0
$37$	143.760	0.638758	0.319379	−	0.947627i	$-0.396526\pi$
$37$	143.760	0.638758	0.319379	+	0.947627i	$0.396526\pi$
$38$	−198.000	−0.845259
$39$	0	0
$40$	−39.0000	−0.154161
$41$	−271.932	−1.03582	−0.517910	−	0.855435i	$-0.673290\pi$
$41$	−271.932	−1.03582	−0.517910	+	0.855435i	$0.673290\pi$
$42$	0	0
$43$	−104.000	−0.368834	−0.184417	−	0.982848i	$-0.559040\pi$
$43$	−104.000	−0.368834	−0.184417	+	0.982848i	$0.559040\pi$
$44$	−69.2820	−0.237379
$45$	0	0
$46$	135.100	0.433030
$47$	−301.377	−0.935326	−0.467663	−	0.883907i	$-0.654904\pi$
$47$	−301.377	−0.935326	−0.467663	+	0.883907i	$0.654904\pi$
$48$	0	0
$49$	−151.000	−0.440233
$50$	211.310	0.597675
$51$	0	0
$52$	0	0
$53$	−93.0000	−0.241029	−0.120514	−	0.992712i	$-0.538454\pi$
$53$	−93.0000	−0.241029	−0.120514	+	0.992712i	$0.538454\pi$
$54$	0	0
$55$	−24.0000	−0.0588393
$56$	−312.000	−0.744513
$57$	0	0
$58$	−244.219	−0.552889
$59$	284.056	0.626796	0.313398	−	0.949622i	$-0.398533\pi$
$59$	284.056	0.626796	0.313398	+	0.949622i	$0.398533\pi$
$60$	0	0
$61$	145.000	0.304350	0.152175	−	0.988354i	$-0.451372\pi$
$61$	145.000	0.304350	0.152175	+	0.988354i	$0.451372\pi$
$62$	270.000	0.553065
$63$	0	0
$64$	307.000	0.599609
$65$	0	0
$66$	0	0
$67$	−786.351	−1.43385	−0.716926	−	0.697149i	$-0.754451\pi$
$67$	−786.351	−1.43385	−0.716926	+	0.697149i	$0.754451\pi$
$68$	−585.000	−1.04326
$69$	0	0
$70$	−41.5692	−0.0709782
$71$	1056.55	1.76605	0.883025	−	0.469326i	$-0.155503\pi$
$71$	1056.55	1.76605	0.883025	+	0.469326i	$0.155503\pi$
$72$	0	0
$73$	458.993	0.735906	0.367953	−	0.929844i	$-0.380059\pi$
$73$	458.993	0.735906	0.367953	+	0.929844i	$0.380059\pi$
$74$	−249.000	−0.391158
$75$	0	0
$76$	−571.577	−0.862689
$77$	−192.000	−0.284161
$78$	0	0
$79$	1276.00	1.81723	0.908615	−	0.417634i	$-0.137141\pi$
$79$	1276.00	1.81723	0.908615	+	0.417634i	$0.137141\pi$
$80$	−1.73205	−0.00242061
$81$	0	0
$82$	471.000	0.634308
$83$	−789.815	−1.04450	−0.522250	−	0.852793i	$-0.674907\pi$
$83$	−789.815	−1.04450	−0.522250	+	0.852793i	$0.674907\pi$
$84$	0	0
$85$	−202.650	−0.258594
$86$	180.133	0.225864
$87$	0	0
$88$	312.000	0.377947
$89$	−976.877	−1.16347	−0.581734	−	0.813379i	$-0.697626\pi$
$89$	−976.877	−1.16347	−0.581734	+	0.813379i	$0.697626\pi$
$90$	0	0
$91$	0	0
$92$	390.000	0.441960
$93$	0	0
$94$	522.000	0.572768
$95$	−198.000	−0.213835
$96$	0	0
$97$	−200.918	−0.210311	−0.105155	−	0.994456i	$-0.533534\pi$
$97$	−200.918	−0.210311	−0.105155	+	0.994456i	$0.533534\pi$
$98$	261.540	0.269587
$99$	0	0
$100$	610.000	0.610000
$101$	429.000	0.422645	0.211322	−	0.977416i	$-0.432223\pi$
$101$	429.000	0.422645	0.211322	+	0.977416i	$0.432223\pi$
$102$	0	0
$103$	−182.000	−0.174107	−0.0870534	−	0.996204i	$-0.527745\pi$
$103$	−182.000	−0.174107	−0.0870534	+	0.996204i	$0.527745\pi$
$104$	0	0
$105$	0	0
$106$	161.081	0.147599
$107$	1506.00	1.36066	0.680330	−	0.732906i	$-0.261837\pi$
$107$	1506.00	1.36066	0.680330	+	0.732906i	$0.261837\pi$
$108$	0	0
$109$	−1551.92	−1.36373	−0.681866	−	0.731477i	$-0.738831\pi$
$109$	−1551.92	−1.36373	−0.681866	+	0.731477i	$0.738831\pi$
$110$	41.5692	0.0360315
$111$	0	0
$112$	−13.8564	−0.0116902
$113$	687.000	0.571925	0.285962	−	0.958241i	$-0.407687\pi$
$113$	687.000	0.571925	0.285962	+	0.958241i	$0.407687\pi$
$114$	0	0
$115$	135.100	0.109549
$116$	−705.000	−0.564290
$117$	0	0
$118$	−492.000	−0.383833
$119$	−1621.20	−1.24887
$120$	0	0
$121$	−1139.00	−0.855748
$122$	−251.147	−0.186376
$123$	0	0
$124$	779.423	0.564470
$125$	427.817	0.306121
$126$	0	0
$127$	−286.000	−0.199830	−0.0999149	−	0.994996i	$-0.531857\pi$
$127$	−286.000	−0.199830	−0.0999149	+	0.994996i	$0.531857\pi$
$128$	923.183	0.637489
$129$	0	0
$130$	0	0
$131$	1974.00	1.31656	0.658279	−	0.752774i	$-0.271285\pi$
$131$	1974.00	1.31656	0.658279	+	0.752774i	$0.271285\pi$
$132$	0	0
$133$	−1584.00	−1.03271
$134$	1362.00	0.878051
$135$	0	0
$136$	2634.45	1.66105
$137$	846.973	0.528188	0.264094	−	0.964497i	$-0.414927\pi$
$137$	846.973	0.528188	0.264094	+	0.964497i	$0.414927\pi$
$138$	0	0
$139$	236.000	0.144009	0.0720045	−	0.997404i	$-0.477060\pi$
$139$	236.000	0.144009	0.0720045	+	0.997404i	$0.477060\pi$
$140$	−120.000	−0.0724418
$141$	0	0
$142$	−1830.00	−1.08148
$143$	0	0
$144$	0	0
$145$	−244.219	−0.139871
$146$	−795.000	−0.450648
$147$	0	0
$148$	−718.801	−0.399224
$149$	−46.7654	−0.0257125	−0.0128563	−	0.999917i	$-0.504092\pi$
$149$	−46.7654	−0.0257125	−0.0128563	+	0.999917i	$0.504092\pi$
$150$	0	0
$151$	−1770.16	−0.953995	−0.476998	−	0.878905i	$-0.658275\pi$
$151$	−1770.16	−0.953995	−0.476998	+	0.878905i	$0.658275\pi$
$152$	2574.00	1.37355
$153$	0	0
