LMFDB - Embedded newform 1521.4.a.x.1.3

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:	$4$
Coefficient field:	4.4.5054412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 29x^{2} + 48 $ x^4 - 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.32750$ 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)

$f(q)$	$=$	$q+1.32750 q^{2} -6.23774 q^{4} +15.4241 q^{5} +7.96501 q^{7} -18.9006 q^{8} +O(q^{10})$	$q+1.32750 q^{2} -6.23774 q^{4} +15.4241 q^{5} +7.96501 q^{7} -18.9006 q^{8} +20.4755 q^{10} +12.7691 q^{11} +10.5736 q^{14} +24.8113 q^{16} +54.0000 q^{17} +84.5794 q^{19} -96.2113 q^{20} +16.9510 q^{22} -122.853 q^{23} +112.902 q^{25} -49.6837 q^{28} -140.853 q^{29} +116.439 q^{31} +184.142 q^{32} +71.6851 q^{34} +122.853 q^{35} +433.898 q^{37} +112.279 q^{38} -291.525 q^{40} -205.823 q^{41} -418.853 q^{43} -79.6501 q^{44} -163.087 q^{46} +485.861 q^{47} -279.559 q^{49} +149.877 q^{50} +674.559 q^{53} +196.951 q^{55} -150.544 q^{56} -186.982 q^{58} +186.226 q^{59} -671.902 q^{61} +154.574 q^{62} +45.9584 q^{64} -14.0364 q^{67} -336.838 q^{68} +163.087 q^{70} -346.789 q^{71} +832.900 q^{73} +576.000 q^{74} -527.584 q^{76} +101.706 q^{77} -335.608 q^{79} +382.691 q^{80} -273.230 q^{82} +568.797 q^{83} +832.900 q^{85} -556.028 q^{86} -241.343 q^{88} -236.671 q^{89} +766.324 q^{92} +644.981 q^{94} +1304.56 q^{95} +1278.94 q^{97} -371.115 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 26 q^{4}+O(q^{10}) $ 4 * q + 26 * q^4	$ 4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} + 1050 q^{64} + 1404 q^{68} + 2304 q^{74} - 816 q^{77} + 288 q^{79} + 28 q^{82} - 2392 q^{88} + 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) $ 4 * q + 26 * q^4 - 20 * q^10 + 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 + 44 * q^25 + 48 * q^29 - 120 * q^35 - 468 * q^38 - 1268 * q^40 - 1064 * q^43 + 716 * q^49 + 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 + 1050 * q^64 + 1404 * q^68 + 2304 * q^74 - 816 * q^77 + 288 * q^79 + 28 * q^82 - 2392 * q^88 + 8568 * q^92 + 6656 * q^94 + 3384 * 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 $

$2$	1.32750	0.469343	0.234671	−	0.972075i	$-0.424599\pi$
$2$	1.32750	0.469343	0.234671	+	0.972075i	$0.424599\pi$
$3$	0	0
$3$	0	0
$4$	−6.23774	−0.779717
$5$	15.4241	1.37957	0.689785	−	0.724014i	$-0.257705\pi$
$5$	15.4241	1.37957	0.689785	+	0.724014i	$0.257705\pi$
$6$	0	0
$7$	7.96501	0.430070	0.215035	−	0.976606i	$-0.431013\pi$
$7$	7.96501	0.430070	0.215035	+	0.976606i	$0.431013\pi$
$8$	−18.9006	−0.835297
$9$	0	0
$10$	20.4755	0.647491
$11$	12.7691	0.350002	0.175001	−	0.984568i	$-0.444007\pi$
$11$	12.7691	0.350002	0.175001	+	0.984568i	$0.444007\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	10.5736	0.201850
$15$	0	0
$16$	24.8113	0.387677
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	84.5794	1.02126	0.510628	−	0.859802i	$-0.329413\pi$
$19$	84.5794	1.02126	0.510628	+	0.859802i	$0.329413\pi$
$20$	−96.2113	−1.07568
$21$	0	0
$22$	16.9510	0.164271
$23$	−122.853	−1.11376	−0.556882	−	0.830591i	$-0.688003\pi$
$23$	−122.853	−1.11376	−0.556882	+	0.830591i	$0.688003\pi$
$24$	0	0
$25$	112.902	0.903215
$26$	0	0
$27$	0	0
$28$	−49.6837	−0.335333
$29$	−140.853	−0.901921	−0.450961	−	0.892544i	$-0.648919\pi$
$29$	−140.853	−0.901921	−0.450961	+	0.892544i	$0.648919\pi$
$30$	0	0
$31$	116.439	0.674617	0.337309	−	0.941394i	$-0.390483\pi$
$31$	116.439	0.674617	0.337309	+	0.941394i	$0.390483\pi$
$32$	184.142	1.01725
$33$	0	0
$34$	71.6851	0.361585
$35$	122.853	0.593312
$36$	0	0
$37$	433.898	1.92790	0.963951	−	0.266081i	$-0.0857289\pi$
$37$	433.898	1.92790	0.963951	+	0.266081i	$0.0857289\pi$
$38$	112.279	0.479319
$39$	0	0
$40$	−291.525	−1.15235
$41$	−205.823	−0.784003	−0.392002	−	0.919965i	$-0.628217\pi$
$41$	−205.823	−0.784003	−0.392002	+	0.919965i	$0.628217\pi$
$42$	0	0
$43$	−418.853	−1.48545	−0.742726	−	0.669595i	$-0.766468\pi$
$43$	−418.853	−1.48545	−0.742726	+	0.669595i	$0.766468\pi$
$44$	−79.6501	−0.272902
$45$	0	0
$46$	−163.087	−0.522737
$47$	485.861	1.50787	0.753937	−	0.656947i	$-0.228152\pi$
$47$	485.861	1.50787	0.753937	+	0.656947i	$0.228152\pi$
$48$	0	0
$49$	−279.559	−0.815040
$50$	149.877	0.423918
$51$	0	0
$52$	0	0
$53$	674.559	1.74826	0.874130	−	0.485693i	$-0.161433\pi$
$53$	674.559	1.74826	0.874130	+	0.485693i	$0.161433\pi$
$54$	0	0
$55$	196.951	0.482852
$56$	−150.544	−0.359236
$57$	0	0
$58$	−186.982	−0.423310
$59$	186.226	0.410925	0.205462	−	0.978665i	$-0.434130\pi$
$59$	186.226	0.410925	0.205462	+	0.978665i	$0.434130\pi$
$60$	0	0
$61$	−671.902	−1.41030	−0.705149	−	0.709059i	$-0.749120\pi$
$61$	−671.902	−1.41030	−0.705149	+	0.709059i	$0.749120\pi$
$62$	154.574	0.316627
$63$	0	0
$64$	45.9584	0.0897626
$65$	0	0
$66$	0	0
$67$	−14.0364	−0.0255944	−0.0127972	−	0.999918i	$-0.504074\pi$
$67$	−14.0364	−0.0255944	−0.0127972	+	0.999918i	$0.504074\pi$
$68$	−336.838	−0.600700
$69$	0	0
$70$	163.087	0.278467
