LMFDB - Embedded newform 1521.4.a.z.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:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 13x^{2} + 14x - 2 $ x^4 - 2x^3 - 13x^2 + 14*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
Root			$-3.29360$ 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)

$f(q)$	$=$	$q-4.12311 q^{2} +9.00000 q^{4} +3.05006 q^{5} -6.68324 q^{7} -4.12311 q^{8} +O(q^{10})$	$q-4.12311 q^{2} +9.00000 q^{4} +3.05006 q^{5} -6.68324 q^{7} -4.12311 q^{8} -12.5757 q^{10} -32.2500 q^{11} +27.5557 q^{14} -55.0000 q^{16} +28.8586 q^{17} -101.194 q^{19} +27.4505 q^{20} +132.970 q^{22} +118.990 q^{23} -115.697 q^{25} -60.1492 q^{28} -160.111 q^{29} -38.0705 q^{31} +259.756 q^{32} -118.987 q^{34} -20.3843 q^{35} -327.568 q^{37} +417.233 q^{38} -12.5757 q^{40} -56.0846 q^{41} +127.879 q^{43} -290.250 q^{44} -490.608 q^{46} +517.983 q^{47} -298.334 q^{49} +477.032 q^{50} +695.546 q^{53} -98.3643 q^{55} +27.5557 q^{56} +660.156 q^{58} -656.523 q^{59} +701.304 q^{61} +156.969 q^{62} -631.000 q^{64} -57.1750 q^{67} +259.727 q^{68} +84.0465 q^{70} -309.226 q^{71} -389.711 q^{73} +1350.60 q^{74} -910.744 q^{76} +215.534 q^{77} +901.820 q^{79} -167.753 q^{80} +231.243 q^{82} -687.095 q^{83} +88.0203 q^{85} -527.257 q^{86} +132.970 q^{88} -1070.54 q^{89} +1070.91 q^{92} -2135.70 q^{94} -308.647 q^{95} +1754.48 q^{97} +1230.06 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{4}+O(q^{10}) $ 4 * q + 36 * q^4	$ 4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) $ 4 * q + 36 * q^4 - 136 * q^10 - 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 - 120 * q^25 - 12 * q^29 + 804 * q^35 + 612 * q^38 - 136 * q^40 + 940 * q^43 + 692 * q^49 + 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 - 2524 * q^64 + 1296 * q^68 + 3060 * q^74 + 2976 * q^77 + 8 * q^79 + 68 * q^82 - 68 * q^88 + 2484 * q^92 - 5372 * q^94 + 108 * 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 $

$2$	−4.12311	−1.45774	−0.728869	−	0.684653i	$-0.759954\pi$
$2$	−4.12311	−1.45774	−0.728869	+	0.684653i	$0.759954\pi$
$3$	0	0
$3$	0	0
$4$	9.00000	1.12500
$5$	3.05006	0.272806	0.136403	−	0.990653i	$-0.456446\pi$
$5$	3.05006	0.272806	0.136403	+	0.990653i	$0.456446\pi$
$6$	0	0
$7$	−6.68324	−0.360861	−0.180431	−	0.983588i	$-0.557749\pi$
$7$	−6.68324	−0.360861	−0.180431	+	0.983588i	$0.557749\pi$
$8$	−4.12311	−0.182217
$9$	0	0
$10$	−12.5757	−0.397679
$11$	−32.2500	−0.883975	−0.441988	−	0.897021i	$-0.645727\pi$
$11$	−32.2500	−0.883975	−0.441988	+	0.897021i	$0.645727\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	27.5557	0.526041
$15$	0	0
$16$	−55.0000	−0.859375
$17$	28.8586	0.411720	0.205860	−	0.978582i	$-0.434001\pi$
$17$	28.8586	0.411720	0.205860	+	0.978582i	$0.434001\pi$
$18$	0	0
$19$	−101.194	−1.22187	−0.610933	−	0.791682i	$-0.709206\pi$
$19$	−101.194	−1.22187	−0.610933	+	0.791682i	$0.709206\pi$
$20$	27.4505	0.306906
$21$	0	0
$22$	132.970	1.28860
$23$	118.990	1.07874	0.539372	−	0.842067i	$-0.318662\pi$
$23$	118.990	1.07874	0.539372	+	0.842067i	$0.318662\pi$
$24$	0	0
$25$	−115.697	−0.925577
$26$	0	0
$27$	0	0
$28$	−60.1492	−0.405969
$29$	−160.111	−1.02524	−0.512620	−	0.858616i	$-0.671325\pi$
$29$	−160.111	−1.02524	−0.512620	+	0.858616i	$0.671325\pi$
$30$	0	0
$31$	−38.0705	−0.220570	−0.110285	−	0.993900i	$-0.535176\pi$
$31$	−38.0705	−0.220570	−0.110285	+	0.993900i	$0.535176\pi$
$32$	259.756	1.43496
$33$	0	0
$34$	−118.987	−0.600179
$35$	−20.3843	−0.0984449
$36$	0	0
$37$	−327.568	−1.45545	−0.727727	−	0.685866i	$-0.759423\pi$
$37$	−327.568	−1.45545	−0.727727	+	0.685866i	$0.759423\pi$
$38$	417.233	1.78116
$39$	0	0
$40$	−12.5757	−0.0497099
$41$	−56.0846	−0.213633	−0.106816	−	0.994279i	$-0.534066\pi$
$41$	−56.0846	−0.213633	−0.106816	+	0.994279i	$0.534066\pi$
$42$	0	0
$43$	127.879	0.453519	0.226759	−	0.973951i	$-0.427187\pi$
$43$	127.879	0.453519	0.226759	+	0.973951i	$0.427187\pi$
$44$	−290.250	−0.994472
$45$	0	0
$46$	−490.608	−1.57253
$47$	517.983	1.60757	0.803783	−	0.594923i	$-0.202817\pi$
$47$	517.983	1.60757	0.803783	+	0.594923i	$0.202817\pi$
$48$	0	0
$49$	−298.334	−0.869779
$50$	477.032	1.34925
$51$	0	0
$52$	0	0
$53$	695.546	1.80265	0.901326	−	0.433141i	$-0.142595\pi$
$53$	695.546	1.80265	0.901326	+	0.433141i	$0.142595\pi$
$54$	0	0
$55$	−98.3643	−0.241153
$56$	27.5557	0.0657551
$57$	0	0
$58$	660.156	1.49453
$59$	−656.523	−1.44868	−0.724339	−	0.689444i	$-0.757855\pi$
$59$	−656.523	−1.44868	−0.724339	+	0.689444i	$0.757855\pi$
$60$	0	0
$61$	701.304	1.47201	0.736007	−	0.676974i	$-0.236709\pi$
$61$	701.304	1.47201	0.736007	+	0.676974i	$0.236709\pi$
$62$	156.969	0.321533
$63$	0	0
$64$	−631.000	−1.23242
$65$	0	0
$66$	0	0
$67$	−57.1750	−0.104254	−0.0521271	−	0.998640i	$-0.516600\pi$
$67$	−57.1750	−0.104254	−0.0521271	+	0.998640i	$0.516600\pi$
$68$	259.727	0.463184
$69$	0	0
$70$	84.0465	0.143507
$71$	−309.226	−0.516879	−0.258440	−	0.966027i	$-0.583208\pi$