$154$	332.554	0.174013
$155$	270.000	0.139916
$156$	0	0
$157$	1211.00	0.615594	0.307797	−	0.951452i	$-0.400408\pi$
$157$	1211.00	0.615594	0.307797	+	0.951452i	$0.400408\pi$
$158$	−2210.10	−1.11282
$159$	0	0
$160$	315.000	0.155643
$161$	1080.80	0.529062
$162$	0	0
$163$	−1004.59	−0.482733	−0.241367	−	0.970434i	$-0.577596\pi$
$163$	−1004.59	−0.482733	−0.241367	+	0.970434i	$0.577596\pi$
$164$	1359.66	0.647388
$165$	0	0
$166$	1368.00	0.639623
$167$	914.523	0.423760	0.211880	−	0.977296i	$-0.432041\pi$
$167$	914.523	0.423760	0.211880	+	0.977296i	$0.432041\pi$
$168$	0	0
$169$	0	0
$170$	351.000	0.158356
$171$	0	0
$172$	520.000	0.230521
$173$	2574.00	1.13120	0.565600	−	0.824680i	$-0.308645\pi$
$173$	2574.00	1.13120	0.565600	+	0.824680i	$0.308645\pi$
$174$	0	0
$175$	1690.48	0.730219
$176$	13.8564	0.00593447
$177$	0	0
$178$	1692.00	0.712476
$179$	3744.00	1.56335	0.781675	−	0.623686i	$-0.214366\pi$
$179$	3744.00	1.56335	0.781675	+	0.623686i	$0.214366\pi$
$180$	0	0
$181$	637.000	0.261590	0.130795	−	0.991409i	$-0.458247\pi$
$181$	637.000	0.261590	0.130795	+	0.991409i	$0.458247\pi$
$182$	0	0
$183$	0	0
$184$	−1756.30	−0.703675
$185$	−249.000	−0.0989559
$186$	0	0
$187$	1621.20	0.633978
$188$	1506.88	0.584579
$189$	0	0
$190$	342.946	0.130947
$191$	2598.00	0.984213	0.492106	−	0.870535i	$-0.336227\pi$
$191$	2598.00	0.984213	0.492106	+	0.870535i	$0.336227\pi$
$192$	0	0
$193$	−1117.17	−0.416662	−0.208331	−	0.978058i	$-0.566803\pi$
$193$	−1117.17	−0.416662	−0.208331	+	0.978058i	$0.566803\pi$
$194$	348.000	0.128788
$195$	0	0
$196$	755.000	0.275146
$197$	−2050.75	−0.741674	−0.370837	−	0.928698i	$-0.620929\pi$
$197$	−2050.75	−0.741674	−0.370837	+	0.928698i	$0.620929\pi$
$198$	0	0
$199$	2522.00	0.898391	0.449196	−	0.893433i	$-0.351711\pi$
$199$	2522.00	0.898391	0.449196	+	0.893433i	$0.351711\pi$
$200$	−2747.03	−0.971223
$201$	0	0
$202$	−743.050	−0.258816
$203$	−1953.75	−0.675500
$204$	0	0
$205$	471.000	0.160469
$206$	315.233	0.106618
$207$	0	0
$208$	0	0
$209$	1584.00	0.524247
$210$	0	0
$211$	1042.00	0.339973	0.169986	−	0.985446i	$-0.445628\pi$
$211$	1042.00	0.339973	0.169986	+	0.985446i	$0.445628\pi$
$212$	465.000	0.150643
$213$	0	0
$214$	−2608.47	−0.833230
$215$	180.133	0.0571395
$216$	0	0
$217$	2160.00	0.675716
$218$	2688.00	0.835112
$219$	0	0
$220$	120.000	0.0367745
$221$	0	0
$222$	0	0
$223$	−2407.55	−0.722966	−0.361483	−	0.932379i	$-0.617730\pi$
$223$	−2407.55	−0.722966	−0.361483	+	0.932379i	$0.617730\pi$
$224$	2520.00	0.751672
$225$	0	0
$226$	−1189.92	−0.350231
$227$	2407.55	0.703942	0.351971	−	0.936011i	$-0.385512\pi$
$227$	2407.55	0.703942	0.351971	+	0.936011i	$0.385512\pi$
$228$	0	0
$229$	2508.01	0.723729	0.361864	−	0.932231i	$-0.382140\pi$
$229$	2508.01	0.723729	0.361864	+	0.932231i	$0.382140\pi$
$230$	−234.000	−0.0670848
$231$	0	0
$232$	3174.85	0.898444
$233$	−5850.00	−1.64483	−0.822417	−	0.568885i	$-0.807375\pi$
$233$	−5850.00	−1.64483	−0.822417	+	0.568885i	$0.807375\pi$
$234$	0	0
$235$	522.000	0.144900
$236$	−1420.28	−0.391748
$237$	0	0
$238$	2808.00	0.764771
$239$	−5383.21	−1.45695	−0.728475	−	0.685072i	$-0.759771\pi$
$239$	−5383.21	−1.45695	−0.728475	+	0.685072i	$0.759771\pi$
$240$	0	0
$241$	4917.29	1.31432	0.657159	−	0.753752i	$-0.271758\pi$
$241$	4917.29	1.31432	0.657159	+	0.753752i	$0.271758\pi$
$242$	1972.81	0.524036
$243$	0	0
$244$	−725.000	−0.190219
$245$	261.540	0.0682006
$246$	0	0
$247$	0	0
$248$	−3510.00	−0.898731
$249$	0	0
$250$	−741.000	−0.187460
$251$	−3978.00	−1.00036	−0.500178	−	0.865923i	$-0.666732\pi$
$251$	−3978.00	−1.00036	−0.500178	+	0.865923i	$0.666732\pi$
$252$	0	0
$253$	−1080.80	−0.268574
$254$	495.367	0.122370
$255$	0	0
$256$	−4055.00	−0.989990
$257$	2067.00	0.501696	0.250848	−	0.968026i	$-0.419291\pi$
$257$	2067.00	0.501696	0.250848	+	0.968026i	$0.419291\pi$
$258$	0	0
$259$	−1992.00	−0.477903
$260$	0	0
$261$	0	0
$262$	−3419.07	−0.806224
$263$	2052.00	0.481109	0.240555	−	0.970636i	$-0.422671\pi$
$263$	2052.00	0.481109	0.240555	+	0.970636i	$0.422671\pi$
$264$	0	0
$265$	161.081	0.0373400
$266$	2743.57	0.632402
$267$	0	0
$268$	3931.76	0.896157
$269$	−3330.00	−0.754772	−0.377386	−	0.926056i	$-0.623177\pi$
$269$	−3330.00	−0.754772	−0.377386	+	0.926056i	$0.623177\pi$
$270$	0	0
$271$	−2805.92	−0.628958	−0.314479	−	0.949264i	$-0.601830\pi$
$271$	−2805.92	−0.628958	−0.314479	+	0.949264i	$0.601830\pi$
$272$	117.000	0.0260815
$273$	0	0
$274$	−1467.00	−0.323448
$275$	−1690.48	−0.370690
$276$	0	0
$277$	−377.000	−0.0817752	−0.0408876	−	0.999164i	$-0.513019\pi$
$277$	−377.000	−0.0817752	−0.0408876	+	0.999164i	$0.513019\pi$
$278$	−408.764	−0.0881872