$71$	−346.789	−0.579665	−0.289833	−	0.957077i	$-0.593600\pi$
$71$	−346.789	−0.579665	−0.289833	+	0.957077i	$0.593600\pi$
$72$	0	0
$73$	832.900	1.33539	0.667695	−	0.744435i	$-0.267281\pi$
$73$	832.900	1.33539	0.667695	+	0.744435i	$0.267281\pi$
$74$	576.000	0.904846
$75$	0	0
$76$	−527.584	−0.796290
$77$	101.706	0.150525
$78$	0	0
$79$	−335.608	−0.477960	−0.238980	−	0.971025i	$-0.576813\pi$
$79$	−335.608	−0.477960	−0.238980	+	0.971025i	$0.576813\pi$
$80$	382.691	0.534827
$81$	0	0
$82$	−273.230	−0.367966
$83$	568.797	0.752212	0.376106	−	0.926577i	$-0.377263\pi$
$83$	568.797	0.752212	0.376106	+	0.926577i	$0.377263\pi$
$84$	0	0
$85$	832.900	1.06283
$86$	−556.028	−0.697186
$87$	0	0
$88$	−241.343	−0.292355
$89$	−236.671	−0.281877	−0.140939	−	0.990018i	$-0.545012\pi$
$89$	−236.671	−0.281877	−0.140939	+	0.990018i	$0.545012\pi$
$90$	0	0
$91$	0	0
$92$	766.324	0.868422
$93$	0	0
$94$	644.981	0.707710
$95$	1304.56	1.40889
$96$	0	0
$97$	1278.94	1.33873	0.669365	−	0.742934i	$-0.266566\pi$
$97$	1278.94	1.33873	0.669365	+	0.742934i	$0.266566\pi$
$98$	−371.115	−0.382533
$99$	0	0
$100$	−704.253	−0.704253
$101$	−632.264	−0.622898	−0.311449	−	0.950263i	$-0.600814\pi$
$101$	−632.264	−0.622898	−0.311449	+	0.950263i	$0.600814\pi$
$102$	0	0
$103$	1506.26	1.44094	0.720469	−	0.693487i	$-0.243927\pi$
$103$	1506.26	1.44094	0.720469	+	0.693487i	$0.243927\pi$
$104$	0	0
$105$	0	0
$106$	895.478	0.820533
$107$	1268.56	1.14613	0.573066	−	0.819509i	$-0.305754\pi$
$107$	1268.56	1.14613	0.573066	+	0.819509i	$0.305754\pi$
$108$	0	0
$109$	−347.425	−0.305296	−0.152648	−	0.988281i	$-0.548780\pi$
$109$	−347.425	−0.305296	−0.152648	+	0.988281i	$0.548780\pi$
$110$	261.453	0.226623
$111$	0	0
$112$	197.622	0.166728
$113$	−659.706	−0.549203	−0.274601	−	0.961558i	$-0.588546\pi$
$113$	−659.706	−0.549203	−0.274601	+	0.961558i	$0.588546\pi$
$114$	0	0
$115$	−1894.89	−1.53652
$116$	878.603	0.703244
$117$	0	0
$118$	247.215	0.192865
$119$	430.111	0.331329
$120$	0	0
$121$	−1167.95	−0.877499
$122$	−891.951	−0.661913
$123$	0	0
$124$	−726.319	−0.526011
$125$	−186.602	−0.133521
$126$	0	0
$127$	275.019	0.192157	0.0960787	−	0.995374i	$-0.469370\pi$
$127$	275.019	0.192157	0.0960787	+	0.995374i	$0.469370\pi$
$128$	−1412.13	−0.975121
$129$	0	0
$130$	0	0
$131$	1183.97	0.789648	0.394824	−	0.918757i	$-0.370805\pi$
$131$	1183.97	0.789648	0.394824	+	0.918757i	$0.370805\pi$
$132$	0	0
$133$	673.676	0.439211
$134$	−18.6334	−0.0120125
$135$	0	0
$136$	−1020.63	−0.643519
$137$	2557.36	1.59482	0.797410	−	0.603438i	$-0.206203\pi$
$137$	2557.36	1.59482	0.797410	+	0.603438i	$0.206203\pi$
$138$	0	0
$139$	545.736	0.333012	0.166506	−	0.986040i	$-0.446751\pi$
$139$	545.736	0.333012	0.166506	+	0.986040i	$0.446751\pi$
$140$	−766.324	−0.462616
$141$	0	0
$142$	−460.362	−0.272062
$143$	0	0
$144$	0	0
$145$	−2172.52	−1.24426
$146$	1105.68	0.626756
$147$	0	0
$148$	−2706.54	−1.50322
$149$	1376.78	0.756981	0.378491	−	0.925605i	$-0.376443\pi$
$149$	1376.78	0.756981	0.378491	+	0.925605i	$0.376443\pi$
$150$	0	0
$151$	−2733.47	−1.47316	−0.736579	−	0.676352i	$-0.763560\pi$
$151$	−2733.47	−1.47316	−0.736579	+	0.676352i	$0.763560\pi$
$152$	−1598.60	−0.853052
$153$	0	0
$154$	135.015	0.0706479
$155$	1795.97	0.930683
$156$	0	0
$157$	1029.97	0.523570	0.261785	−	0.965126i	$-0.415689\pi$
$157$	1029.97	0.523570	0.261785	+	0.965126i	$0.415689\pi$
$158$	−445.520	−0.224327
$159$	0	0
$160$	2840.22	1.40337
$161$	−978.524	−0.478997
$162$	0	0
$163$	2882.91	1.38532	0.692660	−	0.721264i	$-0.256439\pi$
$163$	2882.91	1.38532	0.692660	+	0.721264i	$0.256439\pi$
$164$	1283.87	0.611301
$165$	0	0
$166$	755.079	0.353045
$167$	1153.90	0.534679	0.267340	−	0.963602i	$-0.413855\pi$
$167$	1153.90	0.534679	0.267340	+	0.963602i	$0.413855\pi$
$168$	0	0
$169$	0	0
$170$	1105.68	0.498832
$171$	0	0
$172$	2612.69	1.15823
$173$	1688.85	0.742203	0.371101	−	0.928592i	$-0.378980\pi$
$173$	1688.85	0.742203	0.371101	+	0.928592i	$0.378980\pi$
$174$	0	0
$175$	899.265	0.388446
$176$	316.817	0.135687
$177$	0	0
$178$	−314.181	−0.132297
$179$	942.793	0.393674	0.196837	−	0.980436i	$-0.436933\pi$
$179$	942.793	0.393674	0.196837	+	0.980436i	$0.436933\pi$
$180$	0	0
$181$	482.030	0.197950	0.0989751	−	0.995090i	$-0.468444\pi$
$181$	482.030	0.197950	0.0989751	+	0.995090i	$0.468444\pi$
$182$	0	0
$183$	0	0
$184$	2322.00	0.930325
$185$	6692.47	2.65968
$186$	0	0
$187$	689.530	0.269644
$188$	−3030.67	−1.17572
$189$	0	0
$190$	1731.80	0.661254
$191$	−4223.32	−1.59994	−0.799971	−	0.600039i	$-0.795152\pi$
$191$	−4223.32	−1.59994	−0.799971	+	0.600039i	$0.795152\pi$
$192$	0	0
$193$	229.092	0.0854424	0.0427212	−	0.999087i	$-0.486397\pi$
$193$	229.092	0.0854424	0.0427212	+	0.999087i	$0.486397\pi$
$194$	1697.80	0.628323
$195$	0	0
$196$	1743.81	0.635501
$197$	−228.335	−0.0825798	−0.0412899	−	0.999147i	$-0.513147\pi$
$197$	−228.335	−0.0825798	−0.0412899	+	0.999147i	$0.513147\pi$
$198$	0	0
$199$	2939.02	1.04694	0.523471	−	0.852043i	$-0.324637\pi$