$71$	−309.226	−0.516879	−0.258440	+	0.966027i	$0.583208\pi$
$72$	0	0
$73$	−389.711	−0.624826	−0.312413	−	0.949946i	$-0.601137\pi$
$73$	−389.711	−0.624826	−0.312413	+	0.949946i	$0.601137\pi$
$74$	1350.60	2.12167
$75$	0	0
$76$	−910.744	−1.37460
$77$	215.534	0.318992
$78$	0	0
$79$	901.820	1.28434	0.642169	−	0.766563i	$-0.278035\pi$
$79$	901.820	1.28434	0.642169	+	0.766563i	$0.278035\pi$
$80$	−167.753	−0.234442
$81$	0	0
$82$	231.243	0.311421
$83$	−687.095	−0.908657	−0.454328	−	0.890834i	$-0.650121\pi$
$83$	−687.095	−0.908657	−0.454328	+	0.890834i	$0.650121\pi$
$84$	0	0
$85$	88.0203	0.112319
$86$	−527.257	−0.661111
$87$	0	0
$88$	132.970	0.161076
$89$	−1070.54	−1.27502	−0.637510	−	0.770442i	$-0.720036\pi$
$89$	−1070.54	−1.27502	−0.637510	+	0.770442i	$0.720036\pi$
$90$	0	0
$91$	0	0
$92$	1070.91	1.21359
$93$	0	0
$94$	−2135.70	−2.34341
$95$	−308.647	−0.333332
$96$	0	0
$97$	1754.48	1.83650	0.918251	−	0.395998i	$-0.129601\pi$
$97$	1754.48	1.83650	0.918251	+	0.395998i	$0.129601\pi$
$98$	1230.06	1.26791
$99$	0	0
$100$	−1041.27	−1.04127
$101$	640.840	0.631346	0.315673	−	0.948868i	$-0.397770\pi$
$101$	640.840	0.631346	0.315673	+	0.948868i	$0.397770\pi$
$102$	0	0
$103$	−693.153	−0.663091	−0.331546	−	0.943439i	$-0.607570\pi$
$103$	−693.153	−0.663091	−0.331546	+	0.943439i	$0.607570\pi$
$104$	0	0
$105$	0	0
$106$	−2867.81	−2.62779
$107$	405.676	0.366525	0.183262	−	0.983064i	$-0.441334\pi$
$107$	405.676	0.366525	0.183262	+	0.983064i	$0.441334\pi$
$108$	0	0
$109$	−479.516	−0.421370	−0.210685	−	0.977554i	$-0.567569\pi$
$109$	−479.516	−0.421370	−0.210685	+	0.977554i	$0.567569\pi$
$110$	405.566	0.351538
$111$	0	0
$112$	367.578	0.310115
$113$	1547.20	1.28804	0.644021	−	0.765008i	$-0.277265\pi$
$113$	1547.20	1.28804	0.644021	+	0.765008i	$0.277265\pi$
$114$	0	0
$115$	362.926	0.294288
$116$	−1441.00	−1.15339
$117$	0	0
$118$	2706.91	2.11179
$119$	−192.869	−0.148574
$120$	0	0
$121$	−290.940	−0.218588
$122$	−2891.55	−2.14581
$123$	0	0
$124$	−342.634	−0.248141
$125$	−734.140	−0.525308
$126$	0	0
$127$	−2495.15	−1.74338	−0.871690	−	0.490059i	$-0.836975\pi$
$127$	−2495.15	−1.74338	−0.871690	+	0.490059i	$0.836975\pi$
$128$	523.634	0.361587
$129$	0	0
$130$	0	0
$131$	−43.7571	−0.0291838	−0.0145919	−	0.999894i	$-0.504645\pi$
$131$	−43.7571	−0.0291838	−0.0145919	+	0.999894i	$0.504645\pi$
$132$	0	0
$133$	676.303	0.440924
$134$	235.739	0.151975
$135$	0	0
$136$	−118.987	−0.0750224
$137$	206.194	0.128586	0.0642932	−	0.997931i	$-0.479521\pi$
$137$	206.194	0.128586	0.0642932	+	0.997931i	$0.479521\pi$
$138$	0	0
$139$	−100.000	−0.0610208	−0.0305104	−	0.999534i	$-0.509713\pi$
$139$	−100.000	−0.0610208	−0.0305104	+	0.999534i	$0.509713\pi$
$140$	−183.459	−0.110751
$141$	0	0
$142$	1274.97	0.753474
$143$	0	0
$144$	0	0
$145$	−488.349	−0.279691
$146$	1606.82	0.910832
$147$	0	0
$148$	−2948.11	−1.63739
$149$	−380.451	−0.209179	−0.104590	−	0.994515i	$-0.533353\pi$
$149$	−380.451	−0.209179	−0.104590	+	0.994515i	$0.533353\pi$
$150$	0	0
$151$	−1517.45	−0.817805	−0.408902	−	0.912578i	$-0.634088\pi$
$151$	−1517.45	−0.817805	−0.408902	+	0.912578i	$0.634088\pi$
$152$	417.233	0.222645
$153$	0	0
$154$	−888.671	−0.465007
$155$	−116.117	−0.0601726
$156$	0	0
$157$	1450.16	0.737166	0.368583	−	0.929595i	$-0.379843\pi$
$157$	1450.16	0.737166	0.368583	+	0.929595i	$0.379843\pi$
$158$	−3718.30	−1.87223
$159$	0	0
$160$	792.270	0.391465
$161$	−795.239	−0.389277
$162$	0	0
$163$	2342.36	1.12557	0.562785	−	0.826603i	$-0.309730\pi$
$163$	2342.36	1.12557	0.562785	+	0.826603i	$0.309730\pi$
$164$	−504.762	−0.240337
$165$	0	0
$166$	2832.97	1.32458
$167$	40.0731	0.0185686	0.00928428	−	0.999957i	$-0.497045\pi$
$167$	40.0731	0.0185686	0.00928428	+	0.999957i	$0.497045\pi$
$168$	0	0
$169$	0	0
$170$	−362.917	−0.163732
$171$	0	0
$172$	1150.91	0.510208
$173$	−1909.54	−0.839189	−0.419594	−	0.907712i	$-0.637828\pi$
$173$	−1909.54	−0.839189	−0.419594	+	0.907712i	$0.637828\pi$
$174$	0	0
$175$	773.232	0.334005
$176$	1773.75	0.759666
$177$	0	0
$178$	4413.94	1.85864
$179$	509.959	0.212939	0.106470	−	0.994316i	$-0.466045\pi$
$179$	509.959	0.212939	0.106470	+	0.994316i	$0.466045\pi$
$180$	0	0
$181$	−2136.88	−0.877531	−0.438766	−	0.898602i	$-0.644584\pi$
$181$	−2136.88	−0.877531	−0.438766	+	0.898602i	$0.644584\pi$
$182$	0	0
$183$	0	0
$184$	−490.608	−0.196566
$185$	−999.101	−0.397056
$186$	0	0
$187$	−930.688	−0.363950
$188$	4661.85	1.80851
$189$	0	0
$190$	1272.58	0.485911
$191$	−4057.74	−1.53721	−0.768607	−	0.639721i	$-0.779050\pi$
$191$	−4057.74	−1.53721	−0.768607	+	0.639721i	$0.779050\pi$
$192$	0	0
$193$	873.394	0.325742	0.162871	−	0.986647i	$-0.447925\pi$
$193$	873.394	0.325742	0.162871	+	0.986647i	$0.447925\pi$
$194$	−7233.92	−2.67714
$195$	0	0
$196$	−2685.01	−0.978502
$197$	−4147.25	−1.49989	−0.749947	−	0.661498i	$-0.769921\pi$
$197$	−4147.25	−1.49989	−0.749947	+	0.661498i	$0.769921\pi$
$198$	0	0
$199$	−2404.06	−0.856379	−0.428189	−	0.903689i	$-0.640848\pi$
