LMFDB - Embedded newform 1521.4.a.r.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			$-1.56155$ 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)

$f(q)$	$=$	$q-1.56155 q^{2} -5.56155 q^{4} -3.56155 q^{5} +27.1771 q^{7} +21.1771 q^{8} +5.56155 q^{10} +15.2614 q^{11} -42.4384 q^{14} +11.4233 q^{16} -44.5464 q^{17} -23.9697 q^{19} +19.8078 q^{20} -23.8314 q^{22} -122.739 q^{23} -112.315 q^{25} -151.147 q^{28} +219.909 q^{29} -27.0928 q^{31} -187.255 q^{32} +69.5616 q^{34} -96.7926 q^{35} -94.1922 q^{37} +37.4299 q^{38} -75.4233 q^{40} -160.354 q^{41} -151.302 q^{43} -84.8769 q^{44} +191.663 q^{46} +466.948 q^{47} +395.594 q^{49} +175.386 q^{50} +120.847 q^{53} -54.3542 q^{55} +575.531 q^{56} -343.400 q^{58} -439.633 q^{59} -137.305 q^{61} +42.3068 q^{62} +201.022 q^{64} -512.280 q^{67} +247.747 q^{68} +151.147 q^{70} +410.719 q^{71} +308.004 q^{73} +147.086 q^{74} +133.309 q^{76} +414.759 q^{77} -586.462 q^{79} -40.6847 q^{80} +250.401 q^{82} +1354.20 q^{83} +158.654 q^{85} +236.266 q^{86} +323.191 q^{88} +439.882 q^{89} +682.617 q^{92} -729.164 q^{94} +85.3693 q^{95} +1511.27 q^{97} -617.740 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31}+ \cdots - 650 q^{98}+O(q^{100}) $ 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 + 7 * q^10 + 80 * q^11 - 89 * q^14 - 39 * q^16 - 19 * q^17 + 84 * q^19 + 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 - 125 * q^28 + 44 * q^29 + 86 * q^31 - 123 * q^32 + 135 * q^34 - 107 * q^35 - 209 * q^37 + 314 * q^38 - 89 * q^40 - 230 * q^41 + 287 * q^43 - 178 * q^44 + 4 * q^46 + 435 * q^47 + 383 * q^49 - 144 * q^50 + 118 * q^53 - 18 * q^55 + 1015 * q^56 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 68 * q^67 + 211 * q^68 + 125 * q^70 - 131 * q^71 - 456 * q^73 - 147 * q^74 - 22 * q^76 - 762 * q^77 - 1008 * q^79 - 69 * q^80 + 72 * q^82 + 1958 * q^83 + 173 * q^85 + 1359 * q^86 - 1242 * q^88 - 720 * q^89 + 788 * q^92 - 811 * q^94 + 146 * q^95 + 928 * q^97 - 650 * q^98

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 $

$2$	−1.56155	−0.552092	−0.276046	−	0.961144i	$-0.589024\pi$
$2$	−1.56155	−0.552092	−0.276046	+	0.961144i	$0.589024\pi$
$3$	0	0
$3$	0	0
$4$	−5.56155	−0.695194
$5$	−3.56155	−0.318555	−0.159277	−	0.987234i	$-0.550916\pi$
$5$	−3.56155	−0.318555	−0.159277	+	0.987234i	$0.550916\pi$
$6$	0	0
$7$	27.1771	1.46742	0.733712	−	0.679460i	$-0.237786\pi$
$7$	27.1771	1.46742	0.733712	+	0.679460i	$0.237786\pi$
$8$	21.1771	0.935904
$9$	0	0
$10$	5.56155	0.175872
$11$	15.2614	0.418316	0.209158	−	0.977882i	$-0.432928\pi$
$11$	15.2614	0.418316	0.209158	+	0.977882i	$0.432928\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−42.4384	−0.810154
$15$	0	0
$16$	11.4233	0.178489
$17$	−44.5464	−0.635535	−0.317767	−	0.948169i	$-0.602933\pi$
$17$	−44.5464	−0.635535	−0.317767	+	0.948169i	$0.602933\pi$
$18$	0	0
$19$	−23.9697	−0.289422	−0.144711	−	0.989474i	$-0.546225\pi$
$19$	−23.9697	−0.289422	−0.144711	+	0.989474i	$0.546225\pi$
$20$	19.8078	0.221458
$21$	0	0
$22$	−23.8314	−0.230949
$23$	−122.739	−1.11273	−0.556365	−	0.830938i	$-0.687804\pi$
$23$	−122.739	−1.11273	−0.556365	+	0.830938i	$0.687804\pi$
$24$	0	0
$25$	−112.315	−0.898523
$26$	0	0
$27$	0	0
$28$	−151.147	−1.02014
$29$	219.909	1.40814	0.704071	−	0.710130i	$-0.251364\pi$
$29$	219.909	1.40814	0.704071	+	0.710130i	$0.251364\pi$
$30$	0	0
$31$	−27.0928	−0.156968	−0.0784840	−	0.996915i	$-0.525008\pi$
$31$	−27.0928	−0.156968	−0.0784840	+	0.996915i	$0.525008\pi$
$32$	−187.255	−1.03445
$33$	0	0
$34$	69.5616	0.350874
$35$	−96.7926	−0.467455
$36$	0	0
$37$	−94.1922	−0.418516	−0.209258	−	0.977860i	$-0.567105\pi$
$37$	−94.1922	−0.418516	−0.209258	+	0.977860i	$0.567105\pi$
$38$	37.4299	0.159788
$39$	0	0
$40$	−75.4233	−0.298137
$41$	−160.354	−0.610808	−0.305404	−	0.952223i	$-0.598791\pi$
$41$	−160.354	−0.610808	−0.305404	+	0.952223i	$0.598791\pi$
$42$	0	0
$43$	−151.302	−0.536589	−0.268295	−	0.963337i	$-0.586460\pi$
$43$	−151.302	−0.536589	−0.268295	+	0.963337i	$0.586460\pi$
$44$	−84.8769	−0.290811
$45$	0	0
$46$	191.663	0.614329
$47$	466.948	1.44918	0.724589	−	0.689181i	$-0.242030\pi$
$47$	466.948	1.44918	0.724589	+	0.689181i	$0.242030\pi$
$48$	0	0
$49$	395.594	1.15333
$50$	175.386	0.496067
$51$	0	0
$52$	0	0
$53$	120.847	0.313199	0.156600	−	0.987662i	$-0.449947\pi$
$53$	120.847	0.313199	0.156600	+	0.987662i	$0.449947\pi$
$54$	0	0
$55$	−54.3542	−0.133257
$56$	575.531	1.37337
$57$	0	0
$58$	−343.400	−0.777424
$59$	−439.633	−0.970090	−0.485045	−	0.874489i	$-0.661197\pi$
$59$	−439.633	−0.970090	−0.485045	+	0.874489i	$0.661197\pi$
$60$	0	0
$61$	−137.305	−0.288198	−0.144099	−	0.989563i	$-0.546028\pi$
$61$	−137.305	−0.288198	−0.144099	+	0.989563i	$0.546028\pi$
$62$	42.3068	0.0866609
$63$	0	0
$64$	201.022	0.392621
$65$	0	0
$66$	0	0
$67$	−512.280	−0.934104	−0.467052	−	0.884230i	$-0.654684\pi$
$67$	−512.280	−0.934104	−0.467052	+	0.884230i	$0.654684\pi$
$68$	247.747	0.441820
$69$	0	0
$70$	151.147	0.258078
$71$	410.719	0.686526	0.343263	−	0.939239i	$-0.388468\pi$
$71$	410.719	0.686526	0.343263	+	0.939239i	$0.388468\pi$
$72$	0	0
