LMFDB - Embedded newform 1710.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,4,Mod(1,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1710.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1710.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:	$100.893266110$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{43}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 43 $ x^2 - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.55744$ of defining polynomial
Character	$\chi$	$=$	1710.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.6723 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.6723 q^{7} +8.00000 q^{8} -10.0000 q^{10} +14.6723 q^{11} +86.2467 q^{13} -73.3446 q^{14} +16.0000 q^{16} +15.5405 q^{17} +19.0000 q^{19} -20.0000 q^{20} +29.3446 q^{22} -74.9190 q^{23} +25.0000 q^{25} +172.493 q^{26} -146.689 q^{28} +100.672 q^{29} -78.9190 q^{31} +32.0000 q^{32} +31.0810 q^{34} +183.362 q^{35} -390.672 q^{37} +38.0000 q^{38} -40.0000 q^{40} +272.213 q^{41} -498.051 q^{43} +58.6893 q^{44} -149.838 q^{46} +469.676 q^{47} +1001.86 q^{49} +50.0000 q^{50} +344.987 q^{52} -205.702 q^{53} -73.3616 q^{55} -293.379 q^{56} +201.345 q^{58} -333.736 q^{59} -59.5066 q^{61} -157.838 q^{62} +64.0000 q^{64} -431.234 q^{65} -929.514 q^{67} +62.1620 q^{68} +366.723 q^{70} -276.885 q^{71} +229.217 q^{73} -781.345 q^{74} +76.0000 q^{76} -538.068 q^{77} -666.621 q^{79} -80.0000 q^{80} +544.426 q^{82} +1203.02 q^{83} -77.7025 q^{85} -996.102 q^{86} +117.379 q^{88} +751.599 q^{89} -3162.87 q^{91} -299.676 q^{92} +939.352 q^{94} -95.0000 q^{95} -1380.58 q^{97} +2003.72 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} - 34 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 - 34 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} - 34 q^{7} + 16 q^{8} - 20 q^{10} - 10 q^{11} + 2 q^{13} - 68 q^{14} + 32 q^{16} + 136 q^{17} + 38 q^{19} - 40 q^{20} - 20 q^{22} + 60 q^{23} + 50 q^{25} + 4 q^{26} - 136 q^{28} + 162 q^{29} + 52 q^{31} + 64 q^{32} + 272 q^{34} + 170 q^{35} - 742 q^{37} + 76 q^{38} - 80 q^{40} + 610 q^{41} - 642 q^{43} - 40 q^{44} + 120 q^{46} + 100 q^{47} + 666 q^{49} + 100 q^{50} + 8 q^{52} - 936 q^{53} + 50 q^{55} - 272 q^{56} + 324 q^{58} - 956 q^{59} - 460 q^{61} + 104 q^{62} + 128 q^{64} - 10 q^{65} - 600 q^{67} + 544 q^{68} + 340 q^{70} - 580 q^{71} - 276 q^{73} - 1484 q^{74} + 152 q^{76} - 604 q^{77} - 1648 q^{79} - 160 q^{80} + 1220 q^{82} + 1488 q^{83} - 680 q^{85} - 1284 q^{86} - 80 q^{88} - 346 q^{89} - 3388 q^{91} + 240 q^{92} + 200 q^{94} - 190 q^{95} - 1830 q^{97} + 1332 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 - 34 * q^7 + 16 * q^8 - 20 * q^10 - 10 * q^11 + 2 * q^13 - 68 * q^14 + 32 * q^16 + 136 * q^17 + 38 * q^19 - 40 * q^20 - 20 * q^22 + 60 * q^23 + 50 * q^25 + 4 * q^26 - 136 * q^28 + 162 * q^29 + 52 * q^31 + 64 * q^32 + 272 * q^34 + 170 * q^35 - 742 * q^37 + 76 * q^38 - 80 * q^40 + 610 * q^41 - 642 * q^43 - 40 * q^44 + 120 * q^46 + 100 * q^47 + 666 * q^49 + 100 * q^50 + 8 * q^52 - 936 * q^53 + 50 * q^55 - 272 * q^56 + 324 * q^58 - 956 * q^59 - 460 * q^61 + 104 * q^62 + 128 * q^64 - 10 * q^65 - 600 * q^67 + 544 * q^68 + 340 * q^70 - 580 * q^71 - 276 * q^73 - 1484 * q^74 + 152 * q^76 - 604 * q^77 - 1648 * q^79 - 160 * q^80 + 1220 * q^82 + 1488 * q^83 - 680 * q^85 - 1284 * q^86 - 80 * q^88 - 346 * q^89 - 3388 * q^91 + 240 * q^92 + 200 * q^94 - 190 * q^95 - 1830 * q^97 + 1332 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−36.6723	−1.98012	−0.990059	−	0.140649i	$-0.955081\pi$
$7$	−36.6723	−1.98012	−0.990059	+	0.140649i	$0.955081\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	14.6723	0.402170	0.201085	−	0.979574i	$-0.435553\pi$
$11$	14.6723	0.402170	0.201085	+	0.979574i	$0.435553\pi$
$12$	0	0
$13$	86.2467	1.84004	0.920020	−	0.391870i	$-0.128172\pi$
$13$	86.2467	1.84004	0.920020	+	0.391870i	$0.128172\pi$
$14$	−73.3446	−1.40016
$15$	0	0
$16$	16.0000	0.250000
$17$	15.5405	0.221713	0.110857	−	0.993836i	$-0.464641\pi$
$17$	15.5405	0.221713	0.110857	+	0.993836i	$0.464641\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−20.0000	−0.223607
$21$	0	0
$22$	29.3446	0.284377
$23$	−74.9190	−0.679204	−0.339602	−	0.940569i	$-0.610292\pi$
$23$	−74.9190	−0.679204	−0.339602	+	0.940569i	$0.610292\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	172.493	1.30111
$27$	0	0
$28$	−146.689	−0.990059
$29$	100.672	0.644634	0.322317	−	0.946632i	$-0.395538\pi$
$29$	100.672	0.644634	0.322317	+	0.946632i	$0.395538\pi$
$30$	0	0
$31$	−78.9190	−0.457235	−0.228617	−	0.973516i	$-0.573420\pi$
$31$	−78.9190	−0.457235	−0.228617	+	0.973516i	$0.573420\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	31.0810	0.156775
$35$	183.362	0.885536
$36$	0	0
$37$	−390.672	−1.73584	−0.867921	−	0.496703i	$-0.834544\pi$
$37$	−390.672	−1.73584	−0.867921	+	0.496703i	$0.834544\pi$
$38$	38.0000	0.162221
$39$	0	0
$40$	−40.0000	−0.158114
$41$	272.213	1.03689	0.518445	−	0.855111i	$-0.326511\pi$
$41$	272.213	1.03689	0.518445	+	0.855111i	$0.326511\pi$
$42$	0	0
$43$	−498.051	−1.76633	−0.883163	−	0.469066i	$-0.844591\pi$
$43$	−498.051	−1.76633	−0.883163	+	0.469066i	$0.844591\pi$
$44$	58.6893	0.201085
$45$	0	0
$46$	−149.838	−0.480270
$47$	469.676	1.45764	0.728822	−	0.684703i	$-0.240068\pi$