$199$	−2404.06	−0.856379	−0.428189	+	0.903689i	$0.640848\pi$
$200$	477.032	0.168656
$201$	0	0
$202$	−2642.25	−0.920337
$203$	1070.06	0.369969
$204$	0	0
$205$	−171.061	−0.0582802
$206$	2857.94	0.966613
$207$	0	0
$208$	0	0
$209$	3263.50	1.08010
$210$	0	0
$211$	3868.85	1.26229	0.631144	−	0.775665i	$-0.282586\pi$
$211$	3868.85	1.26229	0.631144	+	0.775665i	$0.282586\pi$
$212$	6259.91	2.02798
$213$	0	0
$214$	−1672.64	−0.534297
$215$	390.037	0.123722
$216$	0	0
$217$	254.434	0.0795950
$218$	1977.09	0.614246
$219$	0	0
$220$	−885.279	−0.271298
$221$	0	0
$222$	0	0
$223$	2813.55	0.844885	0.422443	−	0.906390i	$-0.361173\pi$
$223$	2813.55	0.844885	0.422443	+	0.906390i	$0.361173\pi$
$224$	−1736.01	−0.517822
$225$	0	0
$226$	−6379.29	−1.87763
$227$	4518.37	1.32112	0.660561	−	0.750772i	$-0.270318\pi$
$227$	4518.37	1.32112	0.660561	+	0.750772i	$0.270318\pi$
$228$	0	0
$229$	−1305.27	−0.376658	−0.188329	−	0.982106i	$-0.560307\pi$
$229$	−1305.27	−0.376658	−0.188329	+	0.982106i	$0.560307\pi$
$230$	−1496.38	−0.428994
$231$	0	0
$232$	660.156	0.186816
$233$	3360.55	0.944879	0.472440	−	0.881363i	$-0.343373\pi$
$233$	3360.55	0.944879	0.472440	+	0.881363i	$0.343373\pi$
$234$	0	0
$235$	1579.88	0.438553
$236$	−5908.71	−1.62976
$237$	0	0
$238$	795.219	0.216581
$239$	4737.17	1.28210	0.641050	−	0.767499i	$-0.278499\pi$
$239$	4737.17	1.28210	0.641050	+	0.767499i	$0.278499\pi$
$240$	0	0
$241$	4785.28	1.27903	0.639516	−	0.768778i	$-0.279135\pi$
$241$	4785.28	1.27903	0.639516	+	0.768778i	$0.279135\pi$
$242$	1199.58	0.318643
$243$	0	0
$244$	6311.74	1.65601
$245$	−909.937	−0.237281
$246$	0	0
$247$	0	0
$248$	156.969	0.0401916
$249$	0	0
$250$	3026.94	0.765762
$251$	3273.86	0.823284	0.411642	−	0.911346i	$-0.364955\pi$
$251$	3273.86	0.823284	0.411642	+	0.911346i	$0.364955\pi$
$252$	0	0
$253$	−3837.42	−0.953584
$254$	10287.8	2.54139
$255$	0	0
$256$	2889.00	0.705322
$257$	6545.81	1.58878	0.794390	−	0.607408i	$-0.207791\pi$
$257$	6545.81	1.58878	0.794390	+	0.607408i	$0.207791\pi$
$258$	0	0
$259$	2189.22	0.525217
$260$	0	0
$261$	0	0
$262$	180.415	0.0425424
$263$	−88.2014	−0.0206796	−0.0103398	−	0.999947i	$-0.503291\pi$
$263$	−88.2014	−0.0206796	−0.0103398	+	0.999947i	$0.503291\pi$
$264$	0	0
$265$	2121.46	0.491773
$266$	−2788.47	−0.642752
$267$	0	0
$268$	−514.575	−0.117286
$269$	4527.60	1.02622	0.513109	−	0.858324i	$-0.328494\pi$
$269$	4527.60	1.02622	0.513109	+	0.858324i	$0.328494\pi$
$270$	0	0
$271$	8321.82	1.86537	0.932684	−	0.360695i	$-0.117460\pi$
$271$	8321.82	1.86537	0.932684	+	0.360695i	$0.117460\pi$
$272$	−1587.22	−0.353821
$273$	0	0
$274$	−850.160	−0.187445
$275$	3731.23	0.818187
$276$	0	0
$277$	−2881.31	−0.624986	−0.312493	−	0.949920i	$-0.601164\pi$
$277$	−2881.31	−0.624986	−0.312493	+	0.949920i	$0.601164\pi$
$278$	412.311	0.0889523
$279$	0	0
$280$	84.0465	0.0179384
$281$	2817.99	0.598247	0.299123	−	0.954214i	$-0.403306\pi$
$281$	2817.99	0.598247	0.299123	+	0.954214i	$0.403306\pi$
$282$	0	0
$283$	264.601	0.0555792	0.0277896	−	0.999614i	$-0.491153\pi$
$283$	264.601	0.0555792	0.0277896	+	0.999614i	$0.491153\pi$
$284$	−2783.04	−0.581489
$285$	0	0
$286$	0	0
$287$	374.827	0.0770918
$288$	0	0
$289$	−4080.18	−0.830487
$290$	2013.52	0.407716
$291$	0	0
$292$	−3507.40	−0.702929
$293$	−4292.52	−0.855877	−0.427938	−	0.903808i	$-0.640760\pi$
$293$	−4292.52	−0.855877	−0.427938	+	0.903808i	$0.640760\pi$
$294$	0	0
$295$	−2002.43	−0.395208
$296$	1350.60	0.265209
$297$	0	0
$298$	1568.64	0.304929
$299$	0	0
$300$	0	0
$301$	−854.643	−0.163657
$302$	6256.62	1.19214
$303$	0	0
$304$	5565.66	1.05004
$305$	2139.02	0.401573
$306$	0	0
$307$	7026.26	1.30622	0.653110	−	0.757263i	$-0.273464\pi$
$307$	7026.26	1.30622	0.653110	+	0.757263i	$0.273464\pi$
$308$	1939.81	0.358866
$309$	0	0
$310$	478.763	0.0877159
$311$	1133.21	0.206618	0.103309	−	0.994649i	$-0.467057\pi$
$311$	1133.21	0.206618	0.103309	+	0.994649i	$0.467057\pi$
$312$	0	0
$313$	−5285.95	−0.954566	−0.477283	−	0.878750i	$-0.658378\pi$
$313$	−5285.95	−0.954566	−0.477283	+	0.878750i	$0.658378\pi$
$314$	−5979.15	−1.07459
$315$	0	0
$316$	8116.38	1.44488
$317$	4782.16	0.847296	0.423648	−	0.905827i	$-0.360749\pi$
$317$	4782.16	0.847296	0.423648	+	0.905827i	$0.360749\pi$
$318$	0	0
$319$	5163.59	0.906286
$320$	−1924.59	−0.336212
$321$	0	0
$322$	3278.85	0.567464
$323$	−2920.31	−0.503066
$324$	0	0
$325$	0	0
$326$	−9657.80	−1.64079
$327$	0	0
$328$	231.243	0.0389276
$329$	−3461.81	−0.580108
$330$	0	0
$331$	8669.98	1.43971	0.719857	−	0.694122i	$-0.244207\pi$
$331$	8669.98	1.43971	0.719857	+	0.694122i	$0.244207\pi$
$332$	−6183.86	−1.02224
$333$	0	0
$334$	−165.226	−0.0270681
$335$	−174.387	−0.0284412
$336$	0	0
$337$	8526.59	1.37826	0.689129	−	0.724639i	$-0.257993\pi$
$337$	8526.59	1.37826	0.689129	+	0.724639i	$0.257993\pi$
$338$	0	0
$339$	0	0
$340$	792.183	0.126359
$341$	1227.77	0.194978
$342$	0	0
$343$	4286.19	0.674731
$344$	−527.257	−0.0826389
$345$	0	0
$346$	7873.23	1.22332