$73$	308.004	0.493823	0.246912	−	0.969038i	$-0.420584\pi$
$73$	308.004	0.493823	0.246912	+	0.969038i	$0.420584\pi$
$74$	147.086	0.231060
$75$	0	0
$76$	133.309	0.201205
$77$	414.759	0.613847
$78$	0	0
$79$	−586.462	−0.835217	−0.417608	−	0.908627i	$-0.637132\pi$
$79$	−586.462	−0.835217	−0.417608	+	0.908627i	$0.637132\pi$
$80$	−40.6847	−0.0568585
$81$	0	0
$82$	250.401	0.337222
$83$	1354.20	1.79088	0.895440	−	0.445182i	$-0.146861\pi$
$83$	1354.20	1.79088	0.895440	+	0.445182i	$0.146861\pi$
$84$	0	0
$85$	158.654	0.202453
$86$	236.266	0.296247
$87$	0	0
$88$	323.191	0.391503
$89$	439.882	0.523904	0.261952	−	0.965081i	$-0.415634\pi$
$89$	439.882	0.523904	0.261952	+	0.965081i	$0.415634\pi$
$90$	0	0
$91$	0	0
$92$	682.617	0.773563
$93$	0	0
$94$	−729.164	−0.800080
$95$	85.3693	0.0921969
$96$	0	0
$97$	1511.27	1.58192	0.790959	−	0.611869i	$-0.209582\pi$
$97$	1511.27	1.58192	0.790959	+	0.611869i	$0.209582\pi$
$98$	−617.740	−0.636747
$99$	0	0
$100$	624.648	0.624648
$101$	−336.260	−0.331278	−0.165639	−	0.986186i	$-0.552969\pi$
$101$	−336.260	−0.331278	−0.165639	+	0.986186i	$0.552969\pi$
$102$	0	0
$103$	322.712	0.308716	0.154358	−	0.988015i	$-0.450669\pi$
$103$	322.712	0.308716	0.154358	+	0.988015i	$0.450669\pi$
$104$	0	0
$105$	0	0
$106$	−188.708	−0.172915
$107$	−1434.62	−1.29617	−0.648083	−	0.761570i	$-0.724429\pi$
$107$	−1434.62	−1.29617	−0.648083	+	0.761570i	$0.724429\pi$
$108$	0	0
$109$	−849.147	−0.746179	−0.373089	−	0.927795i	$-0.621702\pi$
$109$	−849.147	−0.746179	−0.373089	+	0.927795i	$0.621702\pi$
$110$	84.8769	0.0735699
$111$	0	0
$112$	310.452	0.261919
$113$	−1614.53	−1.34409	−0.672044	−	0.740511i	$-0.734583\pi$
$113$	−1614.53	−1.34409	−0.672044	+	0.740511i	$0.734583\pi$
$114$	0	0
$115$	437.140	0.354465
$116$	−1223.04	−0.978931
$117$	0	0
$118$	686.509	0.535579
$119$	−1210.64	−0.932599
$120$	0	0
$121$	−1098.09	−0.825012
$122$	214.409	0.159112
$123$	0	0
$124$	150.678	0.109123
$125$	845.211	0.604784
$126$	0	0
$127$	865.174	0.604502	0.302251	−	0.953228i	$-0.402262\pi$
$127$	865.174	0.604502	0.302251	+	0.953228i	$0.402262\pi$
$128$	1184.13	0.817683
$129$	0	0
$130$	0	0
$131$	281.400	0.187680	0.0938400	−	0.995587i	$-0.470086\pi$
$131$	281.400	0.187680	0.0938400	+	0.995587i	$0.470086\pi$
$132$	0	0
$133$	−651.426	−0.424705
$134$	799.953	0.515712
$135$	0	0
$136$	−943.363	−0.594799
$137$	−2641.43	−1.64725	−0.823624	−	0.567137i	$-0.808051\pi$
$137$	−2641.43	−1.64725	−0.823624	+	0.567137i	$0.808051\pi$
$138$	0	0
$139$	−1998.64	−1.21958	−0.609791	−	0.792562i	$-0.708747\pi$
$139$	−1998.64	−1.21958	−0.609791	+	0.792562i	$0.708747\pi$
$140$	538.317	0.324972
$141$	0	0
$142$	−641.359	−0.379026
$143$	0	0
$144$	0	0
$145$	−783.218	−0.448570
$146$	−480.964	−0.272636
$147$	0	0
$148$	523.855	0.290950
$149$	−1752.98	−0.963824	−0.481912	−	0.876220i	$-0.660058\pi$
$149$	−1752.98	−0.963824	−0.481912	+	0.876220i	$0.660058\pi$
$150$	0	0
$151$	2794.64	1.50613	0.753063	−	0.657949i	$-0.228576\pi$
$151$	2794.64	1.50613	0.753063	+	0.657949i	$0.228576\pi$
$152$	−507.608	−0.270871
$153$	0	0
$154$	−647.669	−0.338900
$155$	96.4924	0.0500030
$156$	0	0
$157$	3244.87	1.64949	0.824743	−	0.565508i	$-0.191320\pi$
$157$	3244.87	1.64949	0.824743	+	0.565508i	$0.191320\pi$
$158$	915.792	0.461117
$159$	0	0
$160$	666.918	0.329528
$161$	−3335.68	−1.63285
$162$	0	0
$163$	−3281.47	−1.57684	−0.788418	−	0.615139i	$-0.789100\pi$
$163$	−3281.47	−1.57684	−0.788418	+	0.615139i	$0.789100\pi$
$164$	891.818	0.424630
$165$	0	0
$166$	−2114.66	−0.988731
$167$	−3126.52	−1.44873	−0.724364	−	0.689418i	$-0.757866\pi$
$167$	−3126.52	−1.44873	−0.724364	+	0.689418i	$0.757866\pi$
$168$	0	0
$169$	0	0
$170$	−247.747	−0.111773
$171$	0	0
$172$	841.474	0.373034
$173$	−97.5698	−0.0428792	−0.0214396	−	0.999770i	$-0.506825\pi$
$173$	−97.5698	−0.0428792	−0.0214396	+	0.999770i	$0.506825\pi$
$174$	0	0
$175$	−3052.40	−1.31851
$176$	174.335	0.0746648
$177$	0	0
$178$	−686.900	−0.289243
$179$	34.7150	0.0144956	0.00724782	−	0.999974i	$-0.497693\pi$
$179$	34.7150	0.0144956	0.00724782	+	0.999974i	$0.497693\pi$
$180$	0	0
$181$	−1229.35	−0.504843	−0.252422	−	0.967617i	$-0.581227\pi$
$181$	−1229.35	−0.504843	−0.252422	+	0.967617i	$0.581227\pi$
$182$	0	0
$183$	0	0
$184$	−2599.25	−1.04141
$185$	335.471	0.133320
$186$	0	0
$187$	−679.839	−0.265854
$188$	−2596.96	−1.00746
$189$	0	0
$190$	−133.309	−0.0509012
$191$	−4280.80	−1.62172	−0.810858	−	0.585243i	$-0.800999\pi$
$191$	−4280.80	−1.62172	−0.810858	+	0.585243i	$0.800999\pi$
$192$	0	0
$193$	−472.320	−0.176157	−0.0880786	−	0.996114i	$-0.528073\pi$
$193$	−472.320	−0.176157	−0.0880786	+	0.996114i	$0.528073\pi$
$194$	−2359.93	−0.873365
$195$	0	0
$196$	−2200.12	−0.801791
$197$	−4484.37	−1.62182	−0.810908	−	0.585173i	$-0.801026\pi$
$197$	−4484.37	−1.62182	−0.810908	+	0.585173i	$0.801026\pi$
$198$	0	0
$199$	−366.240	−0.130463	−0.0652314	−	0.997870i	$-0.520779\pi$
$199$	−366.240	−0.130463	−0.0652314	+	0.997870i	$0.520779\pi$
$200$	−2378.51	−0.840931
$201$	0	0
$202$	525.087	0.182896
$203$	5976.49	2.06634
$204$	0	0
$205$	571.110	0.194576
$206$	−503.932	−0.170440
$207$	0	0
$208$	0	0
