LMFDB - Embedded newform 1710.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1710,4,Mod(1,1710)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,-4,0,8,10,0,-10,-16,0,-20,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$100.893266110$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -2.76393 q^{7} -8.00000 q^{8} -10.0000 q^{10} -7.70820 q^{11} -15.9574 q^{13} +5.52786 q^{14} +16.0000 q^{16} -44.0263 q^{17} -19.0000 q^{19} +20.0000 q^{20} +15.4164 q^{22} +82.2492 q^{23} +25.0000 q^{25} +31.9149 q^{26} -11.0557 q^{28} +72.6262 q^{29} -219.108 q^{31} -32.0000 q^{32} +88.0526 q^{34} -13.8197 q^{35} -362.705 q^{37} +38.0000 q^{38} -40.0000 q^{40} +150.154 q^{41} +331.728 q^{43} -30.8328 q^{44} -164.498 q^{46} -267.967 q^{47} -335.361 q^{49} -50.0000 q^{50} -63.8297 q^{52} +759.633 q^{53} -38.5410 q^{55} +22.1115 q^{56} -145.252 q^{58} +721.515 q^{59} +664.014 q^{61} +438.217 q^{62} +64.0000 q^{64} -79.7871 q^{65} +437.686 q^{67} -176.105 q^{68} +27.6393 q^{70} +796.604 q^{71} -994.231 q^{73} +725.410 q^{74} -76.0000 q^{76} +21.3050 q^{77} -1009.93 q^{79} +80.0000 q^{80} -300.308 q^{82} -235.882 q^{83} -220.132 q^{85} -663.457 q^{86} +61.6656 q^{88} -1161.27 q^{89} +44.1052 q^{91} +328.997 q^{92} +535.935 q^{94} -95.0000 q^{95} +1443.28 q^{97} +670.721 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 10 q^{5} - 10 q^{7} - 16 q^{8} - 20 q^{10} - 2 q^{11} + 62 q^{13} + 20 q^{14} + 32 q^{16} + 64 q^{17} - 38 q^{19} + 40 q^{20} + 4 q^{22} + 84 q^{23} + 50 q^{25} - 124 q^{26} - 40 q^{28}+ \cdots + 1252 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 10 * q^5 - 10 * q^7 - 16 * q^8 - 20 * q^10 - 2 * q^11 + 62 * q^13 + 20 * q^14 + 32 * q^16 + 64 * q^17 - 38 * q^19 + 40 * q^20 + 4 * q^22 + 84 * q^23 + 50 * q^25 - 124 * q^26 - 40 * q^28 + 266 * q^29 - 152 * q^31 - 64 * q^32 - 128 * q^34 - 50 * q^35 - 390 * q^37 + 76 * q^38 - 80 * q^40 + 430 * q^41 - 146 * q^43 - 8 * q^44 - 168 * q^46 - 44 * q^47 - 626 * q^49 - 100 * q^50 + 248 * q^52 + 920 * q^53 - 10 * q^55 + 80 * q^56 - 532 * q^58 + 164 * q^59 - 112 * q^61 + 304 * q^62 + 128 * q^64 + 310 * q^65 - 28 * q^67 + 256 * q^68 + 100 * q^70 + 824 * q^71 + 24 * q^73 + 780 * q^74 - 152 * q^76 - 20 * q^77 - 392 * q^79 + 160 * q^80 - 860 * q^82 + 208 * q^83 + 320 * q^85 + 292 * q^86 + 16 * q^88 - 82 * q^89 - 520 * q^91 + 336 * q^92 + 88 * q^94 - 190 * q^95 + 1290 * q^97 + 1252 * 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 $

$2$	−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
$7$	−2.76393	−0.149238	−0.0746192	−	0.997212i	$-0.523774\pi$
$7$	−2.76393	−0.149238	−0.0746192	+	0.997212i	$0.523774\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	−7.70820	−0.211283	−0.105641	−	0.994404i	$-0.533690\pi$
$11$	−7.70820	−0.211283	−0.105641	+	0.994404i	$0.533690\pi$
$12$	0	0
$13$	−15.9574	−0.340446	−0.170223	−	0.985406i	$-0.554449\pi$
$13$	−15.9574	−0.340446	−0.170223	+	0.985406i	$0.554449\pi$
$14$	5.52786	0.105527
$15$	0	0
$16$	16.0000	0.250000
$17$	−44.0263	−0.628115	−0.314057	−	0.949404i	$-0.601688\pi$
$17$	−44.0263	−0.628115	−0.314057	+	0.949404i	$0.601688\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	20.0000	0.223607
$21$	0	0
$22$	15.4164	0.149400
$23$	82.2492	0.745659	0.372829	−	0.927900i	$-0.378388\pi$
$23$	82.2492	0.745659	0.372829	+	0.927900i	$0.378388\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	31.9149	0.240731
$27$	0	0
$28$	−11.0557	−0.0746192
$29$	72.6262	0.465046	0.232523	−	0.972591i	$-0.425302\pi$
$29$	72.6262	0.465046	0.232523	+	0.972591i	$0.425302\pi$
$30$	0	0
$31$	−219.108	−1.26945	−0.634726	−	0.772737i	$-0.718887\pi$
$31$	−219.108	−1.26945	−0.634726	+	0.772737i	$0.718887\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	88.0526	0.444144
$35$	−13.8197	−0.0667414
$36$	0	0
$37$	−362.705	−1.61158	−0.805789	−	0.592203i	$-0.798258\pi$
$37$	−362.705	−1.61158	−0.805789	+	0.592203i	$0.798258\pi$
$38$	38.0000	0.162221
$39$	0	0
$40$	−40.0000	−0.158114
$41$	150.154	0.571954	0.285977	−	0.958236i	$-0.407682\pi$
$41$	150.154	0.571954	0.285977	+	0.958236i	$0.407682\pi$
$42$	0	0
$43$	331.728	1.17647	0.588234	−	0.808691i	$-0.299824\pi$
$43$	331.728	1.17647	0.588234	+	0.808691i	$0.299824\pi$
$44$	−30.8328	−0.105641
$45$	0	0
$46$	−164.498	−0.527260
$47$	−267.967	−0.831640	−0.415820	−	0.909447i	$-0.636505\pi$
$47$	−267.967	−0.831640	−0.415820	+	0.909447i	$0.636505\pi$
$48$	0	0
$49$	−335.361	−0.977728
$50$	−50.0000	−0.141421
$51$	0	0
$52$	−63.8297	−0.170223
$53$	759.633	1.96875	0.984374	−	0.176091i	$-0.0563453\pi$
$53$	759.633	1.96875	0.984374	+	0.176091i	$0.0563453\pi$
$54$	0	0
$55$	−38.5410	−0.0944885
$56$	22.1115	0.0527637
$57$	0	0
$58$	−145.252	−0.328837
$59$	721.515	1.59209	0.796045	−	0.605237i	$-0.206922\pi$
$59$	721.515	1.59209	0.796045	+	0.605237i	$0.206922\pi$
$60$	0	0
$61$	664.014	1.39374	0.696871	−	0.717196i	$-0.254575\pi$
$61$	664.014	1.39374	0.696871	+	0.717196i	$0.254575\pi$
$62$	438.217	0.897638
$63$	0	0
$64$	64.0000	0.125000
$65$	−79.7871	−0.152252
$66$	0	0