$347$	12581.5	1.94643	0.973213	−	0.229907i	$-0.0738421\pi$
$347$	12581.5	1.94643	0.973213	+	0.229907i	$0.0738421\pi$
$348$	0	0
$349$	−8961.18	−1.37444	−0.687222	−	0.726447i	$-0.741170\pi$
$349$	−8961.18	−1.37444	−0.687222	+	0.726447i	$0.741170\pi$
$350$	−3188.12	−0.486892
$351$	0	0
$352$	−8377.11	−1.26847
$353$	5357.99	0.807868	0.403934	−	0.914788i	$-0.367643\pi$
$353$	5357.99	0.807868	0.403934	+	0.914788i	$0.367643\pi$
$354$	0	0
$355$	−943.159	−0.141007
$356$	−9634.84	−1.43440
$357$	0	0
$358$	−2102.61	−0.310409
$359$	2705.40	0.397731	0.198866	−	0.980027i	$-0.436274\pi$
$359$	2705.40	0.397731	0.198866	+	0.980027i	$0.436274\pi$
$360$	0	0
$361$	3381.19	0.492957
$362$	8810.59	1.27921
$363$	0	0
$364$	0	0
$365$	−1188.64	−0.170456
$366$	0	0
$367$	10473.8	1.48972	0.744858	−	0.667223i	$-0.232517\pi$
$367$	10473.8	1.48972	0.744858	+	0.667223i	$0.232517\pi$
$368$	−6544.45	−0.927046
$369$	0	0
$370$	4119.40	0.578804
$371$	−4648.50	−0.650507
$372$	0	0
$373$	−12763.0	−1.77170	−0.885850	−	0.463973i	$-0.846424\pi$
$373$	−12763.0	−1.77170	−0.885850	+	0.463973i	$0.846424\pi$
$374$	3837.32	0.530544
$375$	0	0
$376$	−2135.70	−0.292926
$377$	0	0
$378$	0	0
$379$	−2318.02	−0.314166	−0.157083	−	0.987585i	$-0.550209\pi$
$379$	−2318.02	−0.314166	−0.157083	+	0.987585i	$0.550209\pi$
$380$	−2777.82	−0.374998
$381$	0	0
$382$	16730.5	2.24086
$383$	−1983.34	−0.264606	−0.132303	−	0.991209i	$-0.542237\pi$
$383$	−1983.34	−0.264606	−0.132303	+	0.991209i	$0.542237\pi$
$384$	0	0
$385$	657.392	0.0870229
$386$	−3601.10	−0.474847
$387$	0	0
$388$	15790.3	2.06607
$389$	3244.51	0.422887	0.211444	−	0.977390i	$-0.432184\pi$
$389$	3244.51	0.422887	0.211444	+	0.977390i	$0.432184\pi$
$390$	0	0
$391$	3433.88	0.444140
$392$	1230.06	0.158489
$393$	0	0
$394$	17099.5	2.18645
$395$	2750.60	0.350374
$396$	0	0
$397$	3759.72	0.475302	0.237651	−	0.971351i	$-0.423622\pi$
$397$	3759.72	0.475302	0.237651	+	0.971351i	$0.423622\pi$
$398$	9912.20	1.24838
$399$	0	0
$400$	6363.34	0.795418
$401$	1997.55	0.248760	0.124380	−	0.992235i	$-0.460306\pi$
$401$	1997.55	0.248760	0.124380	+	0.992235i	$0.460306\pi$
$402$	0	0
$403$	0	0
$404$	5767.56	0.710264
$405$	0	0
$406$	−4411.98	−0.539318
$407$	10564.1	1.28659
$408$	0	0
$409$	−5195.23	−0.628087	−0.314044	−	0.949409i	$-0.601684\pi$
$409$	−5195.23	−0.628087	−0.314044	+	0.949409i	$0.601684\pi$
$410$	705.304	0.0849573
$411$	0	0
$412$	−6238.38	−0.745977
$413$	4387.70	0.522772
$414$	0	0
$415$	−2095.68	−0.247887
$416$	0	0
$417$	0	0
$418$	−13455.7	−1.57450
$419$	6822.11	0.795422	0.397711	−	0.917511i	$-0.369805\pi$
$419$	6822.11	0.795422	0.397711	+	0.917511i	$0.369805\pi$
$420$	0	0
$421$	7537.70	0.872601	0.436300	−	0.899801i	$-0.356288\pi$
$421$	7537.70	0.872601	0.436300	+	0.899801i	$0.356288\pi$
$422$	−15951.7	−1.84009
$423$	0	0
$424$	−2867.81	−0.328474
$425$	−3338.85	−0.381078
$426$	0	0
$427$	−4686.99	−0.531192
$428$	3651.08	0.412340
$429$	0	0
$430$	−1608.16	−0.180355
$431$	13404.2	1.49805	0.749023	−	0.662544i	$-0.230523\pi$
$431$	13404.2	1.49805	0.749023	+	0.662544i	$0.230523\pi$
$432$	0	0
$433$	−17715.9	−1.96622	−0.983110	−	0.183014i	$-0.941415\pi$
$433$	−17715.9	−1.96622	−0.983110	+	0.183014i	$0.941415\pi$
$434$	−1049.06	−0.116029
$435$	0	0
$436$	−4315.64	−0.474041
$437$	−12041.1	−1.31808
$438$	0	0
$439$	7163.47	0.778801	0.389401	−	0.921068i	$-0.372682\pi$
$439$	7163.47	0.778801	0.389401	+	0.921068i	$0.372682\pi$
$440$	405.566	0.0439423
$441$	0	0
$442$	0	0
$443$	10169.2	1.09064	0.545321	−	0.838227i	$-0.316408\pi$
$443$	10169.2	1.09064	0.545321	+	0.838227i	$0.316408\pi$
$444$	0	0
$445$	−3265.20	−0.347833
$446$	−11600.6	−1.23162
$447$	0	0
$448$	4217.13	0.444733
$449$	17142.5	1.80179	0.900895	−	0.434037i	$-0.142911\pi$
$449$	17142.5	1.80179	0.900895	+	0.434037i	$0.142911\pi$
$450$	0	0
$451$	1808.73	0.188846
$452$	13924.8	1.44905
$453$	0	0
$454$	−18629.7	−1.92585
$455$	0	0
$456$	0	0
$457$	14091.1	1.44235	0.721177	−	0.692750i	$-0.243601\pi$
$457$	14091.1	1.44235	0.721177	+	0.692750i	$0.243601\pi$
$458$	5381.76	0.549068
$459$	0	0
$460$	3266.34	0.331074
$461$	2922.22	0.295231	0.147616	−	0.989045i	$-0.452840\pi$
$461$	2922.22	0.295231	0.147616	+	0.989045i	$0.452840\pi$
$462$	0	0
$463$	−2072.61	−0.208040	−0.104020	−	0.994575i	$-0.533171\pi$
$463$	−2072.61	−0.208040	−0.104020	+	0.994575i	$0.533171\pi$
$464$	8806.13	0.881065
$465$	0	0
$466$	−13855.9	−1.37739
$467$	2664.19	0.263992	0.131996	−	0.991250i	$-0.457861\pi$
$467$	2664.19	0.263992	0.131996	+	0.991250i	$0.457861\pi$
$468$	0	0
$469$	382.114	0.0376213
$470$	−6514.01	−0.639295
$471$	0	0
$472$	2706.91	0.263974
$473$	−4124.08	−0.400899
$474$	0	0
$475$	11707.8	1.13093
$476$	−1735.82	−0.167145
$477$	0	0
$478$	−19531.8	−1.86897
$479$	5220.70	0.497995	0.248998	−	0.968504i	$-0.419899\pi$
$479$	5220.70	0.497995	0.248998	+	0.968504i	$0.419899\pi$
$480$	0	0
$481$	0	0
$482$	−19730.2	−1.86449
$483$	0	0
$484$	−2618.46	−0.245911
