LMFDB - Embedded newform 342.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [342,4,Mod(1,342)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("342.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,6,0,12,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.1786532220$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.56956.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 31x - 48 $ x^3 - 31*x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.70956$ of defining polynomial
Character	$\chi$	$=$	342.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +4.00000 q^{4} +5.12868 q^{5} -6.23917 q^{7} +8.00000 q^{8} +10.2574 q^{10} -13.8644 q^{11} +67.4714 q^{13} -12.4783 q^{14} +16.0000 q^{16} +77.5706 q^{17} +19.0000 q^{19} +20.5147 q^{20} -27.7287 q^{22} +179.164 q^{23} -98.6967 q^{25} +134.943 q^{26} -24.9567 q^{28} -178.107 q^{29} +302.291 q^{31} +32.0000 q^{32} +155.141 q^{34} -31.9987 q^{35} -280.291 q^{37} +38.0000 q^{38} +41.0294 q^{40} +264.906 q^{41} -47.8197 q^{43} -55.4575 q^{44} +358.327 q^{46} +491.993 q^{47} -304.073 q^{49} -197.393 q^{50} +269.886 q^{52} -56.6073 q^{53} -71.1059 q^{55} -49.9133 q^{56} -356.213 q^{58} +588.974 q^{59} -3.89778 q^{61} +604.582 q^{62} +64.0000 q^{64} +346.039 q^{65} -488.369 q^{67} +310.283 q^{68} -63.9974 q^{70} +447.575 q^{71} -231.943 q^{73} -560.582 q^{74} +76.0000 q^{76} +86.5021 q^{77} -486.736 q^{79} +82.0589 q^{80} +529.813 q^{82} -898.560 q^{83} +397.835 q^{85} -95.6394 q^{86} -110.915 q^{88} +825.497 q^{89} -420.965 q^{91} +716.655 q^{92} +983.987 q^{94} +97.4449 q^{95} -1432.98 q^{97} -608.146 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} + 12 q^{4} + 14 q^{7} + 24 q^{8} + 70 q^{11} + 10 q^{13} + 28 q^{14} + 48 q^{16} + 148 q^{17} + 57 q^{19} + 140 q^{22} + 118 q^{23} + 183 q^{25} + 20 q^{26} + 56 q^{28} + 270 q^{29} + 64 q^{31}+ \cdots - 966 q^{98}+O(q^{100}) $ 3 * q + 6 * q^2 + 12 * q^4 + 14 * q^7 + 24 * q^8 + 70 * q^11 + 10 * q^13 + 28 * q^14 + 48 * q^16 + 148 * q^17 + 57 * q^19 + 140 * q^22 + 118 * q^23 + 183 * q^25 + 20 * q^26 + 56 * q^28 + 270 * q^29 + 64 * q^31 + 96 * q^32 + 296 * q^34 + 312 * q^35 + 2 * q^37 + 114 * q^38 + 406 * q^41 + 70 * q^43 + 280 * q^44 + 236 * q^46 + 310 * q^47 - 483 * q^49 + 366 * q^50 + 40 * q^52 + 950 * q^53 - 1050 * q^55 + 112 * q^56 + 540 * q^58 + 628 * q^59 - 564 * q^61 + 128 * q^62 + 192 * q^64 + 984 * q^65 - 1388 * q^67 + 592 * q^68 + 624 * q^70 + 404 * q^71 - 748 * q^73 + 4 * q^74 + 228 * q^76 + 352 * q^77 - 160 * q^79 + 812 * q^82 + 818 * q^83 - 2922 * q^85 + 140 * q^86 + 560 * q^88 + 1102 * q^89 - 628 * q^91 + 472 * q^92 + 620 * q^94 - 1782 * q^97 - 966 * 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
$5$	5.12868	0.458723	0.229361	−	0.973341i	$-0.426336\pi$
$5$	5.12868	0.458723	0.229361	+	0.973341i	$0.426336\pi$
$6$	0	0
$7$	−6.23917	−0.336883	−0.168442	−	0.985712i	$-0.553873\pi$
$7$	−6.23917	−0.336883	−0.168442	+	0.985712i	$0.553873\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.2574	0.324366
$11$	−13.8644	−0.380024	−0.190012	−	0.981782i	$-0.560853\pi$
$11$	−13.8644	−0.380024	−0.190012	+	0.981782i	$0.560853\pi$
$12$	0	0
$13$	67.4714	1.43948	0.719738	−	0.694246i	$-0.244262\pi$
$13$	67.4714	1.43948	0.719738	+	0.694246i	$0.244262\pi$
$14$	−12.4783	−0.238213
$15$	0	0
$16$	16.0000	0.250000
$17$	77.5706	1.10668	0.553342	−	0.832954i	$-0.313352\pi$
$17$	77.5706	1.10668	0.553342	+	0.832954i	$0.313352\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	20.5147	0.229361
$21$	0	0
$22$	−27.7287	−0.268718
$23$	179.164	1.62427	0.812135	−	0.583469i	$-0.198305\pi$
$23$	179.164	1.62427	0.812135	+	0.583469i	$0.198305\pi$
$24$	0	0
$25$	−98.6967	−0.789573
$26$	134.943	1.01786
$27$	0	0
$28$	−24.9567	−0.168442
$29$	−178.107	−1.14047	−0.570234	−	0.821483i	$-0.693147\pi$
$29$	−178.107	−1.14047	−0.570234	+	0.821483i	$0.693147\pi$
$30$	0	0
$31$	302.291	1.75139	0.875695	−	0.482865i	$-0.160404\pi$
$31$	302.291	1.75139	0.875695	+	0.482865i	$0.160404\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	155.141	0.782544
$35$	−31.9987	−0.154536
$36$	0	0
$37$	−280.291	−1.24539	−0.622697	−	0.782463i	$-0.713963\pi$
$37$	−280.291	−1.24539	−0.622697	+	0.782463i	$0.713963\pi$
$38$	38.0000	0.162221
$39$	0	0
$40$	41.0294	0.162183
$41$	264.906	1.00906	0.504530	−	0.863394i	$-0.331666\pi$
$41$	264.906	1.00906	0.504530	+	0.863394i	$0.331666\pi$
$42$	0	0
$43$	−47.8197	−0.169592	−0.0847958	−	0.996398i	$-0.527024\pi$
$43$	−47.8197	−0.169592	−0.0847958	+	0.996398i	$0.527024\pi$
$44$	−55.4575	−0.190012
$45$	0	0
$46$	358.327	1.14853
$47$	491.993	1.52691	0.763453	−	0.645863i	$-0.223502\pi$
$47$	491.993	1.52691	0.763453	+	0.645863i	$0.223502\pi$
$48$	0	0
$49$	−304.073	−0.886510
$50$	−197.393	−0.558313
$51$	0	0
$52$	269.886	0.719738
$53$	−56.6073	−0.146710	−0.0733549	−	0.997306i	$-0.523371\pi$
$53$	−56.6073	−0.146710	−0.0733549	+	0.997306i	$0.523371\pi$
$54$	0	0
$55$	−71.1059	−0.174326
$56$	−49.9133	−0.119106
$57$	0	0
$58$	−356.213	−0.806432
$59$	588.974	1.29962	0.649812	−	0.760095i	$-0.274847\pi$
$59$	588.974	1.29962	0.649812	+	0.760095i	$0.274847\pi$
