LMFDB - Embedded newform 546.4.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,4,Mod(1,546)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("546.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,3,4,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.2150428631$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +9.00000 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +18.0000 q^{10} +62.0000 q^{11} +12.0000 q^{12} -13.0000 q^{13} -14.0000 q^{14} +27.0000 q^{15} +16.0000 q^{16} -16.0000 q^{17} +18.0000 q^{18} +79.0000 q^{19} +36.0000 q^{20} -21.0000 q^{21} +124.000 q^{22} -155.000 q^{23} +24.0000 q^{24} -44.0000 q^{25} -26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} +51.0000 q^{29} +54.0000 q^{30} +243.000 q^{31} +32.0000 q^{32} +186.000 q^{33} -32.0000 q^{34} -63.0000 q^{35} +36.0000 q^{36} +412.000 q^{37} +158.000 q^{38} -39.0000 q^{39} +72.0000 q^{40} -406.000 q^{41} -42.0000 q^{42} -103.000 q^{43} +248.000 q^{44} +81.0000 q^{45} -310.000 q^{46} +429.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} -88.0000 q^{50} -48.0000 q^{51} -52.0000 q^{52} -169.000 q^{53} +54.0000 q^{54} +558.000 q^{55} -56.0000 q^{56} +237.000 q^{57} +102.000 q^{58} +320.000 q^{59} +108.000 q^{60} -614.000 q^{61} +486.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -117.000 q^{65} +372.000 q^{66} +258.000 q^{67} -64.0000 q^{68} -465.000 q^{69} -126.000 q^{70} -264.000 q^{71} +72.0000 q^{72} -121.000 q^{73} +824.000 q^{74} -132.000 q^{75} +316.000 q^{76} -434.000 q^{77} -78.0000 q^{78} -967.000 q^{79} +144.000 q^{80} +81.0000 q^{81} -812.000 q^{82} -679.000 q^{83} -84.0000 q^{84} -144.000 q^{85} -206.000 q^{86} +153.000 q^{87} +496.000 q^{88} +1059.00 q^{89} +162.000 q^{90} +91.0000 q^{91} -620.000 q^{92} +729.000 q^{93} +858.000 q^{94} +711.000 q^{95} +96.0000 q^{96} -21.0000 q^{97} +98.0000 q^{98} +558.000 q^{99} +O(q^{100})\)

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	18.0000	0.569210
$11$	62.0000	1.69943	0.849714	−	0.527244i	$-0.176775\pi$
$11$	62.0000	1.69943	0.849714	+	0.527244i	$0.176775\pi$
$12$	12.0000	0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−14.0000	−0.267261
$15$	27.0000	0.464758
$16$	16.0000	0.250000
$17$	−16.0000	−0.228269	−0.114134	−	0.993465i	$-0.536409\pi$
$17$	−16.0000	−0.228269	−0.114134	+	0.993465i	$0.536409\pi$
$18$	18.0000	0.235702
$19$	79.0000	0.953886	0.476943	−	0.878934i	$-0.341745\pi$
$19$	79.0000	0.953886	0.476943	+	0.878934i	$0.341745\pi$
$20$	36.0000	0.402492
$21$	−21.0000	−0.218218
$22$	124.000	1.20168
$23$	−155.000	−1.40521	−0.702603	−	0.711582i	$-0.747979\pi$
$23$	−155.000	−1.40521	−0.702603	+	0.711582i	$0.747979\pi$
$24$	24.0000	0.204124