$485$	5351.28	0.501008
$486$	0	0
$487$	−12224.6	−1.13747	−0.568737	−	0.822520i	$-0.692568\pi$
$487$	−12224.6	−1.13747	−0.568737	+	0.822520i	$0.692568\pi$
$488$	−2891.55	−0.268226
$489$	0	0
$490$	3751.77	0.345893
$491$	−19653.2	−1.80639	−0.903195	−	0.429231i	$-0.858784\pi$
$491$	−19653.2	−1.80639	−0.903195	+	0.429231i	$0.858784\pi$
$492$	0	0
$493$	−4620.59	−0.422111
$494$	0	0
$495$	0	0
$496$	2093.88	0.189552
$497$	2066.63	0.186522
$498$	0	0
$499$	−11713.6	−1.05084	−0.525422	−	0.850842i	$-0.676093\pi$
$499$	−11713.6	−1.05084	−0.525422	+	0.850842i	$0.676093\pi$
$500$	−6607.26	−0.590972
$501$	0	0
$502$	−13498.5	−1.20013
$503$	−13003.3	−1.15266	−0.576332	−	0.817216i	$-0.695517\pi$
$503$	−13003.3	−1.15266	−0.576332	+	0.817216i	$0.695517\pi$
$504$	0	0
$505$	1954.60	0.172235
$506$	15822.1	1.39008
$507$	0	0
$508$	−22456.4	−1.96130
$509$	5328.93	0.464049	0.232024	−	0.972710i	$-0.425465\pi$
$509$	5328.93	0.464049	0.232024	+	0.972710i	$0.425465\pi$
$510$	0	0
$511$	2604.54	0.225475
$512$	−16100.7	−1.38976
$513$	0	0
$514$	−26989.1	−2.31603
$515$	−2114.16	−0.180895
$516$	0	0
$517$	−16704.9	−1.42105
$518$	−9026.37	−0.765629
$519$	0	0
$520$	0	0
$521$	11700.3	0.983876	0.491938	−	0.870630i	$-0.336289\pi$
$521$	11700.3	0.983876	0.491938	+	0.870630i	$0.336289\pi$
$522$	0	0
$523$	−4535.04	−0.379165	−0.189583	−	0.981865i	$-0.560714\pi$
$523$	−4535.04	−0.379165	−0.189583	+	0.981865i	$0.560714\pi$
$524$	−393.814	−0.0328318
$525$	0	0
$526$	363.664	0.0301454
$527$	−1098.66	−0.0908128
$528$	0	0
$529$	1991.62	0.163690
$530$	−8746.98	−0.716877
$531$	0	0
$532$	6086.73	0.496040
$533$	0	0
$534$	0	0
$535$	1237.33	0.0999900
$536$	235.739	0.0189969
$537$	0	0
$538$	−18667.8	−1.49596
$539$	9621.27	0.768863
$540$	0	0
$541$	−5184.89	−0.412044	−0.206022	−	0.978547i	$-0.566052\pi$
$541$	−5184.89	−0.412044	−0.206022	+	0.978547i	$0.566052\pi$
$542$	−34311.7	−2.71922
$543$	0	0
$544$	7496.18	0.590801
$545$	−1462.55	−0.114952
$546$	0	0
$547$	5609.12	0.438443	0.219222	−	0.975675i	$-0.429648\pi$
$547$	5609.12	0.438443	0.219222	+	0.975675i	$0.429648\pi$
$548$	1855.75	0.144660
$549$	0	0
$550$	−15384.2	−1.19270
$551$	16202.3	1.25271
$552$	0	0
$553$	−6027.08	−0.463468
$554$	11879.9	0.911066
$555$	0	0
$556$	−900.000	−0.0686484
$557$	20150.5	1.53286	0.766432	−	0.642326i	$-0.222030\pi$
$557$	20150.5	1.53286	0.766432	+	0.642326i	$0.222030\pi$
$558$	0	0
$559$	0	0
$560$	1121.14	0.0846011
$561$	0	0
$562$	−11618.9	−0.872087
$563$	−16292.2	−1.21960	−0.609800	−	0.792556i	$-0.708750\pi$
$563$	−16292.2	−1.21960	−0.609800	+	0.792556i	$0.708750\pi$
$564$	0	0
$565$	4719.06	0.351385
$566$	−1090.98	−0.0810200
$567$	0	0
$568$	1274.97	0.0941843
$569$	−10460.5	−0.770700	−0.385350	−	0.922770i	$-0.625919\pi$
$569$	−10460.5	−0.770700	−0.385350	+	0.922770i	$0.625919\pi$
$570$	0	0
$571$	2225.96	0.163141	0.0815705	−	0.996668i	$-0.474006\pi$
$571$	2225.96	0.163141	0.0815705	+	0.996668i	$0.474006\pi$
$572$	0	0
$573$	0	0
$574$	−1545.45	−0.112380
$575$	−13766.8	−0.998461
$576$	0	0
$577$	4686.23	0.338112	0.169056	−	0.985606i	$-0.445928\pi$
$577$	4686.23	0.338112	0.169056	+	0.985606i	$0.445928\pi$
$578$	16823.0	1.21063
$579$	0	0
$580$	−4395.14	−0.314652
$581$	4592.03	0.327899
$582$	0	0
$583$	−22431.3	−1.59350
$584$	1606.82	0.113854
$585$	0	0
$586$	17698.5	1.24764
$587$	−12090.6	−0.850138	−0.425069	−	0.905161i	$-0.639750\pi$
$587$	−12090.6	−0.850138	−0.425069	+	0.905161i	$0.639750\pi$
$588$	0	0
$589$	3852.50	0.269507
$590$	8256.25	0.576109
$591$	0	0
$592$	18016.2	1.25078
$593$	−6135.97	−0.424914	−0.212457	−	0.977170i	$-0.568147\pi$
$593$	−6135.97	−0.424914	−0.212457	+	0.977170i	$0.568147\pi$
$594$	0	0
$595$	−588.261	−0.0405317
$596$	−3424.06	−0.235327
$597$	0	0
$598$	0	0
$599$	6198.80	0.422831	0.211416	−	0.977396i	$-0.432193\pi$
$599$	6198.80	0.422831	0.211416	+	0.977396i	$0.432193\pi$
$600$	0	0
$601$	18345.4	1.24513	0.622565	−	0.782568i	$-0.286091\pi$
$601$	18345.4	1.24513	0.622565	+	0.782568i	$0.286091\pi$
$602$	3523.79	0.238569
$603$	0	0
$604$	−13657.1	−0.920030
$605$	−887.384	−0.0596319
$606$	0	0
$607$	10388.1	0.694631	0.347315	−	0.937748i	$-0.387093\pi$
$607$	10388.1	0.694631	0.347315	+	0.937748i	$0.387093\pi$
$608$	−26285.7	−1.75333
$609$	0	0
$610$	−8819.40	−0.585389
$611$	0	0
$612$	0	0
$613$	−804.480	−0.0530060	−0.0265030	−	0.999649i	$-0.508437\pi$
$613$	−804.480	−0.0530060	−0.0265030	+	0.999649i	$0.508437\pi$
$614$	−28970.0	−1.90413
$615$	0	0
$616$	−888.671	−0.0581259
$617$	−15218.3	−0.992973	−0.496486	−	0.868044i	$-0.665377\pi$
$617$	−15218.3	−0.992973	−0.496486	+	0.868044i	$0.665377\pi$
$618$	0	0
$619$	−11462.5	−0.744291	−0.372145	−	0.928174i	$-0.621378\pi$
$619$	−11462.5	−0.744291	−0.372145	+	0.928174i	$0.621378\pi$
$620$	−1045.05	−0.0676942
$621$	0	0
$622$	−4672.33	−0.301195
$623$	7154.66	0.460105
$624$	0	0
$625$	12223.0	0.782270
$626$	21794.5	1.39151
$627$	0	0
$628$	13051.4	0.829311
$629$	−9453.14	−0.599239
$630$	0	0