$67$	437.686	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$67$	437.686	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$68$	−176.105	−0.314057
$69$	0	0
$70$	27.6393	0.0471933
$71$	796.604	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$71$	796.604	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$72$	0	0
$73$	−994.231	−1.59405	−0.797027	−	0.603944i	$-0.793595\pi$
$73$	−994.231	−1.59405	−0.797027	+	0.603944i	$0.793595\pi$
$74$	725.410	1.13956
$75$	0	0
$76$	−76.0000	−0.114708
$77$	21.3050	0.0315315
$78$	0	0
$79$	−1009.93	−1.43830	−0.719151	−	0.694854i	$-0.755469\pi$
$79$	−1009.93	−1.43830	−0.719151	+	0.694854i	$0.755469\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−300.308	−0.404433
$83$	−235.882	−0.311945	−0.155973	−	0.987761i	$-0.549851\pi$
$83$	−235.882	−0.311945	−0.155973	+	0.987761i	$0.549851\pi$
$84$	0	0
$85$	−220.132	−0.280901
$86$	−663.457	−0.831888
$87$	0	0
$88$	61.6656	0.0746998
$89$	−1161.27	−1.38308	−0.691541	−	0.722337i	$-0.743068\pi$
$89$	−1161.27	−1.38308	−0.691541	+	0.722337i	$0.743068\pi$
$90$	0	0
$91$	44.1052	0.0508075
$92$	328.997	0.372829
$93$	0	0
$94$	535.935	0.588058
$95$	−95.0000	−0.102598
$96$	0	0
$97$	1443.28	1.51075	0.755374	−	0.655294i	$-0.227455\pi$
$97$	1443.28	1.51075	0.755374	+	0.655294i	$0.227455\pi$
$98$	670.721	0.691358
$99$	0	0
$100$	100.000	0.100000
$101$	1903.41	1.87521	0.937606	−	0.347700i	$-0.113037\pi$
$101$	1903.41	1.87521	0.937606	+	0.347700i	$0.113037\pi$
$102$	0	0
$103$	−844.519	−0.807892	−0.403946	−	0.914783i	$-0.632362\pi$
$103$	−844.519	−0.807892	−0.403946	+	0.914783i	$0.632362\pi$
$104$	127.659	0.120366
$105$	0	0
$106$	−1519.27	−1.39211
$107$	274.597	0.248097	0.124048	−	0.992276i	$-0.460412\pi$
$107$	274.597	0.248097	0.124048	+	0.992276i	$0.460412\pi$
$108$	0	0
$109$	1621.35	1.42475	0.712373	−	0.701801i	$-0.247620\pi$
$109$	1621.35	1.42475	0.712373	+	0.701801i	$0.247620\pi$
$110$	77.0820	0.0668135
$111$	0	0
$112$	−44.2229	−0.0373096
$113$	−693.712	−0.577513	−0.288756	−	0.957403i	$-0.593242\pi$
$113$	−693.712	−0.577513	−0.288756	+	0.957403i	$0.593242\pi$
$114$	0	0
$115$	411.246	0.333469
$116$	290.505	0.232523
$117$	0	0
$118$	−1443.03	−1.12578
$119$	121.686	0.0937388
$120$	0	0
$121$	−1271.58	−0.955360
$122$	−1328.03	−0.985524
$123$	0	0
$124$	−876.433	−0.634726
$125$	125.000	0.0894427
$126$	0	0
$127$	−1310.86	−0.915906	−0.457953	−	0.888976i	$-0.651417\pi$
$127$	−1310.86	−0.915906	−0.457953	+	0.888976i	$0.651417\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	159.574	0.107658
$131$	241.321	0.160949	0.0804746	−	0.996757i	$-0.474356\pi$
$131$	241.321	0.160949	0.0804746	+	0.996757i	$0.474356\pi$
$132$	0	0
$133$	52.5147	0.0342376
$134$	−875.371	−0.564333
$135$	0	0
$136$	352.210	0.222072
$137$	860.610	0.536692	0.268346	−	0.963323i	$-0.413523\pi$
$137$	860.610	0.536692	0.268346	+	0.963323i	$0.413523\pi$
$138$	0	0
$139$	2355.81	1.43754	0.718768	−	0.695250i	$-0.244706\pi$
$139$	2355.81	1.43754	0.718768	+	0.695250i	$0.244706\pi$
$140$	−55.2786	−0.0333707
$141$	0	0
$142$	−1593.21	−0.941542
$143$	123.003	0.0719303
$144$	0	0
$145$	363.131	0.207975
$146$	1988.46	1.12717
$147$	0	0
$148$	−1450.82	−0.805789
$149$	932.079	0.512476	0.256238	−	0.966614i	$-0.417517\pi$
$149$	932.079	0.512476	0.256238	+	0.966614i	$0.417517\pi$
$150$	0	0
$151$	560.263	0.301944	0.150972	−	0.988538i	$-0.451760\pi$
$151$	560.263	0.301944	0.150972	+	0.988538i	$0.451760\pi$
$152$	152.000	0.0811107
$153$	0	0
$154$	−42.6099	−0.0222961
$155$	−1095.54	−0.567716
$156$	0	0
$157$	309.850	0.157508	0.0787538	−	0.996894i	$-0.474906\pi$
$157$	309.850	0.157508	0.0787538	+	0.996894i	$0.474906\pi$
$158$	2019.86	1.01703
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−227.331	−0.111281
$162$	0	0
$163$	3249.65	1.56155	0.780774	−	0.624813i	$-0.214825\pi$
$163$	3249.65	1.56155	0.780774	+	0.624813i	$0.214825\pi$
$164$	600.616	0.285977
$165$	0	0
$166$	471.765	0.220579
$167$	−2159.50	−1.00064	−0.500321	−	0.865840i	$-0.666785\pi$
$167$	−2159.50	−1.00064	−0.500321	+	0.865840i	$0.666785\pi$
$168$	0	0
$169$	−1942.36	−0.884097
$170$	440.263	0.198627
$171$	0	0
$172$	1326.91	0.588234
$173$	−1667.41	−0.732782	−0.366391	−	0.930461i	$-0.619407\pi$
$173$	−1667.41	−0.732782	−0.366391	+	0.930461i	$0.619407\pi$
$174$	0	0
$175$	−69.0983	−0.0298477
$176$	−123.331	−0.0528207
$177$	0	0
$178$	2322.54	0.977987
$179$	−438.865	−0.183253	−0.0916266	−	0.995793i	$-0.529207\pi$
$179$	−438.865	−0.183253	−0.0916266	+	0.995793i	$0.529207\pi$
$180$	0	0
$181$	−3069.52	−1.26053	−0.630265	−	0.776380i	$-0.717054\pi$
$181$	−3069.52	−1.26053	−0.630265	+	0.776380i	$0.717054\pi$
$182$	−88.2105	−0.0359264
$183$	0	0
$184$	−657.994	−0.263630
$185$	−1813.53	−0.720719
$186$	0	0
$187$	339.364	0.132710
$188$	−1071.87	−0.415820
$189$	0	0
$190$	190.000	0.0725476
$191$	628.134	0.237959	0.118980	−	0.992897i	$-0.462038\pi$
$191$	628.134	0.237959	0.118980	+	0.992897i	$0.462038\pi$
$192$	0	0
$193$	2510.75	0.936411	0.468206	−	0.883619i	$-0.344901\pi$
$193$	2510.75	0.936411	0.468206	+	0.883619i	$0.344901\pi$
$194$	−2886.55	−1.06826
$195$	0	0