$47$	469.676	1.45764	0.728822	+	0.684703i	$0.240068\pi$
$48$	0	0
$49$	1001.86	2.92087
$50$	50.0000	0.141421
$51$	0	0
$52$	344.987	0.920020
$53$	−205.702	−0.533121	−0.266560	−	0.963818i	$-0.585887\pi$
$53$	−205.702	−0.533121	−0.266560	+	0.963818i	$0.585887\pi$
$54$	0	0
$55$	−73.3616	−0.179856
$56$	−293.379	−0.700078
$57$	0	0
$58$	201.345	0.455825
$59$	−333.736	−0.736420	−0.368210	−	0.929743i	$-0.620029\pi$
$59$	−333.736	−0.736420	−0.368210	+	0.929743i	$0.620029\pi$
$60$	0	0
$61$	−59.5066	−0.124902	−0.0624511	−	0.998048i	$-0.519892\pi$
$61$	−59.5066	−0.124902	−0.0624511	+	0.998048i	$0.519892\pi$
$62$	−157.838	−0.323314
$63$	0	0
$64$	64.0000	0.125000
$65$	−431.234	−0.822891
$66$	0	0
$67$	−929.514	−1.69490	−0.847449	−	0.530876i	$-0.821863\pi$
$67$	−929.514	−1.69490	−0.847449	+	0.530876i	$0.821863\pi$
$68$	62.1620	0.110857
$69$	0	0
$70$	366.723	0.626169
$71$	−276.885	−0.462820	−0.231410	−	0.972856i	$-0.574334\pi$
$71$	−276.885	−0.462820	−0.231410	+	0.972856i	$0.574334\pi$
$72$	0	0
$73$	229.217	0.367504	0.183752	−	0.982973i	$-0.441176\pi$
$73$	229.217	0.367504	0.183752	+	0.982973i	$0.441176\pi$
$74$	−781.345	−1.22743
$75$	0	0
$76$	76.0000	0.114708
$77$	−538.068	−0.796344
$78$	0	0
$79$	−666.621	−0.949377	−0.474688	−	0.880154i	$-0.657439\pi$
$79$	−666.621	−0.949377	−0.474688	+	0.880154i	$0.657439\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	544.426	0.733192
$83$	1203.02	1.59095	0.795474	−	0.605988i	$-0.207222\pi$
$83$	1203.02	1.59095	0.795474	+	0.605988i	$0.207222\pi$
$84$	0	0
$85$	−77.7025	−0.0991531
$86$	−996.102	−1.24898
$87$	0	0
$88$	117.379	0.142189
$89$	751.599	0.895161	0.447580	−	0.894244i	$-0.352286\pi$
$89$	751.599	0.895161	0.447580	+	0.894244i	$0.352286\pi$
$90$	0	0
$91$	−3162.87	−3.64350
$92$	−299.676	−0.339602
$93$	0	0
$94$	939.352	1.03071
$95$	−95.0000	−0.102598
$96$	0	0
$97$	−1380.58	−1.44512	−0.722559	−	0.691309i	$-0.757034\pi$
$97$	−1380.58	−1.44512	−0.722559	+	0.691309i	$0.757034\pi$
$98$	2003.72	2.06537
$99$	0	0
$100$	100.000	0.100000
$101$	−735.616	−0.724718	−0.362359	−	0.932039i	$-0.618029\pi$
$101$	−735.616	−0.724718	−0.362359	+	0.932039i	$0.618029\pi$
$102$	0	0
$103$	348.663	0.333541	0.166771	−	0.985996i	$-0.446666\pi$
$103$	348.663	0.333541	0.166771	+	0.985996i	$0.446666\pi$
$104$	689.974	0.650553
$105$	0	0
$106$	−411.405	−0.376973
$107$	−292.663	−0.264419	−0.132209	−	0.991222i	$-0.542207\pi$
$107$	−292.663	−0.264419	−0.132209	+	0.991222i	$0.542207\pi$
$108$	0	0
$109$	−683.940	−0.601005	−0.300502	−	0.953781i	$-0.597154\pi$
$109$	−683.940	−0.601005	−0.300502	+	0.953781i	$0.597154\pi$
$110$	−146.723	−0.127177
$111$	0	0
$112$	−586.757	−0.495030
$113$	−2334.21	−1.94322	−0.971611	−	0.236584i	$-0.923972\pi$
$113$	−2334.21	−1.94322	−0.971611	+	0.236584i	$0.923972\pi$
$114$	0	0
$115$	374.595	0.303749
$116$	402.689	0.322317
$117$	0	0
$118$	−667.473	−0.520727
$119$	−569.906	−0.439018
$120$	0	0
$121$	−1115.72	−0.838259
$122$	−119.013	−0.0883193
$123$	0	0
$124$	−315.676	−0.228617
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2409.07	−1.68323	−0.841616	−	0.540077i	$-0.818395\pi$
$127$	−2409.07	−1.68323	−0.841616	+	0.540077i	$0.818395\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−862.467	−0.581872
$131$	−2883.78	−1.92333	−0.961667	−	0.274221i	$-0.911580\pi$
$131$	−2883.78	−1.92333	−0.961667	+	0.274221i	$0.911580\pi$
$132$	0	0
$133$	−696.774	−0.454270
$134$	−1859.03	−1.19847
$135$	0	0
$136$	124.324	0.0783874
$137$	983.311	0.613211	0.306605	−	0.951837i	$-0.400807\pi$
$137$	983.311	0.613211	0.306605	+	0.951837i	$0.400807\pi$
$138$	0	0
$139$	−165.669	−0.101092	−0.0505461	−	0.998722i	$-0.516096\pi$
$139$	−165.669	−0.101092	−0.0505461	+	0.998722i	$0.516096\pi$
$140$	733.446	0.442768
$141$	0	0
$142$	−553.770	−0.327263
$143$	1265.44	0.740009
$144$	0	0
$145$	−503.362	−0.288289
$146$	458.433	0.259864
$147$	0	0
$148$	−1562.69	−0.867921
$149$	−940.945	−0.517351	−0.258675	−	0.965964i	$-0.583286\pi$
$149$	−940.945	−0.517351	−0.258675	+	0.965964i	$0.583286\pi$
$150$	0	0
$151$	−1814.92	−0.978119	−0.489060	−	0.872250i	$-0.662660\pi$
$151$	−1814.92	−0.978119	−0.489060	+	0.872250i	$0.662660\pi$
$152$	152.000	0.0811107
$153$	0	0
$154$	−1076.14	−0.563100
$155$	394.595	0.204482
$156$	0	0
$157$	−589.763	−0.299797	−0.149899	−	0.988701i	$-0.547895\pi$
$157$	−589.763	−0.299797	−0.149899	+	0.988701i	$0.547895\pi$
$158$	−1333.24	−0.671311
$159$	0	0
$160$	−160.000	−0.0790569
$161$	2747.45	1.34490
$162$	0	0
$163$	−867.618	−0.416915	−0.208457	−	0.978031i	$-0.566844\pi$
$163$	−867.618	−0.416915	−0.208457	+	0.978031i	$0.566844\pi$
$164$	1088.85	0.518445
$165$	0	0
$166$	2406.04	1.12497
$167$	−523.533	−0.242588	−0.121294	−	0.992617i	$-0.538704\pi$
$167$	−523.533	−0.242588	−0.121294	+	0.992617i	$0.538704\pi$
$168$	0	0
$169$	5241.49	2.38575
$170$	−155.405	−0.0701118
$171$	0	0
$172$	−1992.20	−0.883163
$173$	3108.25	1.36599	0.682994	−	0.730424i	$-0.260677\pi$
$173$	3108.25	1.36599	0.682994	+	0.730424i	$0.260677\pi$
$174$	0	0
$175$	−916.808	−0.396024
$176$	234.757	0.100542
$177$	0	0
$178$	1503.20	0.632974
$179$	−203.371	−0.0849199	−0.0424600	−	0.999098i	$-0.513519\pi$
$179$	−203.371	−0.0849199	−0.0424600	+	0.999098i	$0.513519\pi$
$180$	0	0
$181$	1450.75	0.595764	0.297882	−	0.954603i	$-0.403720\pi$