$279$	0	0
$280$	540.400	0.115340
$281$	36.3731	0.00772183	0.00386092	−	0.999993i	$-0.498771\pi$
$281$	36.3731	0.00772183	0.00386092	+	0.999993i	$0.498771\pi$
$282$	0	0
$283$	7124.00	1.49639	0.748194	−	0.663480i	$-0.230921\pi$
$283$	7124.00	1.49639	0.748194	+	0.663480i	$0.230921\pi$
$284$	−5282.75	−1.10378
$285$	0	0
$286$	0	0
$287$	3768.00	0.774976
$288$	0	0
$289$	8776.00	1.78628
$290$	423.000	0.0856532
$291$	0	0
$292$	−2294.97	−0.459941
$293$	8322.50	1.65941	0.829703	−	0.558205i	$-0.188510\pi$
$293$	8322.50	1.65941	0.829703	+	0.558205i	$0.188510\pi$
$294$	0	0
$295$	−492.000	−0.0971029
$296$	3237.00	0.635631
$297$	0	0
$298$	81.0000	0.0157457
$299$	0	0
$300$	0	0
$301$	1441.07	0.275952
$302$	3066.00	0.584200
$303$	0	0
$304$	114.315	0.0215672
$305$	−251.147	−0.0471497
$306$	0	0
$307$	−2220.49	−0.412801	−0.206401	−	0.978468i	$-0.566175\pi$
$307$	−2220.49	−0.412801	−0.206401	+	0.978468i	$0.566175\pi$
$308$	960.000	0.177601
$309$	0	0
$310$	−467.654	−0.0856805
$311$	4914.00	0.895972	0.447986	−	0.894041i	$-0.352141\pi$
$311$	4914.00	0.895972	0.447986	+	0.894041i	$0.352141\pi$
$312$	0	0
$313$	−518.000	−0.0935434	−0.0467717	−	0.998906i	$-0.514893\pi$
$313$	−518.000	−0.0935434	−0.0467717	+	0.998906i	$0.514893\pi$
$314$	−2097.51	−0.376973
$315$	0	0
$316$	−6380.00	−1.13577
$317$	3916.17	0.693861	0.346930	−	0.937891i	$-0.387224\pi$
$317$	3916.17	0.693861	0.346930	+	0.937891i	$0.387224\pi$
$318$	0	0
$319$	1953.75	0.342913
$320$	−531.740	−0.0928911
$321$	0	0
$322$	−1872.00	−0.323983
$323$	13374.9	2.30402
$324$	0	0
$325$	0	0
$326$	1740.00	0.295613
$327$	0	0
$328$	−6123.00	−1.03075
$329$	4176.00	0.699788
$330$	0	0
$331$	7454.75	1.23792	0.618958	−	0.785424i	$-0.287555\pi$
$331$	7454.75	1.23792	0.618958	+	0.785424i	$0.287555\pi$
$332$	3949.08	0.652812
$333$	0	0
$334$	−1584.00	−0.259499
$335$	1362.00	0.222131
$336$	0	0
$337$	3575.00	0.577871	0.288936	−	0.957349i	$-0.406699\pi$
$337$	3575.00	0.577871	0.288936	+	0.957349i	$0.406699\pi$
$338$	0	0
$339$	0	0
$340$	1013.25	0.161621
$341$	−2160.00	−0.343022
$342$	0	0
$343$	6845.06	1.07755
$344$	−2341.73	−0.367028
$345$	0	0
$346$	−4458.30	−0.692716
$347$	6966.00	1.07768	0.538839	−	0.842409i	$-0.318863\pi$
$347$	6966.00	1.07768	0.538839	+	0.842409i	$0.318863\pi$
$348$	0	0
$349$	6651.08	1.02013	0.510063	−	0.860137i	$-0.329622\pi$
$349$	6651.08	1.02013	0.510063	+	0.860137i	$0.329622\pi$
$350$	−2928.00	−0.447166
$351$	0	0
$352$	−2520.00	−0.381581
$353$	−5630.90	−0.849015	−0.424508	−	0.905424i	$-0.639553\pi$
$353$	−5630.90	−0.849015	−0.424508	+	0.905424i	$0.639553\pi$
$354$	0	0
$355$	−1830.00	−0.273595
$356$	4884.38	0.727168
$357$	0	0
$358$	−6484.80	−0.957353
$359$	7129.12	1.04808	0.524040	−	0.851694i	$-0.324424\pi$
$359$	7129.12	1.04808	0.524040	+	0.851694i	$0.324424\pi$
$360$	0	0
$361$	6209.00	0.905234
$362$	−1103.32	−0.160191
$363$	0	0
$364$	0	0
$365$	−795.000	−0.114006
$366$	0	0
$367$	2.00000	0.000284466 0	0.000142233	−	1.00000i	$-0.499955\pi$
$367$	2.00000	0.000284466 0	0.000142233		1.00000i	$0.499955\pi$
$368$	−78.0000	−0.0110490
$369$	0	0
$370$	431.281	0.0605979
$371$	1288.65	0.180332
$372$	0	0
$373$	3499.00	0.485714	0.242857	−	0.970062i	$-0.421915\pi$
$373$	3499.00	0.485714	0.242857	+	0.970062i	$0.421915\pi$
$374$	−2808.00	−0.388231
$375$	0	0
$376$	−6786.00	−0.930748
$377$	0	0
$378$	0	0
$379$	5518.31	0.747907	0.373953	−	0.927447i	$-0.378002\pi$
$379$	5518.31	0.747907	0.373953	+	0.927447i	$0.378002\pi$
$380$	990.000	0.133647
$381$	0	0
$382$	−4499.87	−0.602705
$383$	−7364.68	−0.982552	−0.491276	−	0.871004i	$-0.663469\pi$
$383$	−7364.68	−0.982552	−0.491276	+	0.871004i	$0.663469\pi$
$384$	0	0
$385$	332.554	0.0440221
$386$	1935.00	0.255153
$387$	0	0
$388$	1004.59	0.131444
$389$	−1209.00	−0.157580	−0.0787901	−	0.996891i	$-0.525106\pi$
$389$	−1209.00	−0.157580	−0.0787901	+	0.996891i	$0.525106\pi$
$390$	0	0
$391$	−9126.00	−1.18036
$392$	−3400.02	−0.438078
$393$	0	0
$394$	3552.00	0.454181
$395$	−2210.10	−0.281524
$396$	0	0
$397$	11694.8	1.47845	0.739226	−	0.673457i	$-0.235191\pi$
$397$	11694.8	1.47845	0.739226	+	0.673457i	$0.235191\pi$
$398$	−4368.23	−0.550150
$399$	0	0
$400$	−122.000	−0.0152500
$401$	−2980.86	−0.371215	−0.185607	−	0.982624i	$-0.559425\pi$
$401$	−2980.86	−0.371215	−0.185607	+	0.982624i	$0.559425\pi$
$402$	0	0
$403$	0	0
$404$	−2145.00	−0.264153
$405$	0	0
$406$	3384.00	0.413658
$407$	1992.00	0.242604
$408$	0	0
$409$	−43.3013	−0.00523499	−0.00261749	−	0.999997i	$-0.500833\pi$
$409$	−43.3013	−0.00523499	−0.00261749	+	0.999997i	$0.500833\pi$
$410$	−815.796	−0.0982666
$411$	0	0
$412$	910.000	0.108817