$199$	2939.02	1.04694	0.523471	+	0.852043i	$0.324637\pi$
$200$	−2133.92	−0.754453
$201$	0	0
$202$	−839.332	−0.292352
$203$	−1121.89	−0.387889
$204$	0	0
$205$	−3174.63	−1.08159
$206$	1999.57	0.676294
$207$	0	0
$208$	0	0
$209$	1080.00	0.357441
$210$	0	0
$211$	−1607.02	−0.524321	−0.262161	−	0.965024i	$-0.584435\pi$
$211$	−1607.02	−0.524321	−0.262161	+	0.965024i	$0.584435\pi$
$212$	−4207.72	−1.36315
$213$	0	0
$214$	1684.01	0.537929
$215$	−6460.42	−2.04929
$216$	0	0
$217$	927.441	0.290133
$218$	−461.207	−0.143288
$219$	0	0
$220$	−1228.53	−0.376488
$221$	0	0
$222$	0	0
$223$	−130.867	−0.0392981	−0.0196490	−	0.999807i	$-0.506255\pi$
$223$	−130.867	−0.0392981	−0.0196490	+	0.999807i	$0.506255\pi$
$224$	1466.69	0.437489
$225$	0	0
$226$	−875.761	−0.257764
$227$	4325.19	1.26464	0.632319	−	0.774708i	$-0.282103\pi$
$227$	4325.19	1.26464	0.632319	+	0.774708i	$0.282103\pi$
$228$	0	0
$229$	2621.57	0.756499	0.378250	−	0.925704i	$-0.376526\pi$
$229$	2621.57	0.756499	0.378250	+	0.925704i	$0.376526\pi$
$230$	−2515.47	−0.721153
$231$	0	0
$232$	2662.21	0.753373
$233$	4643.12	1.30550	0.652748	−	0.757575i	$-0.273616\pi$
$233$	4643.12	1.30550	0.652748	+	0.757575i	$0.273616\pi$
$234$	0	0
$235$	7493.95	2.08022
$236$	−1161.63	−0.320405
$237$	0	0
$238$	570.972	0.155507
$239$	6696.69	1.81244	0.906219	−	0.422809i	$-0.138956\pi$
$239$	6696.69	1.81244	0.906219	+	0.422809i	$0.138956\pi$
$240$	0	0
$241$	−2301.47	−0.615148	−0.307574	−	0.951524i	$-0.599517\pi$
$241$	−2301.47	−0.615148	−0.307574	+	0.951524i	$0.599517\pi$
$242$	−1550.46	−0.411848
$243$	0	0
$244$	4191.15	1.09963
$245$	−4311.93	−1.12440
$246$	0	0
$247$	0	0
$248$	−2200.78	−0.563506
$249$	0	0
$250$	−247.714	−0.0626673
$251$	−828.000	−0.208219	−0.104109	−	0.994566i	$-0.533199\pi$
$251$	−828.000	−0.208219	−0.104109	+	0.994566i	$0.533199\pi$
$252$	0	0
$253$	−1568.72	−0.389820
$254$	365.088	0.0901877
$255$	0	0
$256$	−2242.27	−0.547429
$257$	−884.763	−0.214747	−0.107374	−	0.994219i	$-0.534244\pi$
$257$	−884.763	−0.214747	−0.107374	+	0.994219i	$0.534244\pi$
$258$	0	0
$259$	3456.00	0.829133
$260$	0	0
$261$	0	0
$262$	1571.72	0.370616
$263$	8343.94	1.95631	0.978155	−	0.207878i	$-0.0666557\pi$
$263$	8343.94	1.95631	0.978155	+	0.207878i	$0.0666557\pi$
$264$	0	0
$265$	10404.4	2.41185
$266$	894.306	0.206141
$267$	0	0
$268$	87.5556	0.0199564
$269$	−2762.56	−0.626157	−0.313078	−	0.949727i	$-0.601360\pi$
$269$	−2762.56	−0.626157	−0.313078	+	0.949727i	$0.601360\pi$
$270$	0	0
$271$	3116.54	0.698585	0.349293	−	0.937014i	$-0.386422\pi$
$271$	3116.54	0.698585	0.349293	+	0.937014i	$0.386422\pi$
$272$	1339.81	0.298669
$273$	0	0
$274$	3394.91	0.748517
$275$	1441.65	0.316127
$276$	0	0
$277$	−502.060	−0.108902	−0.0544510	−	0.998516i	$-0.517341\pi$
$277$	−502.060	−0.108902	−0.0544510	+	0.998516i	$0.517341\pi$
$278$	724.465	0.156297
$279$	0	0
$280$	−2322.00	−0.495592
$281$	6607.56	1.40275	0.701377	−	0.712791i	$-0.252569\pi$
$281$	6607.56	1.40275	0.701377	+	0.712791i	$0.252569\pi$
$282$	0	0
$283$	−4368.98	−0.917699	−0.458850	−	0.888514i	$-0.651738\pi$
$283$	−4368.98	−0.917699	−0.458850	+	0.888514i	$0.651738\pi$
$284$	2163.18	0.451975
$285$	0	0
$286$	0	0
$287$	−1639.38	−0.337176
$288$	0	0
$289$	−1997.00	−0.406473
$290$	−2884.03	−0.583986
$291$	0	0
$292$	−5195.41	−1.04123
$293$	5348.12	1.06635	0.533175	−	0.846005i	$-0.320999\pi$
$293$	5348.12	1.06635	0.533175	+	0.846005i	$0.320999\pi$
$294$	0	0
$295$	2872.36	0.566900
$296$	−8200.94	−1.61037
$297$	0	0
$298$	1827.68	0.355284
$299$	0	0
$300$	0	0
$301$	−3336.17	−0.638849
$302$	−3628.69	−0.691416
$303$	0	0
$304$	2098.53	0.395917
$305$	−10363.5	−1.94561
$306$	0	0
$307$	−4502.46	−0.837032	−0.418516	−	0.908210i	$-0.637450\pi$
$307$	−4502.46	−0.837032	−0.418516	+	0.908210i	$0.637450\pi$
$308$	−634.414	−0.117367
$309$	0	0
$310$	2384.15	0.436809
$311$	7447.20	1.35785	0.678926	−	0.734207i	$-0.262446\pi$
$311$	7447.20	1.35785	0.678926	+	0.734207i	$0.262446\pi$
$312$	0	0
$313$	−6508.93	−1.17542	−0.587710	−	0.809072i	$-0.699970\pi$
$313$	−6508.93	−1.17542	−0.587710	+	0.809072i	$0.699970\pi$
$314$	1367.29	0.245734
$315$	0	0
$316$	2093.43	0.372673
$317$	−2465.57	−0.436846	−0.218423	−	0.975854i	$-0.570091\pi$
$317$	−2465.57	−0.436846	−0.218423	+	0.975854i	$0.570091\pi$
$318$	0	0
$319$	−1798.56	−0.315674
$320$	708.866	0.123834
$321$	0	0
$322$	−1298.99	−0.224814
$323$	4567.29	0.786782
$324$	0	0
$325$	0	0
$326$	3827.07	0.650190
$327$	0	0
$328$	3890.18	0.654876
$329$	3869.89	0.648491
$330$	0	0
$331$	−4114.84	−0.683300	−0.341650	−	0.939827i	$-0.610986\pi$
$331$	−4114.84	−0.683300	−0.341650	+	0.939827i	$0.610986\pi$
$332$	−3548.01	−0.586513
$333$	0	0
$334$	1531.80	0.250948
$335$	−216.499	−0.0353092
$336$	0	0
$337$	4798.05	0.775568	0.387784	−	0.921750i	$-0.373241\pi$
$337$	4798.05	0.775568	0.387784	+	0.921750i	$0.373241\pi$
$338$	0	0
$339$	0	0
$340$	−5195.41	−0.828708
$341$	1486.82	0.236117
$342$	0	0
$343$	−4958.69	−0.780594
$344$	7916.58	1.24079
$345$	0	0
$346$	2241.96	0.348348