$209$	−365.810	−0.121070
$210$	0	0
$211$	2122.55	0.692524	0.346262	−	0.938138i	$-0.387451\pi$
$211$	2122.55	0.692524	0.346262	+	0.938138i	$0.387451\pi$
$212$	−672.095	−0.217734
$213$	0	0
$214$	2240.23	0.715603
$215$	538.870	0.170933
$216$	0	0
$217$	−736.303	−0.230339
$218$	1325.99	0.411960
$219$	0	0
$220$	302.294	0.0926392
$221$	0	0
$222$	0	0
$223$	5926.42	1.77965	0.889826	−	0.456301i	$-0.150826\pi$
$223$	5926.42	1.77965	0.889826	+	0.456301i	$0.150826\pi$
$224$	−5089.04	−1.51797
$225$	0	0
$226$	2521.17	0.742060
$227$	−895.661	−0.261881	−0.130941	−	0.991390i	$-0.541800\pi$
$227$	−895.661	−0.261881	−0.130941	+	0.991390i	$0.541800\pi$
$228$	0	0
$229$	−627.717	−0.181138	−0.0905692	−	0.995890i	$-0.528869\pi$
$229$	−627.717	−0.181138	−0.0905692	+	0.995890i	$0.528869\pi$
$230$	−682.617	−0.195698
$231$	0	0
$232$	4657.03	1.31788
$233$	−2303.72	−0.647734	−0.323867	−	0.946103i	$-0.604983\pi$
$233$	−2303.72	−0.647734	−0.323867	+	0.946103i	$0.604983\pi$
$234$	0	0
$235$	−1663.06	−0.461643
$236$	2445.04	0.674401
$237$	0	0
$238$	1890.48	0.514881
$239$	544.622	0.147400	0.0737001	−	0.997280i	$-0.476519\pi$
$239$	544.622	0.147400	0.0737001	+	0.997280i	$0.476519\pi$
$240$	0	0
$241$	−5426.10	−1.45031	−0.725157	−	0.688584i	$-0.758233\pi$
$241$	−5426.10	−1.45031	−0.725157	+	0.688584i	$0.758233\pi$
$242$	1714.73	0.455483
$243$	0	0
$244$	763.629	0.200354
$245$	−1408.93	−0.367400
$246$	0	0
$247$	0	0
$248$	−573.746	−0.146907
$249$	0	0
$250$	−1319.84	−0.333897
$251$	5221.22	1.31299	0.656494	−	0.754331i	$-0.272039\pi$
$251$	5221.22	1.31299	0.656494	+	0.754331i	$0.272039\pi$
$252$	0	0
$253$	−1873.16	−0.465472
$254$	−1351.02	−0.333741
$255$	0	0
$256$	−3457.26	−0.844057
$257$	−658.206	−0.159758	−0.0798789	−	0.996805i	$-0.525453\pi$
$257$	−658.206	−0.159758	−0.0798789	+	0.996805i	$0.525453\pi$
$258$	0	0
$259$	−2559.87	−0.614141
$260$	0	0
$261$	0	0
$262$	−439.422	−0.103617
$263$	−3246.45	−0.761160	−0.380580	−	0.924748i	$-0.624276\pi$
$263$	−3246.45	−0.761160	−0.380580	+	0.924748i	$0.624276\pi$
$264$	0	0
$265$	−430.401	−0.0997711
$266$	1017.24	0.234477
$267$	0	0
$268$	2849.07	0.649384
$269$	2585.80	0.586093	0.293047	−	0.956098i	$-0.405331\pi$
$269$	2585.80	0.586093	0.293047	+	0.956098i	$0.405331\pi$
$270$	0	0
$271$	−988.933	−0.221673	−0.110836	−	0.993839i	$-0.535353\pi$
$271$	−988.933	−0.221673	−0.110836	+	0.993839i	$0.535353\pi$
$272$	−508.867	−0.113436
$273$	0	0
$274$	4124.74	0.909433
$275$	−1714.09	−0.375866
$276$	0	0
$277$	8142.40	1.76617	0.883086	−	0.469211i	$-0.155462\pi$
$277$	8142.40	1.76617	0.883086	+	0.469211i	$0.155462\pi$
$278$	3120.97	0.673322
$279$	0	0
$280$	−2049.78	−0.437493
$281$	1534.21	0.325705	0.162853	−	0.986650i	$-0.447930\pi$
$281$	1534.21	0.325705	0.162853	+	0.986650i	$0.447930\pi$
$282$	0	0
$283$	−6965.00	−1.46299	−0.731495	−	0.681847i	$-0.761177\pi$
$283$	−6965.00	−1.46299	−0.731495	+	0.681847i	$0.761177\pi$
$284$	−2284.23	−0.477269
$285$	0	0
$286$	0	0
$287$	−4357.96	−0.896314
$288$	0	0
$289$	−2928.62	−0.596096
$290$	1223.04	0.247652
$291$	0	0
$292$	−1712.98	−0.343303
$293$	640.029	0.127614	0.0638070	−	0.997962i	$-0.479676\pi$
$293$	640.029	0.127614	0.0638070	+	0.997962i	$0.479676\pi$
$294$	0	0
$295$	1565.77	0.309027
$296$	−1994.72	−0.391691
$297$	0	0
$298$	2737.37	0.532120
$299$	0	0
$300$	0	0
$301$	−4111.95	−0.787404
$302$	−4363.99	−0.831520
$303$	0	0
$304$	−273.813	−0.0516587
$305$	489.019	0.0918070
$306$	0	0
$307$	100.406	0.0186660	0.00933299	−	0.999956i	$-0.497029\pi$
$307$	100.406	0.0186660	0.00933299	+	0.999956i	$0.497029\pi$
$308$	−2306.71	−0.426743
$309$	0	0
$310$	−150.678	−0.0276062
$311$	3878.92	0.707245	0.353623	−	0.935388i	$-0.384950\pi$
$311$	3878.92	0.707245	0.353623	+	0.935388i	$0.384950\pi$
$312$	0	0
$313$	−3789.39	−0.684311	−0.342155	−	0.939643i	$-0.611157\pi$
$313$	−3789.39	−0.684311	−0.342155	+	0.939643i	$0.611157\pi$
$314$	−5067.04	−0.910668
$315$	0	0
$316$	3261.64	0.580638
$317$	4406.81	0.780791	0.390396	−	0.920647i	$-0.372338\pi$
$317$	4406.81	0.780791	0.390396	+	0.920647i	$0.372338\pi$
$318$	0	0
$319$	3356.11	0.589048
$320$	−715.950	−0.125071
$321$	0	0
$322$	5208.84	0.901482
$323$	1067.76	0.183938
$324$	0	0
$325$	0	0
$326$	5124.19	0.870559
$327$	0	0
$328$	−3395.83	−0.571657
$329$	12690.3	2.12656
$330$	0	0
$331$	4131.49	0.686064	0.343032	−	0.939324i	$-0.388546\pi$
$331$	4131.49	0.686064	0.343032	+	0.939324i	$0.388546\pi$
$332$	−7531.47	−1.24501
$333$	0	0
$334$	4882.23	0.799831
$335$	1824.51	0.297564
$336$	0	0
$337$	−4560.82	−0.737221	−0.368611	−	0.929584i	$-0.620166\pi$
$337$	−4560.82	−0.737221	−0.368611	+	0.929584i	$0.620166\pi$
$338$	0	0
$339$	0	0
$340$	−882.365	−0.140744
$341$	−413.473	−0.0656622
$342$	0	0
$343$	1429.34	0.225007
$344$	−3204.14	−0.502196
$345$	0	0
$346$	152.360	0.0236733
$347$	−10069.4	−1.55779	−0.778896	−	0.627153i	$-0.784220\pi$
$347$	−10069.4	−1.55779	−0.778896	+	0.627153i	$0.784220\pi$
$348$	0	0
$349$	−5879.32	−0.901757	−0.450878	−	0.892585i	$-0.648889\pi$
$349$	−5879.32	−0.901757	−0.450878	+	0.892585i	$0.648889\pi$
$350$	4766.49	0.727942
$351$	0	0
$352$	−2857.76	−0.432725