$60$	0	0
$61$	−3.89778	−0.00818130	−0.00409065	−	0.999992i	$-0.501302\pi$
$61$	−3.89778	−0.00818130	−0.00409065	+	0.999992i	$0.501302\pi$
$62$	604.582	1.23842
$63$	0	0
$64$	64.0000	0.125000
$65$	346.039	0.660321
$66$	0	0
$67$	−488.369	−0.890504	−0.445252	−	0.895405i	$-0.646886\pi$
$67$	−488.369	−0.890504	−0.445252	+	0.895405i	$0.646886\pi$
$68$	310.283	0.553342
$69$	0	0
$70$	−63.9974	−0.109274
$71$	447.575	0.748132	0.374066	−	0.927402i	$-0.377963\pi$
$71$	447.575	0.748132	0.374066	+	0.927402i	$0.377963\pi$
$72$	0	0
$73$	−231.943	−0.371875	−0.185937	−	0.982562i	$-0.559532\pi$
$73$	−231.943	−0.371875	−0.185937	+	0.982562i	$0.559532\pi$
$74$	−560.582	−0.880626
$75$	0	0
$76$	76.0000	0.114708
$77$	86.5021	0.128024
$78$	0	0
$79$	−486.736	−0.693191	−0.346596	−	0.938015i	$-0.612662\pi$
$79$	−486.736	−0.693191	−0.346596	+	0.938015i	$0.612662\pi$
$80$	82.0589	0.114681
$81$	0	0
$82$	529.813	0.713513
$83$	−898.560	−1.18831	−0.594156	−	0.804350i	$-0.702514\pi$
$83$	−898.560	−1.18831	−0.594156	+	0.804350i	$0.702514\pi$
$84$	0	0
$85$	397.835	0.507662
$86$	−95.6394	−0.119919
$87$	0	0
$88$	−110.915	−0.134359
$89$	825.497	0.983174	0.491587	−	0.870829i	$-0.336417\pi$
$89$	825.497	0.983174	0.491587	+	0.870829i	$0.336417\pi$
$90$	0	0
$91$	−420.965	−0.484936
$92$	716.655	0.812135
$93$	0	0
$94$	983.987	1.07969
$95$	97.4449	0.105238
$96$	0	0
$97$	−1432.98	−1.49997	−0.749986	−	0.661453i	$-0.769940\pi$
$97$	−1432.98	−1.49997	−0.749986	+	0.661453i	$0.769940\pi$
$98$	−608.146	−0.626857
$99$	0	0
$100$	−394.787	−0.394787
$101$	290.043	0.285746	0.142873	−	0.989741i	$-0.454366\pi$
$101$	290.043	0.285746	0.142873	+	0.989741i	$0.454366\pi$
$102$	0	0
$103$	−1316.31	−1.25922	−0.629612	−	0.776910i	$-0.716786\pi$
$103$	−1316.31	−1.25922	−0.629612	+	0.776910i	$0.716786\pi$
$104$	539.771	0.508932
$105$	0	0
$106$	−113.215	−0.103739
$107$	−1609.50	−1.45417	−0.727084	−	0.686548i	$-0.759125\pi$
$107$	−1609.50	−1.45417	−0.727084	+	0.686548i	$0.759125\pi$
$108$	0	0
$109$	538.636	0.473321	0.236660	−	0.971592i	$-0.423947\pi$
$109$	538.636	0.473321	0.236660	+	0.971592i	$0.423947\pi$
$110$	−142.212	−0.123267
$111$	0	0
$112$	−99.8267	−0.0842208
$113$	−1294.51	−1.07767	−0.538837	−	0.842410i	$-0.681136\pi$
$113$	−1294.51	−1.07767	−0.538837	+	0.842410i	$0.681136\pi$
$114$	0	0
$115$	918.873	0.745090
$116$	−712.426	−0.570234
$117$	0	0
$118$	1177.95	0.918973
$119$	−483.976	−0.372824
$120$	0	0
$121$	−1138.78	−0.855582
$122$	−7.79555	−0.00578505
$123$	0	0
$124$	1209.16	0.875695
$125$	−1147.27	−0.820918
$126$	0	0
$127$	214.660	0.149984	0.0749922	−	0.997184i	$-0.476107\pi$
$127$	214.660	0.149984	0.0749922	+	0.997184i	$0.476107\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	692.078	0.466917
$131$	921.583	0.614649	0.307325	−	0.951605i	$-0.400566\pi$
$131$	921.583	0.614649	0.307325	+	0.951605i	$0.400566\pi$
$132$	0	0
$133$	−118.544	−0.0772863
$134$	−976.738	−0.629682
$135$	0	0
$136$	620.565	0.391272
$137$	−1484.44	−0.925722	−0.462861	−	0.886431i	$-0.653177\pi$
$137$	−1484.44	−0.925722	−0.462861	+	0.886431i	$0.653177\pi$
$138$	0	0
$139$	154.260	0.0941304	0.0470652	−	0.998892i	$-0.485013\pi$
$139$	154.260	0.0941304	0.0470652	+	0.998892i	$0.485013\pi$
$140$	−127.995	−0.0772681
$141$	0	0
$142$	895.150	0.529010
$143$	−935.448	−0.547036
$144$	0	0
$145$	−913.451	−0.523159
$146$	−463.885	−0.262955
$147$	0	0
$148$	−1121.16	−0.622697
$149$	−1018.24	−0.559848	−0.279924	−	0.960022i	$-0.590309\pi$
$149$	−1018.24	−0.559848	−0.279924	+	0.960022i	$0.590309\pi$
$150$	0	0
$151$	1110.49	0.598479	0.299240	−	0.954178i	$-0.403267\pi$
$151$	1110.49	0.598479	0.299240	+	0.954178i	$0.403267\pi$
$152$	152.000	0.0811107
$153$	0	0
$154$	173.004	0.0905265
$155$	1550.35	0.803403
$156$	0	0
$157$	332.551	0.169048	0.0845238	−	0.996421i	$-0.473063\pi$
$157$	332.551	0.169048	0.0845238	+	0.996421i	$0.473063\pi$
$158$	−973.473	−0.490160
$159$	0	0
$160$	164.118	0.0810915
$161$	−1117.83	−0.547190
$162$	0	0
$163$	3542.26	1.70215	0.851076	−	0.525042i	$-0.175950\pi$
$163$	3542.26	1.70215	0.851076	+	0.525042i	$0.175950\pi$
$164$	1059.63	0.504530
$165$	0	0
$166$	−1797.12	−0.840263
$167$	561.206	0.260044	0.130022	−	0.991511i	$-0.458495\pi$
$167$	561.206	0.260044	0.130022	+	0.991511i	$0.458495\pi$
$168$	0	0
$169$	2355.39	1.07209
$170$	795.670	0.358971
$171$	0	0
$172$	−191.279	−0.0847958
$173$	−1911.12	−0.839883	−0.419942	−	0.907551i	$-0.637949\pi$
$173$	−1911.12	−0.839883	−0.419942	+	0.907551i	$0.637949\pi$
$174$	0	0
$175$	615.785	0.265994
$176$	−221.830	−0.0950060
$177$	0	0
$178$	1650.99	0.695209
$179$	−1690.06	−0.705703	−0.352851	−	0.935679i	$-0.614788\pi$
$179$	−1690.06	−0.705703	−0.352851	+	0.935679i	$0.614788\pi$
$180$	0	0
$181$	−1961.52	−0.805518	−0.402759	−	0.915306i	$-0.631949\pi$
$181$	−1961.52	−0.805518	−0.402759	+	0.915306i	$0.631949\pi$
$182$	−841.931	−0.342901
$183$	0	0
$184$	1433.31	0.574266
$185$	−1437.52	−0.571291
$186$	0	0
$187$	−1075.47	−0.420567
$188$	1967.97	0.763453
$189$	0	0
$190$	194.890	0.0744147
$191$	−3455.41	−1.30903	−0.654514	−	0.756050i	$-0.727127\pi$