$347$	3314.76	0.512812	0.256406	−	0.966569i	$-0.417462\pi$
$347$	3314.76	0.512812	0.256406	+	0.966569i	$0.417462\pi$
$348$	0	0
$349$	−371.740	−0.0570166	−0.0285083	−	0.999594i	$-0.509076\pi$
$349$	−371.740	−0.0570166	−0.0285083	+	0.999594i	$0.509076\pi$
$350$	1193.78	0.182314
$351$	0	0
$352$	2351.32	0.356039
$353$	−7539.10	−1.13673	−0.568365	−	0.822776i	$-0.692424\pi$
$353$	−7539.10	−1.13673	−0.568365	+	0.822776i	$0.692424\pi$
$354$	0	0
$355$	−5348.89	−0.799689
$356$	1476.29	0.219785
$357$	0	0
$358$	1251.56	0.184768
$359$	−12741.5	−1.87317	−0.936586	−	0.350437i	$-0.886033\pi$
$359$	−12741.5	−1.87317	−0.936586	+	0.350437i	$0.886033\pi$
$360$	0	0
$361$	294.676	0.0429619
$362$	639.896	0.0929065
$363$	0	0
$364$	0	0
$365$	12846.7	1.84227
$366$	0	0
$367$	7187.26	1.02227	0.511134	−	0.859501i	$-0.329226\pi$
$367$	7187.26	1.02227	0.511134	+	0.859501i	$0.329226\pi$
$368$	−3048.14	−0.431781
$369$	0	0
$370$	8884.26	1.24830
$371$	5372.87	0.751874
$372$	0	0
$373$	−2087.99	−0.289845	−0.144922	−	0.989443i	$-0.546293\pi$
$373$	−2087.99	−0.289845	−0.144922	+	0.989443i	$0.546293\pi$
$374$	915.352	0.126555
$375$	0	0
$376$	−9183.07	−1.25952
$377$	0	0
$378$	0	0
$379$	3982.08	0.539699	0.269850	−	0.962902i	$-0.413026\pi$
$379$	3982.08	0.539699	0.269850	+	0.962902i	$0.413026\pi$
$380$	−8137.50	−1.09854
$381$	0	0
$382$	−5606.47	−0.750921
$383$	−8638.43	−1.15249	−0.576244	−	0.817278i	$-0.695482\pi$
$383$	−8638.43	−1.15249	−0.576244	+	0.817278i	$0.695482\pi$
$384$	0	0
$385$	1568.72	0.207660
$386$	304.120	0.0401018
$387$	0	0
$388$	−7977.70	−1.04383
$389$	1275.74	0.166279	0.0831393	−	0.996538i	$-0.473505\pi$
$389$	1275.74	0.166279	0.0831393	+	0.996538i	$0.473505\pi$
$390$	0	0
$391$	−6634.06	−0.858053
$392$	5283.83	0.680801
$393$	0	0
$394$	−303.115	−0.0387582
$395$	−5176.44	−0.659379
$396$	0	0
$397$	−4622.65	−0.584394	−0.292197	−	0.956358i	$-0.594386\pi$
$397$	−4622.65	−0.584394	−0.292197	+	0.956358i	$0.594386\pi$
$398$	3901.55	0.491375
$399$	0	0
$400$	2801.24	0.350155
$401$	138.075	0.0171949	0.00859743	−	0.999963i	$-0.497263\pi$
$401$	138.075	0.0171949	0.00859743	+	0.999963i	$0.497263\pi$
$402$	0	0
$403$	0	0
$404$	3943.90	0.485684
$405$	0	0
$406$	−1489.32	−0.182053
$407$	5540.47	0.674769
$408$	0	0
$409$	−1204.64	−0.145637	−0.0728186	−	0.997345i	$-0.523199\pi$
$409$	−1204.64	−0.145637	−0.0728186	+	0.997345i	$0.523199\pi$
$410$	−4214.32	−0.507635
$411$	0	0
$412$	−9395.68	−1.12352
$413$	1483.29	0.176726
$414$	0	0
$415$	8773.16	1.03773
$416$	0	0
$417$	0	0
$418$	1433.70	0.167762
$419$	−5199.85	−0.606275	−0.303138	−	0.952947i	$-0.598034\pi$
$419$	−5199.85	−0.606275	−0.303138	+	0.952947i	$0.598034\pi$
$420$	0	0
$421$	−14136.5	−1.63651	−0.818256	−	0.574854i	$-0.805059\pi$
$421$	−14136.5	−1.63651	−0.818256	+	0.574854i	$0.805059\pi$
$422$	−2133.32	−0.246086
$423$	0	0
$424$	−12749.6	−1.46032
$425$	6096.70	0.695844
$426$	0	0
$427$	−5351.71	−0.606527
$428$	−7912.94	−0.893660
$429$	0	0
$430$	−8576.21	−0.961818
$431$	2279.83	0.254793	0.127396	−	0.991852i	$-0.459338\pi$
$431$	2279.83	0.254793	0.127396	+	0.991852i	$0.459338\pi$
$432$	0	0
$433$	−13298.7	−1.47597	−0.737984	−	0.674819i	$-0.764222\pi$
$433$	−13298.7	−1.47597	−0.737984	+	0.674819i	$0.764222\pi$
$434$	1231.18	0.136172
$435$	0	0
$436$	2167.14	0.238045
$437$	−10390.8	−1.13744
$438$	0	0
$439$	−10452.3	−1.13635	−0.568177	−	0.822907i	$-0.692351\pi$
$439$	−10452.3	−1.13635	−0.568177	+	0.822907i	$0.692351\pi$
$440$	−3722.50	−0.403325
$441$	0	0
$442$	0	0
$443$	−5363.50	−0.575232	−0.287616	−	0.957746i	$-0.592863\pi$
$443$	−5363.50	−0.575232	−0.287616	+	0.957746i	$0.592863\pi$
$444$	0	0
$445$	−3650.43	−0.388870
$446$	−173.726	−0.0184443
$447$	0	0
$448$	366.059	0.0386042
$449$	−9681.73	−1.01762	−0.508808	−	0.860880i	$-0.669914\pi$
$449$	−9681.73	−1.01762	−0.508808	+	0.860880i	$0.669914\pi$
$450$	0	0
$451$	−2628.17	−0.274402
$452$	4115.07	0.428223
$453$	0	0
$454$	5741.69	0.593548
$455$	0	0
$456$	0	0
$457$	−3537.94	−0.362139	−0.181070	−	0.983470i	$-0.557956\pi$
$457$	−3537.94	−0.362139	−0.181070	+	0.983470i	$0.557956\pi$
$458$	3480.14	0.355057
$459$	0	0
$460$	11819.8	1.19805
$461$	15074.9	1.52302	0.761508	−	0.648156i	$-0.224459\pi$
$461$	15074.9	1.52302	0.761508	+	0.648156i	$0.224459\pi$
$462$	0	0
$463$	−11070.0	−1.11116	−0.555580	−	0.831463i	$-0.687504\pi$
$463$	−11070.0	−1.11116	−0.555580	+	0.831463i	$0.687504\pi$
$464$	−3494.74	−0.349654
$465$	0	0
$466$	6163.75	0.612725
$467$	−13252.8	−1.31320	−0.656600	−	0.754239i	$-0.728006\pi$
$467$	−13252.8	−1.31320	−0.656600	+	0.754239i	$0.728006\pi$
$468$	0	0
$469$	−111.800	−0.0110074
$470$	9948.23	0.976335
$471$	0	0
$472$	−3519.79	−0.343244
$473$	−5348.36	−0.519911
$474$	0	0
$475$	9549.18	0.922413
$476$	−2682.92	−0.258343
$477$	0	0
$478$	8889.86	0.850654
$479$	−12241.4	−1.16769	−0.583843	−	0.811866i	$-0.698452\pi$
$479$	−12241.4	−1.16769	−0.583843	+	0.811866i	$0.698452\pi$
$480$	0	0
$481$	0	0
$482$	−3055.20	−0.288715