$353$	−9142.56	−1.37850	−0.689249	−	0.724525i	$-0.742059\pi$
$353$	−9142.56	−1.37850	−0.689249	+	0.724525i	$0.742059\pi$
$354$	0	0
$355$	−1462.80	−0.218696
$356$	−2446.43	−0.364215
$357$	0	0
$358$	−54.2093	−0.00800293
$359$	−2754.32	−0.404924	−0.202462	−	0.979290i	$-0.564894\pi$
$359$	−2754.32	−0.404924	−0.202462	+	0.979290i	$0.564894\pi$
$360$	0	0
$361$	−6284.45	−0.916235
$362$	1919.69	0.278720
$363$	0	0
$364$	0	0
$365$	−1096.97	−0.157310
$366$	0	0
$367$	3040.19	0.432416	0.216208	−	0.976347i	$-0.430631\pi$
$367$	3040.19	0.432416	0.216208	+	0.976347i	$0.430631\pi$
$368$	−1402.08	−0.198610
$369$	0	0
$370$	−523.855	−0.0736052
$371$	3284.26	0.459596
$372$	0	0
$373$	−5384.72	−0.747481	−0.373740	−	0.927533i	$-0.621925\pi$
$373$	−5384.72	−0.747481	−0.373740	+	0.927533i	$0.621925\pi$
$374$	1061.60	0.146776
$375$	0	0
$376$	9888.59	1.35629
$377$	0	0
$378$	0	0
$379$	3424.27	0.464097	0.232049	−	0.972704i	$-0.425457\pi$
$379$	3424.27	0.464097	0.232049	+	0.972704i	$0.425457\pi$
$380$	−474.786	−0.0640948
$381$	0	0
$382$	6684.69	0.895336
$383$	−382.985	−0.0510956	−0.0255478	−	0.999674i	$-0.508133\pi$
$383$	−382.985	−0.0510956	−0.0255478	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	−1477.19	−0.195544
$386$	737.553	0.0972551
$387$	0	0
$388$	−8405.00	−1.09974
$389$	−8588.34	−1.11940	−0.559699	−	0.828696i	$-0.689083\pi$
$389$	−8588.34	−1.11940	−0.559699	+	0.828696i	$0.689083\pi$
$390$	0	0
$391$	5467.56	0.707178
$392$	8377.52	1.07941
$393$	0	0
$394$	7002.57	0.895392
$395$	2088.72	0.266063
$396$	0	0
$397$	7239.16	0.915171	0.457586	−	0.889166i	$-0.348714\pi$
$397$	7239.16	0.915171	0.457586	+	0.889166i	$0.348714\pi$
$398$	571.904	0.0720275
$399$	0	0
$400$	−1283.01	−0.160376
$401$	4269.62	0.531708	0.265854	−	0.964013i	$-0.414346\pi$
$401$	4269.62	0.531708	0.265854	+	0.964013i	$0.414346\pi$
$402$	0	0
$403$	0	0
$404$	1870.12	0.230302
$405$	0	0
$406$	−9332.60	−1.14081
$407$	−1437.50	−0.175072
$408$	0	0
$409$	−13562.5	−1.63967	−0.819834	−	0.572602i	$-0.805934\pi$
$409$	−13562.5	−1.63967	−0.819834	+	0.572602i	$0.805934\pi$
$410$	−891.818	−0.107424
$411$	0	0
$412$	−1794.78	−0.214618
$413$	−11947.9	−1.42353
$414$	0	0
$415$	−4823.06	−0.570494
$416$	0	0
$417$	0	0
$418$	571.232	0.0668418
$419$	14576.9	1.69959	0.849794	−	0.527114i	$-0.176726\pi$
$419$	14576.9	1.69959	0.849794	+	0.527114i	$0.176726\pi$
$420$	0	0
$421$	−15848.4	−1.83469	−0.917343	−	0.398099i	$-0.869670\pi$
$421$	−15848.4	−1.83469	−0.917343	+	0.398099i	$0.869670\pi$
$422$	−3314.48	−0.382337
$423$	0	0
$424$	2559.18	0.293124
$425$	5003.24	0.571042
$426$	0	0
$427$	−3731.55	−0.422909
$428$	7978.70	0.901087
$429$	0	0
$430$	−841.474	−0.0943709
$431$	10694.7	1.19524	0.597618	−	0.801781i	$-0.296114\pi$
$431$	10694.7	1.19524	0.597618	+	0.801781i	$0.296114\pi$
$432$	0	0
$433$	−16079.0	−1.78454	−0.892272	−	0.451498i	$-0.850890\pi$
$433$	−16079.0	−1.78454	−0.892272	+	0.451498i	$0.850890\pi$
$434$	1149.78	0.127168
$435$	0	0
$436$	4722.57	0.518739
$437$	2942.01	0.322049
$438$	0	0
$439$	6035.80	0.656203	0.328101	−	0.944643i	$-0.393591\pi$
$439$	6035.80	0.656203	0.328101	+	0.944643i	$0.393591\pi$
$440$	−1151.06	−0.124715
$441$	0	0
$442$	0	0
$443$	−10201.3	−1.09409	−0.547043	−	0.837105i	$-0.684247\pi$
$443$	−10201.3	−1.09409	−0.547043	+	0.837105i	$0.684247\pi$
$444$	0	0
$445$	−1566.66	−0.166892
$446$	−9254.41	−0.982532
$447$	0	0
$448$	5463.19	0.576141
$449$	−5822.54	−0.611988	−0.305994	−	0.952033i	$-0.598989\pi$
$449$	−5822.54	−0.611988	−0.305994	+	0.952033i	$0.598989\pi$
$450$	0	0
$451$	−2447.22	−0.255511
$452$	8979.27	0.934402
$453$	0	0
$454$	1398.62	0.144583
$455$	0	0
$456$	0	0
$457$	−4621.60	−0.473062	−0.236531	−	0.971624i	$-0.576010\pi$
$457$	−4621.60	−0.473062	−0.236531	+	0.971624i	$0.576010\pi$
$458$	980.213	0.100005
$459$	0	0
$460$	−2431.18	−0.246422
$461$	5127.77	0.518056	0.259028	−	0.965870i	$-0.416598\pi$
$461$	5127.77	0.518056	0.259028	+	0.965870i	$0.416598\pi$
$462$	0	0
$463$	−6486.27	−0.651064	−0.325532	−	0.945531i	$-0.605543\pi$
$463$	−6486.27	−0.651064	−0.325532	+	0.945531i	$0.605543\pi$
$464$	2512.09	0.251338
$465$	0	0
$466$	3597.39	0.357609
$467$	−12978.0	−1.28598	−0.642990	−	0.765875i	$-0.722306\pi$
$467$	−12978.0	−1.28598	−0.642990	+	0.765875i	$0.722306\pi$
$468$	0	0
$469$	−13922.3	−1.37073
$470$	2596.96	0.254869
$471$	0	0
$472$	−9310.13	−0.907910
$473$	−2309.08	−0.224464
$474$	0	0
$475$	2692.16	0.260053
$476$	6733.04	0.648337
$477$	0	0
$478$	−850.456	−0.0813786
$479$	−5808.96	−0.554109	−0.277055	−	0.960854i	$-0.589358\pi$
$479$	−5808.96	−0.554109	−0.277055	+	0.960854i	$0.589358\pi$
$480$	0	0
$481$	0	0
$482$	8473.14	0.800707
$483$	0	0
$484$	6107.09	0.573543
$485$	−5382.46	−0.503928
$486$	0	0
$487$	5387.14	0.501262	0.250631	−	0.968083i	$-0.419362\pi$
$487$	5387.14	0.501262	0.250631	+	0.968083i	$0.419362\pi$
$488$	−2907.72	−0.269726
$489$	0	0
$490$	2200.12	0.202839
$491$	−15259.1	−1.40251	−0.701255	−	0.712911i	$-0.747376\pi$
$491$	−15259.1	−1.40251	−0.701255	+	0.712911i	$0.747376\pi$
$492$	0	0
$493$	−9796.16	−0.894922
$494$	0	0
$495$	0	0
$496$	−309.489	−0.0280171