$413$	−3936.00	−0.468954
$414$	0	0
$415$	1368.00	0.161813
$416$	0	0
$417$	0	0
$418$	−2743.57	−0.321034
$419$	9462.00	1.10322	0.551610	−	0.834102i	$-0.314014\pi$
$419$	9462.00	1.10322	0.551610	+	0.834102i	$0.314014\pi$
$420$	0	0
$421$	−7068.50	−0.818284	−0.409142	−	0.912471i	$-0.634172\pi$
$421$	−7068.50	−0.818284	−0.409142	+	0.912471i	$0.634172\pi$
$422$	−1804.80	−0.208190
$423$	0	0
$424$	−2094.05	−0.239849
$425$	−14274.0	−1.62915
$426$	0	0
$427$	−2009.18	−0.227707
$428$	−7530.00	−0.850412
$429$	0	0
$430$	−312.000	−0.0349906
$431$	9928.12	1.10956	0.554780	−	0.831997i	$-0.312802\pi$
$431$	9928.12	1.10956	0.554780	+	0.831997i	$0.312802\pi$
$432$	0	0
$433$	6617.00	0.734394	0.367197	−	0.930143i	$-0.380317\pi$
$433$	6617.00	0.734394	0.367197	+	0.930143i	$0.380317\pi$
$434$	−3741.23	−0.413790
$435$	0	0
$436$	7759.59	0.852332
$437$	−8916.60	−0.976061
$438$	0	0
$439$	−13988.0	−1.52075	−0.760377	−	0.649482i	$-0.774986\pi$
$439$	−13988.0	−1.52075	−0.760377	+	0.649482i	$0.774986\pi$
$440$	−540.400	−0.0585513
$441$	0	0
$442$	0	0
$443$	−2004.00	−0.214928	−0.107464	−	0.994209i	$-0.534273\pi$
$443$	−2004.00	−0.214928	−0.107464	+	0.994209i	$0.534273\pi$
$444$	0	0
$445$	1692.00	0.180244
$446$	4170.00	0.442725
$447$	0	0
$448$	−4253.92	−0.448613
$449$	−9082.87	−0.954671	−0.477336	−	0.878721i	$-0.658397\pi$
$449$	−9082.87	−0.954671	−0.477336	+	0.878721i	$0.658397\pi$
$450$	0	0
$451$	−3768.00	−0.393411
$452$	−3435.00	−0.357453
$453$	0	0
$454$	−4170.00	−0.431074
$455$	0	0
$456$	0	0
$457$	2523.60	0.258313	0.129156	−	0.991624i	$-0.458773\pi$
$457$	2523.60	0.258313	0.129156	+	0.991624i	$0.458773\pi$
$458$	−4344.00	−0.443192
$459$	0	0
$460$	−675.500	−0.0684681
$461$	19587.8	1.97894	0.989472	−	0.144725i	$-0.0462299\pi$
$461$	19587.8	1.97894	0.989472	+	0.144725i	$0.0462299\pi$
$462$	0	0
$463$	8632.54	0.866497	0.433249	−	0.901274i	$-0.357367\pi$
$463$	8632.54	0.866497	0.433249	+	0.901274i	$0.357367\pi$
$464$	141.000	0.0141072
$465$	0	0
$466$	10132.5	1.00725
$467$	5460.00	0.541025	0.270512	−	0.962716i	$-0.412807\pi$
$467$	5460.00	0.541025	0.270512	+	0.962716i	$0.412807\pi$
$468$	0	0
$469$	10896.0	1.07277
$470$	−904.131	−0.0887328
$471$	0	0
$472$	6396.00	0.623728
$473$	−1441.07	−0.140085
$474$	0	0
$475$	−13946.5	−1.34717
$476$	8106.00	0.780542
$477$	0	0
$478$	9324.00	0.892196
$479$	−2553.04	−0.243531	−0.121766	−	0.992559i	$-0.538856\pi$
$479$	−2553.04	−0.243531	−0.121766	+	0.992559i	$0.538856\pi$
$480$	0	0
$481$	0	0
$482$	−8517.00	−0.804852
$483$	0	0
$484$	5695.00	0.534842
$485$	348.000	0.0325812
$486$	0	0
$487$	−10828.8	−1.00760	−0.503798	−	0.863822i	$-0.668064\pi$
$487$	−10828.8	−1.00760	−0.503798	+	0.863822i	$0.668064\pi$
$488$	3264.92	0.302860
$489$	0	0
$490$	−453.000	−0.0417642
$491$	11388.0	1.04671	0.523354	−	0.852116i	$-0.324681\pi$
$491$	11388.0	1.04671	0.523354	+	0.852116i	$0.324681\pi$
$492$	0	0
$493$	16497.0	1.50707
$494$	0	0
$495$	0	0
$496$	−155.885	−0.0141117
$497$	−14640.0	−1.32132
$498$	0	0
$499$	17677.3	1.58586	0.792931	−	0.609311i	$-0.208554\pi$
$499$	17677.3	1.58586	0.792931	+	0.609311i	$0.208554\pi$
$500$	−2139.08	−0.191325
$501$	0	0
$502$	6890.10	0.612590
$503$	−3876.00	−0.343583	−0.171792	−	0.985133i	$-0.554956\pi$
$503$	−3876.00	−0.343583	−0.171792	+	0.985133i	$0.554956\pi$
$504$	0	0
$505$	−743.050	−0.0654758
$506$	1872.00	0.164467
$507$	0	0
$508$	1430.00	0.124894
$509$	17065.9	1.48612	0.743058	−	0.669228i	$-0.233375\pi$
$509$	17065.9	1.48612	0.743058	+	0.669228i	$0.233375\pi$
$510$	0	0
$511$	−6360.00	−0.550587
$512$	−361.999	−0.0312465
$513$	0	0
$514$	−3580.15	−0.307225
$515$	315.233	0.0269725
$516$	0	0
$517$	−4176.00	−0.355242
$518$	3450.25	0.292655
$519$	0	0
$520$	0	0
$521$	−2121.00	−0.178355	−0.0891773	−	0.996016i	$-0.528424\pi$
$521$	−2121.00	−0.178355	−0.0891773	+	0.996016i	$0.528424\pi$
$522$	0	0
$523$	−11464.0	−0.958481	−0.479241	−	0.877684i	$-0.659088\pi$
$523$	−11464.0	−0.958481	−0.479241	+	0.877684i	$0.659088\pi$
$524$	−9870.00	−0.822849
$525$	0	0
$526$	−3554.17	−0.294618
$527$	−18238.5	−1.50755
$528$	0	0
$529$	−6083.00	−0.499959
$530$	−279.000	−0.0228660
$531$	0	0
$532$	7920.00	0.645443
$533$	0	0
$534$	0	0
$535$	−2608.47	−0.210792
$536$	−17706.0	−1.42683
$537$	0	0
$538$	5767.73	0.462202
$539$	−2092.32	−0.167203
$540$	0	0
$541$	−4764.87	−0.378665	−0.189333	−	0.981913i	$-0.560632\pi$
$541$	−4764.87	−0.378665	−0.189333	+	0.981913i	$0.560632\pi$
$542$	4860.00	0.385157
$543$	0	0
$544$	−21278.2	−1.67702
$545$	2688.00	0.211268
$546$	0	0
$547$	6554.00	0.512301	0.256151	−	0.966637i	$-0.417546\pi$
$547$	6554.00	0.512301	0.256151	+	0.966637i	$0.417546\pi$