$181$	1450.75	0.595764	0.297882	+	0.954603i	$0.403720\pi$
$182$	−6325.73	−2.57634
$183$	0	0
$184$	−599.352	−0.240135
$185$	1953.36	0.776292
$186$	0	0
$187$	228.015	0.0891664
$188$	1878.70	0.728822
$189$	0	0
$190$	−190.000	−0.0725476
$191$	−2860.98	−1.08384	−0.541920	−	0.840430i	$-0.682303\pi$
$191$	−2860.98	−1.08384	−0.541920	+	0.840430i	$0.682303\pi$
$192$	0	0
$193$	718.958	0.268144	0.134072	−	0.990972i	$-0.457195\pi$
$193$	718.958	0.268144	0.134072	+	0.990972i	$0.457195\pi$
$194$	−2761.16	−1.02185
$195$	0	0
$196$	4007.43	1.46044
$197$	−4412.00	−1.59564	−0.797822	−	0.602893i	$-0.794015\pi$
$197$	−4412.00	−1.59564	−0.797822	+	0.602893i	$0.794015\pi$
$198$	0	0
$199$	−326.060	−0.116150	−0.0580749	−	0.998312i	$-0.518496\pi$
$199$	−326.060	−0.116150	−0.0580749	+	0.998312i	$0.518496\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−1471.23	−0.512453
$203$	−3691.89	−1.27645
$204$	0	0
$205$	−1361.06	−0.463711
$206$	697.326	0.235849
$207$	0	0
$208$	1379.95	0.460010
$209$	278.774	0.0922641
$210$	0	0
$211$	1937.58	0.632174	0.316087	−	0.948730i	$-0.397631\pi$
$211$	1937.58	0.632174	0.316087	+	0.948730i	$0.397631\pi$
$212$	−822.810	−0.266560
$213$	0	0
$214$	−585.326	−0.186972
$215$	2490.25	0.789925
$216$	0	0
$217$	2894.14	0.905379
$218$	−1367.88	−0.424975
$219$	0	0
$220$	−293.446	−0.0899279
$221$	1340.32	0.407961
$222$	0	0
$223$	427.714	0.128439	0.0642194	−	0.997936i	$-0.479544\pi$
$223$	427.714	0.128439	0.0642194	+	0.997936i	$0.479544\pi$
$224$	−1173.51	−0.350039
$225$	0	0
$226$	−4668.42	−1.37407
$227$	1403.70	0.410426	0.205213	−	0.978717i	$-0.434211\pi$
$227$	1403.70	0.410426	0.205213	+	0.978717i	$0.434211\pi$
$228$	0	0
$229$	1117.90	0.322590	0.161295	−	0.986906i	$-0.448433\pi$
$229$	1117.90	0.322590	0.161295	+	0.986906i	$0.448433\pi$
$230$	749.190	0.214783
$231$	0	0
$232$	805.379	0.227912
$233$	3906.51	1.09839	0.549193	−	0.835695i	$-0.314935\pi$
$233$	3906.51	1.09839	0.549193	+	0.835695i	$0.314935\pi$
$234$	0	0
$235$	−2348.38	−0.651878
$236$	−1334.95	−0.368210
$237$	0	0
$238$	−1139.81	−0.310433
$239$	6816.14	1.84477	0.922384	−	0.386273i	$-0.126238\pi$
$239$	6816.14	1.84477	0.922384	+	0.386273i	$0.126238\pi$
$240$	0	0
$241$	2928.50	0.782744	0.391372	−	0.920233i	$-0.372001\pi$
$241$	2928.50	0.782744	0.391372	+	0.920233i	$0.372001\pi$
$242$	−2231.45	−0.592739
$243$	0	0
$244$	−238.026	−0.0624511
$245$	−5009.29	−1.30625
$246$	0	0
$247$	1638.69	0.422134
$248$	−631.352	−0.161657
$249$	0	0
$250$	−250.000	−0.0632456
$251$	−2494.67	−0.627340	−0.313670	−	0.949532i	$-0.601559\pi$
$251$	−2494.67	−0.627340	−0.313670	+	0.949532i	$0.601559\pi$
$252$	0	0
$253$	−1099.24	−0.273155
$254$	−4818.14	−1.19022
$255$	0	0
$256$	256.000	0.0625000
$257$	2362.67	0.573460	0.286730	−	0.958011i	$-0.407432\pi$
$257$	2362.67	0.573460	0.286730	+	0.958011i	$0.407432\pi$
$258$	0	0
$259$	14326.9	3.43717
$260$	−1724.93	−0.411446
$261$	0	0
$262$	−5767.55	−1.36000
$263$	−1116.53	−0.261781	−0.130891	−	0.991397i	$-0.541784\pi$
$263$	−1116.53	−0.261781	−0.130891	+	0.991397i	$0.541784\pi$
$264$	0	0
$265$	1028.51	0.238419
$266$	−1393.55	−0.321218
$267$	0	0
$268$	−3718.06	−0.847449
$269$	3252.92	0.737302	0.368651	−	0.929568i	$-0.379820\pi$
$269$	3252.92	0.737302	0.368651	+	0.929568i	$0.379820\pi$
$270$	0	0
$271$	−1232.79	−0.276336	−0.138168	−	0.990409i	$-0.544121\pi$
$271$	−1232.79	−0.276336	−0.138168	+	0.990409i	$0.544121\pi$
$272$	248.648	0.0554283
$273$	0	0
$274$	1966.62	0.433606
$275$	366.808	0.0804340
$276$	0	0
$277$	3531.72	0.766066	0.383033	−	0.923735i	$-0.374880\pi$
$277$	3531.72	0.766066	0.383033	+	0.923735i	$0.374880\pi$
$278$	−331.337	−0.0714830
$279$	0	0
$280$	1466.89	0.313084
$281$	5235.11	1.11139	0.555694	−	0.831387i	$-0.312452\pi$
$281$	5235.11	1.11139	0.555694	+	0.831387i	$0.312452\pi$
$282$	0	0
$283$	−5066.50	−1.06421	−0.532107	−	0.846677i	$-0.678599\pi$
$283$	−5066.50	−1.06421	−0.532107	+	0.846677i	$0.678599\pi$
$284$	−1107.54	−0.231410
$285$	0	0
$286$	2530.88	0.523265
$287$	−9982.67	−2.05317
$288$	0	0
$289$	−4671.49	−0.950843
$290$	−1006.72	−0.203851
$291$	0	0
$292$	916.866	0.183752
$293$	−7387.82	−1.47304	−0.736521	−	0.676415i	$-0.763533\pi$
$293$	−7387.82	−1.47304	−0.736521	+	0.676415i	$0.763533\pi$
$294$	0	0
$295$	1668.68	0.329337
$296$	−3125.38	−0.613713
$297$	0	0
$298$	−1881.89	−0.365822
$299$	−6461.52	−1.24976
$300$	0	0
$301$	18264.7	3.49754
$302$	−3629.84	−0.691635
$303$	0	0
$304$	304.000	0.0573539
$305$	297.533	0.0558580
$306$	0	0
$307$	−1474.47	−0.274112	−0.137056	−	0.990563i	$-0.543764\pi$
$307$	−1474.47	−0.274112	−0.137056	+	0.990563i	$0.543764\pi$
$308$	−2152.27	−0.398172
$309$	0	0
$310$	789.190	0.144590
$311$	−7691.11	−1.40232	−0.701162	−	0.713002i	$-0.747335\pi$
$311$	−7691.11	−1.40232	−0.701162	+	0.713002i	$0.747335\pi$
$312$	0	0
$313$	−4884.48	−0.882068	−0.441034	−	0.897490i	$-0.645388\pi$
$313$	−4884.48	−0.882068	−0.441034	+	0.897490i	$0.645388\pi$
$314$	−1179.53	−0.211989
$315$	0	0
$316$	−2666.49	−0.474688
$317$	−10332.0	−1.83062	−0.915308	−	0.402754i	$-0.868053\pi$
$317$	−10332.0	−1.83062	−0.915308	+	0.402754i	$0.868053\pi$
$318$	0	0
$319$	1477.10	0.259252
$320$	−320.000	−0.0559017
$321$	0	0
$322$	5494.91	0.950991
$323$	295.269	0.0508645
$324$	0	0
$325$	2156.17	0.368008
$326$	−1735.24	−0.294803
$327$	0	0