$497$	11162.1	1.00742
$498$	0	0
$499$	−1856.04	−0.166509	−0.0832544	−	0.996528i	$-0.526531\pi$
$499$	−1856.04	−0.166509	−0.0832544	+	0.996528i	$0.526531\pi$
$500$	−4700.69	−0.420442
$501$	0	0
$502$	−8153.20	−0.724891
$503$	−1049.46	−0.0930283	−0.0465142	−	0.998918i	$-0.514811\pi$
$503$	−1049.46	−0.0930283	−0.0465142	+	0.998918i	$0.514811\pi$
$504$	0	0
$505$	1197.61	0.105530
$506$	2925.04	0.256984
$507$	0	0
$508$	−4811.71	−0.420246
$509$	−551.106	−0.0479909	−0.0239954	−	0.999712i	$-0.507639\pi$
$509$	−551.106	−0.0479909	−0.0239954	+	0.999712i	$0.507639\pi$
$510$	0	0
$511$	8370.64	0.724649
$512$	−4074.36	−0.351686
$513$	0	0
$514$	1027.82	0.0882010
$515$	−1149.36	−0.0983431
$516$	0	0
$517$	7126.26	0.606214
$518$	3997.37	0.339063
$519$	0	0
$520$	0	0
$521$	8995.30	0.756413	0.378206	−	0.925721i	$-0.376541\pi$
$521$	8995.30	0.756413	0.378206	+	0.925721i	$0.376541\pi$
$522$	0	0
$523$	2663.91	0.222724	0.111362	−	0.993780i	$-0.464479\pi$
$523$	2663.91	0.222724	0.111362	+	0.993780i	$0.464479\pi$
$524$	−1565.02	−0.130474
$525$	0	0
$526$	5069.51	0.420230
$527$	1206.89	0.0997586
$528$	0	0
$529$	2897.77	0.238167
$530$	672.095	0.0550829
$531$	0	0
$532$	3622.94	0.295253
$533$	0	0
$534$	0	0
$535$	5109.47	0.412900
$536$	−10848.6	−0.874232
$537$	0	0
$538$	−4037.86	−0.323577
$539$	6037.30	0.482458
$540$	0	0
$541$	6169.23	0.490270	0.245135	−	0.969489i	$-0.421168\pi$
$541$	6169.23	0.490270	0.245135	+	0.969489i	$0.421168\pi$
$542$	1544.27	0.122384
$543$	0	0
$544$	8341.52	0.657426
$545$	3024.28	0.237699
$546$	0	0
$547$	5140.42	0.401807	0.200904	−	0.979611i	$-0.435612\pi$
$547$	5140.42	0.401807	0.200904	+	0.979611i	$0.435612\pi$
$548$	14690.5	1.14516
$549$	0	0
$550$	2676.64	0.207513
$551$	−5271.15	−0.407547
$552$	0	0
$553$	−15938.3	−1.22562
$554$	−12714.8	−0.975090
$555$	0	0
$556$	11115.5	0.847847
$557$	2778.56	0.211367	0.105683	−	0.994400i	$-0.466297\pi$
$557$	2778.56	0.211367	0.105683	+	0.994400i	$0.466297\pi$
$558$	0	0
$559$	0	0
$560$	−1105.69	−0.0834356
$561$	0	0
$562$	−2395.75	−0.179819
$563$	−4906.14	−0.367263	−0.183632	−	0.982995i	$-0.558785\pi$
$563$	−4906.14	−0.367263	−0.183632	+	0.982995i	$0.558785\pi$
$564$	0	0
$565$	5750.22	0.428166
$566$	10876.2	0.807706
$567$	0	0
$568$	8697.82	0.642522
$569$	9363.15	0.689849	0.344924	−	0.938631i	$-0.387905\pi$
$569$	9363.15	0.689849	0.344924	+	0.938631i	$0.387905\pi$
$570$	0	0
$571$	7199.32	0.527640	0.263820	−	0.964572i	$-0.415018\pi$
$571$	7199.32	0.527640	0.263820	+	0.964572i	$0.415018\pi$
$572$	0	0
$573$	0	0
$574$	6805.18	0.494848
$575$	13785.4	0.999813
$576$	0	0
$577$	11449.6	0.826086	0.413043	−	0.910711i	$-0.364466\pi$
$577$	11449.6	0.826086	0.413043	+	0.910711i	$0.364466\pi$
$578$	4573.19	0.329100
$579$	0	0
$580$	4355.91	0.311843
$581$	36803.3	2.62798
$582$	0	0
$583$	1844.28	0.131016
$584$	6522.62	0.462171
$585$	0	0
$586$	−999.439	−0.0704547
$587$	−5439.39	−0.382466	−0.191233	−	0.981545i	$-0.561249\pi$
$587$	−5439.39	−0.382466	−0.191233	+	0.981545i	$0.561249\pi$
$588$	0	0
$589$	649.406	0.0454301
$590$	−2445.04	−0.170611
$591$	0	0
$592$	−1075.99	−0.0747006
$593$	−28405.8	−1.96709	−0.983547	−	0.180651i	$-0.942180\pi$
$593$	−28405.8	−1.96709	−0.983547	+	0.180651i	$0.942180\pi$
$594$	0	0
$595$	4311.76	0.297084
$596$	9749.30	0.670045
$597$	0	0
$598$	0	0
$599$	10482.3	0.715020	0.357510	−	0.933909i	$-0.383626\pi$
$599$	10482.3	0.715020	0.357510	+	0.933909i	$0.383626\pi$
$600$	0	0
$601$	3199.54	0.217158	0.108579	−	0.994088i	$-0.465370\pi$
$601$	3199.54	0.217158	0.108579	+	0.994088i	$0.465370\pi$
$602$	6421.02	0.434720
$603$	0	0
$604$	−15542.6	−1.04705
$605$	3910.91	0.262812
$606$	0	0
$607$	11342.8	0.758468	0.379234	−	0.925301i	$-0.376188\pi$
$607$	11342.8	0.758468	0.379234	+	0.925301i	$0.376188\pi$
$608$	4488.44	0.299392
$609$	0	0
$610$	−763.629	−0.0506859
$611$	0	0
$612$	0	0
$613$	−14385.4	−0.947831	−0.473916	−	0.880570i	$-0.657160\pi$
$613$	−14385.4	−0.947831	−0.473916	+	0.880570i	$0.657160\pi$
$614$	−156.789	−0.0103053
$615$	0	0
$616$	8783.39	0.574502
$617$	22056.8	1.43918	0.719588	−	0.694401i	$-0.244331\pi$
$617$	22056.8	1.43918	0.719588	+	0.694401i	$0.244331\pi$
$618$	0	0
$619$	−13621.4	−0.884477	−0.442238	−	0.896898i	$-0.645815\pi$
$619$	−13621.4	−0.884477	−0.442238	+	0.896898i	$0.645815\pi$
$620$	−536.648	−0.0347618
$621$	0	0
$622$	−6057.14	−0.390465
$623$	11954.7	0.768789
$624$	0	0
$625$	11029.2	0.705866
$626$	5917.34	0.377803
$627$	0	0
$628$	−18046.5	−1.14671
$629$	4195.92	0.265982
$630$	0	0
$631$	18737.5	1.18214	0.591068	−	0.806622i	$-0.298707\pi$
$631$	18737.5	1.18214	0.591068	+	0.806622i	$0.298707\pi$
$632$	−12419.6	−0.781683
$633$	0	0
$634$	−6881.46	−0.431069
$635$	−3081.36	−0.192567
$636$	0	0
$637$	0	0
$638$	−5240.75	−0.325209
$639$	0	0
$640$	−4217.35	−0.260477
$641$	−29798.7	−1.83616	−0.918081	−	0.396394i	$-0.870261\pi$
$641$	−29798.7	−1.83616	−0.918081	+	0.396394i	$0.870261\pi$
$642$	0	0
$643$	22983.5	1.40961	0.704807	−	0.709399i	$-0.251034\pi$
$643$	22983.5	1.40961	0.704807	+	0.709399i	$0.251034\pi$
$644$	18551.5	1.13515
$645$	0	0
$646$	−1667.37	−0.101551