$548$	−4234.86	−0.330118
$549$	0	0
$550$	2928.00	0.227001
$551$	16118.5	1.24622
$552$	0	0
$553$	−17680.8	−1.35961
$554$	652.983	0.0500769
$555$	0	0
$556$	−1180.00	−0.0900057
$557$	−18112.1	−1.37780	−0.688898	−	0.724858i	$-0.741905\pi$
$557$	−18112.1	−1.37780	−0.688898	+	0.724858i	$0.741905\pi$
$558$	0	0
$559$	0	0
$560$	24.0000	0.00181104
$561$	0	0
$562$	−63.0000	−0.00472864
$563$	−12168.0	−0.910870	−0.455435	−	0.890269i	$-0.650516\pi$
$563$	−12168.0	−0.910870	−0.455435	+	0.890269i	$0.650516\pi$
$564$	0	0
$565$	−1189.92	−0.0886022
$566$	−12339.1	−0.916347
$567$	0	0
$568$	23790.0	1.75741
$569$	−7722.00	−0.568933	−0.284467	−	0.958686i	$-0.591817\pi$
$569$	−7722.00	−0.568933	−0.284467	+	0.958686i	$0.591817\pi$
$570$	0	0
$571$	−11440.0	−0.838440	−0.419220	−	0.907885i	$-0.637696\pi$
$571$	−11440.0	−0.838440	−0.419220	+	0.907885i	$0.637696\pi$
$572$	0	0
$573$	0	0
$574$	−6526.37	−0.474574
$575$	9516.00	0.690165
$576$	0	0
$577$	−15444.7	−1.11433	−0.557167	−	0.830400i	$-0.688112\pi$
$577$	−15444.7	−1.11433	−0.557167	+	0.830400i	$0.688112\pi$
$578$	−15200.5	−1.09387
$579$	0	0
$580$	1221.10	0.0874194
$581$	10944.0	0.781469
$582$	0	0
$583$	−1288.65	−0.0915442
$584$	10335.0	0.732304
$585$	0	0
$586$	−14415.0	−1.01617
$587$	14071.2	0.989403	0.494702	−	0.869063i	$-0.335277\pi$
$587$	14071.2	0.989403	0.494702	+	0.869063i	$0.335277\pi$
$588$	0	0
$589$	−17820.0	−1.24662
$590$	852.169	0.0594631
$591$	0	0
$592$	143.760	0.00998059
$593$	−26938.6	−1.86549	−0.932745	−	0.360538i	$-0.882593\pi$
$593$	−26938.6	−1.86549	−0.932745	+	0.360538i	$0.882593\pi$
$594$	0	0
$595$	2808.00	0.193474
$596$	233.827	0.0160703
$597$	0	0
$598$	0	0
$599$	10554.0	0.719908	0.359954	−	0.932970i	$-0.382792\pi$
$599$	10554.0	0.719908	0.359954	+	0.932970i	$0.382792\pi$
$600$	0	0
$601$	−14831.0	−1.00660	−0.503302	−	0.864111i	$-0.667882\pi$
$601$	−14831.0	−1.00660	−0.503302	+	0.864111i	$0.667882\pi$
$602$	−2496.00	−0.168986
$603$	0	0
$604$	8850.78	0.596247
$605$	1972.81	0.132572
$606$	0	0
$607$	−7954.00	−0.531866	−0.265933	−	0.963991i	$-0.585680\pi$
$607$	−7954.00	−0.531866	−0.265933	+	0.963991i	$0.585680\pi$
$608$	−20790.0	−1.38675
$609$	0	0
$610$	435.000	0.0288732
$611$	0	0
$612$	0	0
$613$	25220.4	1.66173	0.830866	−	0.556472i	$-0.187845\pi$
$613$	25220.4	1.66173	0.830866	+	0.556472i	$0.187845\pi$
$614$	3846.00	0.252788
$615$	0	0
$616$	−4323.20	−0.282771
$617$	−17384.6	−1.13432	−0.567162	−	0.823607i	$-0.691959\pi$
$617$	−17384.6	−1.13432	−0.567162	+	0.823607i	$0.691959\pi$
$618$	0	0
$619$	−8209.92	−0.533093	−0.266547	−	0.963822i	$-0.585883\pi$
$619$	−8209.92	−0.533093	−0.266547	+	0.963822i	$0.585883\pi$
$620$	−1350.00	−0.0874473
$621$	0	0
$622$	−8511.30	−0.548669
$623$	13536.0	0.870479
$624$	0	0
$625$	14509.0	0.928576
$626$	897.202	0.0572834
$627$	0	0
$628$	−6055.00	−0.384747
$629$	16819.9	1.06622
$630$	0	0
$631$	12865.7	0.811687	0.405843	−	0.913943i	$-0.366978\pi$
$631$	12865.7	0.811687	0.405843	+	0.913943i	$0.366978\pi$
$632$	28731.3	1.80834
$633$	0	0
$634$	−6783.00	−0.424901
$635$	495.367	0.0309575
$636$	0	0
$637$	0	0
$638$	−3384.00	−0.209990
$639$	0	0
$640$	−1599.00	−0.0987594
$641$	6201.00	0.382098	0.191049	−	0.981581i	$-0.438811\pi$
$641$	6201.00	0.382098	0.191049	+	0.981581i	$0.438811\pi$
$642$	0	0
$643$	16821.7	1.03170	0.515849	−	0.856679i	$-0.327476\pi$
$643$	16821.7	1.03170	0.515849	+	0.856679i	$0.327476\pi$
$644$	−5404.00	−0.330664
$645$	0	0
$646$	−23166.0	−1.41092
$647$	−13494.0	−0.819944	−0.409972	−	0.912098i	$-0.634462\pi$
$647$	−13494.0	−0.819944	−0.409972	+	0.912098i	$0.634462\pi$
$648$	0	0
$649$	3936.00	0.238061
$650$	0	0
$651$	0	0
$652$	5022.95	0.301708
$653$	11334.0	0.679225	0.339612	−	0.940566i	$-0.389704\pi$
$653$	11334.0	0.679225	0.339612	+	0.940566i	$0.389704\pi$
$654$	0	0
$655$	−3419.07	−0.203960
$656$	−271.932	−0.0161847
$657$	0	0
$658$	−7233.04	−0.428531
$659$	−13236.0	−0.782400	−0.391200	−	0.920306i	$-0.627940\pi$
$659$	−13236.0	−0.782400	−0.391200	+	0.920306i	$0.627940\pi$
$660$	0	0
$661$	11852.4	0.697437	0.348718	−	0.937228i	$-0.386617\pi$
$661$	11852.4	0.697437	0.348718	+	0.937228i	$0.386617\pi$
$662$	−12912.0	−0.758065
$663$	0	0
$664$	−17784.0	−1.03939
$665$	2743.57	0.159986
$666$	0	0
$667$	−10998.0	−0.638447
$668$	−4572.61	−0.264850
$669$	0	0
$670$	−2359.05	−0.136027
$671$	2009.18	0.115594
$672$	0	0
$673$	−8021.00	−0.459416	−0.229708	−	0.973260i	$-0.573777\pi$
$673$	−8021.00	−0.459416	−0.229708	+	0.973260i	$0.573777\pi$
$674$	−6192.08	−0.353873
$675$	0	0
$676$	0	0
$677$	21630.0	1.22793	0.613965	−	0.789333i	$-0.289574\pi$