$328$	2177.70	0.366596
$329$	−17224.1	−2.88631
$330$	0	0
$331$	6737.02	1.11873	0.559366	−	0.828921i	$-0.311045\pi$
$331$	6737.02	1.11873	0.559366	+	0.828921i	$0.311045\pi$
$332$	4812.08	0.795474
$333$	0	0
$334$	−1047.07	−0.171536
$335$	4647.57	0.757982
$336$	0	0
$337$	1794.73	0.290104	0.145052	−	0.989424i	$-0.453665\pi$
$337$	1794.73	0.290104	0.145052	+	0.989424i	$0.453665\pi$
$338$	10483.0	1.68698
$339$	0	0
$340$	−310.810	−0.0495766
$341$	−1157.92	−0.183886
$342$	0	0
$343$	−24161.9	−3.80355
$344$	−3984.41	−0.624491
$345$	0	0
$346$	6216.50	0.965900
$347$	11438.0	1.76953	0.884763	−	0.466042i	$-0.154320\pi$
$347$	11438.0	1.76953	0.884763	+	0.466042i	$0.154320\pi$
$348$	0	0
$349$	−2069.92	−0.317480	−0.158740	−	0.987320i	$-0.550743\pi$
$349$	−2069.92	−0.317480	−0.158740	+	0.987320i	$0.550743\pi$
$350$	−1833.62	−0.280031
$351$	0	0
$352$	469.514	0.0710943
$353$	309.518	0.0466685	0.0233342	−	0.999728i	$-0.492572\pi$
$353$	309.518	0.0466685	0.0233342	+	0.999728i	$0.492572\pi$
$354$	0	0
$355$	1384.43	0.206979
$356$	3006.40	0.447580
$357$	0	0
$358$	−406.742	−0.0600474
$359$	−5692.53	−0.836881	−0.418441	−	0.908244i	$-0.637423\pi$
$359$	−5692.53	−0.836881	−0.418441	+	0.908244i	$0.637423\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	2901.50	0.421269
$363$	0	0
$364$	−12651.5	−1.82175
$365$	−1146.08	−0.164353
$366$	0	0
$367$	−5525.76	−0.785946	−0.392973	−	0.919550i	$-0.628553\pi$
$367$	−5525.76	−0.785946	−0.392973	+	0.919550i	$0.628553\pi$
$368$	−1198.70	−0.169801
$369$	0	0
$370$	3906.72	0.548921
$371$	7543.59	1.05564
$372$	0	0
$373$	−4442.73	−0.616717	−0.308359	−	0.951270i	$-0.599780\pi$
$373$	−4442.73	−0.616717	−0.308359	+	0.951270i	$0.599780\pi$
$374$	456.030	0.0630501
$375$	0	0
$376$	3757.41	0.515355
$377$	8682.66	1.18615
$378$	0	0
$379$	7728.52	1.04746	0.523730	−	0.851884i	$-0.324540\pi$
$379$	7728.52	1.04746	0.523730	+	0.851884i	$0.324540\pi$
$380$	−380.000	−0.0512989
$381$	0	0
$382$	−5721.97	−0.766391
$383$	6823.02	0.910287	0.455144	−	0.890418i	$-0.349588\pi$
$383$	6823.02	0.910287	0.455144	+	0.890418i	$0.349588\pi$
$384$	0	0
$385$	2690.34	0.356136
$386$	1437.92	0.189606
$387$	0	0
$388$	−5522.31	−0.722559
$389$	5072.27	0.661117	0.330559	−	0.943785i	$-0.392763\pi$
$389$	5072.27	0.661117	0.330559	+	0.943785i	$0.392763\pi$
$390$	0	0
$391$	−1164.28	−0.150588
$392$	8014.87	1.03268
$393$	0	0
$394$	−8824.00	−1.12829
$395$	3333.11	0.424574
$396$	0	0
$397$	6018.92	0.760909	0.380455	−	0.924800i	$-0.375768\pi$
$397$	6018.92	0.760909	0.380455	+	0.924800i	$0.375768\pi$
$398$	−652.121	−0.0821303
$399$	0	0
$400$	400.000	0.0500000
$401$	−13290.9	−1.65515	−0.827574	−	0.561357i	$-0.810279\pi$
$401$	−13290.9	−1.65515	−0.827574	+	0.561357i	$0.810279\pi$
$402$	0	0
$403$	−6806.50	−0.841330
$404$	−2942.46	−0.362359
$405$	0	0
$406$	−7383.77	−0.902588
$407$	−5732.07	−0.698103
$408$	0	0
$409$	−13481.7	−1.62990	−0.814950	−	0.579532i	$-0.803235\pi$
$409$	−13481.7	−1.62990	−0.814950	+	0.579532i	$0.803235\pi$
$410$	−2722.13	−0.327893
$411$	0	0
$412$	1394.65	0.166771
$413$	12238.9	1.45820
$414$	0	0
$415$	−6015.10	−0.711494
$416$	2759.89	0.325276
$417$	0	0
$418$	557.548	0.0652406
$419$	−3748.89	−0.437101	−0.218550	−	0.975826i	$-0.570133\pi$
$419$	−3748.89	−0.437101	−0.218550	+	0.975826i	$0.570133\pi$
$420$	0	0
$421$	319.695	0.0370094	0.0185047	−	0.999829i	$-0.494109\pi$
$421$	319.695	0.0370094	0.0185047	+	0.999829i	$0.494109\pi$
$422$	3875.16	0.447014
$423$	0	0
$424$	−1645.62	−0.188487
$425$	388.512	0.0443426
$426$	0	0
$427$	2182.24	0.247321
$428$	−1170.65	−0.132209
$429$	0	0
$430$	4980.51	0.558561
$431$	−10955.8	−1.22441	−0.612204	−	0.790700i	$-0.709717\pi$
$431$	−10955.8	−1.22441	−0.612204	+	0.790700i	$0.709717\pi$
$432$	0	0
$433$	−14023.5	−1.55641	−0.778206	−	0.628009i	$-0.783870\pi$
$433$	−14023.5	−1.55641	−0.778206	+	0.628009i	$0.783870\pi$
$434$	5788.29	0.640200
$435$	0	0
$436$	−2735.76	−0.300502
$437$	−1423.46	−0.155820
$438$	0	0
$439$	11548.1	1.25550	0.627748	−	0.778417i	$-0.283977\pi$
$439$	11548.1	1.25550	0.627748	+	0.778417i	$0.283977\pi$
$440$	−586.893	−0.0635887
$441$	0	0
$442$	2680.63	0.288472
$443$	−13092.8	−1.40419	−0.702096	−	0.712083i	$-0.747752\pi$
$443$	−13092.8	−1.40419	−0.702096	+	0.712083i	$0.747752\pi$
$444$	0	0
$445$	−3757.99	−0.400328
$446$	855.428	0.0908199
$447$	0	0
$448$	−2347.03	−0.247515
$449$	5739.91	0.603303	0.301652	−	0.953418i	$-0.402462\pi$
$449$	5739.91	0.603303	0.301652	+	0.953418i	$0.402462\pi$
$450$	0	0
$451$	3993.99	0.417006
$452$	−9336.84	−0.971611
$453$	0	0
$454$	2807.40	0.290215
$455$	15814.3	1.62942
$456$	0	0
$457$	9936.47	1.01709	0.508543	−	0.861037i	$-0.330184\pi$
$457$	9936.47	1.01709	0.508543	+	0.861037i	$0.330184\pi$
$458$	2235.80	0.228105
$459$	0	0
$460$	1498.38	0.151875
$461$	11072.8	1.11869	0.559343	−	0.828937i	$-0.311054\pi$
$461$	11072.8	1.11869	0.559343	+	0.828937i	$0.311054\pi$
$462$	0	0
$463$	4973.53	0.499221	0.249611	−	0.968346i	$-0.419697\pi$
$463$	4973.53	0.499221	0.249611	+	0.968346i	$0.419697\pi$
$464$	1610.76	0.161158
$465$	0	0
$466$	7813.02	0.776677
$467$	16160.1	1.60129	0.800644	−	0.599141i	$-0.204491\pi$
$467$	16160.1	1.60129	0.800644	+	0.599141i	$0.204491\pi$
$468$	0	0
$469$	34087.4	3.35610
$470$	−4696.76	−0.460948
$471$	0	0
$472$	−2669.89	−0.260364