$191$	−3455.41	−1.30903	−0.654514	+	0.756050i	$0.727127\pi$
$192$	0	0
$193$	−5163.92	−1.92595	−0.962973	−	0.269599i	$-0.913109\pi$
$193$	−5163.92	−1.92595	−0.962973	+	0.269599i	$0.913109\pi$
$194$	−2865.96	−1.06064
$195$	0	0
$196$	−1216.29	−0.443255
$197$	4896.38	1.77082	0.885412	−	0.464806i	$-0.153876\pi$
$197$	4896.38	1.77082	0.885412	+	0.464806i	$0.153876\pi$
$198$	0	0
$199$	−3439.87	−1.22536	−0.612678	−	0.790333i	$-0.709908\pi$
$199$	−3439.87	−1.22536	−0.612678	+	0.790333i	$0.709908\pi$
$200$	−789.573	−0.279156
$201$	0	0
$202$	580.086	0.202053
$203$	1111.24	0.384204
$204$	0	0
$205$	1358.62	0.462879
$206$	−2632.62	−0.890406
$207$	0	0
$208$	1079.54	0.359869
$209$	−263.423	−0.0871835
$210$	0	0
$211$	2501.99	0.816323	0.408161	−	0.912910i	$-0.366170\pi$
$211$	2501.99	0.816323	0.408161	+	0.912910i	$0.366170\pi$
$212$	−226.429	−0.0733549
$213$	0	0
$214$	−3219.00	−1.02825
$215$	−245.252	−0.0777955
$216$	0	0
$217$	−1886.04	−0.590014
$218$	1077.27	0.334688
$219$	0	0
$220$	−284.424	−0.0871629
$221$	5233.80	1.59305
$222$	0	0
$223$	3067.08	0.921019	0.460509	−	0.887655i	$-0.347667\pi$
$223$	3067.08	0.921019	0.460509	+	0.887655i	$0.347667\pi$
$224$	−199.653	−0.0595531
$225$	0	0
$226$	−2589.02	−0.762031
$227$	112.623	0.0329299	0.0164649	−	0.999864i	$-0.494759\pi$
$227$	112.623	0.0329299	0.0164649	+	0.999864i	$0.494759\pi$
$228$	0	0
$229$	−4159.86	−1.20040	−0.600200	−	0.799850i	$-0.704912\pi$
$229$	−4159.86	−1.20040	−0.600200	+	0.799850i	$0.704912\pi$
$230$	1837.75	0.526858
$231$	0	0
$232$	−1424.85	−0.403216
$233$	149.875	0.0421401	0.0210701	−	0.999778i	$-0.493293\pi$
$233$	149.875	0.0421401	0.0210701	+	0.999778i	$0.493293\pi$
$234$	0	0
$235$	2523.28	0.700427
$236$	2355.90	0.649812
$237$	0	0
$238$	−967.952	−0.263626
$239$	−205.107	−0.0555117	−0.0277559	−	0.999615i	$-0.508836\pi$
$239$	−205.107	−0.0555117	−0.0277559	+	0.999615i	$0.508836\pi$
$240$	0	0
$241$	5813.07	1.55375	0.776873	−	0.629657i	$-0.216805\pi$
$241$	5813.07	1.55375	0.776873	+	0.629657i	$0.216805\pi$
$242$	−2277.56	−0.604988
$243$	0	0
$244$	−15.5911	−0.00409065
$245$	−1559.49	−0.406662
$246$	0	0
$247$	1281.96	0.330239
$248$	2418.33	0.619210
$249$	0	0
$250$	−2294.54	−0.580477
$251$	−6244.43	−1.57030	−0.785149	−	0.619307i	$-0.787414\pi$
$251$	−6244.43	−1.57030	−0.785149	+	0.619307i	$0.787414\pi$
$252$	0	0
$253$	−2483.99	−0.617262
$254$	429.320	0.106055
$255$	0	0
$256$	256.000	0.0625000
$257$	6047.76	1.46789	0.733947	−	0.679207i	$-0.237676\pi$
$257$	6047.76	1.46789	0.733947	+	0.679207i	$0.237676\pi$
$258$	0	0
$259$	1748.78	0.419552
$260$	1384.16	0.330160
$261$	0	0
$262$	1843.17	0.434623
$263$	5302.97	1.24333	0.621664	−	0.783284i	$-0.286457\pi$
$263$	5302.97	1.24333	0.621664	+	0.783284i	$0.286457\pi$
$264$	0	0
$265$	−290.321	−0.0672991
$266$	−237.088	−0.0546497
$267$	0	0
$268$	−1953.48	−0.445252
$269$	−4450.53	−1.00875	−0.504375	−	0.863485i	$-0.668277\pi$
$269$	−4450.53	−1.00875	−0.504375	+	0.863485i	$0.668277\pi$
$270$	0	0
$271$	4337.79	0.972331	0.486165	−	0.873867i	$-0.338395\pi$
$271$	4337.79	0.972331	0.486165	+	0.873867i	$0.338395\pi$
$272$	1241.13	0.276671
$273$	0	0
$274$	−2968.87	−0.654584
$275$	1368.37	0.300057
$276$	0	0
$277$	−3501.12	−0.759429	−0.379714	−	0.925104i	$-0.623978\pi$
$277$	−3501.12	−0.759429	−0.379714	+	0.925104i	$0.623978\pi$
$278$	308.519	0.0665602
$279$	0	0
$280$	−255.989	−0.0546368
$281$	−3580.65	−0.760156	−0.380078	−	0.924955i	$-0.624103\pi$
$281$	−3580.65	−0.760156	−0.380078	+	0.924955i	$0.624103\pi$
$282$	0	0
$283$	−5899.36	−1.23915	−0.619577	−	0.784936i	$-0.712696\pi$
$283$	−5899.36	−1.23915	−0.619577	+	0.784936i	$0.712696\pi$
$284$	1790.30	0.374066
$285$	0	0
$286$	−1870.90	−0.386813
$287$	−1652.80	−0.339935
$288$	0	0
$289$	1104.20	0.224751
$290$	−1826.90	−0.369929
$291$	0	0
$292$	−927.771	−0.185937
$293$	6333.11	1.26274	0.631372	−	0.775480i	$-0.282492\pi$
$293$	6333.11	1.26274	0.631372	+	0.775480i	$0.282492\pi$
$294$	0	0
$295$	3020.66	0.596168
$296$	−2242.33	−0.440313
$297$	0	0
$298$	−2036.48	−0.395872
$299$	12088.4	2.33810
$300$	0	0
$301$	298.355	0.0571326
$302$	2220.98	0.423189
$303$	0	0
$304$	304.000	0.0573539
$305$	−19.9904	−0.00375295
$306$	0	0
$307$	6092.56	1.13264	0.566321	−	0.824185i	$-0.308366\pi$
$307$	6092.56	1.13264	0.566321	+	0.824185i	$0.308366\pi$
$308$	346.009	0.0640119
$309$	0	0
$310$	3100.71	0.568092
$311$	2086.40	0.380415	0.190207	−	0.981744i	$-0.439084\pi$
$311$	2086.40	0.380415	0.190207	+	0.981744i	$0.439084\pi$
$312$	0	0
$313$	7369.87	1.33089	0.665447	−	0.746445i	$-0.268241\pi$
$313$	7369.87	1.33089	0.665447	+	0.746445i	$0.268241\pi$
$314$	665.103	0.119535
$315$	0	0
$316$	−1946.95	−0.346596
$317$	10787.0	1.91123	0.955613	−	0.294624i	$-0.0951942\pi$
$317$	10787.0	1.91123	0.955613	+	0.294624i	$0.0951942\pi$
$318$	0	0
$319$	2469.33	0.433405
$320$	328.235	0.0573404
$321$	0	0
$322$	−2235.67	−0.386922
$323$	1473.84	0.253891
$324$	0	0
$325$	−6659.20	−1.13657
$326$	7084.51	1.20360
$327$	0	0
$328$	2119.25	0.356756
$329$	−3069.63	−0.514389
$330$	0	0
$331$	−3269.08	−0.542854	−0.271427	−	0.962459i	$-0.587496\pi$