$647$	24905.4	1.51334	0.756672	−	0.653794i	$-0.226824\pi$
$647$	24905.4	1.51334	0.756672	+	0.653794i	$0.226824\pi$
$648$	0	0
$649$	−6709.39	−0.405804
$650$	0	0
$651$	0	0
$652$	18250.1	1.09621
$653$	−10077.8	−0.603946	−0.301973	−	0.953316i	$-0.597645\pi$
$653$	−10077.8	−0.603946	−0.301973	+	0.953316i	$0.597645\pi$
$654$	0	0
$655$	−1002.22	−0.0597864
$656$	−1831.77	−0.109022
$657$	0	0
$658$	−19816.5	−1.17406
$659$	−12334.6	−0.729116	−0.364558	−	0.931181i	$-0.618780\pi$
$659$	−12334.6	−0.729116	−0.364558	+	0.931181i	$0.618780\pi$
$660$	0	0
$661$	12749.1	0.750202	0.375101	−	0.926984i	$-0.377608\pi$
$661$	12749.1	0.750202	0.375101	+	0.926984i	$0.377608\pi$
$662$	−6451.54	−0.378771
$663$	0	0
$664$	28678.1	1.67609
$665$	2320.09	0.135292
$666$	0	0
$667$	−26991.3	−1.56688
$668$	17388.3	1.00715
$669$	0	0
$670$	−2849.07	−0.164283
$671$	−2095.46	−0.120558
$672$	0	0
$673$	−13618.2	−0.780007	−0.390004	−	0.920813i	$-0.627526\pi$
$673$	−13618.2	−0.780007	−0.390004	+	0.920813i	$0.627526\pi$
$674$	7121.96	0.407014
$675$	0	0
$676$	0	0
$677$	−9655.67	−0.548150	−0.274075	−	0.961708i	$-0.588372\pi$
$677$	−9655.67	−0.548150	−0.274075	+	0.961708i	$0.588372\pi$
$678$	0	0
$679$	41071.9	2.32135
$680$	3359.84	0.189476
$681$	0	0
$682$	645.660	0.0362516
$683$	16316.8	0.914119	0.457060	−	0.889436i	$-0.348903\pi$
$683$	16316.8	0.914119	0.457060	+	0.889436i	$0.348903\pi$
$684$	0	0
$685$	9407.61	0.524739
$686$	−2232.00	−0.124225
$687$	0	0
$688$	−1728.37	−0.0957753
$689$	0	0
$690$	0	0
$691$	−2350.84	−0.129421	−0.0647106	−	0.997904i	$-0.520612\pi$
$691$	−2350.84	−0.129421	−0.0647106	+	0.997904i	$0.520612\pi$
$692$	542.640	0.0298093
$693$	0	0
$694$	15723.9	0.860045
$695$	7118.24	0.388504
$696$	0	0
$697$	7143.20	0.388189
$698$	9180.88	0.497853
$699$	0	0
$700$	16976.1	0.916623
$701$	8076.90	0.435179	0.217589	−	0.976040i	$-0.430181\pi$
$701$	8076.90	0.435179	0.217589	+	0.976040i	$0.430181\pi$
$702$	0	0
$703$	2257.76	0.121128
$704$	3067.87	0.164239
$705$	0	0
$706$	14276.6	0.761058
$707$	−9138.55	−0.486125
$708$	0	0
$709$	13624.9	0.721712	0.360856	−	0.932622i	$-0.382485\pi$
$709$	13624.9	0.721712	0.360856	+	0.932622i	$0.382485\pi$
$710$	2284.23	0.120741
$711$	0	0
$712$	9315.43	0.490324
$713$	3325.33	0.174663
$714$	0	0
$715$	0	0
$716$	−193.069	−0.0100773
$717$	0	0
$718$	4301.02	0.223555
$719$	−16235.8	−0.842131	−0.421066	−	0.907030i	$-0.638344\pi$
$719$	−16235.8	−0.842131	−0.421066	+	0.907030i	$0.638344\pi$
$720$	0	0
$721$	8770.37	0.453018
$722$	9813.51	0.505846
$723$	0	0
$724$	6837.08	0.350964
$725$	−24699.2	−1.26525
$726$	0	0
$727$	24181.2	1.23361	0.616803	−	0.787118i	$-0.288428\pi$
$727$	24181.2	1.23361	0.616803	+	0.787118i	$0.288428\pi$
$728$	0	0
$729$	0	0
$730$	1712.98	0.0868496
$731$	6739.96	0.341021
$732$	0	0
$733$	−3053.70	−0.153876	−0.0769379	−	0.997036i	$-0.524514\pi$
$733$	−3053.70	−0.153876	−0.0769379	+	0.997036i	$0.524514\pi$
$734$	−4747.41	−0.238733
$735$	0	0
$736$	22983.4	1.15106
$737$	−7818.10	−0.390751
$738$	0	0
$739$	8033.62	0.399894	0.199947	−	0.979807i	$-0.435923\pi$
$739$	8033.62	0.399894	0.199947	+	0.979807i	$0.435923\pi$
$740$	−1865.74	−0.0926836
$741$	0	0
$742$	−5128.54	−0.253739
$743$	−16139.6	−0.796912	−0.398456	−	0.917187i	$-0.630454\pi$
$743$	−16139.6	−0.796912	−0.398456	+	0.917187i	$0.630454\pi$
$744$	0	0
$745$	6243.33	0.307031
$746$	8408.53	0.412678
$747$	0	0
$748$	3780.96	0.184820
$749$	−38988.7	−1.90202
$750$	0	0
$751$	−18491.1	−0.898469	−0.449235	−	0.893414i	$-0.648303\pi$
$751$	−18491.1	−0.898469	−0.449235	+	0.893414i	$0.648303\pi$
$752$	5334.08	0.258662
$753$	0	0
$754$	0	0
$755$	−9953.28	−0.479784
$756$	0	0
$757$	160.630	0.00771227	0.00385613	−	0.999993i	$-0.498773\pi$
$757$	160.630	0.00771227	0.00385613	+	0.999993i	$0.498773\pi$
$758$	−5347.17	−0.256224
$759$	0	0
$760$	1807.87	0.0862874
$761$	26799.1	1.27656	0.638282	−	0.769803i	$-0.279645\pi$
$761$	26799.1	1.27656	0.638282	+	0.769803i	$0.279645\pi$
$762$	0	0
$763$	−23077.3	−1.09496
$764$	23807.9	1.12741
$765$	0	0
$766$	598.052	0.0282095
$767$	0	0
$768$	0	0
$769$	5145.82	0.241304	0.120652	−	0.992695i	$-0.461501\pi$
$769$	5145.82	0.241304	0.120652	+	0.992695i	$0.461501\pi$
$770$	2306.71	0.107958
$771$	0	0
$772$	2626.83	0.122463
$773$	12810.6	0.596072	0.298036	−	0.954555i	$-0.403668\pi$
$773$	12810.6	0.596072	0.298036	+	0.954555i	$0.403668\pi$
$774$	0	0
$775$	3042.94	0.141039
$776$	32004.3	1.48052
$777$	0	0
$778$	13411.1	0.618011
$779$	3843.64	0.176781
$780$	0	0
$781$	6268.13	0.287185
$782$	−8537.89	−0.390428
$783$	0	0
$784$	4518.98	0.205857
$785$	−11556.8	−0.525452
$786$	0	0
$787$	−28073.0	−1.27153	−0.635764	−	0.771883i	$-0.719315\pi$
$787$	−28073.0	−1.27153	−0.635764	+	0.771883i	$0.719315\pi$
$788$	24940.0	1.12748
$789$	0	0
$790$	−3261.64	−0.146891
$791$	−43878.1	−1.97235
$792$	0	0
$793$	0	0
$794$	−11304.3	−0.505259
$795$	0	0
$796$	2036.87	0.0906970
$797$	−30093.1	−1.33746	−0.668729	−	0.743507i	$-0.733161\pi$
$797$	−30093.1	−1.33746	−0.668729	+	0.743507i	$0.733161\pi$
$798$	0	0
$799$	−20800.8	−0.921003
$800$	21031.6	0.929473
$801$	0	0
$802$	−6667.24	−0.293552
$803$	4700.56	0.206574