$196$	−1341.44	−0.488864
$197$	−235.723	−0.0852516	−0.0426258	−	0.999091i	$-0.513572\pi$
$197$	−235.723	−0.0852516	−0.0426258	+	0.999091i	$0.513572\pi$
$198$	0	0
$199$	4469.36	1.59208	0.796041	−	0.605243i	$-0.206924\pi$
$199$	4469.36	1.59208	0.796041	+	0.605243i	$0.206924\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−3806.82	−1.32597
$203$	−200.734	−0.0694027
$204$	0	0
$205$	750.770	0.255786
$206$	1689.04	0.571266
$207$	0	0
$208$	−255.319	−0.0851114
$209$	146.456	0.0484716
$210$	0	0
$211$	−1282.95	−0.418588	−0.209294	−	0.977853i	$-0.567117\pi$
$211$	−1282.95	−0.418588	−0.209294	+	0.977853i	$0.567117\pi$
$212$	3038.53	0.984374
$213$	0	0
$214$	−549.195	−0.175431
$215$	1658.64	0.526132
$216$	0	0
$217$	605.601	0.189451
$218$	−3242.70	−1.00745
$219$	0	0
$220$	−154.164	−0.0472443
$221$	702.547	0.213839
$222$	0	0
$223$	−1613.19	−0.484427	−0.242213	−	0.970223i	$-0.577873\pi$
$223$	−1613.19	−0.484427	−0.242213	+	0.970223i	$0.577873\pi$
$224$	88.4458	0.0263819
$225$	0	0
$226$	1387.42	0.408363
$227$	−5274.03	−1.54207	−0.771035	−	0.636793i	$-0.780261\pi$
$227$	−5274.03	−1.54207	−0.771035	+	0.636793i	$0.780261\pi$
$228$	0	0
$229$	2251.16	0.649611	0.324805	−	0.945781i	$-0.394701\pi$
$229$	2251.16	0.649611	0.324805	+	0.945781i	$0.394701\pi$
$230$	−822.492	−0.235798
$231$	0	0
$232$	−581.009	−0.164419
$233$	2543.84	0.715246	0.357623	−	0.933866i	$-0.383587\pi$
$233$	2543.84	0.715246	0.357623	+	0.933866i	$0.383587\pi$
$234$	0	0
$235$	−1339.84	−0.371921
$236$	2886.06	0.796045
$237$	0	0
$238$	−243.371	−0.0662833
$239$	−1207.99	−0.326939	−0.163470	−	0.986548i	$-0.552269\pi$
$239$	−1207.99	−0.326939	−0.163470	+	0.986548i	$0.552269\pi$
$240$	0	0
$241$	4561.52	1.21923	0.609613	−	0.792699i	$-0.291325\pi$
$241$	4561.52	1.21923	0.609613	+	0.792699i	$0.291325\pi$
$242$	2543.17	0.675541
$243$	0	0
$244$	2656.06	0.696871
$245$	−1676.80	−0.437253
$246$	0	0
$247$	303.191	0.0781036
$248$	1752.87	0.448819
$249$	0	0
$250$	−250.000	−0.0632456
$251$	6059.90	1.52389	0.761947	−	0.647639i	$-0.224244\pi$
$251$	6059.90	1.52389	0.761947	+	0.647639i	$0.224244\pi$
$252$	0	0
$253$	−633.994	−0.157545
$254$	2621.72	0.647643
$255$	0	0
$256$	256.000	0.0625000
$257$	6363.59	1.54455	0.772276	−	0.635287i	$-0.219118\pi$
$257$	6363.59	1.54455	0.772276	+	0.635287i	$0.219118\pi$
$258$	0	0
$259$	1002.49	0.240509
$260$	−319.149	−0.0761260
$261$	0	0
$262$	−482.642	−0.113808
$263$	−2908.69	−0.681967	−0.340984	−	0.940069i	$-0.610760\pi$
$263$	−2908.69	−0.681967	−0.340984	+	0.940069i	$0.610760\pi$
$264$	0	0
$265$	3798.17	0.880451
$266$	−105.029	−0.0242096
$267$	0	0
$268$	1750.74	0.399043
$269$	2663.82	0.603778	0.301889	−	0.953343i	$-0.402383\pi$
$269$	2663.82	0.603778	0.301889	+	0.953343i	$0.402383\pi$
$270$	0	0
$271$	4262.87	0.955537	0.477769	−	0.878486i	$-0.341446\pi$
$271$	4262.87	0.955537	0.477769	+	0.878486i	$0.341446\pi$
$272$	−704.421	−0.157029
$273$	0	0
$274$	−1721.22	−0.379499
$275$	−192.705	−0.0422566
$276$	0	0
$277$	907.280	0.196798	0.0983992	−	0.995147i	$-0.468628\pi$
$277$	907.280	0.196798	0.0983992	+	0.995147i	$0.468628\pi$
$278$	−4711.63	−1.01649
$279$	0	0
$280$	110.557	0.0235966
$281$	5597.15	1.18825	0.594124	−	0.804373i	$-0.297499\pi$
$281$	5597.15	1.18825	0.594124	+	0.804373i	$0.297499\pi$
$282$	0	0
$283$	8473.01	1.77975	0.889873	−	0.456208i	$-0.150793\pi$
$283$	8473.01	1.77975	0.889873	+	0.456208i	$0.150793\pi$
$284$	3186.41	0.665771
$285$	0	0
$286$	−246.006	−0.0508624
$287$	−415.016	−0.0853575
$288$	0	0
$289$	−2974.68	−0.605472
$290$	−726.262	−0.147061
$291$	0	0
$292$	−3976.92	−0.797027
$293$	241.594	0.0481710	0.0240855	−	0.999710i	$-0.492333\pi$
$293$	241.594	0.0481710	0.0240855	+	0.999710i	$0.492333\pi$
$294$	0	0
$295$	3607.58	0.712004
$296$	2901.64	0.569779
$297$	0	0
$298$	−1864.16	−0.362375
$299$	−1312.49	−0.253856
$300$	0	0
$301$	−916.874	−0.175574
$302$	−1120.53	−0.213507
$303$	0	0
$304$	−304.000	−0.0573539
$305$	3320.07	0.623300
$306$	0	0
$307$	987.717	0.183622	0.0918110	−	0.995776i	$-0.470734\pi$
$307$	987.717	0.183622	0.0918110	+	0.995776i	$0.470734\pi$
$308$	85.2198	0.0157657
$309$	0	0
$310$	2191.08	0.401436
$311$	810.772	0.147828	0.0739142	−	0.997265i	$-0.476451\pi$
$311$	810.772	0.147828	0.0739142	+	0.997265i	$0.476451\pi$
$312$	0	0
$313$	265.589	0.0479615	0.0239808	−	0.999712i	$-0.492366\pi$
$313$	265.589	0.0479615	0.0239808	+	0.999712i	$0.492366\pi$
$314$	−619.700	−0.111375
$315$	0	0
$316$	−4039.71	−0.719151
$317$	6449.54	1.14272	0.571360	−	0.820700i	$-0.306416\pi$
$317$	6449.54	1.14272	0.571360	+	0.820700i	$0.306416\pi$
$318$	0	0
$319$	−559.817	−0.0982563
$320$	320.000	0.0559017
$321$	0	0
$322$	454.663	0.0786874
$323$	836.500	0.144099
$324$	0	0
$325$	−398.936	−0.0680891
$326$	−6499.30	−1.10418
$327$	0	0
$328$	−1201.23	−0.202216
$329$	740.644	0.124113
$330$	0	0
$331$	−1482.89	−0.246244	−0.123122	−	0.992392i	$-0.539291\pi$
$331$	−1482.89	−0.246244	−0.123122	+	0.992392i	$0.539291\pi$
$332$	−943.529	−0.155973
$333$	0	0
$334$	4319.00	0.707561
$335$	2188.43	0.356915
$336$	0	0
$337$	2444.54	0.395141	0.197571	−	0.980289i	$-0.436695\pi$