$473$	−7307.56	−0.710363
$474$	0	0
$475$	475.000	0.0458831
$476$	−2279.62	−0.219509
$477$	0	0
$478$	13632.3	1.30445
$479$	17457.3	1.66523	0.832613	−	0.553856i	$-0.186844\pi$
$479$	17457.3	1.66523	0.832613	+	0.553856i	$0.186844\pi$
$480$	0	0
$481$	−33694.2	−3.19402
$482$	5857.00	0.553484
$483$	0	0
$484$	−4462.89	−0.419130
$485$	6902.89	0.646277
$486$	0	0
$487$	14092.4	1.31126	0.655632	−	0.755080i	$-0.272402\pi$
$487$	14092.4	1.31126	0.655632	+	0.755080i	$0.272402\pi$
$488$	−476.053	−0.0441596
$489$	0	0
$490$	−10018.6	−0.923660
$491$	5263.81	0.483814	0.241907	−	0.970299i	$-0.422227\pi$
$491$	5263.81	0.483814	0.241907	+	0.970299i	$0.422227\pi$
$492$	0	0
$493$	1564.50	0.142924
$494$	3277.37	0.298494
$495$	0	0
$496$	−1262.70	−0.114309
$497$	10154.0	0.916439
$498$	0	0
$499$	−7946.67	−0.712909	−0.356455	−	0.934313i	$-0.616015\pi$
$499$	−7946.67	−0.712909	−0.356455	+	0.934313i	$0.616015\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	−4989.34	−0.443596
$503$	−19625.7	−1.73969	−0.869846	−	0.493324i	$-0.835782\pi$
$503$	−19625.7	−1.73969	−0.869846	+	0.493324i	$0.835782\pi$
$504$	0	0
$505$	3678.08	0.324104
$506$	−2198.47	−0.193150
$507$	0	0
$508$	−9636.28	−0.841616
$509$	−16656.4	−1.45045	−0.725227	−	0.688510i	$-0.758265\pi$
$509$	−16656.4	−1.45045	−0.725227	+	0.688510i	$0.758265\pi$
$510$	0	0
$511$	−8405.90	−0.727701
$512$	512.000	0.0441942
$513$	0	0
$514$	4725.34	0.405498
$515$	−1743.31	−0.149164
$516$	0	0
$517$	6891.24	0.586221
$518$	28653.7	2.43045
$519$	0	0
$520$	−3449.87	−0.290936
$521$	−14204.1	−1.19442	−0.597211	−	0.802084i	$-0.703725\pi$
$521$	−14204.1	−1.19442	−0.597211	+	0.802084i	$0.703725\pi$
$522$	0	0
$523$	8810.56	0.736632	0.368316	−	0.929701i	$-0.379934\pi$
$523$	8810.56	0.736632	0.368316	+	0.929701i	$0.379934\pi$
$524$	−11535.1	−0.961667
$525$	0	0
$526$	−2233.07	−0.185107
$527$	−1226.44	−0.101375
$528$	0	0
$529$	−6554.14	−0.538682
$530$	2057.02	0.168588
$531$	0	0
$532$	−2787.10	−0.227135
$533$	23477.5	1.90792
$534$	0	0
$535$	1463.31	0.118252
$536$	−7436.11	−0.599237
$537$	0	0
$538$	6505.85	0.521351
$539$	14699.6	1.17469
$540$	0	0
$541$	11957.8	0.950291	0.475146	−	0.879907i	$-0.342395\pi$
$541$	11957.8	0.950291	0.475146	+	0.879907i	$0.342395\pi$
$542$	−2465.59	−0.195399
$543$	0	0
$544$	497.296	0.0391937
$545$	3419.70	0.268778
$546$	0	0
$547$	−9152.37	−0.715406	−0.357703	−	0.933835i	$-0.616440\pi$
$547$	−9152.37	−0.715406	−0.357703	+	0.933835i	$0.616440\pi$
$548$	3933.24	0.306605
$549$	0	0
$550$	733.616	0.0568754
$551$	1912.77	0.147889
$552$	0	0
$553$	24446.6	1.87988
$554$	7063.43	0.541691
$555$	0	0
$556$	−662.674	−0.0505461
$557$	−17620.6	−1.34041	−0.670205	−	0.742176i	$-0.733794\pi$
$557$	−17620.6	−1.34041	−0.670205	+	0.742176i	$0.733794\pi$
$558$	0	0
$559$	−42955.2	−3.25011
$560$	2933.79	0.221384
$561$	0	0
$562$	10470.2	0.785870
$563$	9746.76	0.729622	0.364811	−	0.931082i	$-0.381134\pi$
$563$	9746.76	0.729622	0.364811	+	0.931082i	$0.381134\pi$
$564$	0	0
$565$	11671.1	0.869035
$566$	−10133.0	−0.752512
$567$	0	0
$568$	−2215.08	−0.163632
$569$	24713.6	1.82082	0.910410	−	0.413706i	$-0.135766\pi$
$569$	24713.6	1.82082	0.910410	+	0.413706i	$0.135766\pi$
$570$	0	0
$571$	13597.5	0.996564	0.498282	−	0.867015i	$-0.333964\pi$
$571$	13597.5	0.996564	0.498282	+	0.867015i	$0.333964\pi$
$572$	5061.76	0.370005
$573$	0	0
$574$	−19965.3	−1.45181
$575$	−1872.98	−0.135841
$576$	0	0
$577$	1638.67	0.118230	0.0591152	−	0.998251i	$-0.481172\pi$
$577$	1638.67	0.118230	0.0591152	+	0.998251i	$0.481172\pi$
$578$	−9342.99	−0.672348
$579$	0	0
$580$	−2013.45	−0.144144
$581$	−44117.6	−3.15027
$582$	0	0
$583$	−3018.13	−0.214405
$584$	1833.73	0.129932
$585$	0	0
$586$	−14775.6	−1.04160
$587$	14672.1	1.03166	0.515830	−	0.856691i	$-0.327484\pi$
$587$	14672.1	1.03166	0.515830	+	0.856691i	$0.327484\pi$
$588$	0	0
$589$	−1499.46	−0.104897
$590$	3337.36	0.232876
$591$	0	0
$592$	−6250.76	−0.433960
$593$	−10486.6	−0.726195	−0.363098	−	0.931751i	$-0.618281\pi$
$593$	−10486.6	−0.726195	−0.363098	+	0.931751i	$0.618281\pi$
$594$	0	0
$595$	2849.53	0.196335
$596$	−3763.78	−0.258675
$597$	0	0
$598$	−12923.0	−0.883716
$599$	−3289.45	−0.224379	−0.112190	−	0.993687i	$-0.535786\pi$
$599$	−3289.45	−0.224379	−0.112190	+	0.993687i	$0.535786\pi$
$600$	0	0
$601$	18617.4	1.26359	0.631797	−	0.775134i	$-0.282317\pi$
$601$	18617.4	1.26359	0.631797	+	0.775134i	$0.282317\pi$
$602$	36529.4	2.47313
$603$	0	0
$604$	−7259.68	−0.489060
$605$	5578.62	0.374881
$606$	0	0
$607$	−1829.89	−0.122361	−0.0611805	−	0.998127i	$-0.519487\pi$
$607$	−1829.89	−0.122361	−0.0611805	+	0.998127i	$0.519487\pi$
$608$	608.000	0.0405554
$609$	0	0
$610$	595.066	0.0394976
$611$	40508.0	2.68213
$612$	0	0
$613$	16151.4	1.06419	0.532095	−	0.846685i	$-0.321405\pi$
$613$	16151.4	1.06419	0.532095	+	0.846685i	$0.321405\pi$
$614$	−2948.93	−0.193826
$615$	0	0
$616$	−4304.54	−0.281550
$617$	6155.98	0.401670	0.200835	−	0.979625i	$-0.435634\pi$
$617$	6155.98	0.401670	0.200835	+	0.979625i	$0.435634\pi$
$618$	0	0
$619$	−8516.24	−0.552983	−0.276492	−	0.961016i	$-0.589172\pi$
$619$	−8516.24	−0.552983	−0.276492	+	0.961016i	$0.589172\pi$
$620$	1578.38	0.102241
$621$	0	0
$622$	−15382.2	−0.991592
$623$	−27562.9	−1.77252
$624$	0	0
$625$	625.000	0.0400000
$626$	−9768.96	−0.623716
$627$	0	0