$483$	0	0
$484$	7285.37	0.684201
$485$	19726.5	1.84687
$486$	0	0
$487$	−13413.3	−1.24808	−0.624041	−	0.781392i	$-0.714510\pi$
$487$	−13413.3	−1.24808	−0.624041	+	0.781392i	$0.714510\pi$
$488$	12699.4	1.17802
$489$	0	0
$490$	−5724.10	−0.527731
$491$	737.885	0.0678214	0.0339107	−	0.999425i	$-0.489204\pi$
$491$	737.885	0.0678214	0.0339107	+	0.999425i	$0.489204\pi$
$492$	0	0
$493$	−7606.06	−0.694847
$494$	0	0
$495$	0	0
$496$	2889.01	0.261533
$497$	−2762.17	−0.249297
$498$	0	0
$499$	1865.65	0.167370	0.0836852	−	0.996492i	$-0.473331\pi$
$499$	1865.65	0.167370	0.0836852	+	0.996492i	$0.473331\pi$
$500$	1163.97	0.104109
$501$	0	0
$502$	−1099.17	−0.0977259
$503$	16632.0	1.47432	0.737161	−	0.675717i	$-0.236166\pi$
$503$	16632.0	1.47432	0.737161	+	0.675717i	$0.236166\pi$
$504$	0	0
$505$	−9752.09	−0.859331
$506$	−2082.47	−0.182959
$507$	0	0
$508$	−1715.50	−0.149829
$509$	−4128.91	−0.359549	−0.179775	−	0.983708i	$-0.557537\pi$
$509$	−4128.91	−0.359549	−0.179775	+	0.983708i	$0.557537\pi$
$510$	0	0
$511$	6634.06	0.574312
$512$	8320.40	0.718190
$513$	0	0
$514$	−1174.52	−0.100790
$515$	23232.7	1.98788
$516$	0	0
$517$	6203.99	0.527758
$518$	4587.85	0.389147
$519$	0	0
$520$	0	0
$521$	988.234	0.0831005	0.0415502	−	0.999136i	$-0.486770\pi$
$521$	988.234	0.0831005	0.0415502	+	0.999136i	$0.486770\pi$
$522$	0	0
$523$	−9441.62	−0.789394	−0.394697	−	0.918811i	$-0.629150\pi$
$523$	−9441.62	−0.789394	−0.394697	+	0.918811i	$0.629150\pi$
$524$	−7385.30	−0.615703
$525$	0	0
$526$	11076.6	0.918180
$527$	6287.73	0.519730
$528$	0	0
$529$	2925.83	0.240472
$530$	13811.9	1.13198
$531$	0	0
$532$	−4202.21	−0.342461
$533$	0	0
$534$	0	0
$535$	19566.3	1.58117
$536$	265.297	0.0213789
$537$	0	0
$538$	−3667.30	−0.293882
$539$	−3569.70	−0.285265
$540$	0	0
$541$	14001.5	1.11270	0.556351	−	0.830948i	$-0.312201\pi$
$541$	14001.5	1.11270	0.556351	+	0.830948i	$0.312201\pi$
$542$	4137.22	0.327876
$543$	0	0
$544$	9943.67	0.783697
$545$	−5358.70	−0.421177
$546$	0	0
$547$	−4244.85	−0.331804	−0.165902	−	0.986142i	$-0.553053\pi$
$547$	−4244.85	−0.331804	−0.165902	+	0.986142i	$0.553053\pi$
$548$	−15952.2	−1.24351
$549$	0	0
$550$	1913.80	0.148372
$551$	−11913.3	−0.921092
$552$	0	0
$553$	−2673.12	−0.205556
$554$	−666.485	−0.0511124
$555$	0	0
$556$	−3404.16	−0.259655
$557$	11732.2	0.892473	0.446236	−	0.894915i	$-0.352764\pi$
$557$	11732.2	0.892473	0.446236	+	0.894915i	$0.352764\pi$
$558$	0	0
$559$	0	0
$560$	3048.14	0.230013
$561$	0	0
$562$	8771.54	0.658372
$563$	9941.80	0.744222	0.372111	−	0.928188i	$-0.378634\pi$
$563$	9941.80	0.744222	0.372111	+	0.928188i	$0.378634\pi$
$564$	0	0
$565$	−10175.3	−0.757664
$566$	−5799.83	−0.430716
$567$	0	0
$568$	6554.52	0.484193
$569$	−3690.77	−0.271925	−0.135962	−	0.990714i	$-0.543413\pi$
$569$	−3690.77	−0.271925	−0.135962	+	0.990714i	$0.543413\pi$
$570$	0	0
$571$	−5685.09	−0.416661	−0.208331	−	0.978058i	$-0.566803\pi$
$571$	−5685.09	−0.416661	−0.208331	+	0.978058i	$0.566803\pi$
$572$	0	0
$573$	0	0
$574$	−2176.28	−0.158251
$575$	−13870.3	−1.00597
$576$	0	0
$577$	7746.50	0.558910	0.279455	−	0.960159i	$-0.409846\pi$
$577$	7746.50	0.558910	0.279455	+	0.960159i	$0.409846\pi$
$578$	−2651.02	−0.190775
$579$	0	0
$580$	13551.6	0.970175
$581$	4530.47	0.323504
$582$	0	0
$583$	8613.48	0.611894
$584$	−15742.3	−1.11545
$585$	0	0
$586$	7099.64	0.500484
$587$	2766.54	0.194527	0.0972635	−	0.995259i	$-0.468991\pi$
$587$	2766.54	0.194527	0.0972635	+	0.995259i	$0.468991\pi$
$588$	0	0
$589$	9848.38	0.688957
$590$	3813.07	0.266070
$591$	0	0
$592$	10765.6	0.747402
$593$	1440.79	0.0997743	0.0498871	−	0.998755i	$-0.484114\pi$
$593$	1440.79	0.0997743	0.0498871	+	0.998755i	$0.484114\pi$
$594$	0	0
$595$	6634.06	0.457092
$596$	−8587.99	−0.590231
$597$	0	0
$598$	0	0
$599$	−23837.5	−1.62600	−0.813001	−	0.582263i	$-0.802168\pi$
$599$	−23837.5	−1.62600	−0.813001	+	0.582263i	$0.802168\pi$
$600$	0	0
$601$	−6694.23	−0.454348	−0.227174	−	0.973854i	$-0.572949\pi$
$601$	−6694.23	−0.454348	−0.227174	+	0.973854i	$0.572949\pi$
$602$	−4428.77	−0.299839
$603$	0	0
$604$	17050.7	1.14865
$605$	−18014.6	−1.21057
$606$	0	0
$607$	3330.50	0.222703	0.111352	−	0.993781i	$-0.464482\pi$
$607$	3330.50	0.222703	0.111352	+	0.993781i	$0.464482\pi$
$608$	15574.6	1.03887
$609$	0	0
$610$	−13757.5	−0.913156
$611$	0	0
$612$	0	0
$613$	−13490.3	−0.888857	−0.444428	−	0.895814i	$-0.646593\pi$
$613$	−13490.3	−0.888857	−0.444428	+	0.895814i	$0.646593\pi$
$614$	−5977.02	−0.392855
$615$	0	0
$616$	−1922.30	−0.125733
$617$	−7470.76	−0.487458	−0.243729	−	0.969843i	$-0.578371\pi$
$617$	−7470.76	−0.487458	−0.243729	+	0.969843i	$0.578371\pi$
$618$	0	0
$619$	24806.9	1.61078	0.805389	−	0.592746i	$-0.201956\pi$
$619$	24806.9	1.61078	0.805389	+	0.592746i	$0.201956\pi$
$620$	−11202.8	−0.725669
$621$	0	0
$622$	9886.17	0.637298
$623$	−1885.09	−0.121227
$624$	0	0
$625$	−16990.9	−1.08742
$626$	−8640.61	−0.551675
$627$	0	0
$628$	−6424.68	−0.408237
$629$	23430.5	1.48527
$630$	0	0