$331$	−3269.08	−0.542854	−0.271427	+	0.962459i	$0.587496\pi$
$332$	−3594.24	−0.594156
$333$	0	0
$334$	1122.41	0.183879
$335$	−2504.69	−0.408495
$336$	0	0
$337$	−1903.68	−0.307716	−0.153858	−	0.988093i	$-0.549170\pi$
$337$	−1903.68	−0.307716	−0.153858	+	0.988093i	$0.549170\pi$
$338$	4710.78	0.758084
$339$	0	0
$340$	1591.34	0.253831
$341$	−4191.08	−0.665570
$342$	0	0
$343$	4037.20	0.635534
$344$	−382.558	−0.0599597
$345$	0	0
$346$	−3822.24	−0.593887
$347$	−1580.07	−0.244446	−0.122223	−	0.992503i	$-0.539002\pi$
$347$	−1580.07	−0.244446	−0.122223	+	0.992503i	$0.539002\pi$
$348$	0	0
$349$	5266.11	0.807702	0.403851	−	0.914825i	$-0.367671\pi$
$349$	5266.11	0.807702	0.403851	+	0.914825i	$0.367671\pi$
$350$	1231.57	0.188086
$351$	0	0
$352$	−443.660	−0.0671794
$353$	7057.79	1.06416	0.532079	−	0.846694i	$-0.321411\pi$
$353$	7057.79	1.06416	0.532079	+	0.846694i	$0.321411\pi$
$354$	0	0
$355$	2295.47	0.343186
$356$	3301.99	0.491587
$357$	0	0
$358$	−3380.12	−0.499007
$359$	563.567	0.0828522	0.0414261	−	0.999142i	$-0.486810\pi$
$359$	563.567	0.0828522	0.0414261	+	0.999142i	$0.486810\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−3923.04	−0.569587
$363$	0	0
$364$	−1683.86	−0.242468
$365$	−1189.56	−0.170587
$366$	0	0
$367$	4952.13	0.704357	0.352178	−	0.935933i	$-0.385441\pi$
$367$	4952.13	0.704357	0.352178	+	0.935933i	$0.385441\pi$
$368$	2866.62	0.406068
$369$	0	0
$370$	−2875.05	−0.403964
$371$	353.183	0.0494241
$372$	0	0
$373$	453.156	0.0629049	0.0314524	−	0.999505i	$-0.489987\pi$
$373$	453.156	0.0629049	0.0314524	+	0.999505i	$0.489987\pi$
$374$	−2150.94	−0.297386
$375$	0	0
$376$	3935.95	0.539843
$377$	−12017.1	−1.64168
$378$	0	0
$379$	7544.45	1.02251	0.511256	−	0.859428i	$-0.329180\pi$
$379$	7544.45	1.02251	0.511256	+	0.859428i	$0.329180\pi$
$380$	389.780	0.0526191
$381$	0	0
$382$	−6910.81	−0.925623
$383$	6217.70	0.829529	0.414765	−	0.909929i	$-0.363864\pi$
$383$	6217.70	0.829529	0.414765	+	0.909929i	$0.363864\pi$
$384$	0	0
$385$	443.642	0.0587275
$386$	−10327.8	−1.36185
$387$	0	0
$388$	−5731.93	−0.749986
$389$	2147.15	0.279858	0.139929	−	0.990162i	$-0.455313\pi$
$389$	2147.15	0.279858	0.139929	+	0.990162i	$0.455313\pi$
$390$	0	0
$391$	13897.8	1.79756
$392$	−2432.58	−0.313428
$393$	0	0
$394$	9792.76	1.25216
$395$	−2496.31	−0.317983
$396$	0	0
$397$	−13761.4	−1.73971	−0.869856	−	0.493305i	$-0.835789\pi$
$397$	−13761.4	−1.73971	−0.869856	+	0.493305i	$0.835789\pi$
$398$	−6879.73	−0.866457
$399$	0	0
$400$	−1579.15	−0.197393
$401$	8937.99	1.11307	0.556536	−	0.830824i	$-0.312130\pi$
$401$	8937.99	1.11307	0.556536	+	0.830824i	$0.312130\pi$
$402$	0	0
$403$	20396.0	2.52108
$404$	1160.17	0.142873
$405$	0	0
$406$	2222.47	0.271674
$407$	3886.06	0.473280
$408$	0	0
$409$	993.253	0.120081	0.0600406	−	0.998196i	$-0.480877\pi$
$409$	993.253	0.120081	0.0600406	+	0.998196i	$0.480877\pi$
$410$	2717.24	0.327305
$411$	0	0
$412$	−5265.25	−0.629612
$413$	−3674.71	−0.437822
$414$	0	0
$415$	−4608.43	−0.545106
$416$	2159.08	0.254466
$417$	0	0
$418$	−526.846	−0.0616481
$419$	1358.74	0.158422	0.0792109	−	0.996858i	$-0.474760\pi$
$419$	1358.74	0.158422	0.0792109	+	0.996858i	$0.474760\pi$
$420$	0	0
$421$	−10002.0	−1.15789	−0.578943	−	0.815368i	$-0.696535\pi$
$421$	−10002.0	−1.15789	−0.578943	+	0.815368i	$0.696535\pi$
$422$	5003.98	0.577227
$423$	0	0
$424$	−452.859	−0.0518697
$425$	−7655.96	−0.873809
$426$	0	0
$427$	24.3189	0.00275614
$428$	−6437.99	−0.727084
$429$	0	0
$430$	−490.504	−0.0550097
$431$	−11218.8	−1.25381	−0.626904	−	0.779097i	$-0.715678\pi$
$431$	−11218.8	−1.25381	−0.626904	+	0.779097i	$0.715678\pi$
$432$	0	0
$433$	−1572.19	−0.174491	−0.0872456	−	0.996187i	$-0.527806\pi$
$433$	−1572.19	−0.174491	−0.0872456	+	0.996187i	$0.527806\pi$
$434$	−3772.09	−0.417203
$435$	0	0
$436$	2154.54	0.236660
$437$	3404.11	0.372633
$438$	0	0
$439$	−9499.44	−1.03276	−0.516382	−	0.856358i	$-0.672722\pi$
$439$	−9499.44	−1.03276	−0.516382	+	0.856358i	$0.672722\pi$
$440$	−568.847	−0.0616335
$441$	0	0
$442$	10467.6	1.12645
$443$	−10703.4	−1.14793	−0.573967	−	0.818878i	$-0.694596\pi$
$443$	−10703.4	−1.14793	−0.573967	+	0.818878i	$0.694596\pi$
$444$	0	0
$445$	4233.71	0.451004
$446$	6134.17	0.651259
$447$	0	0
$448$	−399.307	−0.0421104
$449$	−11068.5	−1.16338	−0.581689	−	0.813411i	$-0.697608\pi$
$449$	−11068.5	−1.16338	−0.581689	+	0.813411i	$0.697608\pi$
$450$	0	0
$451$	−3672.76	−0.383467
$452$	−5178.04	−0.538837
$453$	0	0
$454$	225.247	0.0232849
$455$	−2159.00	−0.222451
$456$	0	0
$457$	−15993.1	−1.63704	−0.818519	−	0.574479i	$-0.805205\pi$
$457$	−15993.1	−1.63704	−0.818519	+	0.574479i	$0.805205\pi$
$458$	−8319.73	−0.848811
$459$	0	0
$460$	3675.49	0.372545
$461$	−211.411	−0.0213588	−0.0106794	−	0.999943i	$-0.503399\pi$
$461$	−211.411	−0.0213588	−0.0106794	+	0.999943i	$0.503399\pi$
$462$	0	0
$463$	8714.30	0.874704	0.437352	−	0.899291i	$-0.355916\pi$
$463$	8714.30	0.874704	0.437352	+	0.899291i	$0.355916\pi$
$464$	−2849.70	−0.285117
$465$	0	0
$466$	299.750	0.0297976
$467$	−2200.09	−0.218004	−0.109002	−	0.994042i	$-0.534766\pi$
$467$	−2200.09	−0.218004	−0.109002	+	0.994042i	$0.534766\pi$
$468$	0	0