$337$	2444.54	0.395141	0.197571	+	0.980289i	$0.436695\pi$
$338$	3884.72	0.625151
$339$	0	0
$340$	−880.526	−0.140451
$341$	1688.93	0.268213
$342$	0	0
$343$	1874.94	0.295153
$344$	−2653.83	−0.415944
$345$	0	0
$346$	3334.83	0.518155
$347$	−4943.22	−0.764743	−0.382372	−	0.924009i	$-0.624893\pi$
$347$	−4943.22	−0.764743	−0.382372	+	0.924009i	$0.624893\pi$
$348$	0	0
$349$	12750.4	1.95563	0.977814	−	0.209475i	$-0.0671754\pi$
$349$	12750.4	1.95563	0.977814	+	0.209475i	$0.0671754\pi$
$350$	138.197	0.0211055
$351$	0	0
$352$	246.663	0.0373499
$353$	10348.1	1.56027	0.780133	−	0.625613i	$-0.215151\pi$
$353$	10348.1	1.56027	0.780133	+	0.625613i	$0.215151\pi$
$354$	0	0
$355$	3983.02	0.595484
$356$	−4645.08	−0.691541
$357$	0	0
$358$	877.731	0.129580
$359$	4091.93	0.601570	0.300785	−	0.953692i	$-0.402751\pi$
$359$	4091.93	0.601570	0.300785	+	0.953692i	$0.402751\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	6139.05	0.891329
$363$	0	0
$364$	176.421	0.0254038
$365$	−4971.15	−0.712882
$366$	0	0
$367$	−5089.12	−0.723842	−0.361921	−	0.932209i	$-0.617879\pi$
$367$	−5089.12	−0.723842	−0.361921	+	0.932209i	$0.617879\pi$
$368$	1315.99	0.186415
$369$	0	0
$370$	3627.05	0.509625
$371$	−2099.57	−0.293813
$372$	0	0
$373$	4163.90	0.578012	0.289006	−	0.957327i	$-0.406675\pi$
$373$	4163.90	0.578012	0.289006	+	0.957327i	$0.406675\pi$
$374$	−678.728	−0.0938400
$375$	0	0
$376$	2143.74	0.294029
$377$	−1158.93	−0.158323
$378$	0	0
$379$	11585.9	1.57026	0.785130	−	0.619331i	$-0.212596\pi$
$379$	11585.9	1.57026	0.785130	+	0.619331i	$0.212596\pi$
$380$	−380.000	−0.0512989
$381$	0	0
$382$	−1256.27	−0.168262
$383$	3345.13	0.446288	0.223144	−	0.974785i	$-0.428368\pi$
$383$	3345.13	0.446288	0.223144	+	0.974785i	$0.428368\pi$
$384$	0	0
$385$	106.525	0.0141013
$386$	−5021.49	−0.662143
$387$	0	0
$388$	5773.11	0.755374
$389$	4067.36	0.530137	0.265069	−	0.964230i	$-0.414605\pi$
$389$	4067.36	0.530137	0.265069	+	0.964230i	$0.414605\pi$
$390$	0	0
$391$	−3621.13	−0.468359
$392$	2682.89	0.345679
$393$	0	0
$394$	471.446	0.0602820
$395$	−5049.64	−0.643228
$396$	0	0
$397$	−3996.26	−0.505205	−0.252602	−	0.967570i	$-0.581286\pi$
$397$	−3996.26	−0.505205	−0.252602	+	0.967570i	$0.581286\pi$
$398$	−8938.72	−1.12577
$399$	0	0
$400$	400.000	0.0500000
$401$	−4211.11	−0.524421	−0.262211	−	0.965011i	$-0.584451\pi$
$401$	−4211.11	−0.524421	−0.262211	+	0.965011i	$0.584451\pi$
$402$	0	0
$403$	3496.41	0.432180
$404$	7613.64	0.937606
$405$	0	0
$406$	401.468	0.0490751
$407$	2795.80	0.340499
$408$	0	0
$409$	−9617.74	−1.16276	−0.581378	−	0.813634i	$-0.697486\pi$
$409$	−9617.74	−1.16276	−0.581378	+	0.813634i	$0.697486\pi$
$410$	−1501.54	−0.180868
$411$	0	0
$412$	−3378.07	−0.403946
$413$	−1994.22	−0.237601
$414$	0	0
$415$	−1179.41	−0.139506
$416$	510.638	0.0601829
$417$	0	0
$418$	−292.912	−0.0342746
$419$	6831.50	0.796517	0.398259	−	0.917273i	$-0.369615\pi$
$419$	6831.50	0.796517	0.398259	+	0.917273i	$0.369615\pi$
$420$	0	0
$421$	−11478.5	−1.32881	−0.664404	−	0.747373i	$-0.731315\pi$
$421$	−11478.5	−1.32881	−0.664404	+	0.747373i	$0.731315\pi$
$422$	2565.91	0.295987
$423$	0	0
$424$	−6077.06	−0.696057
$425$	−1100.66	−0.125623
$426$	0	0
$427$	−1835.29	−0.208000
$428$	1098.39	0.124048
$429$	0	0
$430$	−3317.28	−0.372032
$431$	5513.31	0.616165	0.308082	−	0.951360i	$-0.400313\pi$
$431$	5513.31	0.616165	0.308082	+	0.951360i	$0.400313\pi$
$432$	0	0
$433$	17227.8	1.91204	0.956020	−	0.293302i	$-0.0947542\pi$
$433$	17227.8	1.91204	0.956020	+	0.293302i	$0.0947542\pi$
$434$	−1211.20	−0.133962
$435$	0	0
$436$	6485.41	0.712373
$437$	−1562.74	−0.171066
$438$	0	0
$439$	16157.5	1.75662	0.878309	−	0.478092i	$-0.158672\pi$
$439$	16157.5	1.75662	0.878309	+	0.478092i	$0.158672\pi$
$440$	308.328	0.0334067
$441$	0	0
$442$	−1405.09	−0.151207
$443$	−8749.51	−0.938379	−0.469189	−	0.883098i	$-0.655454\pi$
$443$	−8749.51	−0.938379	−0.469189	+	0.883098i	$0.655454\pi$
$444$	0	0
$445$	−5806.35	−0.618533
$446$	3226.38	0.342542
$447$	0	0
$448$	−176.892	−0.0186548
$449$	15036.6	1.58045	0.790224	−	0.612817i	$-0.209964\pi$
$449$	15036.6	1.58045	0.790224	+	0.612817i	$0.209964\pi$
$450$	0	0
$451$	−1157.42	−0.120844
$452$	−2774.85	−0.288756
$453$	0	0
$454$	10548.1	1.09041
$455$	220.526	0.0227218
$456$	0	0
$457$	2054.13	0.210258	0.105129	−	0.994459i	$-0.466474\pi$
$457$	2054.13	0.210258	0.105129	+	0.994459i	$0.466474\pi$
$458$	−4502.32	−0.459344
$459$	0	0
$460$	1644.98	0.166734
$461$	18233.8	1.84215	0.921076	−	0.389383i	$-0.127312\pi$
$461$	18233.8	1.84215	0.921076	+	0.389383i	$0.127312\pi$
$462$	0	0
$463$	−13997.7	−1.40503	−0.702513	−	0.711671i	$-0.747939\pi$
$463$	−13997.7	−1.40503	−0.702513	+	0.711671i	$0.747939\pi$
$464$	1162.02	0.116262
$465$	0	0
$466$	−5087.67	−0.505755
$467$	−9986.52	−0.989552	−0.494776	−	0.869020i	$-0.664750\pi$
$467$	−9986.52	−0.989552	−0.494776	+	0.869020i	$0.664750\pi$
$468$	0	0
$469$	−1209.73	−0.119105
$470$	2679.67	0.262988
$471$	0	0
$472$	−5772.12	−0.562889
$473$	−2557.03	−0.248567
$474$	0	0
$475$	−475.000	−0.0458831
$476$	486.743	0.0468694