$631$	314.333	0.0198311	0.00991554	−	0.999951i	$-0.496844\pi$
$631$	314.333	0.0198311	0.00991554	+	0.999951i	$0.496844\pi$
$632$	6343.19	0.399238
$633$	0	0
$634$	−3273.05	−0.205031
$635$	4241.91	0.265095
$636$	0	0
$637$	0	0
$638$	−2387.59	−0.148159
$639$	0	0
$640$	−21780.7	−1.34525
$641$	−5550.41	−0.342009	−0.171005	−	0.985270i	$-0.554701\pi$
$641$	−5550.41	−0.342009	−0.171005	+	0.985270i	$0.554701\pi$
$642$	0	0
$643$	5479.48	0.336065	0.168032	−	0.985781i	$-0.446259\pi$
$643$	5479.48	0.336065	0.168032	+	0.985781i	$0.446259\pi$
$644$	6103.78	0.373482
$645$	0	0
$646$	6063.08	0.369271
$647$	−4724.83	−0.287098	−0.143549	−	0.989643i	$-0.545851\pi$
$647$	−4724.83	−0.287098	−0.143549	+	0.989643i	$0.545851\pi$
$648$	0	0
$649$	2377.93	0.143824
$650$	0	0
$651$	0	0
$652$	−17982.9	−1.08016
$653$	−3463.91	−0.207585	−0.103793	−	0.994599i	$-0.533098\pi$
$653$	−3463.91	−0.207585	−0.103793	+	0.994599i	$0.533098\pi$
$654$	0	0
$655$	18261.6	1.08938
$656$	−5106.73	−0.303940
$657$	0	0
$658$	5137.28	0.304365
$659$	2606.35	0.154065	0.0770327	−	0.997029i	$-0.475455\pi$
$659$	2606.35	0.154065	0.0770327	+	0.997029i	$0.475455\pi$
$660$	0	0
$661$	−22436.7	−1.32025	−0.660127	−	0.751154i	$-0.729498\pi$
$661$	−22436.7	−1.32025	−0.660127	+	0.751154i	$0.729498\pi$
$662$	−5462.46	−0.320702
$663$	0	0
$664$	−10750.6	−0.628321
$665$	10390.8	0.605923
$666$	0	0
$667$	17304.2	1.00453
$668$	−7197.73	−0.416899
$669$	0	0
$670$	−287.403	−0.0165721
$671$	−8579.56	−0.493607
$672$	0	0
$673$	−633.970	−0.0363117	−0.0181558	−	0.999835i	$-0.505779\pi$
$673$	−633.970	−0.0363117	−0.0181558	+	0.999835i	$0.505779\pi$
$674$	6369.42	0.364007
$675$	0	0
$676$	0	0
$677$	−24457.4	−1.38844	−0.694221	−	0.719762i	$-0.744251\pi$
$677$	−24457.4	−1.38844	−0.694221	+	0.719762i	$0.744251\pi$
$678$	0	0
$679$	10186.8	0.575747
$680$	−15742.3	−0.887780
$681$	0	0
$682$	1973.76	0.110820
$683$	12367.6	0.692875	0.346437	−	0.938073i	$-0.387391\pi$
$683$	12367.6	0.692875	0.346437	+	0.938073i	$0.387391\pi$
$684$	0	0
$685$	39445.0	2.20017
$686$	−6582.66	−0.366366
$687$	0	0
$688$	−10392.3	−0.575875
$689$	0	0
$690$	0	0
$691$	1050.99	0.0578605	0.0289302	−	0.999581i	$-0.490790\pi$
$691$	1050.99	0.0578605	0.0289302	+	0.999581i	$0.490790\pi$
$692$	−10534.6	−0.578709
$693$	0	0
$694$	4400.35	0.240685
$695$	8417.46	0.459414
$696$	0	0
$697$	−11114.4	−0.604002
$698$	−493.486	−0.0267603
$699$	0	0
$700$	−5609.38	−0.302878
$701$	24294.1	1.30895	0.654476	−	0.756083i	$-0.272889\pi$
$701$	24294.1	1.30895	0.654476	+	0.756083i	$0.272889\pi$
$702$	0	0
$703$	36698.8	1.96888
$704$	586.846	0.0314170
$705$	0	0
$706$	−10008.2	−0.533516
$707$	−5035.99	−0.267890
$708$	0	0
$709$	−27465.9	−1.45487	−0.727436	−	0.686176i	$-0.759288\pi$
$709$	−27465.9	−1.45487	−0.727436	+	0.686176i	$0.759288\pi$
$710$	−7100.66	−0.375328
$711$	0	0
$712$	4473.23	0.235451
$713$	−14304.9	−0.751365
$714$	0	0
$715$	0	0
$716$	−5880.90	−0.306955
$717$	0	0
$718$	−16914.3	−0.879160
$719$	−36433.5	−1.88976	−0.944882	−	0.327411i	$-0.893824\pi$
$719$	−36433.5	−1.88976	−0.944882	+	0.327411i	$0.893824\pi$
$720$	0	0
$721$	11997.4	0.619704
$722$	391.183	0.0201639
$723$	0	0
$724$	−3006.78	−0.154345
$725$	−15902.6	−0.814629
$726$	0	0
$727$	551.608	0.0281403	0.0140701	−	0.999901i	$-0.495521\pi$
$727$	551.608	0.0281403	0.0140701	+	0.999901i	$0.495521\pi$
$728$	0	0
$729$	0	0
$730$	17054.0	0.864654
$731$	−22618.1	−1.14440
$732$	0	0
$733$	−20317.2	−1.02379	−0.511893	−	0.859049i	$-0.671055\pi$
$733$	−20317.2	−1.02379	−0.511893	+	0.859049i	$0.671055\pi$
$734$	9541.10	0.479794
$735$	0	0
$736$	−22622.4	−1.13298
$737$	−179.232	−0.00895807
$738$	0	0
$739$	29931.6	1.48992	0.744962	−	0.667107i	$-0.232468\pi$
$739$	29931.6	1.48992	0.744962	+	0.667107i	$0.232468\pi$
$740$	−41745.9	−2.07380
$741$	0	0
$742$	7132.49	0.352887
$743$	24376.4	1.20361	0.601805	−	0.798643i	$-0.294448\pi$
$743$	24376.4	1.20361	0.601805	+	0.798643i	$0.294448\pi$
$744$	0	0
$745$	21235.5	1.04431
$746$	−2771.81	−0.136037
$747$	0	0
$748$	−4301.11	−0.210246
$749$	10104.1	0.492917
$750$	0	0
$751$	22692.2	1.10260	0.551298	−	0.834308i	$-0.314133\pi$
$751$	22692.2	1.10260	0.551298	+	0.834308i	$0.314133\pi$
$752$	12054.8	0.584567
$753$	0	0
$754$	0	0
$755$	−42161.3	−2.03232
$756$	0	0
$757$	−33063.5	−1.58747	−0.793734	−	0.608265i	$-0.791866\pi$
$757$	−33063.5	−1.58747	−0.793734	+	0.608265i	$0.791866\pi$
$758$	5286.22	0.253304
$759$	0	0
$760$	−24657.0	−1.17685
$761$	−216.324	−0.0103045	−0.00515226	−	0.999987i	$-0.501640\pi$
$761$	−216.324	−0.0103045	−0.00515226	+	0.999987i	$0.501640\pi$
$762$	0	0
$763$	−2767.24	−0.131299
$764$	26344.0	1.24750
$765$	0	0
$766$	−11467.5	−0.540912
$767$	0	0
$768$	0	0
$769$	−19214.4	−0.901025	−0.450512	−	0.892770i	$-0.648759\pi$
$769$	−19214.4	−0.901025	−0.450512	+	0.892770i	$0.648759\pi$
$770$	2082.47	0.0974638
$771$	0	0
$772$	−1429.01	−0.0666209
$773$	33175.6	1.54365	0.771826	−	0.635834i	$-0.219344\pi$
$773$	33175.6	1.54365	0.771826	+	0.635834i	$0.219344\pi$