$677$	21630.0	1.22793	0.613965	+	0.789333i	$0.289574\pi$
$678$	0	0
$679$	2784.00	0.157349
$680$	−4563.00	−0.257328
$681$	0	0
$682$	3741.23	0.210057
$683$	26538.5	1.48677	0.743387	−	0.668861i	$-0.233218\pi$
$683$	26538.5	1.48677	0.743387	+	0.668861i	$0.233218\pi$
$684$	0	0
$685$	−1467.00	−0.0818266
$686$	−11856.0	−0.659860
$687$	0	0
$688$	−104.000	−0.00576303
$689$	0	0
$690$	0	0
$691$	−831.384	−0.0457704	−0.0228852	−	0.999738i	$-0.507285\pi$
$691$	−831.384	−0.0457704	−0.0228852	+	0.999738i	$0.507285\pi$
$692$	−12870.0	−0.707000
$693$	0	0
$694$	−12065.5	−0.659941
$695$	−408.764	−0.0223098
$696$	0	0
$697$	−31816.0	−1.72901
$698$	−11520.0	−0.624697
$699$	0	0
$700$	−8452.41	−0.456387
$701$	30186.0	1.62640	0.813202	−	0.581981i	$-0.197722\pi$
$701$	30186.0	1.62640	0.813202	+	0.581981i	$0.197722\pi$
$702$	0	0
$703$	16434.0	0.881679
$704$	4253.92	0.227735
$705$	0	0
$706$	9753.00	0.519914
$707$	−5944.40	−0.316212
$708$	0	0
$709$	−11880.1	−0.629292	−0.314646	−	0.949209i	$-0.601886\pi$
$709$	−11880.1	−0.629292	−0.314646	+	0.949209i	$0.601886\pi$
$710$	3169.65	0.167542
$711$	0	0
$712$	−21996.0	−1.15777
$713$	12159.0	0.638651
$714$	0	0
$715$	0	0
$716$	−18720.0	−0.977094
$717$	0	0
$718$	−12348.0	−0.641815
$719$	18408.0	0.954802	0.477401	−	0.878686i	$-0.341579\pi$
$719$	18408.0	0.954802	0.477401	+	0.878686i	$0.341579\pi$
$720$	0	0
$721$	2521.87	0.130262
$722$	−10754.3	−0.554340
$723$	0	0
$724$	−3185.00	−0.163494
$725$	−17202.0	−0.881195
$726$	0	0
$727$	−21112.0	−1.07703	−0.538515	−	0.842616i	$-0.681014\pi$
$727$	−21112.0	−1.07703	−0.538515	+	0.842616i	$0.681014\pi$
$728$	0	0
$729$	0	0
$730$	1376.98	0.0698142
$731$	−12168.0	−0.615663
$732$	0	0
$733$	23959.5	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$733$	23959.5	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$734$	−3.46410	−0.000174199 0
$735$	0	0
$736$	14185.5	0.710441
$737$	−10896.0	−0.544585
$738$	0	0
$739$	−3166.19	−0.157605	−0.0788025	−	0.996890i	$-0.525110\pi$
$739$	−3166.19	−0.157605	−0.0788025	+	0.996890i	$0.525110\pi$
$740$	1245.00	0.0618474
$741$	0	0
$742$	−2232.00	−0.110430
$743$	30103.0	1.48637	0.743185	−	0.669086i	$-0.233314\pi$
$743$	30103.0	1.48637	0.743185	+	0.669086i	$0.233314\pi$
$744$	0	0
$745$	81.0000	0.00398337
$746$	−6060.45	−0.297438
$747$	0	0
$748$	−8106.00	−0.396236
$749$	−20867.7	−1.01801
$750$	0	0
$751$	−28496.0	−1.38460	−0.692299	−	0.721610i	$-0.743402\pi$
$751$	−28496.0	−1.38460	−0.692299	+	0.721610i	$0.743402\pi$
$752$	−301.377	−0.0146145
$753$	0	0
$754$	0	0
$755$	3066.00	0.147792
$756$	0	0
$757$	17422.0	0.836477	0.418239	−	0.908337i	$-0.362648\pi$
$757$	17422.0	0.836477	0.418239	+	0.908337i	$0.362648\pi$
$758$	−9558.00	−0.457998
$759$	0	0
$760$	−4458.30	−0.212789
$761$	−41326.7	−1.96858	−0.984292	−	0.176547i	$-0.943507\pi$
$761$	−41326.7	−1.96858	−0.984292	+	0.176547i	$0.943507\pi$
$762$	0	0
$763$	21504.0	1.02031
$764$	−12990.0	−0.615133
$765$	0	0
$766$	12756.0	0.601688
$767$	0	0
$768$	0	0
$769$	−14071.2	−0.659844	−0.329922	−	0.944008i	$-0.607022\pi$
$769$	−14071.2	−0.659844	−0.329922	+	0.944008i	$0.607022\pi$
$770$	−576.000	−0.0269579
$771$	0	0
$772$	5585.86	0.260414
$773$	200.918	0.00934866	0.00467433	−	0.999989i	$-0.498512\pi$
$773$	200.918	0.00934866	0.00467433	+	0.999989i	$0.498512\pi$
$774$	0	0
$775$	19017.9	0.881476
$776$	−4524.00	−0.209281
$777$	0	0
$778$	2094.05	0.0964978
$779$	−31086.0	−1.42975
$780$	0	0
$781$	14640.0	0.670756
$782$	15806.7	0.722821
$783$	0	0
$784$	−151.000	−0.00687864
$785$	−2097.51	−0.0953675
$786$	0	0
$787$	−6903.95	−0.312706	−0.156353	−	0.987701i	$-0.549974\pi$
$787$	−6903.95	−0.312706	−0.156353	+	0.987701i	$0.549974\pi$
$788$	10253.7	0.463546
$789$	0	0
$790$	3828.00	0.172398
$791$	−9519.35	−0.427900
$792$	0	0
$793$	0	0
$794$	−20256.0	−0.905363
$795$	0	0
$796$	−12610.0	−0.561494
$797$	−31278.0	−1.39012	−0.695059	−	0.718953i	$-0.744622\pi$
$797$	−31278.0	−1.39012	−0.695059	+	0.718953i	$0.744622\pi$
$798$	0	0
$799$	−35261.1	−1.56126
$800$	22187.6	0.980561
$801$	0	0
$802$	5163.00	0.227322
$803$	6360.00	0.279501
$804$	0	0
$805$	−1872.00	−0.0819619
$806$	0	0
$807$	0	0
$808$	9659.65	0.420576
$809$	−8049.00	−0.349799	−0.174900	−	0.984586i	$-0.555960\pi$
$809$	−8049.00	−0.349799	−0.174900	+	0.984586i	$0.555960\pi$
$810$	0	0
$811$	−14026.1	−0.607305	−0.303653	−	0.952783i	$-0.598206\pi$
$811$	−14026.1	−0.607305	−0.303653	+	0.952783i	$0.598206\pi$
$812$	9768.77	0.422188
$813$	0	0
$814$	−3450.25	−0.148564
$815$	1740.00	0.0747847
$816$	0	0
$817$	−11888.8	−0.509102
$818$	75.0000	0.00320576
$819$	0	0
$820$	−2355.00	−0.100293