$469$	3047.02	0.299996
$470$	5046.55	0.495277
$471$	0	0
$472$	4711.79	0.459487
$473$	662.990	0.0644489
$474$	0	0
$475$	−1875.24	−0.181141
$476$	−1935.90	−0.186412
$477$	0	0
$478$	−410.215	−0.0392527
$479$	−12464.6	−1.18898	−0.594492	−	0.804102i	$-0.702647\pi$
$479$	−12464.6	−1.18898	−0.594492	+	0.804102i	$0.702647\pi$
$480$	0	0
$481$	−18911.6	−1.79272
$482$	11626.1	1.09866
$483$	0	0
$484$	−4555.12	−0.427791
$485$	−7349.31	−0.688072
$486$	0	0
$487$	−9659.55	−0.898801	−0.449401	−	0.893330i	$-0.648362\pi$
$487$	−9659.55	−0.898801	−0.449401	+	0.893330i	$0.648362\pi$
$488$	−31.1822	−0.00289253
$489$	0	0
$490$	−3118.98	−0.287554
$491$	−9209.41	−0.846466	−0.423233	−	0.906021i	$-0.639105\pi$
$491$	−9209.41	−0.846466	−0.423233	+	0.906021i	$0.639105\pi$
$492$	0	0
$493$	−13815.8	−1.26214
$494$	2563.91	0.233514
$495$	0	0
$496$	4836.66	0.437847
$497$	−2792.50	−0.252033
$498$	0	0
$499$	−5030.99	−0.451339	−0.225669	−	0.974204i	$-0.572457\pi$
$499$	−5030.99	−0.451339	−0.225669	+	0.974204i	$0.572457\pi$
$500$	−4589.07	−0.410459
$501$	0	0
$502$	−12488.9	−1.11037
$503$	8172.48	0.724439	0.362220	−	0.932093i	$-0.382019\pi$
$503$	8172.48	0.724439	0.362220	+	0.932093i	$0.382019\pi$
$504$	0	0
$505$	1487.54	0.131078
$506$	−4967.99	−0.436470
$507$	0	0
$508$	858.641	0.0749922
$509$	−19218.2	−1.67354	−0.836771	−	0.547553i	$-0.815560\pi$
$509$	−19218.2	−1.67354	−0.836771	+	0.547553i	$0.815560\pi$
$510$	0	0
$511$	1447.13	0.125278
$512$	512.000	0.0441942
$513$	0	0
$514$	12095.5	1.03796
$515$	−6750.94	−0.577635
$516$	0	0
$517$	−6821.18	−0.580261
$518$	3497.57	0.296668
$519$	0	0
$520$	2768.31	0.233459
$521$	12193.1	1.02532	0.512660	−	0.858592i	$-0.328660\pi$
$521$	12193.1	1.02532	0.512660	+	0.858592i	$0.328660\pi$
$522$	0	0
$523$	16254.2	1.35898	0.679490	−	0.733685i	$-0.262201\pi$
$523$	16254.2	1.35898	0.679490	+	0.733685i	$0.262201\pi$
$524$	3686.33	0.307325
$525$	0	0
$526$	10605.9	0.879165
$527$	23448.9	1.93824
$528$	0	0
$529$	19932.6	1.63825
$530$	−580.642	−0.0475877
$531$	0	0
$532$	−474.177	−0.0386432
$533$	17873.6	1.45252
$534$	0	0
$535$	−8254.60	−0.667061
$536$	−3906.95	−0.314841
$537$	0	0
$538$	−8901.06	−0.713293
$539$	4215.78	0.336895
$540$	0	0
$541$	−22038.0	−1.75136	−0.875680	−	0.482892i	$-0.839586\pi$
$541$	−22038.0	−1.75136	−0.875680	+	0.482892i	$0.839586\pi$
$542$	8675.57	0.687542
$543$	0	0
$544$	2482.26	0.195636
$545$	2762.49	0.217123
$546$	0	0
$547$	657.058	0.0513597	0.0256798	−	0.999670i	$-0.491825\pi$
$547$	657.058	0.0513597	0.0256798	+	0.999670i	$0.491825\pi$
$548$	−5937.74	−0.462861
$549$	0	0
$550$	2736.73	0.212172
$551$	−3384.02	−0.261641
$552$	0	0
$553$	3036.83	0.233525
$554$	−7002.23	−0.536997
$555$	0	0
$556$	617.038	0.0470652
$557$	−16627.1	−1.26484	−0.632419	−	0.774627i	$-0.717938\pi$
$557$	−16627.1	−1.26484	−0.632419	+	0.774627i	$0.717938\pi$
$558$	0	0
$559$	−3226.46	−0.244123
$560$	−511.979	−0.0386340
$561$	0	0
$562$	−7161.30	−0.537511
$563$	2379.19	0.178101	0.0890506	−	0.996027i	$-0.471617\pi$
$563$	2379.19	0.178101	0.0890506	+	0.996027i	$0.471617\pi$
$564$	0	0
$565$	−6639.13	−0.494354
$566$	−11798.7	−0.876215
$567$	0	0
$568$	3580.60	0.264505
$569$	−9386.07	−0.691537	−0.345769	−	0.938320i	$-0.612382\pi$
$569$	−9386.07	−0.691537	−0.345769	+	0.938320i	$0.612382\pi$
$570$	0	0
$571$	−13268.7	−0.972463	−0.486231	−	0.873830i	$-0.661629\pi$
$571$	−13268.7	−0.972463	−0.486231	+	0.873830i	$0.661629\pi$
$572$	−3741.79	−0.273518
$573$	0	0
$574$	−3305.59	−0.240371
$575$	−17682.9	−1.28248
$576$	0	0
$577$	12823.8	0.925238	0.462619	−	0.886557i	$-0.346910\pi$
$577$	12823.8	0.925238	0.462619	+	0.886557i	$0.346910\pi$
$578$	2208.41	0.158923
$579$	0	0
$580$	−3653.80	−0.261579
$581$	5606.27	0.400322
$582$	0	0
$583$	784.825	0.0557532
$584$	−1855.54	−0.131478
$585$	0	0
$586$	12666.2	0.892895
$587$	−13425.6	−0.944009	−0.472005	−	0.881596i	$-0.656469\pi$
$587$	−13425.6	−0.944009	−0.472005	+	0.881596i	$0.656469\pi$
$588$	0	0
$589$	5743.53	0.401796
$590$	6041.32	0.421554
$591$	0	0
$592$	−4484.66	−0.311348
$593$	−24121.5	−1.67041	−0.835204	−	0.549941i	$-0.814650\pi$
$593$	−24121.5	−1.67041	−0.835204	+	0.549941i	$0.814650\pi$
$594$	0	0
$595$	−2482.16	−0.171023
$596$	−4072.96	−0.279924
$597$	0	0
$598$	24176.9	1.65329
$599$	11853.3	0.808534	0.404267	−	0.914641i	$-0.367527\pi$
$599$	11853.3	0.808534	0.404267	+	0.914641i	$0.367527\pi$
$600$	0	0
$601$	−4097.82	−0.278126	−0.139063	−	0.990284i	$-0.544409\pi$
$601$	−4097.82	−0.278126	−0.139063	+	0.990284i	$0.544409\pi$
$602$	596.710	0.0403988
$603$	0	0
$604$	4441.96	0.299240
$605$	−5840.43	−0.392475
$606$	0	0
$607$	−1836.71	−0.122817	−0.0614085	−	0.998113i	$-0.519559\pi$
$607$	−1836.71	−0.122817	−0.0614085	+	0.998113i	$0.519559\pi$
$608$	608.000	0.0405554
$609$	0	0
$610$	−39.9809	−0.00265374
$611$	33195.5	2.19795
$612$	0	0
$613$	2088.93	0.137636	0.0688182	−	0.997629i	$-0.478077\pi$
$613$	2088.93	0.137636	0.0688182	+	0.997629i	$0.478077\pi$
$614$	12185.1	0.800898
$615$	0	0
$616$	692.017	0.0452632
$617$	257.433	0.0167972	0.00839860	−	0.999965i	$-0.497327\pi$
$617$	257.433	0.0167972	0.00839860	+	0.999965i	$0.497327\pi$