$477$	0	0
$478$	2415.98	0.231181
$479$	2988.28	0.285048	0.142524	−	0.989791i	$-0.454478\pi$
$479$	2988.28	0.285048	0.142524	+	0.989791i	$0.454478\pi$
$480$	0	0
$481$	5787.84	0.548654
$482$	−9123.04	−0.862123
$483$	0	0
$484$	−5086.33	−0.477680
$485$	7216.38	0.675627
$486$	0	0
$487$	5064.87	0.471275	0.235638	−	0.971841i	$-0.424282\pi$
$487$	5064.87	0.471275	0.235638	+	0.971841i	$0.424282\pi$
$488$	−5312.11	−0.492762
$489$	0	0
$490$	3353.61	0.309185
$491$	8108.65	0.745292	0.372646	−	0.927974i	$-0.378451\pi$
$491$	8108.65	0.745292	0.372646	+	0.927974i	$0.378451\pi$
$492$	0	0
$493$	−3197.46	−0.292102
$494$	−606.382	−0.0552276
$495$	0	0
$496$	−3505.73	−0.317363
$497$	−2201.76	−0.198717
$498$	0	0
$499$	−4913.49	−0.440798	−0.220399	−	0.975410i	$-0.570736\pi$
$499$	−4913.49	−0.440798	−0.220399	+	0.975410i	$0.570736\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−12119.8	−1.07756
$503$	−11233.6	−0.995785	−0.497893	−	0.867239i	$-0.665893\pi$
$503$	−11233.6	−0.995785	−0.497893	+	0.867239i	$0.665893\pi$
$504$	0	0
$505$	9517.05	0.838620
$506$	1267.99	0.111401
$507$	0	0
$508$	−5243.44	−0.457953
$509$	17195.0	1.49736	0.748678	−	0.662934i	$-0.230689\pi$
$509$	17195.0	1.49736	0.748678	+	0.662934i	$0.230689\pi$
$510$	0	0
$511$	2747.99	0.237894
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−12727.2	−1.09216
$515$	−4222.59	−0.361300
$516$	0	0
$517$	2065.55	0.175711
$518$	−2004.98	−0.170066
$519$	0	0
$520$	638.297	0.0538292
$521$	9873.44	0.830256	0.415128	−	0.909763i	$-0.363737\pi$
$521$	9873.44	0.830256	0.415128	+	0.909763i	$0.363737\pi$
$522$	0	0
$523$	−8048.30	−0.672902	−0.336451	−	0.941701i	$-0.609227\pi$
$523$	−8048.30	−0.672902	−0.336451	+	0.941701i	$0.609227\pi$
$524$	965.285	0.0804746
$525$	0	0
$526$	5817.37	0.482224
$527$	9646.53	0.797362
$528$	0	0
$529$	−5402.07	−0.443993
$530$	−7596.33	−0.622573
$531$	0	0
$532$	210.059	0.0171188
$533$	−2396.07	−0.194719
$534$	0	0
$535$	1372.99	0.110952
$536$	−3501.49	−0.282166
$537$	0	0
$538$	−5327.65	−0.426935
$539$	2585.03	0.206577
$540$	0	0
$541$	−10650.4	−0.846393	−0.423196	−	0.906038i	$-0.639092\pi$
$541$	−10650.4	−0.846393	−0.423196	+	0.906038i	$0.639092\pi$
$542$	−8525.73	−0.675667
$543$	0	0
$544$	1408.84	0.111036
$545$	8106.76	0.637166
$546$	0	0
$547$	−21827.6	−1.70618	−0.853090	−	0.521764i	$-0.825274\pi$
$547$	−21827.6	−1.70618	−0.853090	+	0.521764i	$0.825274\pi$
$548$	3442.44	0.268346
$549$	0	0
$550$	385.410	0.0298799
$551$	−1379.90	−0.106689
$552$	0	0
$553$	2791.37	0.214650
$554$	−1814.56	−0.139158
$555$	0	0
$556$	9423.25	0.718768
$557$	20564.0	1.56432	0.782159	−	0.623079i	$-0.214119\pi$
$557$	20564.0	1.56432	0.782159	+	0.623079i	$0.214119\pi$
$558$	0	0
$559$	−5293.53	−0.400523
$560$	−221.115	−0.0166853
$561$	0	0
$562$	−11194.3	−0.840219
$563$	−10760.6	−0.805517	−0.402759	−	0.915306i	$-0.631949\pi$
$563$	−10760.6	−0.805517	−0.402759	+	0.915306i	$0.631949\pi$
$564$	0	0
$565$	−3468.56	−0.258272
$566$	−16946.0	−1.25847
$567$	0	0
$568$	−6372.83	−0.470771
$569$	−21954.7	−1.61755	−0.808777	−	0.588115i	$-0.799870\pi$
$569$	−21954.7	−1.61755	−0.808777	+	0.588115i	$0.799870\pi$
$570$	0	0
$571$	−15598.2	−1.14319	−0.571597	−	0.820534i	$-0.693676\pi$
$571$	−15598.2	−1.14319	−0.571597	+	0.820534i	$0.693676\pi$
$572$	492.012	0.0359652
$573$	0	0
$574$	830.031	0.0603568
$575$	2056.23	0.149132
$576$	0	0
$577$	−17249.7	−1.24457	−0.622284	−	0.782791i	$-0.713795\pi$
$577$	−17249.7	−1.24457	−0.622284	+	0.782791i	$0.713795\pi$
$578$	5949.37	0.428133
$579$	0	0
$580$	1452.52	0.103987
$581$	651.963	0.0465542
$582$	0	0
$583$	−5855.41	−0.415963
$584$	7953.84	0.563583
$585$	0	0
$586$	−483.189	−0.0340620
$587$	−14572.0	−1.02462	−0.512308	−	0.858802i	$-0.671209\pi$
$587$	−14572.0	−1.02462	−0.512308	+	0.858802i	$0.671209\pi$
$588$	0	0
$589$	4163.06	0.291232
$590$	−7215.15	−0.503463
$591$	0	0
$592$	−5803.28	−0.402894
$593$	20116.4	1.39305	0.696526	−	0.717531i	$-0.254728\pi$
$593$	20116.4	1.39305	0.696526	+	0.717531i	$0.254728\pi$
$594$	0	0
$595$	608.429	0.0419212
$596$	3728.32	0.256238
$597$	0	0
$598$	2624.97	0.179503
$599$	13556.6	0.924718	0.462359	−	0.886693i	$-0.347003\pi$
$599$	13556.6	0.924718	0.462359	+	0.886693i	$0.347003\pi$
$600$	0	0
$601$	−27787.7	−1.88600	−0.942998	−	0.332799i	$-0.892007\pi$
$601$	−27787.7	−1.88600	−0.942998	+	0.332799i	$0.892007\pi$
$602$	1833.75	0.124150
$603$	0	0
$604$	2241.05	0.150972
$605$	−6357.92	−0.427250
$606$	0	0
$607$	20870.6	1.39557	0.697784	−	0.716308i	$-0.254169\pi$
$607$	20870.6	1.39557	0.697784	+	0.716308i	$0.254169\pi$
$608$	608.000	0.0405554
$609$	0	0
$610$	−6640.14	−0.440740
$611$	4276.07	0.283128
$612$	0	0
$613$	8289.90	0.546209	0.273104	−	0.961984i	$-0.411950\pi$
$613$	8289.90	0.546209	0.273104	+	0.961984i	$0.411950\pi$
$614$	−1975.43	−0.129840
$615$	0	0
$616$	−170.440	−0.0111481
$617$	−16006.8	−1.04443	−0.522213	−	0.852815i	$-0.674893\pi$
$617$	−16006.8	−1.04443	−0.522213	+	0.852815i	$0.674893\pi$
$618$	0	0
$619$	−9823.47	−0.637865	−0.318933	−	0.947777i	$-0.603324\pi$
$619$	−9823.47	−0.637865	−0.318933	+	0.947777i	$0.603324\pi$