$804$	0	0
$805$	11880.2	0.520151
$806$	0	0
$807$	0	0
$808$	−7120.99	−0.310044
$809$	24337.1	1.05766	0.528831	−	0.848727i	$-0.322631\pi$
$809$	24337.1	1.05766	0.528831	+	0.848727i	$0.322631\pi$
$810$	0	0
$811$	−19078.7	−0.826071	−0.413035	−	0.910715i	$-0.635531\pi$
$811$	−19078.7	−0.826071	−0.413035	+	0.910715i	$0.635531\pi$
$812$	−33238.5	−1.43651
$813$	0	0
$814$	2244.74	0.0966559
$815$	11687.1	0.502309
$816$	0	0
$817$	3626.66	0.155301
$818$	21178.6	0.905248
$819$	0	0
$820$	−3176.26	−0.135268
$821$	2013.92	0.0856104	0.0428052	−	0.999083i	$-0.486371\pi$
$821$	2013.92	0.0856104	0.0428052	+	0.999083i	$0.486371\pi$
$822$	0	0
$823$	−7692.10	−0.325795	−0.162898	−	0.986643i	$-0.552084\pi$
$823$	−7692.10	−0.325795	−0.162898	+	0.986643i	$0.552084\pi$
$824$	6834.10	0.288929
$825$	0	0
$826$	18657.3	0.785922
$827$	−4762.76	−0.200263	−0.100131	−	0.994974i	$-0.531926\pi$
$827$	−4762.76	−0.200263	−0.100131	+	0.994974i	$0.531926\pi$
$828$	0	0
$829$	19977.7	0.836976	0.418488	−	0.908222i	$-0.362560\pi$
$829$	19977.7	0.836976	0.418488	+	0.908222i	$0.362560\pi$
$830$	7531.47	0.314965
$831$	0	0
$832$	0	0
$833$	−17622.3	−0.732984
$834$	0	0
$835$	11135.3	0.461499
$836$	2034.47	0.0841671
$837$	0	0
$838$	−22762.6	−0.938330
$839$	−30615.8	−1.25980	−0.629901	−	0.776676i	$-0.716905\pi$
$839$	−30615.8	−1.25980	−0.629901	+	0.776676i	$0.716905\pi$
$840$	0	0
$841$	23971.0	0.982861
$842$	24748.1	1.01292
$843$	0	0
$844$	−11804.7	−0.481439
$845$	0	0
$846$	0	0
$847$	−29842.9	−1.21064
$848$	1380.47	0.0559026
$849$	0	0
$850$	−7812.83	−0.315268
$851$	11561.0	0.465696
$852$	0	0
$853$	−5660.88	−0.227227	−0.113614	−	0.993525i	$-0.536243\pi$
$853$	−5660.88	−0.227227	−0.113614	+	0.993525i	$0.536243\pi$
$854$	5827.01	0.233485
$855$	0	0
$856$	−30381.0	−1.21309
$857$	−41346.1	−1.64802	−0.824012	−	0.566572i	$-0.808269\pi$
$857$	−41346.1	−1.64802	−0.824012	+	0.566572i	$0.808269\pi$
$858$	0	0
$859$	−34810.5	−1.38268	−0.691339	−	0.722530i	$-0.742979\pi$
$859$	−34810.5	−1.38268	−0.691339	+	0.722530i	$0.742979\pi$
$860$	−2996.96	−0.118832
$861$	0	0
$862$	−16700.4	−0.659880
$863$	−8360.51	−0.329774	−0.164887	−	0.986312i	$-0.552726\pi$
$863$	−8360.51	−0.329774	−0.164887	+	0.986312i	$0.552726\pi$
$864$	0	0
$865$	347.500	0.0136594
$866$	25108.2	0.985233
$867$	0	0
$868$	4094.99	0.160130
$869$	−8950.21	−0.349385
$870$	0	0
$871$	0	0
$872$	−17982.4	−0.698352
$873$	0	0
$874$	−4594.10	−0.177801
$875$	22970.4	0.887475
$876$	0	0
$877$	40579.3	1.56245	0.781223	−	0.624251i	$-0.214596\pi$
$877$	40579.3	1.56245	0.781223	+	0.624251i	$0.214596\pi$
$878$	−9425.22	−0.362285
$879$	0	0
$880$	−620.903	−0.0237848
$881$	−10445.2	−0.399442	−0.199721	−	0.979853i	$-0.564004\pi$
$881$	−10445.2	−0.399442	−0.199721	+	0.979853i	$0.564004\pi$
$882$	0	0
$883$	18227.6	0.694685	0.347343	−	0.937738i	$-0.387084\pi$
$883$	18227.6	0.694685	0.347343	+	0.937738i	$0.387084\pi$
$884$	0	0
$885$	0	0
$886$	15929.9	0.604036
$887$	23517.7	0.890245	0.445122	−	0.895470i	$-0.353160\pi$
$887$	23517.7	0.890245	0.445122	+	0.895470i	$0.353160\pi$
$888$	0	0
$889$	23512.9	0.887061
$890$	2446.43	0.0921399
$891$	0	0
$892$	−32960.1	−1.23720
$893$	−11192.6	−0.419424
$894$	0	0
$895$	−123.639	−0.00461766
$896$	32181.2	1.19989
$897$	0	0
$898$	9092.21	0.337874
$899$	−5957.95	−0.221033
$900$	0	0
$901$	−5383.28	−0.199049
$902$	3821.47	0.141065
$903$	0	0
$904$	−34191.0	−1.25794
$905$	4378.38	0.160820
$906$	0	0
$907$	−30564.6	−1.11894	−0.559471	−	0.828850i	$-0.688996\pi$
$907$	−30564.6	−1.11894	−0.559471	+	0.828850i	$0.688996\pi$
$908$	4981.26	0.182058
$909$	0	0
$910$	0	0
$911$	32766.5	1.19166	0.595831	−	0.803110i	$-0.296823\pi$
$911$	32766.5	1.19166	0.595831	+	0.803110i	$0.296823\pi$
$912$	0	0
$913$	20667.0	0.749154
$914$	7216.87	0.261174
$915$	0	0
$916$	3491.08	0.125926
$917$	7647.64	0.275406
$918$	0	0
$919$	20686.7	0.742538	0.371269	−	0.928525i	$-0.378923\pi$
$919$	20686.7	0.742538	0.371269	+	0.928525i	$0.378923\pi$
$920$	9257.35	0.331745
$921$	0	0
$922$	−8007.28	−0.286015
$923$	0	0
$924$	0	0
$925$	10579.2	0.376047
$926$	10128.6	0.359447
$927$	0	0
$928$	−41179.0	−1.45665
$929$	45632.2	1.61156	0.805782	−	0.592212i	$-0.201745\pi$
$929$	45632.2	1.61156	0.805782	+	0.592212i	$0.201745\pi$
$930$	0	0
$931$	−9482.26	−0.333801
$932$	12812.3	0.450301
$933$	0	0
$934$	20265.9	0.709979
$935$	2421.28	0.0846892
$936$	0	0
$937$	−17761.4	−0.619253	−0.309626	−	0.950858i	$-0.600204\pi$
$937$	−17761.4	−0.619253	−0.309626	+	0.950858i	$0.600204\pi$
$938$	21740.4	0.756768
$939$	0	0
$940$	9249.19	0.320931
$941$	−44888.3	−1.55507	−0.777534	−	0.628841i	$-0.783530\pi$
$941$	−44888.3	−1.55507	−0.777534	+	0.628841i	$0.783530\pi$
$942$	0	0
$943$	19681.7	0.679664
$944$	−5022.05	−0.173150
$945$	0	0
$946$	3605.74	0.123925
$947$	16069.6	0.551415	0.275708	−	0.961242i	$-0.411088\pi$
$947$	16069.6	0.551415	0.275708	+	0.961242i	$0.411088\pi$
$948$	0	0
$949$	0	0
$950$	−4203.96	−0.143573
$951$	0	0
$952$	−25637.8	−0.872823
$953$	−3512.03	−0.119377	−0.0596883	−	0.998217i	$-0.519011\pi$
$953$	−3512.03	−0.119377	−0.0596883	+	0.998217i	$0.519011\pi$
$954$	0	0
$955$	15246.3	0.516605