$618$	0	0
$619$	6453.03	0.419013	0.209506	−	0.977807i	$-0.432814\pi$
$619$	6453.03	0.419013	0.209506	+	0.977807i	$0.432814\pi$
$620$	6201.42	0.401701
$621$	0	0
$622$	4172.80	0.268994
$623$	−5150.41	−0.331215
$624$	0	0
$625$	6453.11	0.412999
$626$	14739.7	0.941084
$627$	0	0
$628$	1330.21	0.0845238
$629$	−21742.4	−1.37826
$630$	0	0
$631$	4879.66	0.307854	0.153927	−	0.988082i	$-0.450808\pi$
$631$	4879.66	0.307854	0.153927	+	0.988082i	$0.450808\pi$
$632$	−3893.89	−0.245080
$633$	0	0
$634$	21574.0	1.35144
$635$	1100.92	0.0688013
$636$	0	0
$637$	−20516.2	−1.27611
$638$	4938.67	0.306464
$639$	0	0
$640$	656.471	0.0405458
$641$	32225.5	1.98570	0.992848	−	0.119387i	$-0.0380930\pi$
$641$	32225.5	1.98570	0.992848	+	0.119387i	$0.0380930\pi$
$642$	0	0
$643$	4756.76	0.291739	0.145870	−	0.989304i	$-0.453402\pi$
$643$	4756.76	0.291739	0.145870	+	0.989304i	$0.453402\pi$
$644$	−4471.33	−0.273595
$645$	0	0
$646$	2947.68	0.179528
$647$	−1565.49	−0.0951246	−0.0475623	−	0.998868i	$-0.515145\pi$
$647$	−1565.49	−0.0951246	−0.0475623	+	0.998868i	$0.515145\pi$
$648$	0	0
$649$	−8165.75	−0.493889
$650$	−13318.4	−0.803678
$651$	0	0
$652$	14169.0	0.851076
$653$	26483.2	1.58708	0.793542	−	0.608515i	$-0.208235\pi$
$653$	26483.2	1.58708	0.793542	+	0.608515i	$0.208235\pi$
$654$	0	0
$655$	4726.50	0.281954
$656$	4238.50	0.252265
$657$	0	0
$658$	−6139.26	−0.363728
$659$	19012.9	1.12388	0.561940	−	0.827178i	$-0.310055\pi$
$659$	19012.9	1.12388	0.561940	+	0.827178i	$0.310055\pi$
$660$	0	0
$661$	14742.7	0.867511	0.433756	−	0.901030i	$-0.357188\pi$
$661$	14742.7	0.867511	0.433756	+	0.901030i	$0.357188\pi$
$662$	−6538.16	−0.383856
$663$	0	0
$664$	−7188.48	−0.420131
$665$	−607.975	−0.0354530
$666$	0	0
$667$	−31910.2	−1.85243
$668$	2244.82	0.130022
$669$	0	0
$670$	−5009.38	−0.288849
$671$	54.0402	0.00310909
$672$	0	0
$673$	−2127.97	−0.121883	−0.0609416	−	0.998141i	$-0.519410\pi$
$673$	−2127.97	−0.121883	−0.0609416	+	0.998141i	$0.519410\pi$
$674$	−3807.36	−0.217588
$675$	0	0
$676$	9421.55	0.536046
$677$	−18032.2	−1.02369	−0.511843	−	0.859079i	$-0.671037\pi$
$677$	−18032.2	−1.02369	−0.511843	+	0.859079i	$0.671037\pi$
$678$	0	0
$679$	8940.62	0.505316
$680$	3182.68	0.179486
$681$	0	0
$682$	−8382.15	−0.470629
$683$	21854.0	1.22433	0.612166	−	0.790729i	$-0.290298\pi$
$683$	21854.0	1.22433	0.612166	+	0.790729i	$0.290298\pi$
$684$	0	0
$685$	−7613.19	−0.424650
$686$	8074.39	0.449390
$687$	0	0
$688$	−765.115	−0.0423979
$689$	−3819.38	−0.211185
$690$	0	0
$691$	3323.35	0.182961	0.0914806	−	0.995807i	$-0.470840\pi$
$691$	3323.35	0.182961	0.0914806	+	0.995807i	$0.470840\pi$
$692$	−7644.48	−0.419942
$693$	0	0
$694$	−3160.15	−0.172849
$695$	791.148	0.0431798
$696$	0	0
$697$	20549.0	1.11671
$698$	10532.2	0.571132
$699$	0	0
$700$	2463.14	0.132997
$701$	33127.4	1.78489	0.892443	−	0.451161i	$-0.148990\pi$
$701$	33127.4	1.78489	0.892443	+	0.451161i	$0.148990\pi$
$702$	0	0
$703$	−5325.53	−0.285713
$704$	−887.320	−0.0475030
$705$	0	0
$706$	14115.6	0.752474
$707$	−1809.63	−0.0962631
$708$	0	0
$709$	9647.34	0.511021	0.255510	−	0.966806i	$-0.417757\pi$
$709$	9647.34	0.511021	0.255510	+	0.966806i	$0.417757\pi$
$710$	4590.94	0.242669
$711$	0	0
$712$	6603.97	0.347604
$713$	54159.6	2.84473
$714$	0	0
$715$	−4797.61	−0.250938
$716$	−6760.23	−0.352851
$717$	0	0
$718$	1127.13	0.0585854
$719$	−35313.7	−1.83168	−0.915840	−	0.401543i	$-0.868474\pi$
$719$	−35313.7	−1.83168	−0.915840	+	0.401543i	$0.868474\pi$
$720$	0	0
$721$	8212.69	0.424211
$722$	722.000	0.0372161
$723$	0	0
$724$	−7846.08	−0.402759
$725$	17578.5	0.900482
$726$	0	0
$727$	31494.8	1.60671	0.803354	−	0.595501i	$-0.203047\pi$
$727$	31494.8	1.60671	0.803354	+	0.595501i	$0.203047\pi$
$728$	−3367.72	−0.171451
$729$	0	0
$730$	−2379.12	−0.120624
$731$	−3709.40	−0.187684
$732$	0	0
$733$	−12254.6	−0.617510	−0.308755	−	0.951142i	$-0.599912\pi$
$733$	−12254.6	−0.617510	−0.308755	+	0.951142i	$0.599912\pi$
$734$	9904.25	0.498055
$735$	0	0
$736$	5733.24	0.287133
$737$	6770.93	0.338413
$738$	0	0
$739$	−14573.9	−0.725451	−0.362726	−	0.931896i	$-0.618154\pi$
$739$	−14573.9	−0.725451	−0.362726	+	0.931896i	$0.618154\pi$
$740$	−5750.09	−0.285645
$741$	0	0
$742$	706.365	0.0349481
$743$	−34917.7	−1.72410	−0.862050	−	0.506824i	$-0.830819\pi$
$743$	−34917.7	−1.72410	−0.862050	+	0.506824i	$0.830819\pi$
$744$	0	0
$745$	−5222.22	−0.256815
$746$	906.312	0.0444805
$747$	0	0
$748$	−4301.87	−0.210283
$749$	10041.9	0.489885
$750$	0	0
$751$	16411.2	0.797410	0.398705	−	0.917079i	$-0.369460\pi$
$751$	16411.2	0.797410	0.398705	+	0.917079i	$0.369460\pi$
$752$	7871.89	0.381727
$753$	0	0
$754$	−24034.2	−1.16084
$755$	5695.35	0.274536
$756$	0	0
$757$	−23798.7	−1.14264	−0.571321	−	0.820727i	$-0.693569\pi$
$757$	−23798.7	−1.14264	−0.571321	+	0.820727i	$0.693569\pi$
$758$	15088.9	0.723026
$759$	0	0
$760$	779.559	0.0372073
$761$	−644.338	−0.0306928	−0.0153464	−	0.999882i	$-0.504885\pi$
$761$	−644.338	−0.0306928	−0.0153464	+	0.999882i	$0.504885\pi$
$762$	0	0
$763$	−3360.64	−0.159454
$764$	−13821.6	−0.654514
$765$	0	0
$766$	12435.4	0.586566
$767$	39738.9	1.87078
$768$	0	0