$620$	−4382.17	−0.283858
$621$	0	0
$622$	−1621.54	−0.104530
$623$	3209.67	0.206409
$624$	0	0
$625$	625.000	0.0400000
$626$	−531.177	−0.0339139
$627$	0	0
$628$	1239.40	0.0787538
$629$	15968.6	1.01226
$630$	0	0
$631$	7114.01	0.448818	0.224409	−	0.974495i	$-0.427955\pi$
$631$	7114.01	0.448818	0.224409	+	0.974495i	$0.427955\pi$
$632$	8079.43	0.508517
$633$	0	0
$634$	−12899.1	−0.808025
$635$	−6554.30	−0.409606
$636$	0	0
$637$	5351.49	0.332863
$638$	1119.63	0.0694777
$639$	0	0
$640$	−640.000	−0.0395285
$641$	9831.58	0.605809	0.302905	−	0.953021i	$-0.402044\pi$
$641$	9831.58	0.605809	0.302905	+	0.953021i	$0.402044\pi$
$642$	0	0
$643$	16619.8	1.01931	0.509657	−	0.860378i	$-0.329772\pi$
$643$	16619.8	1.01931	0.509657	+	0.860378i	$0.329772\pi$
$644$	−909.325	−0.0556404
$645$	0	0
$646$	−1673.00	−0.101894
$647$	−11214.6	−0.681441	−0.340721	−	0.940165i	$-0.610671\pi$
$647$	−11214.6	−0.681441	−0.340721	+	0.940165i	$0.610671\pi$
$648$	0	0
$649$	−5561.59	−0.336381
$650$	797.871	0.0481463
$651$	0	0
$652$	12998.6	0.780774
$653$	−16751.8	−1.00391	−0.501953	−	0.864895i	$-0.667385\pi$
$653$	−16751.8	−1.00391	−0.501953	+	0.864895i	$0.667385\pi$
$654$	0	0
$655$	1206.61	0.0719786
$656$	2402.46	0.142989
$657$	0	0
$658$	−1481.29	−0.0877608
$659$	6661.91	0.393796	0.196898	−	0.980424i	$-0.436913\pi$
$659$	6661.91	0.393796	0.196898	+	0.980424i	$0.436913\pi$
$660$	0	0
$661$	−7200.89	−0.423725	−0.211862	−	0.977299i	$-0.567953\pi$
$661$	−7200.89	−0.423725	−0.211862	+	0.977299i	$0.567953\pi$
$662$	2965.77	0.174121
$663$	0	0
$664$	1887.06	0.110289
$665$	262.574	0.0153115
$666$	0	0
$667$	5973.45	0.346766
$668$	−8638.01	−0.500321
$669$	0	0
$670$	−4376.86	−0.252377
$671$	−5118.35	−0.294474
$672$	0	0
$673$	−14431.5	−0.826587	−0.413293	−	0.910598i	$-0.635622\pi$
$673$	−14431.5	−0.826587	−0.413293	+	0.910598i	$0.635622\pi$
$674$	−4889.08	−0.279407
$675$	0	0
$676$	−7769.44	−0.442048
$677$	−30133.8	−1.71069	−0.855345	−	0.518059i	$-0.826655\pi$
$677$	−30133.8	−1.71069	−0.855345	+	0.518059i	$0.826655\pi$
$678$	0	0
$679$	−3989.12	−0.225461
$680$	1761.05	0.0993136
$681$	0	0
$682$	−3377.86	−0.189656
$683$	−6029.49	−0.337792	−0.168896	−	0.985634i	$-0.554020\pi$
$683$	−6029.49	−0.337792	−0.168896	+	0.985634i	$0.554020\pi$
$684$	0	0
$685$	4303.05	0.240016
$686$	−3749.89	−0.208705
$687$	0	0
$688$	5307.65	0.294117
$689$	−12121.8	−0.670252
$690$	0	0
$691$	29317.3	1.61401	0.807005	−	0.590544i	$-0.201087\pi$
$691$	29317.3	1.61401	0.807005	+	0.590544i	$0.201087\pi$
$692$	−6669.66	−0.366391
$693$	0	0
$694$	9886.44	0.540755
$695$	11779.1	0.642885
$696$	0	0
$697$	−6610.73	−0.359253
$698$	−25500.8	−1.38284
$699$	0	0
$700$	−276.393	−0.0149238
$701$	−21824.4	−1.17589	−0.587944	−	0.808902i	$-0.700062\pi$
$701$	−21824.4	−1.17589	−0.587944	+	0.808902i	$0.700062\pi$
$702$	0	0
$703$	6891.40	0.369721
$704$	−493.325	−0.0264104
$705$	0	0
$706$	−20696.2	−1.10327
$707$	−5260.90	−0.279853
$708$	0	0
$709$	16303.6	0.863601	0.431801	−	0.901969i	$-0.357878\pi$
$709$	16303.6	0.863601	0.431801	+	0.901969i	$0.357878\pi$
$710$	−7966.04	−0.421070
$711$	0	0
$712$	9290.16	0.488994
$713$	−18021.5	−0.946578
$714$	0	0
$715$	615.016	0.0321682
$716$	−1755.46	−0.0916266
$717$	0	0
$718$	−8183.86	−0.425374
$719$	3999.89	0.207470	0.103735	−	0.994605i	$-0.466921\pi$
$719$	3999.89	0.207470	0.103735	+	0.994605i	$0.466921\pi$
$720$	0	0
$721$	2334.19	0.120568
$722$	−722.000	−0.0372161
$723$	0	0
$724$	−12278.1	−0.630265
$725$	1815.65	0.0930092
$726$	0	0
$727$	−4347.43	−0.221784	−0.110892	−	0.993832i	$-0.535371\pi$
$727$	−4347.43	−0.221784	−0.110892	+	0.993832i	$0.535371\pi$
$728$	−352.842	−0.0179632
$729$	0	0
$730$	9942.31	0.504084
$731$	−14604.8	−0.738956
$732$	0	0
$733$	10361.7	0.522125	0.261063	−	0.965322i	$-0.415927\pi$
$733$	10361.7	0.522125	0.261063	+	0.965322i	$0.415927\pi$
$734$	10178.2	0.511834
$735$	0	0
$736$	−2631.98	−0.131815
$737$	−3373.77	−0.168622
$738$	0	0
$739$	19730.6	0.982143	0.491071	−	0.871119i	$-0.336605\pi$
$739$	19730.6	0.982143	0.491071	+	0.871119i	$0.336605\pi$
$740$	−7254.10	−0.360360
$741$	0	0
$742$	4199.15	0.207757
$743$	−19837.0	−0.979475	−0.489737	−	0.871870i	$-0.662907\pi$
$743$	−19837.0	−0.979475	−0.489737	+	0.871870i	$0.662907\pi$
$744$	0	0
$745$	4660.39	0.229186
$746$	−8327.80	−0.408716
$747$	0	0
$748$	1357.46	0.0663549
$749$	−758.969	−0.0370255
$750$	0	0
$751$	−20446.5	−0.993480	−0.496740	−	0.867900i	$-0.665470\pi$
$751$	−20446.5	−0.993480	−0.496740	+	0.867900i	$0.665470\pi$
$752$	−4287.48	−0.207910
$753$	0	0
$754$	2317.85	0.111951
$755$	2801.32	0.135034
$756$	0	0
$757$	8965.43	0.430455	0.215227	−	0.976564i	$-0.430951\pi$
$757$	8965.43	0.430455	0.215227	+	0.976564i	$0.430951\pi$
$758$	−23171.8	−1.11034
$759$	0	0
$760$	760.000	0.0362738
$761$	6434.83	0.306521	0.153261	−	0.988186i	$-0.451023\pi$
$761$	6434.83	0.306521	0.153261	+	0.988186i	$0.451023\pi$
$762$	0	0
$763$	−4481.30	−0.212627
$764$	2512.54	0.118980
$765$	0	0
$766$	−6690.27	−0.315573
$767$	−11513.5	−0.542020
$768$	0	0
$769$	9299.93	0.436104	0.218052	−	0.975937i	$-0.430030\pi$