$821$	8036.72	0.341636	0.170818	−	0.985303i	$-0.445359\pi$
$821$	8036.72	0.341636	0.170818	+	0.985303i	$0.445359\pi$
$822$	0	0
$823$	40300.0	1.70689	0.853445	−	0.521184i	$-0.174509\pi$
$823$	40300.0	1.70689	0.853445	+	0.521184i	$0.174509\pi$
$824$	−4098.03	−0.173255
$825$	0	0
$826$	6817.35	0.287174
$827$	−39525.4	−1.66195	−0.830975	−	0.556310i	$-0.812217\pi$
$827$	−39525.4	−1.66195	−0.830975	+	0.556310i	$0.812217\pi$
$828$	0	0
$829$	12311.0	0.515776	0.257888	−	0.966175i	$-0.416973\pi$
$829$	12311.0	0.515776	0.257888	+	0.966175i	$0.416973\pi$
$830$	−2369.45	−0.0990899
$831$	0	0
$832$	0	0
$833$	−17667.0	−0.734844
$834$	0	0
$835$	−1584.00	−0.0656486
$836$	−7920.00	−0.327654
$837$	0	0
$838$	−16388.7	−0.675581
$839$	−21467.0	−0.883343	−0.441671	−	0.897177i	$-0.645614\pi$
$839$	−21467.0	−0.883343	−0.441671	+	0.897177i	$0.645614\pi$
$840$	0	0
$841$	−4508.00	−0.184837
$842$	12243.0	0.501095
$843$	0	0
$844$	−5210.00	−0.212483
$845$	0	0
$846$	0	0
$847$	15782.4	0.640249
$848$	−93.0000	−0.00376608
$849$	0	0
$850$	24723.3	0.997649
$851$	−11213.3	−0.451688
$852$	0	0
$853$	774.227	0.0310774	0.0155387	−	0.999879i	$-0.495054\pi$
$853$	774.227	0.0310774	0.0155387	+	0.999879i	$0.495054\pi$
$854$	3480.00	0.139442
$855$	0	0
$856$	33910.1	1.35400
$857$	13923.0	0.554960	0.277480	−	0.960731i	$-0.410501\pi$
$857$	13923.0	0.554960	0.277480	+	0.960731i	$0.410501\pi$
$858$	0	0
$859$	−22358.0	−0.888062	−0.444031	−	0.896011i	$-0.646452\pi$
$859$	−22358.0	−0.888062	−0.444031	+	0.896011i	$0.646452\pi$
$860$	−900.666	−0.0357122
$861$	0	0
$862$	−17196.0	−0.679464
$863$	−2230.88	−0.0879955	−0.0439977	−	0.999032i	$-0.514009\pi$
$863$	−2230.88	−0.0879955	−0.0439977	+	0.999032i	$0.514009\pi$
$864$	0	0
$865$	−4458.30	−0.175245
$866$	−11461.0	−0.449723
$867$	0	0
$868$	−10800.0	−0.422322
$869$	17680.8	0.690195
$870$	0	0
$871$	0	0
$872$	−34944.0	−1.35706
$873$	0	0
$874$	15444.0	0.597713
$875$	−5928.00	−0.229032
$876$	0	0
$877$	16754.1	0.645093	0.322547	−	0.946554i	$-0.395461\pi$
$877$	16754.1	0.645093	0.322547	+	0.946554i	$0.395461\pi$
$878$	24227.9	0.931268
$879$	0	0
$880$	−24.0000	−0.000919363 0
$881$	17355.0	0.663683	0.331842	−	0.943335i	$-0.392330\pi$
$881$	17355.0	0.663683	0.331842	+	0.943335i	$0.392330\pi$
$882$	0	0
$883$	46982.0	1.79057	0.895283	−	0.445497i	$-0.146973\pi$
$883$	46982.0	1.79057	0.895283	+	0.445497i	$0.146973\pi$
$884$	0	0
$885$	0	0
$886$	3471.03	0.131616
$887$	8916.00	0.337508	0.168754	−	0.985658i	$-0.446026\pi$
$887$	8916.00	0.337508	0.168754	+	0.985658i	$0.446026\pi$
$888$	0	0
$889$	3962.93	0.149508
$890$	−2930.63	−0.110376
$891$	0	0
$892$	12037.8	0.451854
$893$	−34452.0	−1.29103
$894$	0	0
$895$	−6484.80	−0.242193
$896$	−12792.0	−0.476954
$897$	0	0
$898$	15732.0	0.584614
$899$	−21979.7	−0.815423
$900$	0	0
$901$	−10881.0	−0.402329
$902$	6526.37	0.240914
$903$	0	0
$904$	15468.9	0.569126
$905$	−1103.32	−0.0405254
$906$	0	0
$907$	30836.0	1.12888	0.564439	−	0.825475i	$-0.309092\pi$
$907$	30836.0	1.12888	0.564439	+	0.825475i	$0.309092\pi$
$908$	−12037.8	−0.439964
$909$	0	0
$910$	0	0
$911$	27480.0	0.999400	0.499700	−	0.866199i	$-0.333444\pi$
$911$	27480.0	0.999400	0.499700	+	0.866199i	$0.333444\pi$
$912$	0	0
$913$	−10944.0	−0.396707
$914$	−4371.00	−0.158184
$915$	0	0
$916$	−12540.0	−0.452331
$917$	−27352.5	−0.985017
$918$	0	0
$919$	−28442.0	−1.02091	−0.510454	−	0.859905i	$-0.670523\pi$
$919$	−28442.0	−1.02091	−0.510454	+	0.859905i	$0.670523\pi$
$920$	3042.00	0.109013
$921$	0	0
$922$	−33927.0	−1.21185
$923$	0	0
$924$	0	0
$925$	−17538.7	−0.623427
$926$	−14952.0	−0.530619
$927$	0	0
$928$	−25643.0	−0.907083
$929$	6978.43	0.246453	0.123227	−	0.992379i	$-0.460676\pi$
$929$	6978.43	0.246453	0.123227	+	0.992379i	$0.460676\pi$
$930$	0	0
$931$	−17261.6	−0.607655
$932$	29250.0	1.02802
$933$	0	0
$934$	−9457.00	−0.331309
$935$	−2808.00	−0.0982154
$936$	0	0
$937$	−38465.0	−1.34109	−0.670543	−	0.741871i	$-0.733939\pi$
$937$	−38465.0	−1.34109	−0.670543	+	0.741871i	$0.733939\pi$
$938$	−18872.4	−0.656937
$939$	0	0
$940$	−2610.00	−0.0905626
$941$	4884.38	0.169210	0.0846049	−	0.996415i	$-0.473037\pi$
$941$	4884.38	0.169210	0.0846049	+	0.996415i	$0.473037\pi$
$942$	0	0
$943$	21210.7	0.732466
$944$	284.056	0.00979369
$945$	0	0
$946$	2496.00	0.0857843
$947$	21765.0	0.746849	0.373424	−	0.927661i	$-0.378184\pi$
$947$	21765.0	0.746849	0.373424	+	0.927661i	$0.378184\pi$
$948$	0	0
$949$	0	0
$950$	24156.0	0.824973
$951$	0	0
$952$	−36504.0	−1.24275
$953$	6474.00	0.220056	0.110028	−	0.993928i	$-0.464906\pi$
$953$	6474.00	0.220056	0.110028	+	0.993928i	$0.464906\pi$
$954$	0	0
$955$	−4499.87	−0.152474