$769$	−37826.3	−1.77380	−0.886900	−	0.461961i	$-0.847146\pi$
$769$	−37826.3	−1.77380	−0.886900	+	0.461961i	$0.847146\pi$
$770$	887.283	0.0415266
$771$	0	0
$772$	−20655.7	−0.962973
$773$	−25716.5	−1.19658	−0.598292	−	0.801278i	$-0.704154\pi$
$773$	−25716.5	−1.19658	−0.598292	+	0.801278i	$0.704154\pi$
$774$	0	0
$775$	−29835.1	−1.38285
$776$	−11463.9	−0.530320
$777$	0	0
$778$	4294.30	0.197889
$779$	5033.22	0.231494
$780$	0	0
$781$	−6205.35	−0.284308
$782$	27795.7	1.27106
$783$	0	0
$784$	−4865.16	−0.221627
$785$	1705.55	0.0775460
$786$	0	0
$787$	10892.3	0.493354	0.246677	−	0.969098i	$-0.420661\pi$
$787$	10892.3	0.493354	0.246677	+	0.969098i	$0.420661\pi$
$788$	19585.5	0.885412
$789$	0	0
$790$	−4992.63	−0.224848
$791$	8076.66	0.363051
$792$	0	0
$793$	−262.988	−0.0117768
$794$	−27522.8	−1.23016
$795$	0	0
$796$	−13759.5	−0.612678
$797$	22513.1	1.00057	0.500286	−	0.865860i	$-0.333228\pi$
$797$	22513.1	1.00057	0.500286	+	0.865860i	$0.333228\pi$
$798$	0	0
$799$	38164.2	1.68980
$800$	−3158.29	−0.139578
$801$	0	0
$802$	17876.0	0.787061
$803$	3215.74	0.141321
$804$	0	0
$805$	−5733.00	−0.251009
$806$	40792.0	1.78268
$807$	0	0
$808$	2320.34	0.101026
$809$	26389.4	1.14685	0.573426	−	0.819258i	$-0.305614\pi$
$809$	26389.4	1.14685	0.573426	+	0.819258i	$0.305614\pi$
$810$	0	0
$811$	−1587.61	−0.0687406	−0.0343703	−	0.999409i	$-0.510943\pi$
$811$	−1587.61	−0.0687406	−0.0343703	+	0.999409i	$0.510943\pi$
$812$	4444.95	0.192102
$813$	0	0
$814$	7772.12	0.334659
$815$	18167.1	0.780817
$816$	0	0
$817$	−908.574	−0.0389070
$818$	1986.51	0.0849102
$819$	0	0
$820$	5434.48	0.231439
$821$	−29155.2	−1.23937	−0.619685	−	0.784851i	$-0.712739\pi$
$821$	−29155.2	−1.23937	−0.619685	+	0.784851i	$0.712739\pi$
$822$	0	0
$823$	−17328.6	−0.733947	−0.366973	−	0.930231i	$-0.619606\pi$
$823$	−17328.6	−0.733947	−0.366973	+	0.930231i	$0.619606\pi$
$824$	−10530.5	−0.445203
$825$	0	0
$826$	−7349.41	−0.309587
$827$	−14899.3	−0.626482	−0.313241	−	0.949674i	$-0.601415\pi$
$827$	−14899.3	−0.626482	−0.313241	+	0.949674i	$0.601415\pi$
$828$	0	0
$829$	47237.1	1.97903	0.989513	−	0.144445i	$-0.0461397\pi$
$829$	47237.1	1.97903	0.989513	+	0.144445i	$0.0461397\pi$
$830$	−9216.86	−0.385448
$831$	0	0
$832$	4318.17	0.179935
$833$	−23587.1	−0.981087
$834$	0	0
$835$	2878.24	0.119288
$836$	−1053.69	−0.0435918
$837$	0	0
$838$	2717.48	0.112021
$839$	−18848.6	−0.775596	−0.387798	−	0.921744i	$-0.626764\pi$
$839$	−18848.6	−0.775596	−0.387798	+	0.921744i	$0.626764\pi$
$840$	0	0
$841$	7332.93	0.300665
$842$	−20004.1	−0.818749
$843$	0	0
$844$	10008.0	0.408161
$845$	12080.0	0.491794
$846$	0	0
$847$	7105.03	0.288231
$848$	−905.717	−0.0366774
$849$	0	0
$850$	−15311.9	−0.617876
$851$	−50218.0	−2.02286
$852$	0	0
$853$	−10105.7	−0.405640	−0.202820	−	0.979216i	$-0.565011\pi$
$853$	−10105.7	−0.405640	−0.202820	+	0.979216i	$0.565011\pi$
$854$	48.6378	0.00194889
$855$	0	0
$856$	−12876.0	−0.514126
$857$	−41405.4	−1.65039	−0.825193	−	0.564851i	$-0.808934\pi$
$857$	−41405.4	−1.65039	−0.825193	+	0.564851i	$0.808934\pi$
$858$	0	0
$859$	29850.9	1.18568	0.592840	−	0.805320i	$-0.298007\pi$
$859$	29850.9	1.18568	0.592840	+	0.805320i	$0.298007\pi$
$860$	−981.007	−0.0388978
$861$	0	0
$862$	−22437.6	−0.886576
$863$	28597.1	1.12799	0.563996	−	0.825778i	$-0.309263\pi$
$863$	28597.1	1.12799	0.563996	+	0.825778i	$0.309263\pi$
$864$	0	0
$865$	−9801.52	−0.385274
$866$	−3144.38	−0.123384
$867$	0	0
$868$	−7544.18	−0.295007
$869$	6748.29	0.263429
$870$	0	0
$871$	−32950.9	−1.28186
$872$	4309.09	0.167344
$873$	0	0
$874$	6808.22	0.263491
$875$	7158.00	0.276554
$876$	0	0
$877$	16659.9	0.641465	0.320733	−	0.947170i	$-0.396071\pi$
$877$	16659.9	0.641465	0.320733	+	0.947170i	$0.396071\pi$
$878$	−18998.9	−0.730275
$879$	0	0
$880$	−1137.69	−0.0435814
$881$	4020.70	0.153758	0.0768790	−	0.997040i	$-0.475504\pi$
$881$	4020.70	0.153758	0.0768790	+	0.997040i	$0.475504\pi$
$882$	0	0
$883$	−7727.54	−0.294510	−0.147255	−	0.989099i	$-0.547044\pi$
$883$	−7727.54	−0.294510	−0.147255	+	0.989099i	$0.547044\pi$
$884$	20935.2	0.796523
$885$	0	0
$886$	−21406.8	−0.811712
$887$	30222.8	1.14406	0.572030	−	0.820233i	$-0.306156\pi$
$887$	30222.8	1.14406	0.572030	+	0.820233i	$0.306156\pi$
$888$	0	0
$889$	−1339.30	−0.0505272
$890$	8467.41	0.318908
$891$	0	0
$892$	12268.3	0.460509
$893$	9347.87	0.350296
$894$	0	0
$895$	−8667.76	−0.323722
$896$	−798.613	−0.0297766
$897$	0	0
$898$	−22137.1	−0.822632
$899$	−53840.0	−1.99740
$900$	0	0
$901$	−4391.07	−0.162361
$902$	−7345.52	−0.271152
$903$	0	0
$904$	−10356.1	−0.381016
$905$	−10060.0	−0.369509
$906$	0	0
$907$	−9450.13	−0.345961	−0.172980	−	0.984925i	$-0.555340\pi$
$907$	−9450.13	−0.345961	−0.172980	+	0.984925i	$0.555340\pi$
$908$	450.494	0.0164649
$909$	0	0
$910$	−4317.99	−0.157297
$911$	−19545.5	−0.710836	−0.355418	−	0.934707i	$-0.615661\pi$
$911$	−19545.5	−0.710836	−0.355418	+	0.934707i	$0.615661\pi$
$912$	0	0
$913$	12458.0	0.451587
$914$	−31986.2	−1.15756
$915$	0	0
$916$	−16639.5	−0.600200
$917$	−5749.91	−0.207065
$918$	0	0
$919$	−14437.2	−0.518214	−0.259107	−	0.965849i	$-0.583428\pi$