$774$	0	0
$775$	13146.2	0.609325
$776$	−24172.8	−1.11824
$777$	0	0
$778$	1693.54	0.0780416
$779$	−17408.4	−0.800667
$780$	0	0
$781$	−4428.17	−0.202884
$782$	−8806.72	−0.402721
$783$	0	0
$784$	−6936.21	−0.315972
$785$	15886.3	0.722302
$786$	0	0
$787$	14887.9	0.674330	0.337165	−	0.941446i	$-0.390532\pi$
$787$	14887.9	0.674330	0.337165	+	0.941446i	$0.390532\pi$
$788$	1424.30	0.0643889
$789$	0	0
$790$	−6871.73	−0.309475
$791$	−5254.56	−0.236196
$792$	0	0
$793$	0	0
$794$	−6136.58	−0.274281
$795$	0	0
$796$	−18332.8	−0.816319
$797$	−12954.5	−0.575751	−0.287876	−	0.957668i	$-0.592949\pi$
$797$	−12954.5	−0.575751	−0.287876	+	0.957668i	$0.592949\pi$
$798$	0	0
$799$	26236.5	1.16168
$800$	20790.0	0.918796
$801$	0	0
$802$	183.295	0.00807028
$803$	10635.4	0.467389
$804$	0	0
$805$	−15092.8	−0.660810
$806$	0	0
$807$	0	0
$808$	11950.2	0.520305
$809$	−8275.59	−0.359647	−0.179823	−	0.983699i	$-0.557553\pi$
$809$	−8275.59	−0.359647	−0.179823	+	0.983699i	$0.557553\pi$
$810$	0	0
$811$	−26327.1	−1.13991	−0.569956	−	0.821675i	$-0.693040\pi$
$811$	−26327.1	−1.13991	−0.569956	+	0.821675i	$0.693040\pi$
$812$	6998.09	0.302444
$813$	0	0
$814$	7354.98	0.316698
$815$	44466.3	1.91115
$816$	0	0
$817$	−35426.3	−1.51703
$818$	−1599.16	−0.0683538
$819$	0	0
$820$	19802.5	0.843333
$821$	−34439.0	−1.46398	−0.731992	−	0.681314i	$-0.761409\pi$
$821$	−34439.0	−1.46398	−0.731992	+	0.681314i	$0.761409\pi$
$822$	0	0
$823$	−13870.5	−0.587479	−0.293739	−	0.955886i	$-0.594900\pi$
$823$	−13870.5	−0.587479	−0.293739	+	0.955886i	$0.594900\pi$
$824$	−28469.3	−1.20361
$825$	0	0
$826$	1969.07	0.0829453
$827$	2132.30	0.0896583	0.0448292	−	0.998995i	$-0.485726\pi$
$827$	2132.30	0.0896583	0.0448292	+	0.998995i	$0.485726\pi$
$828$	0	0
$829$	−6212.39	−0.260272	−0.130136	−	0.991496i	$-0.541541\pi$
$829$	−6212.39	−0.260272	−0.130136	+	0.991496i	$0.541541\pi$
$830$	11646.4	0.487051
$831$	0	0
$832$	0	0
$833$	−15096.2	−0.627913
$834$	0	0
$835$	17797.8	0.737628
$836$	−6736.76	−0.278703
$837$	0	0
$838$	−6902.81	−0.284551
$839$	−4550.52	−0.187248	−0.0936242	−	0.995608i	$-0.529845\pi$
$839$	−4550.52	−0.187248	−0.0936242	+	0.995608i	$0.529845\pi$
$840$	0	0
$841$	−4549.47	−0.186538
$842$	−18766.2	−0.768085
$843$	0	0
$844$	10024.2	0.408822
$845$	0	0
$846$	0	0
$847$	−9302.74	−0.377386
$848$	16736.7	0.677759
$849$	0	0
$850$	8093.38	0.326589
$851$	−53305.6	−2.14723
$852$	0	0
$853$	−12262.8	−0.492228	−0.246114	−	0.969241i	$-0.579154\pi$
$853$	−12262.8	−0.492228	−0.246114	+	0.969241i	$0.579154\pi$
$854$	−7104.40	−0.284669
$855$	0	0
$856$	−23976.5	−0.957362
$857$	34949.1	1.39304	0.696521	−	0.717536i	$-0.254730\pi$
$857$	34949.1	1.39304	0.696521	+	0.717536i	$0.254730\pi$
$858$	0	0
$859$	−21762.1	−0.864394	−0.432197	−	0.901779i	$-0.642261\pi$
$859$	−21762.1	−0.864394	−0.432197	+	0.901779i	$0.642261\pi$
$860$	40298.4	1.59786
$861$	0	0
$862$	3026.48	0.119585
$863$	−19811.5	−0.781450	−0.390725	−	0.920507i	$-0.627776\pi$
$863$	−19811.5	−0.781450	−0.390725	+	0.920507i	$0.627776\pi$
$864$	0	0
$865$	26049.0	1.02392
$866$	−17654.0	−0.692735
$867$	0	0
$868$	−5785.14	−0.226222
$869$	−4285.40	−0.167287
$870$	0	0
$871$	0	0
$872$	6566.54	0.255013
$873$	0	0
$874$	−13793.8	−0.533848
$875$	−1486.28	−0.0574235
$876$	0	0
$877$	25716.8	0.990186	0.495093	−	0.868840i	$-0.335134\pi$
$877$	25716.8	0.990186	0.495093	+	0.868840i	$0.335134\pi$
$878$	−13875.4	−0.533339
$879$	0	0
$880$	4886.61	0.187190
$881$	−34709.6	−1.32735	−0.663676	−	0.748020i	$-0.731005\pi$
$881$	−34709.6	−1.32735	−0.663676	+	0.748020i	$0.731005\pi$
$882$	0	0
$883$	−3848.68	−0.146680	−0.0733400	−	0.997307i	$-0.523366\pi$
$883$	−3848.68	−0.146680	−0.0733400	+	0.997307i	$0.523366\pi$
$884$	0	0
$885$	0	0
$886$	−7120.06	−0.269981
$887$	−32804.8	−1.24180	−0.620900	−	0.783890i	$-0.713233\pi$
$887$	−32804.8	−1.24180	−0.620900	+	0.783890i	$0.713233\pi$
$888$	0	0
$889$	2190.53	0.0826412
$890$	−4845.95	−0.182513
$891$	0	0
$892$	816.311	0.0306414
$893$	41093.8	1.53992
$894$	0	0
$895$	14541.7	0.543101
$896$	−11247.6	−0.419371
$897$	0	0
$898$	−12852.5	−0.477610
$899$	−16400.8	−0.608452
$900$	0	0
$901$	36426.2	1.34687
$902$	−3488.90	−0.128789
$903$	0	0
$904$	12468.8	0.458748
$905$	7434.86	0.273086
$906$	0	0
$907$	−22262.2	−0.814999	−0.407500	−	0.913205i	$-0.633599\pi$
$907$	−22262.2	−0.814999	−0.407500	+	0.913205i	$0.633599\pi$
$908$	−26979.4	−0.986060
$909$	0	0
$910$	0	0
$911$	13515.3	0.491528	0.245764	−	0.969330i	$-0.420961\pi$
$911$	13515.3	0.491528	0.245764	+	0.969330i	$0.420961\pi$
$912$	0	0
$913$	7263.01	0.263275
$914$	−4696.62	−0.169968
$915$	0	0
$916$	−16352.7	−0.589855
$917$	9430.33	0.339604
$918$	0	0
$919$	26600.6	0.954811	0.477405	−	0.878683i	$-0.341577\pi$
$919$	26600.6	0.954811	0.477405	+	0.878683i	$0.341577\pi$
$920$	35814.6	1.28345
$921$	0	0
$922$	20012.0	0.714816
$923$	0	0
$924$	0	0
$925$	48987.9	1.74131
$926$	−14695.5	−0.521515
$927$	0	0
$928$	−25936.9	−0.917480