$631$	−4468.68	−0.281926	−0.140963	−	0.990015i	$-0.545020\pi$
$631$	−4468.68	−0.281926	−0.140963	+	0.990015i	$0.545020\pi$
$632$	−3718.30	−0.234028
$633$	0	0
$634$	−19717.3	−1.23514
$635$	−7610.37	−0.475603
$636$	0	0
$637$	0	0
$638$	−21290.0	−1.32113
$639$	0	0
$640$	1597.12	0.0986430
$641$	−6142.36	−0.378484	−0.189242	−	0.981930i	$-0.560603\pi$
$641$	−6142.36	−0.378484	−0.189242	+	0.981930i	$0.560603\pi$
$642$	0	0
$643$	20738.2	1.27190	0.635951	−	0.771729i	$-0.280608\pi$
$643$	20738.2	1.27190	0.635951	+	0.771729i	$0.280608\pi$
$644$	−7157.15	−0.437937
$645$	0	0
$646$	12040.7	0.733339
$647$	−852.757	−0.0518166	−0.0259083	−	0.999664i	$-0.508248\pi$
$647$	−852.757	−0.0518166	−0.0259083	+	0.999664i	$0.508248\pi$
$648$	0	0
$649$	21172.8	1.28060
$650$	0	0
$651$	0	0
$652$	21081.3	1.26627
$653$	7345.75	0.440217	0.220108	−	0.975475i	$-0.429359\pi$
$653$	7345.75	0.440217	0.220108	+	0.975475i	$0.429359\pi$
$654$	0	0
$655$	−133.462	−0.00796151
$656$	3084.65	0.183591
$657$	0	0
$658$	14273.4	0.845645
$659$	−12540.7	−0.741297	−0.370648	−	0.928773i	$-0.620864\pi$
$659$	−12540.7	−0.741297	−0.370648	+	0.928773i	$0.620864\pi$
$660$	0	0
$661$	−2242.95	−0.131983	−0.0659915	−	0.997820i	$-0.521021\pi$
$661$	−2242.95	−0.131983	−0.0659915	+	0.997820i	$0.521021\pi$
$662$	−35747.3	−2.09873
$663$	0	0
$664$	2832.97	0.165573
$665$	2062.76	0.120287
$666$	0	0
$667$	−19051.7	−1.10597
$668$	360.658	0.0208896
$669$	0	0
$670$	719.016	0.0414597
$671$	−22617.0	−1.30122
$672$	0	0
$673$	−4776.46	−0.273579	−0.136790	−	0.990600i	$-0.543678\pi$
$673$	−4776.46	−0.273579	−0.136790	+	0.990600i	$0.543678\pi$
$674$	−35156.0	−2.00914
$675$	0	0
$676$	0	0
$677$	7933.57	0.450387	0.225193	−	0.974314i	$-0.427699\pi$
$677$	7933.57	0.450387	0.225193	+	0.974314i	$0.427699\pi$
$678$	0	0
$679$	−11725.6	−0.662723
$680$	−362.917	−0.0204665
$681$	0	0
$682$	−5062.23	−0.284227
$683$	−23573.8	−1.32068	−0.660340	−	0.750967i	$-0.729588\pi$
$683$	−23573.8	−1.32068	−0.660340	+	0.750967i	$0.729588\pi$
$684$	0	0
$685$	628.904	0.0350791
$686$	−17672.4	−0.983581
$687$	0	0
$688$	−7033.32	−0.389743
$689$	0	0
$690$	0	0
$691$	−12543.8	−0.690575	−0.345288	−	0.938497i	$-0.612219\pi$
$691$	−12543.8	−0.690575	−0.345288	+	0.938497i	$0.612219\pi$
$692$	−17185.9	−0.944087
$693$	0	0
$694$	−51874.8	−2.83738
$695$	−305.006	−0.0166468
$696$	0	0
$697$	−1618.52	−0.0879568
$698$	36947.9	2.00358
$699$	0	0
$700$	6959.09	0.375755
$701$	581.786	0.0313463	0.0156731	−	0.999877i	$-0.495011\pi$
$701$	581.786	0.0313463	0.0156731	+	0.999877i	$0.495011\pi$
$702$	0	0
$703$	33147.9	1.77837
$704$	20349.7	1.08943
$705$	0	0
$706$	−22091.6	−1.17766
$707$	−4282.89	−0.227828
$708$	0	0
$709$	−20742.0	−1.09871	−0.549353	−	0.835590i	$-0.685126\pi$
$709$	−20742.0	−1.09871	−0.549353	+	0.835590i	$0.685126\pi$
$710$	3888.74	0.205552
$711$	0	0
$712$	4413.94	0.232331
$713$	−4530.01	−0.237938
$714$	0	0
$715$	0	0
$716$	4589.63	0.239556
$717$	0	0
$718$	−11154.6	−0.579788
$719$	−25350.2	−1.31489	−0.657443	−	0.753504i	$-0.728362\pi$
$719$	−25350.2	−1.31489	−0.657443	+	0.753504i	$0.728362\pi$
$720$	0	0
$721$	4632.51	0.239284
$722$	−13941.0	−0.718602
$723$	0	0
$724$	−19231.9	−0.987223
$725$	18524.4	0.948938
$726$	0	0
$727$	33428.2	1.70534	0.852672	−	0.522447i	$-0.174981\pi$
$727$	33428.2	1.70534	0.852672	+	0.522447i	$0.174981\pi$
$728$	0	0
$729$	0	0
$730$	4900.90	0.248480
$731$	3690.39	0.186722
$732$	0	0
$733$	−3842.67	−0.193632	−0.0968160	−	0.995302i	$-0.530866\pi$
$733$	−3842.67	−0.193632	−0.0968160	+	0.995302i	$0.530866\pi$
$734$	−43184.4	−2.17162
$735$	0	0
$736$	30908.3	1.54796
$737$	1843.89	0.0921582
$738$	0	0
$739$	29029.6	1.44502	0.722511	−	0.691359i	$-0.242988\pi$
$739$	29029.6	1.44502	0.722511	+	0.691359i	$0.242988\pi$
$740$	−8991.91	−0.446688
$741$	0	0
$742$	19166.3	0.948269
$743$	34996.7	1.72800	0.864000	−	0.503492i	$-0.167952\pi$
$743$	34996.7	1.72800	0.864000	+	0.503492i	$0.167952\pi$
$744$	0	0
$745$	−1160.40	−0.0570653
$746$	52623.2	2.58267
$747$	0	0
$748$	−8376.19	−0.409444
$749$	−2711.23	−0.132265
$750$	0	0
$751$	10454.1	0.507957	0.253979	−	0.967210i	$-0.418261\pi$
$751$	10454.1	0.507957	0.253979	+	0.967210i	$0.418261\pi$
$752$	−28489.1	−1.38150
$753$	0	0
$754$	0	0
$755$	−4628.32	−0.223102
$756$	0	0
$757$	−28130.4	−1.35062	−0.675308	−	0.737536i	$-0.735989\pi$
$757$	−28130.4	−1.35062	−0.675308	+	0.737536i	$0.735989\pi$
$758$	9557.45	0.457971
$759$	0	0
$760$	1272.58	0.0607388
$761$	−21087.0	−1.00447	−0.502236	−	0.864731i	$-0.667489\pi$
$761$	−21087.0	−1.00447	−0.502236	+	0.864731i	$0.667489\pi$
$762$	0	0
$763$	3204.72	0.152056
$764$	−36519.7	−1.72937
$765$	0	0
$766$	8177.54	0.385727
$767$	0	0
$768$	0	0
$769$	19527.9	0.915728	0.457864	−	0.889022i	$-0.348615\pi$
$769$	19527.9	0.915728	0.457864	+	0.889022i	$0.348615\pi$
$770$	−2710.50	−0.126857
$771$	0	0
$772$	7860.55	0.366460
$773$	29352.0	1.36574	0.682870	−	0.730540i	$-0.260731\pi$
$773$	29352.0	1.36574	0.682870	+	0.730540i	$0.260731\pi$