$919$	−14437.2	−0.518214	−0.259107	+	0.965849i	$0.583428\pi$
$920$	7350.99	0.263429
$921$	0	0
$922$	−422.822	−0.0151029
$923$	30198.5	1.07692
$924$	0	0
$925$	27663.8	0.983330
$926$	17428.6	0.618509
$927$	0	0
$928$	−5699.41	−0.201608
$929$	−47411.7	−1.67441	−0.837205	−	0.546890i	$-0.815812\pi$
$929$	−47411.7	−1.67441	−0.837205	+	0.546890i	$0.815812\pi$
$930$	0	0
$931$	−5777.38	−0.203379
$932$	599.501	0.0210701
$933$	0	0
$934$	−4400.18	−0.154152
$935$	−5515.73	−0.192924
$936$	0	0
$937$	3980.28	0.138773	0.0693864	−	0.997590i	$-0.477896\pi$
$937$	3980.28	0.138773	0.0693864	+	0.997590i	$0.477896\pi$
$938$	6094.03	0.212129
$939$	0	0
$940$	10093.1	0.350214
$941$	−17823.8	−0.617469	−0.308734	−	0.951148i	$-0.599905\pi$
$941$	−17823.8	−0.617469	−0.308734	+	0.951148i	$0.599905\pi$
$942$	0	0
$943$	47461.6	1.63899
$944$	9423.58	0.324906
$945$	0	0
$946$	1325.98	0.0455722
$947$	−45769.4	−1.57054	−0.785272	−	0.619151i	$-0.787477\pi$
$947$	−45769.4	−1.57054	−0.785272	+	0.619151i	$0.787477\pi$
$948$	0	0
$949$	−15649.5	−0.535305
$950$	−3750.47	−0.128086
$951$	0	0
$952$	−3871.81	−0.131813
$953$	38065.7	1.29388	0.646941	−	0.762540i	$-0.276048\pi$
$953$	38065.7	1.29388	0.646941	+	0.762540i	$0.276048\pi$
$954$	0	0
$955$	−17721.7	−0.600482
$956$	−820.430	−0.0277559
$957$	0	0
$958$	−24929.2	−0.840738
$959$	9261.64	0.311860
$960$	0	0
$961$	61588.9	2.06737
$962$	−37823.3	−1.26764
$963$	0	0
$964$	23252.3	0.776873
$965$	−26484.1	−0.883475
$966$	0	0
$967$	11933.6	0.396854	0.198427	−	0.980116i	$-0.436417\pi$
$967$	11933.6	0.396854	0.198427	+	0.980116i	$0.436417\pi$
$968$	−9110.23	−0.302494
$969$	0	0
$970$	−14698.6	−0.486540
$971$	33005.3	1.09083	0.545413	−	0.838168i	$-0.316373\pi$
$971$	33005.3	1.09083	0.545413	+	0.838168i	$0.316373\pi$
$972$	0	0
$973$	−962.451	−0.0317110
$974$	−19319.1	−0.635548
$975$	0	0
$976$	−62.3644	−0.00204532
$977$	−8727.11	−0.285778	−0.142889	−	0.989739i	$-0.545639\pi$
$977$	−8727.11	−0.285778	−0.142889	+	0.989739i	$0.545639\pi$
$978$	0	0
$979$	−11445.0	−0.373630
$980$	−6237.97	−0.203331
$981$	0	0
$982$	−18418.8	−0.598542
$983$	38175.0	1.23865	0.619326	−	0.785134i	$-0.287406\pi$
$983$	38175.0	1.23865	0.619326	+	0.785134i	$0.287406\pi$
$984$	0	0
$985$	25112.0	0.812318
$986$	−27631.7	−0.892466
$987$	0	0
$988$	5127.83	0.165119
$989$	−8567.56	−0.275463
$990$	0	0
$991$	−37770.2	−1.21071	−0.605354	−	0.795957i	$-0.706968\pi$
$991$	−37770.2	−1.21071	−0.605354	+	0.795957i	$0.706968\pi$
$992$	9673.32	0.309605
$993$	0	0
$994$	−5584.99	−0.178215
$995$	−17642.0	−0.562099
$996$	0	0
$997$	9465.53	0.300678	0.150339	−	0.988634i	$-0.451963\pi$
$997$	9465.53	0.300678	0.150339	+	0.988634i	$0.451963\pi$
$998$	−10062.0	−0.319145
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.a.m.1.2	yes	3
3.2	odd	2		342.4.a.l.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.4.a.l.1.2	✓	3	3.2	odd	2
342.4.a.m.1.2	yes	3	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.12868 q^{5} -6.23917 q^{7} +8.00000 q^{8} +10.2574 q^{10} -13.8644 q^{11} +67.4714 q^{13} -12.4783 q^{14} +16.0000 q^{16} +77.5706 q^{17} +19.0000 q^{19} +20.5147 q^{20} -27.7287 q^{22} +179.164 q^{23} -98.6967 q^{25} +134.943 q^{26} -24.9567 q^{28} -178.107 q^{29} +302.291 q^{31} +32.0000 q^{32} +155.141 q^{34} -31.9987 q^{35} -280.291 q^{37} +38.0000 q^{38} +41.0294 q^{40} +264.906 q^{41} -47.8197 q^{43} -55.4575 q^{44} +358.327 q^{46} +491.993 q^{47} -304.073 q^{49} -197.393 q^{50} +269.886 q^{52} -56.6073 q^{53} -71.1059 q^{55} -49.9133 q^{56} -356.213 q^{58} +588.974 q^{59} -3.89778 q^{61} +604.582 q^{62} +64.0000 q^{64} +346.039 q^{65} -488.369 q^{67} +310.283 q^{68} -63.9974 q^{70} +447.575 q^{71} -231.943 q^{73} -560.582 q^{74} +76.0000 q^{76} +86.5021 q^{77} -486.736 q^{79} +82.0589 q^{80} +529.813 q^{82} -898.560 q^{83} +397.835 q^{85} -95.6394 q^{86} -110.915 q^{88} +825.497 q^{89} -420.965 q^{91} +716.655 q^{92} +983.987 q^{94} +97.4449 q^{95} -1432.98 q^{97} -608.146 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} + 12 q^{4} + 14 q^{7} + 24 q^{8} + 70 q^{11} + 10 q^{13} + 28 q^{14} + 48 q^{16} + 148 q^{17} + 57 q^{19} + 140 q^{22} + 118 q^{23} + 183 q^{25} + 20 q^{26} + 56 q^{28} + 270 q^{29} + 64 q^{31}+ \cdots - 966 q^{98}+O(q^{100}) \) 3 * q + 6 * q^2 + 12 * q^4 + 14 * q^7 + 24 * q^8 + 70 * q^11 + 10 * q^13 + 28 * q^14 + 48 * q^16 + 148 * q^17 + 57 * q^19 + 140 * q^22 + 118 * q^23 + 183 * q^25 + 20 * q^26 + 56 * q^28 + 270 * q^29 + 64 * q^31 + 96 * q^32 + 296 * q^34 + 312 * q^35 + 2 * q^37 + 114 * q^38 + 406 * q^41 + 70 * q^43 + 280 * q^44 + 236 * q^46 + 310 * q^47 - 483 * q^49 + 366 * q^50 + 40 * q^52 + 950 * q^53 - 1050 * q^55 + 112 * q^56 + 540 * q^58 + 628 * q^59 - 564 * q^61 + 128 * q^62 + 192 * q^64 + 984 * q^65 - 1388 * q^67 + 592 * q^68 + 624 * q^70 + 404 * q^71 - 748 * q^73 + 4 * q^74 + 228 * q^76 + 352 * q^77 - 160 * q^79 + 812 * q^82 + 818 * q^83 - 2922 * q^85 + 140 * q^86 + 560 * q^88 + 1102 * q^89 - 628 * q^91 + 472 * q^92 + 620 * q^94 - 1782 * q^97 - 966 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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