$769$	9299.93	0.436104	0.218052	+	0.975937i	$0.430030\pi$
$770$	−213.050	−0.00997113
$771$	0	0
$772$	10043.0	0.468206
$773$	−6499.31	−0.302411	−0.151206	−	0.988502i	$-0.548316\pi$
$773$	−6499.31	−0.302411	−0.151206	+	0.988502i	$0.548316\pi$
$774$	0	0
$775$	−5477.71	−0.253890
$776$	−11546.2	−0.534130
$777$	0	0
$778$	−8134.72	−0.374863
$779$	−2852.93	−0.131215
$780$	0	0
$781$	−6140.38	−0.281332
$782$	7242.26	0.331180
$783$	0	0
$784$	−5365.77	−0.244432
$785$	1549.25	0.0704396
$786$	0	0
$787$	7779.28	0.352352	0.176176	−	0.984359i	$-0.443627\pi$
$787$	7779.28	0.352352	0.176176	+	0.984359i	$0.443627\pi$
$788$	−942.892	−0.0426258
$789$	0	0
$790$	10099.3	0.454831
$791$	1917.37	0.0861870
$792$	0	0
$793$	−10596.0	−0.474493
$794$	7992.51	0.357234
$795$	0	0
$796$	17877.4	0.796041
$797$	−13787.2	−0.612757	−0.306378	−	0.951910i	$-0.599117\pi$
$797$	−13787.2	−0.612757	−0.306378	+	0.951910i	$0.599117\pi$
$798$	0	0
$799$	11797.6	0.522365
$800$	−800.000	−0.0353553
$801$	0	0
$802$	8422.22	0.370822
$803$	7663.73	0.336796
$804$	0	0
$805$	−1136.66	−0.0497663
$806$	−6992.81	−0.305597
$807$	0	0
$808$	−15227.3	−0.662987
$809$	−21123.7	−0.918011	−0.459005	−	0.888434i	$-0.651794\pi$
$809$	−21123.7	−0.918011	−0.459005	+	0.888434i	$0.651794\pi$
$810$	0	0
$811$	−19832.8	−0.858723	−0.429361	−	0.903133i	$-0.641261\pi$
$811$	−19832.8	−0.858723	−0.429361	+	0.903133i	$0.641261\pi$
$812$	−802.935	−0.0347014
$813$	0	0
$814$	−5591.61	−0.240769
$815$	16248.3	0.698346
$816$	0	0
$817$	−6302.84	−0.269900
$818$	19235.5	0.822192
$819$	0	0
$820$	3003.08	0.127893
$821$	−10001.6	−0.425164	−0.212582	−	0.977143i	$-0.568187\pi$
$821$	−10001.6	−0.425164	−0.212582	+	0.977143i	$0.568187\pi$
$822$	0	0
$823$	−22153.4	−0.938296	−0.469148	−	0.883120i	$-0.655439\pi$
$823$	−22153.4	−0.938296	−0.469148	+	0.883120i	$0.655439\pi$
$824$	6756.15	0.285633
$825$	0	0
$826$	3988.44	0.168009
$827$	−5242.15	−0.220420	−0.110210	−	0.993908i	$-0.535152\pi$
$827$	−5242.15	−0.220420	−0.110210	+	0.993908i	$0.535152\pi$
$828$	0	0
$829$	36538.3	1.53079	0.765396	−	0.643559i	$-0.222543\pi$
$829$	36538.3	1.53079	0.765396	+	0.643559i	$0.222543\pi$
$830$	2358.82	0.0986457
$831$	0	0
$832$	−1021.28	−0.0425557
$833$	14764.7	0.614125
$834$	0	0
$835$	−10797.5	−0.447501
$836$	585.823	0.0242358
$837$	0	0
$838$	−13663.0	−0.563223
$839$	−4670.82	−0.192199	−0.0960993	−	0.995372i	$-0.530637\pi$
$839$	−4670.82	−0.192199	−0.0960993	+	0.995372i	$0.530637\pi$
$840$	0	0
$841$	−19114.4	−0.783732
$842$	22957.0	0.939610
$843$	0	0
$844$	−5131.81	−0.209294
$845$	−9711.80	−0.395380
$846$	0	0
$847$	3514.57	0.142576
$848$	12154.1	0.492187
$849$	0	0
$850$	2201.32	0.0888288
$851$	−29832.2	−1.20169
$852$	0	0
$853$	34935.8	1.40232	0.701160	−	0.713004i	$-0.252666\pi$
$853$	34935.8	1.40232	0.701160	+	0.713004i	$0.252666\pi$
$854$	3670.58	0.147078
$855$	0	0
$856$	−2196.78	−0.0877154
$857$	−24249.8	−0.966577	−0.483289	−	0.875461i	$-0.660558\pi$
$857$	−24249.8	−0.966577	−0.483289	+	0.875461i	$0.660558\pi$
$858$	0	0
$859$	40826.7	1.62164	0.810820	−	0.585295i	$-0.199021\pi$
$859$	40826.7	1.62164	0.810820	+	0.585295i	$0.199021\pi$
$860$	6634.57	0.263066
$861$	0	0
$862$	−11026.6	−0.435694
$863$	−21236.3	−0.837650	−0.418825	−	0.908067i	$-0.637558\pi$
$863$	−21236.3	−0.837650	−0.418825	+	0.908067i	$0.637558\pi$
$864$	0	0
$865$	−8337.07	−0.327710
$866$	−34455.5	−1.35202
$867$	0	0
$868$	2422.40	0.0947255
$869$	7784.74	0.303888
$870$	0	0
$871$	−6984.34	−0.271705
$872$	−12970.8	−0.503724
$873$	0	0
$874$	3125.47	0.120962
$875$	−345.492	−0.0133483
$876$	0	0
$877$	569.390	0.0219235	0.0109618	−	0.999940i	$-0.496511\pi$
$877$	569.390	0.0219235	0.0109618	+	0.999940i	$0.496511\pi$
$878$	−32315.0	−1.24212
$879$	0	0
$880$	−616.656	−0.0236221
$881$	23645.0	0.904222	0.452111	−	0.891962i	$-0.350671\pi$
$881$	23645.0	0.904222	0.452111	+	0.891962i	$0.350671\pi$
$882$	0	0
$883$	7994.42	0.304681	0.152341	−	0.988328i	$-0.451319\pi$
$883$	7994.42	0.304681	0.152341	+	0.988328i	$0.451319\pi$
$884$	2810.19	0.106919
$885$	0	0
$886$	17499.0	0.663534
$887$	−49130.4	−1.85980	−0.929898	−	0.367817i	$-0.880105\pi$
$887$	−49130.4	−1.85980	−0.929898	+	0.367817i	$0.880105\pi$
$888$	0	0
$889$	3623.13	0.136688
$890$	11612.7	0.437369
$891$	0	0
$892$	−6452.76	−0.242213
$893$	5091.38	0.190791
$894$	0	0
$895$	−2194.33	−0.0819534
$896$	353.783	0.0131909
$897$	0	0
$898$	−30073.2	−1.11755
$899$	−15913.0	−0.590354
$900$	0	0
$901$	−33443.8	−1.23660
$902$	2314.84	0.0854497
$903$	0	0
$904$	5549.70	0.204182
$905$	−15347.6	−0.563726
$906$	0	0
$907$	−46407.4	−1.69893	−0.849466	−	0.527643i	$-0.823076\pi$
$907$	−46407.4	−1.69893	−0.849466	+	0.527643i	$0.823076\pi$
$908$	−21096.1	−0.771035
$909$	0	0
$910$	−441.052	−0.0160668
$911$	−35383.9	−1.28685	−0.643426	−	0.765508i	$-0.722488\pi$
$911$	−35383.9	−1.28685	−0.643426	+	0.765508i	$0.722488\pi$
$912$	0	0
$913$	1818.23	0.0659087
$914$	−4108.26	−0.148675
$915$	0	0
$916$	9004.64	0.324805
$917$	−666.995	−0.0240198
$918$	0	0
$919$	−28252.4	−1.01410	−0.507052	−	0.861916i	$-0.669265\pi$