$628$	−2359.05	−0.149899
$629$	−6071.24	−0.384859
$630$	0	0
$631$	7705.45	0.486131	0.243066	−	0.970010i	$-0.421847\pi$
$631$	7705.45	0.486131	0.243066	+	0.970010i	$0.421847\pi$
$632$	−5332.97	−0.335655
$633$	0	0
$634$	−20664.1	−1.29444
$635$	12045.3	0.752764
$636$	0	0
$637$	86407.0	5.37452
$638$	2954.19	0.183319
$639$	0	0
$640$	−640.000	−0.0395285
$641$	−5000.54	−0.308127	−0.154063	−	0.988061i	$-0.549236\pi$
$641$	−5000.54	−0.308127	−0.154063	+	0.988061i	$0.549236\pi$
$642$	0	0
$643$	575.821	0.0353159	0.0176580	−	0.999844i	$-0.494379\pi$
$643$	575.821	0.0353159	0.0176580	+	0.999844i	$0.494379\pi$
$644$	10989.8	0.672452
$645$	0	0
$646$	590.539	0.0359666
$647$	6994.03	0.424983	0.212491	−	0.977163i	$-0.431842\pi$
$647$	6994.03	0.424983	0.212491	+	0.977163i	$0.431842\pi$
$648$	0	0
$649$	−4896.69	−0.296166
$650$	4312.34	0.260221
$651$	0	0
$652$	−3470.47	−0.208457
$653$	226.109	0.0135503	0.00677514	−	0.999977i	$-0.497843\pi$
$653$	226.109	0.0135503	0.00677514	+	0.999977i	$0.497843\pi$
$654$	0	0
$655$	14418.9	0.860141
$656$	4355.40	0.259223
$657$	0	0
$658$	−34448.2	−2.04093
$659$	7732.85	0.457100	0.228550	−	0.973532i	$-0.426602\pi$
$659$	7732.85	0.457100	0.228550	+	0.973532i	$0.426602\pi$
$660$	0	0
$661$	8785.71	0.516981	0.258491	−	0.966014i	$-0.416775\pi$
$661$	8785.71	0.516981	0.258491	+	0.966014i	$0.416775\pi$
$662$	13474.0	0.791063
$663$	0	0
$664$	9624.17	0.562485
$665$	3483.87	0.203156
$666$	0	0
$667$	−7542.27	−0.437838
$668$	−2094.13	−0.121294
$669$	0	0
$670$	9295.14	0.535974
$671$	−873.100	−0.0502319
$672$	0	0
$673$	−22887.0	−1.31089	−0.655445	−	0.755243i	$-0.727519\pi$
$673$	−22887.0	−1.31089	−0.655445	+	0.755243i	$0.727519\pi$
$674$	3589.45	0.205134
$675$	0	0
$676$	20966.0	1.19288
$677$	−11801.7	−0.669979	−0.334990	−	0.942222i	$-0.608733\pi$
$677$	−11801.7	−0.669979	−0.334990	+	0.942222i	$0.608733\pi$
$678$	0	0
$679$	50629.0	2.86151
$680$	−621.620	−0.0350559
$681$	0	0
$682$	−2315.85	−0.130027
$683$	−23596.0	−1.32192	−0.660962	−	0.750419i	$-0.729852\pi$
$683$	−23596.0	−1.32192	−0.660962	+	0.750419i	$0.729852\pi$
$684$	0	0
$685$	−4916.55	−0.274236
$686$	−48323.8	−2.68952
$687$	0	0
$688$	−7968.81	−0.441582
$689$	−17741.2	−0.980964
$690$	0	0
$691$	−2954.17	−0.162636	−0.0813182	−	0.996688i	$-0.525913\pi$
$691$	−2954.17	−0.162636	−0.0813182	+	0.996688i	$0.525913\pi$
$692$	12433.0	0.682994
$693$	0	0
$694$	22876.1	1.25124
$695$	828.343	0.0452098
$696$	0	0
$697$	4230.32	0.229892
$698$	−4139.85	−0.224492
$699$	0	0
$700$	−3667.23	−0.198012
$701$	24781.0	1.33519	0.667594	−	0.744525i	$-0.267324\pi$
$701$	24781.0	1.33519	0.667594	+	0.744525i	$0.267324\pi$
$702$	0	0
$703$	−7422.77	−0.398229
$704$	939.028	0.0502712
$705$	0	0
$706$	619.035	0.0329996
$707$	26976.7	1.43503
$708$	0	0
$709$	−14077.7	−0.745698	−0.372849	−	0.927892i	$-0.621619\pi$
$709$	−14077.7	−0.745698	−0.372849	+	0.927892i	$0.621619\pi$
$710$	2768.85	0.146357
$711$	0	0
$712$	6012.79	0.316487
$713$	5912.54	0.310556
$714$	0	0
$715$	−6327.19	−0.330942
$716$	−813.484	−0.0424600
$717$	0	0
$718$	−11385.1	−0.591764
$719$	−2275.65	−0.118035	−0.0590177	−	0.998257i	$-0.518797\pi$
$719$	−2275.65	−0.118035	−0.0590177	+	0.998257i	$0.518797\pi$
$720$	0	0
$721$	−12786.3	−0.660452
$722$	722.000	0.0372161
$723$	0	0
$724$	5803.00	0.297882
$725$	2516.81	0.128927
$726$	0	0
$727$	−20542.6	−1.04798	−0.523991	−	0.851724i	$-0.675557\pi$
$727$	−20542.6	−1.04798	−0.523991	+	0.851724i	$0.675557\pi$
$728$	−25302.9	−1.28817
$729$	0	0
$730$	−2292.17	−0.116215
$731$	−7739.96	−0.391618
$732$	0	0
$733$	−8477.26	−0.427169	−0.213584	−	0.976925i	$-0.568514\pi$
$733$	−8477.26	−0.427169	−0.213584	+	0.976925i	$0.568514\pi$
$734$	−11051.5	−0.555748
$735$	0	0
$736$	−2397.41	−0.120067
$737$	−13638.1	−0.681637
$738$	0	0
$739$	15515.6	0.772326	0.386163	−	0.922431i	$-0.373800\pi$
$739$	15515.6	0.772326	0.386163	+	0.922431i	$0.373800\pi$
$740$	7813.45	0.388146
$741$	0	0
$742$	15087.2	0.746452
$743$	13547.4	0.668917	0.334459	−	0.942410i	$-0.391447\pi$
$743$	13547.4	0.668917	0.334459	+	0.942410i	$0.391447\pi$
$744$	0	0
$745$	4704.73	0.231366
$746$	−8885.45	−0.436085
$747$	0	0
$748$	912.060	0.0445832
$749$	10732.6	0.523580
$750$	0	0
$751$	−31210.0	−1.51647	−0.758234	−	0.651983i	$-0.773938\pi$
$751$	−31210.0	−1.51647	−0.758234	+	0.651983i	$0.773938\pi$
$752$	7514.82	0.364411
$753$	0	0
$754$	17365.3	0.838736
$755$	9074.60	0.437428
$756$	0	0
$757$	18812.4	0.903233	0.451616	−	0.892212i	$-0.350848\pi$
$757$	18812.4	0.903233	0.451616	+	0.892212i	$0.350848\pi$
$758$	15457.0	0.740666
$759$	0	0
$760$	−760.000	−0.0362738
$761$	11694.6	0.557071	0.278535	−	0.960426i	$-0.410151\pi$
$761$	11694.6	0.557071	0.278535	+	0.960426i	$0.410151\pi$
$762$	0	0
$763$	25081.7	1.19006
$764$	−11443.9	−0.541920
$765$	0	0
$766$	13646.0	0.643670
$767$	−28783.7	−1.35504
$768$	0	0
$769$	−1511.75	−0.0708911	−0.0354456	−	0.999372i	$-0.511285\pi$
$769$	−1511.75	−0.0708911	−0.0354456	+	0.999372i	$0.511285\pi$
$770$	5380.68	0.251826
$771$	0	0
$772$	2875.83	0.134072
$773$	−2726.10	−0.126845	−0.0634224	−	0.997987i	$-0.520202\pi$
$773$	−2726.10	−0.126845	−0.0634224	+	0.997987i	$0.520202\pi$
$774$	0	0
$775$	−1972.98	−0.0914469
$776$	−11044.6	−0.510926
$777$	0	0
$778$	10144.5	0.467480
$779$	5172.04	0.237879
$780$	0	0
$781$	−4062.55	−0.186132