$774$	0	0
$775$	4404.64	0.204154
$776$	−7233.92	−0.334643
$777$	0	0
$778$	−13377.5	−0.616459
$779$	5675.42	0.261031
$780$	0	0
$781$	9972.54	0.456908
$782$	−14158.3	−0.647440
$783$	0	0
$784$	16408.4	0.747467
$785$	4423.06	0.201103
$786$	0	0
$787$	4463.12	0.202151	0.101076	−	0.994879i	$-0.467772\pi$
$787$	4463.12	0.202151	0.101076	+	0.994879i	$0.467772\pi$
$788$	−37325.2	−1.68738
$789$	0	0
$790$	−11341.0	−0.510754
$791$	−10340.3	−0.464804
$792$	0	0
$793$	0	0
$794$	−15501.7	−0.692866
$795$	0	0
$796$	−21636.6	−0.963426
$797$	34785.5	1.54600	0.773002	−	0.634404i	$-0.218754\pi$
$797$	34785.5	1.54600	0.773002	+	0.634404i	$0.218754\pi$
$798$	0	0
$799$	14948.2	0.661866
$800$	−30053.0	−1.32817
$801$	0	0
$802$	−8236.09	−0.362627
$803$	12568.2	0.552330
$804$	0	0
$805$	−2425.53	−0.106197
$806$	0	0
$807$	0	0
$808$	−2642.25	−0.115042
$809$	10620.0	0.461530	0.230765	−	0.973010i	$-0.425877\pi$
$809$	10620.0	0.461530	0.230765	+	0.973010i	$0.425877\pi$
$810$	0	0
$811$	5497.87	0.238047	0.119024	−	0.992891i	$-0.462024\pi$
$811$	5497.87	0.238047	0.119024	+	0.992891i	$0.462024\pi$
$812$	9630.57	0.416215
$813$	0	0
$814$	−43556.7	−1.87551
$815$	7144.34	0.307062
$816$	0	0
$817$	−12940.5	−0.554139
$818$	21420.5	0.915587
$819$	0	0
$820$	−1539.55	−0.0655653
$821$	−21305.3	−0.905678	−0.452839	−	0.891592i	$-0.649589\pi$
$821$	−21305.3	−0.905678	−0.452839	+	0.891592i	$0.649589\pi$
$822$	0	0
$823$	17342.6	0.734537	0.367268	−	0.930115i	$-0.380293\pi$
$823$	17342.6	0.734537	0.367268	+	0.930115i	$0.380293\pi$
$824$	2857.94	0.120827
$825$	0	0
$826$	−18091.0	−0.762064
$827$	−5129.96	−0.215703	−0.107851	−	0.994167i	$-0.534397\pi$
$827$	−5129.96	−0.215703	−0.107851	+	0.994167i	$0.534397\pi$
$828$	0	0
$829$	−8471.81	−0.354931	−0.177466	−	0.984127i	$-0.556790\pi$
$829$	−8471.81	−0.354931	−0.177466	+	0.984127i	$0.556790\pi$
$830$	8640.72	0.361354
$831$	0	0
$832$	0	0
$833$	−8609.50	−0.358105
$834$	0	0
$835$	122.225	0.00506561
$836$	29371.5	1.21511
$837$	0	0
$838$	−28128.3	−1.15952
$839$	19155.0	0.788207	0.394103	−	0.919066i	$-0.371055\pi$
$839$	19155.0	0.788207	0.394103	+	0.919066i	$0.371055\pi$
$840$	0	0
$841$	1246.67	0.0511160
$842$	−31078.7	−1.27202
$843$	0	0
$844$	34819.7	1.42007
$845$	0	0
$846$	0	0
$847$	1944.42	0.0788797
$848$	−38255.0	−1.54915
$849$	0	0
$850$	13766.4	0.555512
$851$	−38977.3	−1.57006
$852$	0	0
$853$	−18075.1	−0.725532	−0.362766	−	0.931880i	$-0.618168\pi$
$853$	−18075.1	−0.725532	−0.362766	+	0.931880i	$0.618168\pi$
$854$	19324.9	0.774339
$855$	0	0
$856$	−1672.64	−0.0667871
$857$	21054.6	0.839219	0.419609	−	0.907705i	$-0.362167\pi$
$857$	21054.6	0.839219	0.419609	+	0.907705i	$0.362167\pi$
$858$	0	0
$859$	920.322	0.0365553	0.0182776	−	0.999833i	$-0.494182\pi$
$859$	920.322	0.0365553	0.0182776	+	0.999833i	$0.494182\pi$
$860$	3510.33	0.139188
$861$	0	0
$862$	−55267.0	−2.18376
$863$	19427.5	0.766304	0.383152	−	0.923685i	$-0.374839\pi$
$863$	19427.5	0.766304	0.383152	+	0.923685i	$0.374839\pi$
$864$	0	0
$865$	−5824.21	−0.228935
$866$	73044.7	2.86623
$867$	0	0
$868$	2289.91	0.0895444
$869$	−29083.7	−1.13532
$870$	0	0
$871$	0	0
$872$	1977.09	0.0767808
$873$	0	0
$874$	49646.5	1.92142
$875$	4906.44	0.189563
$876$	0	0
$877$	14872.2	0.572632	0.286316	−	0.958135i	$-0.407569\pi$
$877$	14872.2	0.572632	0.286316	+	0.958135i	$0.407569\pi$
$878$	−29535.7	−1.13529
$879$	0	0
$880$	5410.04	0.207241
$881$	12940.6	0.494870	0.247435	−	0.968905i	$-0.420412\pi$
$881$	12940.6	0.494870	0.247435	+	0.968905i	$0.420412\pi$
$882$	0	0
$883$	−25585.5	−0.975108	−0.487554	−	0.873093i	$-0.662111\pi$
$883$	−25585.5	−0.975108	−0.487554	+	0.873093i	$0.662111\pi$
$884$	0	0
$885$	0	0
$886$	−41928.8	−1.58987
$887$	−3716.46	−0.140684	−0.0703418	−	0.997523i	$-0.522409\pi$
$887$	−3716.46	−0.140684	−0.0703418	+	0.997523i	$0.522409\pi$
$888$	0	0
$889$	16675.7	0.629118
$890$	13462.8	0.507049
$891$	0	0
$892$	25322.0	0.950496
$893$	−52416.7	−1.96423
$894$	0	0
$895$	1555.40	0.0580910
$896$	−3499.58	−0.130483
$897$	0	0
$898$	−70680.3	−2.62654
$899$	6095.52	0.226137
$900$	0	0
$901$	20072.5	0.742187
$902$	−7457.57	−0.275288
$903$	0	0
$904$	−6379.29	−0.234703
$905$	−6517.61	−0.239395
$906$	0	0
$907$	−12960.4	−0.474469	−0.237235	−	0.971452i	$-0.576241\pi$
$907$	−12960.4	−0.474469	−0.237235	+	0.971452i	$0.576241\pi$
$908$	40665.3	1.48626
$909$	0	0
$910$	0	0
$911$	36607.1	1.33134	0.665668	−	0.746248i	$-0.268147\pi$
$911$	36607.1	1.33134	0.665668	+	0.746248i	$0.268147\pi$
$912$	0	0
$913$	22158.8	0.803230
$914$	−58099.3	−2.10258
$915$	0	0
$916$	−11747.4	−0.423740
$917$	292.440	0.0105313
$918$	0	0
$919$	20356.3	0.730676	0.365338	−	0.930875i	$-0.380953\pi$
$919$	20356.3	0.730676	0.365338	+	0.930875i	$0.380953\pi$
$920$	−1496.38	−0.0536243
$921$	0	0
$922$	−12048.6	−0.430370
$923$	0	0
$924$	0	0
$925$	37898.7	1.34714
$926$	8545.61	0.303268
$927$	0	0
$928$	−41589.8	−1.47118
$929$	45069.7	1.59170	0.795849	−	0.605495i	$-0.207025\pi$