$929$	−32887.7	−1.16148	−0.580738	−	0.814090i	$-0.697236\pi$
$929$	−32887.7	−1.16148	−0.580738	+	0.814090i	$0.697236\pi$
$930$	0	0
$931$	−23644.9	−0.832363
$932$	−28962.6	−1.01792
$933$	0	0
$934$	−17593.1	−0.616341
$935$	10635.4	0.371993
$936$	0	0
$937$	−9261.78	−0.322913	−0.161456	−	0.986880i	$-0.551619\pi$
$937$	−9261.78	−0.322913	−0.161456	+	0.986880i	$0.551619\pi$
$938$	−148.415	−0.00516623
$939$	0	0
$940$	−46745.3	−1.62198
$941$	12054.9	0.417619	0.208810	−	0.977956i	$-0.433041\pi$
$941$	12054.9	0.417619	0.208810	+	0.977956i	$0.433041\pi$
$942$	0	0
$943$	25285.9	0.873195
$944$	4620.51	0.159306
$945$	0	0
$946$	−7099.96	−0.244016
$947$	20221.4	0.693885	0.346942	−	0.937886i	$-0.387220\pi$
$947$	20221.4	0.693885	0.346942	+	0.937886i	$0.387220\pi$
$948$	0	0
$949$	0	0
$950$	12676.5	0.432928
$951$	0	0
$952$	−8129.36	−0.276758
$953$	20331.5	0.691083	0.345542	−	0.938403i	$-0.387695\pi$
$953$	20331.5	0.691083	0.345542	+	0.938403i	$0.387695\pi$
$954$	0	0
$955$	−65140.8	−2.20723
$956$	−41772.2	−1.41319
$957$	0	0
$958$	−16250.4	−0.548045
$959$	20369.4	0.685885
$960$	0	0
$961$	−16232.9	−0.544891
$962$	0	0
$963$	0	0
$964$	14356.0	0.479641
$965$	3533.53	0.117874
$966$	0	0
$967$	11082.2	0.368541	0.184270	−	0.982876i	$-0.441008\pi$
$967$	11082.2	0.368541	0.184270	+	0.982876i	$0.441008\pi$
$968$	22075.0	0.732973
$969$	0	0
$970$	26186.9	0.866816
$971$	36694.1	1.21274	0.606369	−	0.795184i	$-0.292626\pi$
$971$	36694.1	1.21274	0.606369	+	0.795184i	$0.292626\pi$
$972$	0	0
$973$	4346.79	0.143219
$974$	−17806.2	−0.585778
$975$	0	0
$976$	−16670.8	−0.546740
$977$	17155.3	0.561767	0.280883	−	0.959742i	$-0.409373\pi$
$977$	17155.3	0.561767	0.280883	+	0.959742i	$0.409373\pi$
$978$	0	0
$979$	−3022.07	−0.0986575
$980$	26896.7	0.876718
$981$	0	0
$982$	979.544	0.0318315
$983$	−38419.7	−1.24659	−0.623296	−	0.781986i	$-0.714207\pi$
$983$	−38419.7	−1.24659	−0.623296	+	0.781986i	$0.714207\pi$
$984$	0	0
$985$	−3521.86	−0.113925
$986$	−10097.1	−0.326121
$987$	0	0
$988$	0	0
$989$	51457.3	1.65445
$990$	0	0
$991$	51728.9	1.65815	0.829073	−	0.559140i	$-0.188869\pi$
$991$	51728.9	1.65815	0.829073	+	0.559140i	$0.188869\pi$
$992$	21441.4	0.686255
$993$	0	0
$994$	−3666.79	−0.117006
$995$	45331.6	1.44433
$996$	0	0
$997$	−26846.8	−0.852805	−0.426403	−	0.904533i	$-0.640219\pi$
$997$	−26846.8	−0.852805	−0.426403	+	0.904533i	$0.640219\pi$
$998$	2476.65	0.0785541
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.x.1.3		4
3.2	odd	2		507.4.a.j.1.2		4
13.5	odd	4		117.4.b.d.64.2		4
13.8	odd	4		117.4.b.d.64.3		4
13.12	even	2	inner	1521.4.a.x.1.2		4
39.5	even	4		39.4.b.a.25.3	yes	4
39.8	even	4		39.4.b.a.25.2	✓	4
39.38	odd	2		507.4.a.j.1.3		4
156.47	odd	4		624.4.c.e.337.1		4
156.83	odd	4		624.4.c.e.337.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.a.25.2	✓	4	39.8	even	4
39.4.b.a.25.3	yes	4	39.5	even	4
117.4.b.d.64.2		4	13.5	odd	4
117.4.b.d.64.3		4	13.8	odd	4
507.4.a.j.1.2		4	3.2	odd	2
507.4.a.j.1.3		4	39.38	odd	2
624.4.c.e.337.1		4	156.47	odd	4
624.4.c.e.337.4		4	156.83	odd	4
1521.4.a.x.1.2		4	13.12	even	2	inner
1521.4.a.x.1.3		4	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32750 q^{2} -6.23774 q^{4} +15.4241 q^{5} +7.96501 q^{7} -18.9006 q^{8} +O(q^{10})\)	\(q+1.32750 q^{2} -6.23774 q^{4} +15.4241 q^{5} +7.96501 q^{7} -18.9006 q^{8} +20.4755 q^{10} +12.7691 q^{11} +10.5736 q^{14} +24.8113 q^{16} +54.0000 q^{17} +84.5794 q^{19} -96.2113 q^{20} +16.9510 q^{22} -122.853 q^{23} +112.902 q^{25} -49.6837 q^{28} -140.853 q^{29} +116.439 q^{31} +184.142 q^{32} +71.6851 q^{34} +122.853 q^{35} +433.898 q^{37} +112.279 q^{38} -291.525 q^{40} -205.823 q^{41} -418.853 q^{43} -79.6501 q^{44} -163.087 q^{46} +485.861 q^{47} -279.559 q^{49} +149.877 q^{50} +674.559 q^{53} +196.951 q^{55} -150.544 q^{56} -186.982 q^{58} +186.226 q^{59} -671.902 q^{61} +154.574 q^{62} +45.9584 q^{64} -14.0364 q^{67} -336.838 q^{68} +163.087 q^{70} -346.789 q^{71} +832.900 q^{73} +576.000 q^{74} -527.584 q^{76} +101.706 q^{77} -335.608 q^{79} +382.691 q^{80} -273.230 q^{82} +568.797 q^{83} +832.900 q^{85} -556.028 q^{86} -241.343 q^{88} -236.671 q^{89} +766.324 q^{92} +644.981 q^{94} +1304.56 q^{95} +1278.94 q^{97} -371.115 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 26 q^{4}+O(q^{10}) \) 4 * q + 26 * q^4	\( 4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} + 1050 q^{64} + 1404 q^{68} + 2304 q^{74} - 816 q^{77} + 288 q^{79} + 28 q^{82} - 2392 q^{88} + 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100}) \) 4 * q + 26 * q^4 - 20 * q^10 + 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 + 44 * q^25 + 48 * q^29 - 120 * q^35 - 468 * q^38 - 1268 * q^40 - 1064 * q^43 + 716 * q^49 + 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 + 1050 * q^64 + 1404 * q^68 + 2304 * q^74 - 816 * q^77 + 288 * q^79 + 28 * q^82 - 2392 * q^88 + 8568 * q^92 + 6656 * q^94 + 3384 * q^95

Introduction

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