$782$	−2328.56	−0.106482
$783$	0	0
$784$	16029.7	0.730218
$785$	2948.81	0.134073
$786$	0	0
$787$	−25993.6	−1.17735	−0.588673	−	0.808371i	$-0.700350\pi$
$787$	−25993.6	−1.17735	−0.588673	+	0.808371i	$0.700350\pi$
$788$	−17648.0	−0.797822
$789$	0	0
$790$	6666.21	0.300219
$791$	85600.9	3.84781
$792$	0	0
$793$	−5132.25	−0.229825
$794$	12037.8	0.538044
$795$	0	0
$796$	−1304.24	−0.0580749
$797$	−33045.8	−1.46869	−0.734343	−	0.678778i	$-0.762510\pi$
$797$	−33045.8	−1.46869	−0.734343	+	0.678778i	$0.762510\pi$
$798$	0	0
$799$	7299.00	0.323179
$800$	800.000	0.0353553
$801$	0	0
$802$	−26581.7	−1.17037
$803$	3363.14	0.147799
$804$	0	0
$805$	−13737.3	−0.601460
$806$	−13613.0	−0.594910
$807$	0	0
$808$	−5884.93	−0.256226
$809$	−20828.5	−0.905179	−0.452590	−	0.891719i	$-0.649500\pi$
$809$	−20828.5	−0.905179	−0.452590	+	0.891719i	$0.649500\pi$
$810$	0	0
$811$	5875.03	0.254378	0.127189	−	0.991879i	$-0.459405\pi$
$811$	5875.03	0.254378	0.127189	+	0.991879i	$0.459405\pi$
$812$	−14767.5	−0.638226
$813$	0	0
$814$	−11464.1	−0.493634
$815$	4338.09	0.186450
$816$	0	0
$817$	−9462.97	−0.405223
$818$	−26963.5	−1.15251
$819$	0	0
$820$	−5444.26	−0.231856
$821$	−32893.9	−1.39830	−0.699151	−	0.714974i	$-0.746439\pi$
$821$	−32893.9	−1.39830	−0.699151	+	0.714974i	$0.746439\pi$
$822$	0	0
$823$	−11122.9	−0.471104	−0.235552	−	0.971862i	$-0.575690\pi$
$823$	−11122.9	−0.471104	−0.235552	+	0.971862i	$0.575690\pi$
$824$	2789.30	0.117925
$825$	0	0
$826$	24477.8	1.03110
$827$	6660.32	0.280051	0.140025	−	0.990148i	$-0.455282\pi$
$827$	6660.32	0.280051	0.140025	+	0.990148i	$0.455282\pi$
$828$	0	0
$829$	15622.0	0.654493	0.327247	−	0.944939i	$-0.393879\pi$
$829$	15622.0	0.654493	0.327247	+	0.944939i	$0.393879\pi$
$830$	−12030.2	−0.503102
$831$	0	0
$832$	5519.79	0.230005
$833$	15569.4	0.647595
$834$	0	0
$835$	2617.66	0.108489
$836$	1115.10	0.0461321
$837$	0	0
$838$	−7497.78	−0.309077
$839$	17641.4	0.725921	0.362961	−	0.931804i	$-0.381766\pi$
$839$	17641.4	0.725921	0.362961	+	0.931804i	$0.381766\pi$
$840$	0	0
$841$	−14254.1	−0.584447
$842$	639.389	0.0261696
$843$	0	0
$844$	7750.33	0.316087
$845$	−26207.5	−1.06694
$846$	0	0
$847$	40916.2	1.65985
$848$	−3291.24	−0.133280
$849$	0	0
$850$	777.025	0.0313550
$851$	29268.8	1.17899
$852$	0	0
$853$	−46294.4	−1.85825	−0.929127	−	0.369760i	$-0.879440\pi$
$853$	−46294.4	−1.85825	−0.929127	+	0.369760i	$0.879440\pi$
$854$	4364.49	0.174883
$855$	0	0
$856$	−2341.30	−0.0934861
$857$	−8139.72	−0.324443	−0.162221	−	0.986754i	$-0.551866\pi$
$857$	−8139.72	−0.324443	−0.162221	+	0.986754i	$0.551866\pi$
$858$	0	0
$859$	−8902.12	−0.353593	−0.176796	−	0.984247i	$-0.556573\pi$
$859$	−8902.12	−0.353593	−0.176796	+	0.984247i	$0.556573\pi$
$860$	9961.02	0.394963
$861$	0	0
$862$	−21911.5	−0.865788
$863$	−44952.6	−1.77312	−0.886561	−	0.462612i	$-0.846912\pi$
$863$	−44952.6	−1.77312	−0.886561	+	0.462612i	$0.846912\pi$
$864$	0	0
$865$	−15541.3	−0.610889
$866$	−28047.0	−1.10055
$867$	0	0
$868$	11576.6	0.452689
$869$	−9780.88	−0.381811
$870$	0	0
$871$	−80167.5	−3.11868
$872$	−5471.52	−0.212487
$873$	0	0
$874$	−2846.92	−0.110181
$875$	4584.04	0.177107
$876$	0	0
$877$	−17490.2	−0.673433	−0.336716	−	0.941606i	$-0.609316\pi$
$877$	−17490.2	−0.673433	−0.336716	+	0.941606i	$0.609316\pi$
$878$	23096.3	0.887770
$879$	0	0
$880$	−1173.79	−0.0449640
$881$	33061.3	1.26432	0.632158	−	0.774840i	$-0.282169\pi$
$881$	33061.3	1.26432	0.632158	+	0.774840i	$0.282169\pi$
$882$	0	0
$883$	3002.47	0.114429	0.0572147	−	0.998362i	$-0.481778\pi$
$883$	3002.47	0.114429	0.0572147	+	0.998362i	$0.481778\pi$
$884$	5361.26	0.203981
$885$	0	0
$886$	−26185.6	−0.992913
$887$	−17065.4	−0.645999	−0.323000	−	0.946399i	$-0.604691\pi$
$887$	−17065.4	−0.645999	−0.323000	+	0.946399i	$0.604691\pi$
$888$	0	0
$889$	88346.2	3.33300
$890$	−7515.99	−0.283075
$891$	0	0
$892$	1710.86	0.0642194
$893$	8923.85	0.334407
$894$	0	0
$895$	1016.86	0.0379773
$896$	−4694.06	−0.175019
$897$	0	0
$898$	11479.8	0.426600
$899$	−7944.96	−0.294749
$900$	0	0
$901$	−3196.72	−0.118200
$902$	7987.98	0.294868
$903$	0	0
$904$	−18673.7	−0.687033
$905$	−7253.75	−0.266434
$906$	0	0
$907$	5491.57	0.201041	0.100521	−	0.994935i	$-0.467949\pi$
$907$	5491.57	0.201041	0.100521	+	0.994935i	$0.467949\pi$
$908$	5614.79	0.205213
$909$	0	0
$910$	31628.7	1.15218
$911$	40901.2	1.48751	0.743753	−	0.668454i	$-0.233044\pi$
$911$	40901.2	1.48751	0.743753	+	0.668454i	$0.233044\pi$
$912$	0	0
$913$	17651.1	0.639831
$914$	19872.9	0.719188
$915$	0	0
$916$	4471.61	0.161295
$917$	105755.	3.80843
$918$	0	0
$919$	8198.91	0.294295	0.147148	−	0.989115i	$-0.452991\pi$
$919$	8198.91	0.294295	0.147148	+	0.989115i	$0.452991\pi$
$920$	2996.76	0.107392
$921$	0	0
$922$	22145.7	0.791030
$923$	−23880.4	−0.851608
$924$	0	0
$925$	−9766.81	−0.347168
$926$	9947.05	0.353003
$927$	0	0
$928$	3221.51	0.113956
$929$	−9605.08	−0.339217	−0.169609	−	0.985512i	$-0.554250\pi$
$929$	−9605.08	−0.339217	−0.169609	+	0.985512i	$0.554250\pi$
$930$	0	0
$931$	19035.3	0.670094
$932$	15626.0	0.549193
$933$	0	0
$934$	32320.2	1.13228
$935$	−1140.07	−0.0398764
$936$	0	0
$937$	43484.7	1.51610	0.758049	−	0.652197i	$-0.226153\pi$
$937$	43484.7	1.51610	0.758049	+	0.652197i	$0.226153\pi$
$938$	68174.9	2.37312
$939$	0	0
$940$	−9393.52	−0.325939