$929$	45069.7	1.59170	0.795849	+	0.605495i	$0.207025\pi$
$930$	0	0
$931$	30189.6	1.06275
$932$	30244.9	1.06299
$933$	0	0
$934$	−10984.7	−0.384831
$935$	−2838.65	−0.0992876
$936$	0	0
$937$	−6771.10	−0.236075	−0.118037	−	0.993009i	$-0.537660\pi$
$937$	−6771.10	−0.236075	−0.118037	+	0.993009i	$0.537660\pi$
$938$	−1575.50	−0.0548420
$939$	0	0
$940$	14218.9	0.493372
$941$	−36690.7	−1.27108	−0.635538	−	0.772070i	$-0.719222\pi$
$941$	−36690.7	−1.27108	−0.635538	+	0.772070i	$0.719222\pi$
$942$	0	0
$943$	−6673.51	−0.230455
$944$	36108.8	1.24496
$945$	0	0
$946$	17004.0	0.584406
$947$	50861.2	1.74527	0.872634	−	0.488375i	$-0.162410\pi$
$947$	50861.2	1.74527	0.872634	+	0.488375i	$0.162410\pi$
$948$	0	0
$949$	0	0
$950$	−48272.6	−1.64860
$951$	0	0
$952$	795.219	0.0270727
$953$	−11855.6	−0.402980	−0.201490	−	0.979491i	$-0.564578\pi$
$953$	−11855.6	−0.402980	−0.201490	+	0.979491i	$0.564578\pi$
$954$	0	0
$955$	−12376.4	−0.419361
$956$	42634.5	1.44236
$957$	0	0
$958$	−21525.5	−0.725947
$959$	−1378.04	−0.0464019
$960$	0	0
$961$	−28341.6	−0.951349
$962$	0	0
$963$	0	0
$964$	43067.5	1.43891
$965$	2663.90	0.0888643
$966$	0	0
$967$	40661.7	1.35221	0.676107	−	0.736803i	$-0.263666\pi$
$967$	40661.7	1.35221	0.676107	+	0.736803i	$0.263666\pi$
$968$	1199.58	0.0398304
$969$	0	0
$970$	−22063.9	−0.730339
$971$	−57318.3	−1.89437	−0.947184	−	0.320690i	$-0.896085\pi$
$971$	−57318.3	−1.89437	−0.947184	+	0.320690i	$0.896085\pi$
$972$	0	0
$973$	668.324	0.0220200
$974$	50403.3	1.65814
$975$	0	0
$976$	−38571.7	−1.26501
$977$	3026.73	0.0991134	0.0495567	−	0.998771i	$-0.484219\pi$
$977$	3026.73	0.0991134	0.0495567	+	0.998771i	$0.484219\pi$
$978$	0	0
$979$	34524.8	1.12709
$980$	−8189.43	−0.266941
$981$	0	0
$982$	81032.3	2.63324
$983$	−33942.5	−1.10132	−0.550659	−	0.834730i	$-0.685624\pi$
$983$	−33942.5	−1.10132	−0.550659	+	0.834730i	$0.685624\pi$
$984$	0	0
$985$	−12649.3	−0.409179
$986$	19051.2	0.615327
$987$	0	0
$988$	0	0
$989$	15216.3	0.489231
$990$	0	0
$991$	21637.5	0.693580	0.346790	−	0.937943i	$-0.387272\pi$
$991$	21637.5	0.693580	0.346790	+	0.937943i	$0.387272\pi$
$992$	−9889.02	−0.316509
$993$	0	0
$994$	−8520.95	−0.271900
$995$	−7332.53	−0.233625
$996$	0	0
$997$	19624.6	0.623388	0.311694	−	0.950182i	$-0.399104\pi$
$997$	19624.6	0.623388	0.311694	+	0.950182i	$0.399104\pi$
$998$	48296.3	1.53186
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.z.1.1		4
3.2	odd	2		507.4.a.k.1.4		4
13.2	odd	12		117.4.q.d.82.1		4
13.7	odd	12		117.4.q.d.10.1		4
13.12	even	2	inner	1521.4.a.z.1.4		4
39.2	even	12		39.4.j.b.4.2	✓	4
39.5	even	4		507.4.b.e.337.1		4
39.8	even	4		507.4.b.e.337.4		4
39.20	even	12		39.4.j.b.10.2	yes	4
39.38	odd	2		507.4.a.k.1.1		4
156.59	odd	12		624.4.bv.c.49.1		4
156.119	odd	12		624.4.bv.c.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.b.4.2	✓	4	39.2	even	12
39.4.j.b.10.2	yes	4	39.20	even	12
117.4.q.d.10.1		4	13.7	odd	12
117.4.q.d.82.1		4	13.2	odd	12
507.4.a.k.1.1		4	39.38	odd	2
507.4.a.k.1.4		4	3.2	odd	2
507.4.b.e.337.1		4	39.5	even	4
507.4.b.e.337.4		4	39.8	even	4
624.4.bv.c.49.1		4	156.59	odd	12
624.4.bv.c.433.2		4	156.119	odd	12
1521.4.a.z.1.1		4	1.1	even	1	trivial
1521.4.a.z.1.4		4	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)

\(f(q)\)	\(=\)	\(q-4.12311 q^{2} +9.00000 q^{4} +3.05006 q^{5} -6.68324 q^{7} -4.12311 q^{8} +O(q^{10})\)	\(q-4.12311 q^{2} +9.00000 q^{4} +3.05006 q^{5} -6.68324 q^{7} -4.12311 q^{8} -12.5757 q^{10} -32.2500 q^{11} +27.5557 q^{14} -55.0000 q^{16} +28.8586 q^{17} -101.194 q^{19} +27.4505 q^{20} +132.970 q^{22} +118.990 q^{23} -115.697 q^{25} -60.1492 q^{28} -160.111 q^{29} -38.0705 q^{31} +259.756 q^{32} -118.987 q^{34} -20.3843 q^{35} -327.568 q^{37} +417.233 q^{38} -12.5757 q^{40} -56.0846 q^{41} +127.879 q^{43} -290.250 q^{44} -490.608 q^{46} +517.983 q^{47} -298.334 q^{49} +477.032 q^{50} +695.546 q^{53} -98.3643 q^{55} +27.5557 q^{56} +660.156 q^{58} -656.523 q^{59} +701.304 q^{61} +156.969 q^{62} -631.000 q^{64} -57.1750 q^{67} +259.727 q^{68} +84.0465 q^{70} -309.226 q^{71} -389.711 q^{73} +1350.60 q^{74} -910.744 q^{76} +215.534 q^{77} +901.820 q^{79} -167.753 q^{80} +231.243 q^{82} -687.095 q^{83} +88.0203 q^{85} -527.257 q^{86} +132.970 q^{88} -1070.54 q^{89} +1070.91 q^{92} -2135.70 q^{94} -308.647 q^{95} +1754.48 q^{97} +1230.06 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{4}+O(q^{10}) \) 4 * q + 36 * q^4	\( 4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) 4 * q + 36 * q^4 - 136 * q^10 - 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 - 120 * q^25 - 12 * q^29 + 804 * q^35 + 612 * q^38 - 136 * q^40 + 940 * q^43 + 692 * q^49 + 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 - 2524 * q^64 + 1296 * q^68 + 3060 * q^74 + 2976 * q^77 + 8 * q^79 + 68 * q^82 - 68 * q^88 + 2484 * q^92 - 5372 * q^94 + 108 * q^95

Introduction

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