$25$	−44.0000	−0.352000
$26$	−26.0000	−0.196116
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	51.0000	0.326568	0.163284	−	0.986579i	$-0.447791\pi$
$29$	51.0000	0.326568	0.163284	+	0.986579i	$0.447791\pi$
$30$	54.0000	0.328634
$31$	243.000	1.40787	0.703937	−	0.710263i	$-0.251424\pi$
$31$	243.000	1.40787	0.703937	+	0.710263i	$0.251424\pi$
$32$	32.0000	0.176777
$33$	186.000	0.981165
$34$	−32.0000	−0.161410
$35$	−63.0000	−0.304256
$36$	36.0000	0.166667
$37$	412.000	1.83060	0.915302	−	0.402767i	$-0.131952\pi$
$37$	412.000	1.83060	0.915302	+	0.402767i	$0.131952\pi$
$38$	158.000	0.674500
$39$	−39.0000	−0.160128
$40$	72.0000	0.284605
$41$	−406.000	−1.54650	−0.773251	−	0.634101i	$-0.781371\pi$
$41$	−406.000	−1.54650	−0.773251	+	0.634101i	$0.781371\pi$
$42$	−42.0000	−0.154303
$43$	−103.000	−0.365287	−0.182644	−	0.983179i	$-0.558465\pi$
$43$	−103.000	−0.365287	−0.182644	+	0.983179i	$0.558465\pi$
$44$	248.000	0.849714
$45$	81.0000	0.268328
$46$	−310.000	−0.993631
$47$	429.000	1.33141	0.665703	−	0.746217i	$-0.268132\pi$
$47$	429.000	1.33141	0.665703	+	0.746217i	$0.268132\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−88.0000	−0.248902
$51$	−48.0000	−0.131791
$52$	−52.0000	−0.138675
$53$	−169.000	−0.437999	−0.218999	−	0.975725i	$-0.570279\pi$
$53$	−169.000	−0.437999	−0.218999	+	0.975725i	$0.570279\pi$
$54$	54.0000	0.136083
$55$	558.000	1.36801
$56$	−56.0000	−0.133631
$57$	237.000	0.550727
$58$	102.000	0.230918
$59$	320.000	0.706109	0.353055	−	0.935603i	$-0.385143\pi$
$59$	320.000	0.706109	0.353055	+	0.935603i	$0.385143\pi$
$60$	108.000	0.232379
$61$	−614.000	−1.28876	−0.644382	−	0.764703i	$-0.722885\pi$
$61$	−614.000	−1.28876	−0.644382	+	0.764703i	$0.722885\pi$
$62$	486.000	0.995517
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−117.000	−0.223263
$66$	372.000	0.693788
$67$	258.000	0.470444	0.235222	−	0.971942i	$-0.424418\pi$
$67$	258.000	0.470444	0.235222	+	0.971942i	$0.424418\pi$
$68$	−64.0000	−0.114134
$69$	−465.000	−0.811296
$70$	−126.000	−0.215141
$71$	−264.000	−0.441282	−0.220641	−	0.975355i	$-0.570815\pi$
$71$	−264.000	−0.441282	−0.220641	+	0.975355i	$0.570815\pi$
$72$	72.0000	0.117851
$73$	−121.000	−0.194000	−0.0969999	−	0.995284i	$-0.530925\pi$
$73$	−121.000	−0.194000	−0.0969999	+	0.995284i	$0.530925\pi$
$74$	824.000	1.29443
$75$	−132.000	−0.203227
$76$	316.000	0.476943
$77$	−434.000	−0.642323
$78$	−78.0000	−0.113228
$79$	−967.000	−1.37716	−0.688582	−	0.725158i	$-0.741767\pi$
$79$	−967.000	−1.37716	−0.688582	+	0.725158i	$0.741767\pi$
$80$	144.000	0.201246
$81$	81.0000	0.111111
$82$	−812.000	−1.09354
$83$	−679.000	−0.897951	−0.448975	−	0.893544i	$-0.648211\pi$
$83$	−679.000	−0.897951	−0.448975	+	0.893544i	$0.648211\pi$