$956$	26916.1	0.910594
$957$	0	0
$958$	4422.00	0.149132
$959$	−11736.0	−0.395177
$960$	0	0
$961$	−5491.00	−0.184317
$962$	0	0
$963$	0	0
$964$	−24586.5	−0.821449
$965$	1935.00	0.0645491
$966$	0	0
$967$	−7541.35	−0.250789	−0.125395	−	0.992107i	$-0.540020\pi$
$967$	−7541.35	−0.250789	−0.125395	+	0.992107i	$0.540020\pi$
$968$	−25646.5	−0.851559
$969$	0	0
$970$	−602.754	−0.0199518
$971$	−34998.0	−1.15668	−0.578342	−	0.815795i	$-0.696300\pi$
$971$	−34998.0	−1.15668	−0.578342	+	0.815795i	$0.696300\pi$
$972$	0	0
$973$	−3270.11	−0.107744
$974$	18756.0	0.617024
$975$	0	0
$976$	145.000	0.00475547
$977$	25216.9	0.825753	0.412877	−	0.910787i	$-0.364524\pi$
$977$	25216.9	0.825753	0.412877	+	0.910787i	$0.364524\pi$
$978$	0	0
$979$	−13536.0	−0.441892
$980$	−1307.70	−0.0426254
$981$	0	0
$982$	−19724.6	−0.640975
$983$	56440.6	1.83131	0.915654	−	0.401967i	$-0.131673\pi$
$983$	56440.6	1.83131	0.915654	+	0.401967i	$0.131673\pi$
$984$	0	0
$985$	3552.00	0.114900
$986$	−28573.6	−0.922891
$987$	0	0
$988$	0	0
$989$	8112.00	0.260816
$990$	0	0
$991$	59282.0	1.90026	0.950129	−	0.311859i	$-0.100951\pi$
$991$	59282.0	1.90026	0.950129	+	0.311859i	$0.100951\pi$
$992$	28350.0	0.907372
$993$	0	0
$994$	25357.2	0.809137
$995$	−4368.23	−0.139178
$996$	0	0
$997$	−37711.0	−1.19791	−0.598957	−	0.800782i	$-0.704418\pi$
$997$	−37711.0	−1.19791	−0.598957	+	0.800782i	$0.704418\pi$
$998$	−30618.0	−0.971138
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.o.1.1		2
3.2	odd	2		169.4.a.i.1.2		2
13.2	odd	12		117.4.q.a.82.1		2
13.7	odd	12		117.4.q.a.10.1		2
13.12	even	2	inner	1521.4.a.o.1.2		2
39.2	even	12		13.4.e.b.4.1	✓	2
39.5	even	4		169.4.b.d.168.1		2
39.8	even	4		169.4.b.d.168.2		2
39.11	even	12		169.4.e.a.147.1		2
39.17	odd	6		169.4.c.h.146.2		4
39.20	even	12		13.4.e.b.10.1	yes	2
39.23	odd	6		169.4.c.h.22.2		4
39.29	odd	6		169.4.c.h.22.1		4
39.32	even	12		169.4.e.a.23.1		2
39.35	odd	6		169.4.c.h.146.1		4
39.38	odd	2		169.4.a.i.1.1		2
156.59	odd	12		208.4.w.b.49.1		2
156.119	odd	12		208.4.w.b.17.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.e.b.4.1	✓	2	39.2	even	12
13.4.e.b.10.1	yes	2	39.20	even	12
117.4.q.a.10.1		2	13.7	odd	12
117.4.q.a.82.1		2	13.2	odd	12
169.4.a.i.1.1		2	39.38	odd	2
169.4.a.i.1.2		2	3.2	odd	2
169.4.b.d.168.1		2	39.5	even	4
169.4.b.d.168.2		2	39.8	even	4
169.4.c.h.22.1		4	39.29	odd	6
169.4.c.h.22.2		4	39.23	odd	6
169.4.c.h.146.1		4	39.35	odd	6
169.4.c.h.146.2		4	39.17	odd	6
169.4.e.a.23.1		2	39.32	even	12
169.4.e.a.147.1		2	39.11	even	12
208.4.w.b.17.1		2	156.119	odd	12
208.4.w.b.49.1		2	156.59	odd	12
1521.4.a.o.1.1		2	1.1	even	1	trivial
1521.4.a.o.1.2		2	13.12	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} -5.00000 q^{4} -1.73205 q^{5} -13.8564 q^{7} +22.5167 q^{8} +3.00000 q^{10} +13.8564 q^{11} +24.0000 q^{14} +1.00000 q^{16} +117.000 q^{17} +114.315 q^{19} +8.66025 q^{20} -24.0000 q^{22} -78.0000 q^{23} -122.000 q^{25} +69.2820 q^{28} +141.000 q^{29} -155.885 q^{31} -181.865 q^{32} -202.650 q^{34} +24.0000 q^{35} +143.760 q^{37} -198.000 q^{38} -39.0000 q^{40} -271.932 q^{41} -104.000 q^{43} -69.2820 q^{44} +135.100 q^{46} -301.377 q^{47} -151.000 q^{49} +211.310 q^{50} -93.0000 q^{53} -24.0000 q^{55} -312.000 q^{56} -244.219 q^{58} +284.056 q^{59} +145.000 q^{61} +270.000 q^{62} +307.000 q^{64} -786.351 q^{67} -585.000 q^{68} -41.5692 q^{70} +1056.55 q^{71} +458.993 q^{73} -249.000 q^{74} -571.577 q^{76} -192.000 q^{77} +1276.00 q^{79} -1.73205 q^{80} +471.000 q^{82} -789.815 q^{83} -202.650 q^{85} +180.133 q^{86} +312.000 q^{88} -976.877 q^{89} +390.000 q^{92} +522.000 q^{94} -198.000 q^{95} -200.918 q^{97} +261.540 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{4}+O(q^{10}) \) 2 * q - 10 * q^4	\( 2 q - 10 q^{4} + 6 q^{10} + 48 q^{14} + 2 q^{16} + 234 q^{17} - 48 q^{22} - 156 q^{23} - 244 q^{25} + 282 q^{29} + 48 q^{35} - 396 q^{38} - 78 q^{40} - 208 q^{43} - 302 q^{49} - 186 q^{53} - 48 q^{55} - 624 q^{56} + 290 q^{61} + 540 q^{62} + 614 q^{64} - 1170 q^{68} - 498 q^{74} - 384 q^{77} + 2552 q^{79} + 942 q^{82} + 624 q^{88} + 780 q^{92} + 1044 q^{94} - 396 q^{95}+O(q^{100}) \) 2 * q - 10 * q^4 + 6 * q^10 + 48 * q^14 + 2 * q^16 + 234 * q^17 - 48 * q^22 - 156 * q^23 - 244 * q^25 + 282 * q^29 + 48 * q^35 - 396 * q^38 - 78 * q^40 - 208 * q^43 - 302 * q^49 - 186 * q^53 - 48 * q^55 - 624 * q^56 + 290 * q^61 + 540 * q^62 + 614 * q^64 - 1170 * q^68 - 498 * q^74 - 384 * q^77 + 2552 * q^79 + 942 * q^82 + 624 * q^88 + 780 * q^92 + 1044 * q^94 - 396 * q^95

Introduction

L-functions

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