$919$	−28252.4	−1.01410	−0.507052	+	0.861916i	$0.669265\pi$
$920$	−3289.97	−0.117899
$921$	0	0
$922$	−36467.6	−1.30260
$923$	−12711.7	−0.453318
$924$	0	0
$925$	−9067.63	−0.322315
$926$	27995.4	0.993504
$927$	0	0
$928$	−2324.04	−0.0822093
$929$	8118.71	0.286724	0.143362	−	0.989670i	$-0.454209\pi$
$929$	8118.71	0.286724	0.143362	+	0.989670i	$0.454209\pi$
$930$	0	0
$931$	6371.85	0.224306
$932$	10175.3	0.357623
$933$	0	0
$934$	19973.0	0.699719
$935$	1696.82	0.0593496
$936$	0	0
$937$	−7845.06	−0.273519	−0.136759	−	0.990604i	$-0.543669\pi$
$937$	−7845.06	−0.273519	−0.136759	+	0.990604i	$0.543669\pi$
$938$	2419.47	0.0842200
$939$	0	0
$940$	−5359.35	−0.185960
$941$	26653.8	0.923369	0.461685	−	0.887044i	$-0.347245\pi$
$941$	26653.8	0.923369	0.461685	+	0.887044i	$0.347245\pi$
$942$	0	0
$943$	12350.1	0.426483
$944$	11544.2	0.398022
$945$	0	0
$946$	5114.06	0.175764
$947$	31616.3	1.08489	0.542445	−	0.840091i	$-0.317499\pi$
$947$	31616.3	1.08489	0.542445	+	0.840091i	$0.317499\pi$
$948$	0	0
$949$	15865.4	0.542689
$950$	950.000	0.0324443
$951$	0	0
$952$	−973.486	−0.0331417
$953$	−8396.16	−0.285392	−0.142696	−	0.989767i	$-0.545577\pi$
$953$	−8396.16	−0.285392	−0.142696	+	0.989767i	$0.545577\pi$
$954$	0	0
$955$	3140.67	0.106419
$956$	−4831.97	−0.163470
$957$	0	0
$958$	−5976.57	−0.201560
$959$	−2378.67	−0.0800951
$960$	0	0
$961$	18217.5	0.611509
$962$	−11575.7	−0.387957
$963$	0	0
$964$	18246.1	0.609613
$965$	12553.7	0.418776
$966$	0	0
$967$	53070.8	1.76488	0.882442	−	0.470422i	$-0.155898\pi$
$967$	53070.8	1.76488	0.882442	+	0.470422i	$0.155898\pi$
$968$	10172.7	0.337771
$969$	0	0
$970$	−14432.8	−0.477740
$971$	2589.03	0.0855675	0.0427837	−	0.999084i	$-0.486377\pi$
$971$	2589.03	0.0855675	0.0427837	+	0.999084i	$0.486377\pi$
$972$	0	0
$973$	−6511.31	−0.214535
$974$	−10129.7	−0.333242
$975$	0	0
$976$	10624.2	0.348436
$977$	8727.18	0.285780	0.142890	−	0.989739i	$-0.454360\pi$
$977$	8727.18	0.285780	0.142890	+	0.989739i	$0.454360\pi$
$978$	0	0
$979$	8951.31	0.292222
$980$	−6707.21	−0.218627
$981$	0	0
$982$	−16217.3	−0.527001
$983$	39732.2	1.28918	0.644588	−	0.764530i	$-0.277029\pi$
$983$	39732.2	1.28918	0.644588	+	0.764530i	$0.277029\pi$
$984$	0	0
$985$	−1178.61	−0.0381257
$986$	6394.92	0.206548
$987$	0	0
$988$	1212.76	0.0390518
$989$	27284.4	0.877243
$990$	0	0
$991$	−32045.2	−1.02719	−0.513597	−	0.858031i	$-0.671687\pi$
$991$	−32045.2	−1.02719	−0.513597	+	0.858031i	$0.671687\pi$
$992$	7011.47	0.224410
$993$	0	0
$994$	4403.52	0.140514
$995$	22346.8	0.712001
$996$	0	0
$997$	43042.7	1.36728	0.683638	−	0.729821i	$-0.260397\pi$
$997$	43042.7	1.36728	0.683638	+	0.729821i	$0.260397\pi$
$998$	9826.98	0.311691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.a.k.1.2		2
3.2	odd	2		570.4.a.m.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.4.a.m.1.2	✓	2	3.2	odd	2
1710.4.a.k.1.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -2.76393 q^{7} -8.00000 q^{8} -10.0000 q^{10} -7.70820 q^{11} -15.9574 q^{13} +5.52786 q^{14} +16.0000 q^{16} -44.0263 q^{17} -19.0000 q^{19} +20.0000 q^{20} +15.4164 q^{22} +82.2492 q^{23} +25.0000 q^{25} +31.9149 q^{26} -11.0557 q^{28} +72.6262 q^{29} -219.108 q^{31} -32.0000 q^{32} +88.0526 q^{34} -13.8197 q^{35} -362.705 q^{37} +38.0000 q^{38} -40.0000 q^{40} +150.154 q^{41} +331.728 q^{43} -30.8328 q^{44} -164.498 q^{46} -267.967 q^{47} -335.361 q^{49} -50.0000 q^{50} -63.8297 q^{52} +759.633 q^{53} -38.5410 q^{55} +22.1115 q^{56} -145.252 q^{58} +721.515 q^{59} +664.014 q^{61} +438.217 q^{62} +64.0000 q^{64} -79.7871 q^{65} +437.686 q^{67} -176.105 q^{68} +27.6393 q^{70} +796.604 q^{71} -994.231 q^{73} +725.410 q^{74} -76.0000 q^{76} +21.3050 q^{77} -1009.93 q^{79} +80.0000 q^{80} -300.308 q^{82} -235.882 q^{83} -220.132 q^{85} -663.457 q^{86} +61.6656 q^{88} -1161.27 q^{89} +44.1052 q^{91} +328.997 q^{92} +535.935 q^{94} -95.0000 q^{95} +1443.28 q^{97} +670.721 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 10 q^{5} - 10 q^{7} - 16 q^{8} - 20 q^{10} - 2 q^{11} + 62 q^{13} + 20 q^{14} + 32 q^{16} + 64 q^{17} - 38 q^{19} + 40 q^{20} + 4 q^{22} + 84 q^{23} + 50 q^{25} - 124 q^{26} - 40 q^{28}+ \cdots + 1252 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 10 * q^5 - 10 * q^7 - 16 * q^8 - 20 * q^10 - 2 * q^11 + 62 * q^13 + 20 * q^14 + 32 * q^16 + 64 * q^17 - 38 * q^19 + 40 * q^20 + 4 * q^22 + 84 * q^23 + 50 * q^25 - 124 * q^26 - 40 * q^28 + 266 * q^29 - 152 * q^31 - 64 * q^32 - 128 * q^34 - 50 * q^35 - 390 * q^37 + 76 * q^38 - 80 * q^40 + 430 * q^41 - 146 * q^43 - 8 * q^44 - 168 * q^46 - 44 * q^47 - 626 * q^49 - 100 * q^50 + 248 * q^52 + 920 * q^53 - 10 * q^55 + 80 * q^56 - 532 * q^58 + 164 * q^59 - 112 * q^61 + 304 * q^62 + 128 * q^64 + 310 * q^65 - 28 * q^67 + 256 * q^68 + 100 * q^70 + 824 * q^71 + 24 * q^73 + 780 * q^74 - 152 * q^76 - 20 * q^77 - 392 * q^79 + 160 * q^80 - 860 * q^82 + 208 * q^83 + 320 * q^85 + 292 * q^86 + 16 * q^88 - 82 * q^89 - 520 * q^91 + 336 * q^92 + 88 * q^94 - 190 * q^95 + 1290 * q^97 + 1252 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more