$941$	13138.1	0.455142	0.227571	−	0.973761i	$-0.426922\pi$
$941$	13138.1	0.455142	0.227571	+	0.973761i	$0.426922\pi$
$942$	0	0
$943$	−20393.9	−0.704260
$944$	−5339.78	−0.184105
$945$	0	0
$946$	−14615.1	−0.502303
$947$	−34354.8	−1.17886	−0.589430	−	0.807820i	$-0.700647\pi$
$947$	−34354.8	−1.17886	−0.589430	+	0.807820i	$0.700647\pi$
$948$	0	0
$949$	19769.2	0.676222
$950$	950.000	0.0324443
$951$	0	0
$952$	−4559.25	−0.155216
$953$	16359.3	0.556064	0.278032	−	0.960572i	$-0.410318\pi$
$953$	16359.3	0.556064	0.278032	+	0.960572i	$0.410318\pi$
$954$	0	0
$955$	14304.9	0.484708
$956$	27264.6	0.922384
$957$	0	0
$958$	34914.5	1.17749
$959$	−36060.3	−1.21423
$960$	0	0
$961$	−23562.8	−0.790936
$962$	−67388.4	−2.25851
$963$	0	0
$964$	11714.0	0.391372
$965$	−3594.79	−0.119918
$966$	0	0
$967$	−12501.4	−0.415736	−0.207868	−	0.978157i	$-0.566653\pi$
$967$	−12501.4	−0.415736	−0.207868	+	0.978157i	$0.566653\pi$
$968$	−8925.79	−0.296369
$969$	0	0
$970$	13805.8	0.456987
$971$	−2645.39	−0.0874299	−0.0437150	−	0.999044i	$-0.513919\pi$
$971$	−2645.39	−0.0874299	−0.0437150	+	0.999044i	$0.513919\pi$
$972$	0	0
$973$	6075.45	0.200175
$974$	28184.7	0.927204
$975$	0	0
$976$	−952.106	−0.0312256
$977$	19046.7	0.623703	0.311852	−	0.950131i	$-0.399051\pi$
$977$	19046.7	0.623703	0.311852	+	0.950131i	$0.399051\pi$
$978$	0	0
$979$	11027.7	0.360007
$980$	−20037.2	−0.653127
$981$	0	0
$982$	10527.6	0.342108
$983$	−9676.99	−0.313986	−0.156993	−	0.987600i	$-0.550180\pi$
$983$	−9676.99	−0.313986	−0.156993	+	0.987600i	$0.550180\pi$
$984$	0	0
$985$	22060.0	0.713594
$986$	3128.99	0.101062
$987$	0	0
$988$	6554.75	0.211067
$989$	37313.5	1.19970
$990$	0	0
$991$	39653.0	1.27106	0.635529	−	0.772077i	$-0.280782\pi$
$991$	39653.0	1.27106	0.635529	+	0.772077i	$0.280782\pi$
$992$	−2525.41	−0.0808284
$993$	0	0
$994$	20308.0	0.648020
$995$	1630.30	0.0519437
$996$	0	0
$997$	−12635.3	−0.401367	−0.200683	−	0.979656i	$-0.564316\pi$
$997$	−12635.3	−0.401367	−0.200683	+	0.979656i	$0.564316\pi$
$998$	−15893.3	−0.504103
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.a.o.1.1		2
3.2	odd	2		570.4.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.4.a.k.1.1	✓	2	3.2	odd	2
1710.4.a.o.1.1		2	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:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.6723 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} -36.6723 q^{7} +8.00000 q^{8} -10.0000 q^{10} +14.6723 q^{11} +86.2467 q^{13} -73.3446 q^{14} +16.0000 q^{16} +15.5405 q^{17} +19.0000 q^{19} -20.0000 q^{20} +29.3446 q^{22} -74.9190 q^{23} +25.0000 q^{25} +172.493 q^{26} -146.689 q^{28} +100.672 q^{29} -78.9190 q^{31} +32.0000 q^{32} +31.0810 q^{34} +183.362 q^{35} -390.672 q^{37} +38.0000 q^{38} -40.0000 q^{40} +272.213 q^{41} -498.051 q^{43} +58.6893 q^{44} -149.838 q^{46} +469.676 q^{47} +1001.86 q^{49} +50.0000 q^{50} +344.987 q^{52} -205.702 q^{53} -73.3616 q^{55} -293.379 q^{56} +201.345 q^{58} -333.736 q^{59} -59.5066 q^{61} -157.838 q^{62} +64.0000 q^{64} -431.234 q^{65} -929.514 q^{67} +62.1620 q^{68} +366.723 q^{70} -276.885 q^{71} +229.217 q^{73} -781.345 q^{74} +76.0000 q^{76} -538.068 q^{77} -666.621 q^{79} -80.0000 q^{80} +544.426 q^{82} +1203.02 q^{83} -77.7025 q^{85} -996.102 q^{86} +117.379 q^{88} +751.599 q^{89} -3162.87 q^{91} -299.676 q^{92} +939.352 q^{94} -95.0000 q^{95} -1380.58 q^{97} +2003.72 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} - 34 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 - 34 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} - 34 q^{7} + 16 q^{8} - 20 q^{10} - 10 q^{11} + 2 q^{13} - 68 q^{14} + 32 q^{16} + 136 q^{17} + 38 q^{19} - 40 q^{20} - 20 q^{22} + 60 q^{23} + 50 q^{25} + 4 q^{26} - 136 q^{28} + 162 q^{29} + 52 q^{31} + 64 q^{32} + 272 q^{34} + 170 q^{35} - 742 q^{37} + 76 q^{38} - 80 q^{40} + 610 q^{41} - 642 q^{43} - 40 q^{44} + 120 q^{46} + 100 q^{47} + 666 q^{49} + 100 q^{50} + 8 q^{52} - 936 q^{53} + 50 q^{55} - 272 q^{56} + 324 q^{58} - 956 q^{59} - 460 q^{61} + 104 q^{62} + 128 q^{64} - 10 q^{65} - 600 q^{67} + 544 q^{68} + 340 q^{70} - 580 q^{71} - 276 q^{73} - 1484 q^{74} + 152 q^{76} - 604 q^{77} - 1648 q^{79} - 160 q^{80} + 1220 q^{82} + 1488 q^{83} - 680 q^{85} - 1284 q^{86} - 80 q^{88} - 346 q^{89} - 3388 q^{91} + 240 q^{92} + 200 q^{94} - 190 q^{95} - 1830 q^{97} + 1332 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 - 34 * q^7 + 16 * q^8 - 20 * q^10 - 10 * q^11 + 2 * q^13 - 68 * q^14 + 32 * q^16 + 136 * q^17 + 38 * q^19 - 40 * q^20 - 20 * q^22 + 60 * q^23 + 50 * q^25 + 4 * q^26 - 136 * q^28 + 162 * q^29 + 52 * q^31 + 64 * q^32 + 272 * q^34 + 170 * q^35 - 742 * q^37 + 76 * q^38 - 80 * q^40 + 610 * q^41 - 642 * q^43 - 40 * q^44 + 120 * q^46 + 100 * q^47 + 666 * q^49 + 100 * q^50 + 8 * q^52 - 936 * q^53 + 50 * q^55 - 272 * q^56 + 324 * q^58 - 956 * q^59 - 460 * q^61 + 104 * q^62 + 128 * q^64 - 10 * q^65 - 600 * q^67 + 544 * q^68 + 340 * q^70 - 580 * q^71 - 276 * q^73 - 1484 * q^74 + 152 * q^76 - 604 * q^77 - 1648 * q^79 - 160 * q^80 + 1220 * q^82 + 1488 * q^83 - 680 * q^85 - 1284 * q^86 - 80 * q^88 - 346 * q^89 - 3388 * q^91 + 240 * q^92 + 200 * q^94 - 190 * q^95 - 1830 * q^97 + 1332 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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