$956$	−3028.94	−0.102472
$957$	0	0
$958$	9071.00	0.305919
$959$	−71786.5	−2.41721
$960$	0	0
$961$	−29057.0	−0.975361
$962$	0	0
$963$	0	0
$964$	30177.5	1.00825
$965$	1682.19	0.0561158
$966$	0	0
$967$	37011.9	1.23084	0.615421	−	0.788199i	$-0.288986\pi$
$967$	37011.9	1.23084	0.615421	+	0.788199i	$0.288986\pi$
$968$	−23254.4	−0.772132
$969$	0	0
$970$	8405.00	0.278215
$971$	−19532.3	−0.645542	−0.322771	−	0.946477i	$-0.604614\pi$
$971$	−19532.3	−0.645542	−0.322771	+	0.946477i	$0.604614\pi$
$972$	0	0
$973$	−54317.1	−1.78965
$974$	−8412.30	−0.276743
$975$	0	0
$976$	−1568.47	−0.0514402
$977$	30201.2	0.988970	0.494485	−	0.869186i	$-0.335357\pi$
$977$	30201.2	0.988970	0.494485	+	0.869186i	$0.335357\pi$
$978$	0	0
$979$	6713.21	0.219157
$980$	7835.83	0.255415
$981$	0	0
$982$	23827.8	0.774315
$983$	−38774.9	−1.25812	−0.629058	−	0.777359i	$-0.716559\pi$
$983$	−38774.9	−1.25812	−0.629058	+	0.777359i	$0.716559\pi$
$984$	0	0
$985$	15971.3	0.516638
$986$	15297.2	0.494080
$987$	0	0
$988$	0	0
$989$	18570.6	0.597079
$990$	0	0
$991$	−27728.9	−0.888838	−0.444419	−	0.895819i	$-0.646590\pi$
$991$	−27728.9	−0.888838	−0.444419	+	0.895819i	$0.646590\pi$
$992$	5073.25	0.162375
$993$	0	0
$994$	−17430.3	−0.556192
$995$	1304.38	0.0415596
$996$	0	0
$997$	−48918.2	−1.55392	−0.776958	−	0.629552i	$-0.783239\pi$
$997$	−48918.2	−1.55392	−0.776958	+	0.629552i	$0.783239\pi$
$998$	2898.31	0.0919283
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.r.1.1		2
3.2	odd	2		169.4.a.g.1.2		2
13.12	even	2		117.4.a.d.1.2		2
39.2	even	12		169.4.e.f.147.3		8
39.5	even	4		169.4.b.f.168.2		4
39.8	even	4		169.4.b.f.168.3		4
39.11	even	12		169.4.e.f.147.2		8
39.17	odd	6		169.4.c.g.146.2		4
39.20	even	12		169.4.e.f.23.3		8
39.23	odd	6		169.4.c.g.22.2		4
39.29	odd	6		169.4.c.j.22.1		4
39.32	even	12		169.4.e.f.23.2		8
39.35	odd	6		169.4.c.j.146.1		4
39.38	odd	2		13.4.a.b.1.1	✓	2
52.51	odd	2		1872.4.a.bb.1.2		2
156.155	even	2		208.4.a.h.1.1		2
195.38	even	4		325.4.b.e.274.3		4
195.77	even	4		325.4.b.e.274.2		4
195.194	odd	2		325.4.a.f.1.2		2
273.272	even	2		637.4.a.b.1.1		2
312.77	odd	2		832.4.a.s.1.1		2
312.155	even	2		832.4.a.z.1.2		2
429.428	even	2		1573.4.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.1	✓	2	39.38	odd	2
117.4.a.d.1.2		2	13.12	even	2
169.4.a.g.1.2		2	3.2	odd	2
169.4.b.f.168.2		4	39.5	even	4
169.4.b.f.168.3		4	39.8	even	4
169.4.c.g.22.2		4	39.23	odd	6
169.4.c.g.146.2		4	39.17	odd	6
169.4.c.j.22.1		4	39.29	odd	6
169.4.c.j.146.1		4	39.35	odd	6
169.4.e.f.23.2		8	39.32	even	12
169.4.e.f.23.3		8	39.20	even	12
169.4.e.f.147.2		8	39.11	even	12
169.4.e.f.147.3		8	39.2	even	12
208.4.a.h.1.1		2	156.155	even	2
325.4.a.f.1.2		2	195.194	odd	2
325.4.b.e.274.2		4	195.77	even	4
325.4.b.e.274.3		4	195.38	even	4
637.4.a.b.1.1		2	273.272	even	2
832.4.a.s.1.1		2	312.77	odd	2
832.4.a.z.1.2		2	312.155	even	2
1521.4.a.r.1.1		2	1.1	even	1	trivial
1573.4.a.b.1.2		2	429.428	even	2
1872.4.a.bb.1.2		2	52.51	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} -5.56155 q^{4} -3.56155 q^{5} +27.1771 q^{7} +21.1771 q^{8} +5.56155 q^{10} +15.2614 q^{11} -42.4384 q^{14} +11.4233 q^{16} -44.5464 q^{17} -23.9697 q^{19} +19.8078 q^{20} -23.8314 q^{22} -122.739 q^{23} -112.315 q^{25} -151.147 q^{28} +219.909 q^{29} -27.0928 q^{31} -187.255 q^{32} +69.5616 q^{34} -96.7926 q^{35} -94.1922 q^{37} +37.4299 q^{38} -75.4233 q^{40} -160.354 q^{41} -151.302 q^{43} -84.8769 q^{44} +191.663 q^{46} +466.948 q^{47} +395.594 q^{49} +175.386 q^{50} +120.847 q^{53} -54.3542 q^{55} +575.531 q^{56} -343.400 q^{58} -439.633 q^{59} -137.305 q^{61} +42.3068 q^{62} +201.022 q^{64} -512.280 q^{67} +247.747 q^{68} +151.147 q^{70} +410.719 q^{71} +308.004 q^{73} +147.086 q^{74} +133.309 q^{76} +414.759 q^{77} -586.462 q^{79} -40.6847 q^{80} +250.401 q^{82} +1354.20 q^{83} +158.654 q^{85} +236.266 q^{86} +323.191 q^{88} +439.882 q^{89} +682.617 q^{92} -729.164 q^{94} +85.3693 q^{95} +1511.27 q^{97} -617.740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 7 q^{4} - 3 q^{5} + 9 q^{7} - 3 q^{8} + 7 q^{10} + 80 q^{11} - 89 q^{14} - 39 q^{16} - 19 q^{17} + 84 q^{19} + 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} - 125 q^{28} + 44 q^{29} + 86 q^{31}+ \cdots - 650 q^{98}+O(q^{100}) \) 2 * q + q^2 - 7 * q^4 - 3 * q^5 + 9 * q^7 - 3 * q^8 + 7 * q^10 + 80 * q^11 - 89 * q^14 - 39 * q^16 - 19 * q^17 + 84 * q^19 + 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 - 125 * q^28 + 44 * q^29 + 86 * q^31 - 123 * q^32 + 135 * q^34 - 107 * q^35 - 209 * q^37 + 314 * q^38 - 89 * q^40 - 230 * q^41 + 287 * q^43 - 178 * q^44 + 4 * q^46 + 435 * q^47 + 383 * q^49 - 144 * q^50 + 118 * q^53 - 18 * q^55 + 1015 * q^56 - 794 * q^58 - 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 68 * q^67 + 211 * q^68 + 125 * q^70 - 131 * q^71 - 456 * q^73 - 147 * q^74 - 22 * q^76 - 762 * q^77 - 1008 * q^79 - 69 * q^80 + 72 * q^82 + 1958 * q^83 + 173 * q^85 + 1359 * q^86 - 1242 * q^88 - 720 * q^89 + 788 * q^92 - 811 * q^94 + 146 * q^95 + 928 * q^97 - 650 * q^98

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