$84$	−84.0000	−0.109109
$85$	−144.000	−0.183753
$86$	−206.000	−0.258297
$87$	153.000	0.188544
$88$	496.000	0.600838
$89$	1059.00	1.26128	0.630639	−	0.776076i	$-0.282793\pi$
$89$	1059.00	1.26128	0.630639	+	0.776076i	$0.282793\pi$
$90$	162.000	0.189737
$91$	91.0000	0.104828
$92$	−620.000	−0.702603
$93$	729.000	0.812836
$94$	858.000	0.941446
$95$	711.000	0.767864
$96$	96.0000	0.102062
$97$	−21.0000	−0.0219817	−0.0109909	−	0.999940i	$-0.503499\pi$
$97$	−21.0000	−0.0219817	−0.0109909	+	0.999940i	$0.503499\pi$
$98$	98.0000	0.101015
$99$	558.000	0.566476
$100$	−176.000	−0.176000
$101$	894.000	0.880756	0.440378	−	0.897813i	$-0.354844\pi$
$101$	894.000	0.880756	0.440378	+	0.897813i	$0.354844\pi$
$102$	−96.0000	−0.0931904
$103$	128.000	0.122449	0.0612243	−	0.998124i	$-0.480499\pi$
$103$	128.000	0.122449	0.0612243	+	0.998124i	$0.480499\pi$
$104$	−104.000	−0.0980581
$105$	−189.000	−0.175662
$106$	−338.000	−0.309712
$107$	−708.000	−0.639672	−0.319836	−	0.947473i	$-0.603628\pi$
$107$	−708.000	−0.639672	−0.319836	+	0.947473i	$0.603628\pi$
$108$	108.000	0.0962250
$109$	1238.00	1.08788	0.543940	−	0.839124i	$-0.316932\pi$
$109$	1238.00	1.08788	0.543940	+	0.839124i	$0.316932\pi$
$110$	1116.00	0.967331
$111$	1236.00	1.05690
$112$	−112.000	−0.0944911
$113$	1073.00	0.893269	0.446634	−	0.894717i	$-0.352623\pi$
$113$	1073.00	0.893269	0.446634	+	0.894717i	$0.352623\pi$
$114$	474.000	0.389423
$115$	−1395.00	−1.13117
$116$	204.000	0.163284
$117$	−117.000	−0.0924500
$118$	640.000	0.499295
$119$	112.000	0.0862775
$120$	216.000	0.164317
$121$	2513.00	1.88805
$122$	−1228.00	−0.911294
$123$	−1218.00	−0.892873
$124$	972.000	0.703937
$125$	−1521.00	−1.08834
$126$	−126.000	−0.0890871
$127$	−1748.00	−1.22134	−0.610669	−	0.791886i	$-0.709099\pi$
$127$	−1748.00	−1.22134	−0.610669	+	0.791886i	$0.709099\pi$
$128$	128.000	0.0883883
$129$	−309.000	−0.210899
$130$	−234.000	−0.157870
$131$	−936.000	−0.624265	−0.312132	−	0.950039i	$-0.601043\pi$
$131$	−936.000	−0.624265	−0.312132	+	0.950039i	$0.601043\pi$
$132$	744.000	0.490582
$133$	−553.000	−0.360535
$134$	516.000	0.332654
$135$	243.000	0.154919
$136$	−128.000	−0.0807052
$137$	−1128.00	−0.703442	−0.351721	−	0.936105i	$-0.614403\pi$
$137$	−1128.00	−0.703442	−0.351721	+	0.936105i	$0.614403\pi$
$138$	−930.000	−0.573673
$139$	−854.000	−0.521118	−0.260559	−	0.965458i	$-0.583907\pi$
$139$	−854.000	−0.521118	−0.260559	+	0.965458i	$0.583907\pi$
$140$	−252.000	−0.152128
$141$	1287.00	0.768688
$142$	−528.000	−0.312034
$143$	−806.000	−0.471336
$144$	144.000	0.0833333
$145$	459.000	0.262882
$146$	−242.000	−0.137179
$147$	147.000	0.0824786
$148$	1648.00	0.915302
$149$	−442.000	−0.243020	−0.121510	−	0.992590i	$-0.538774\pi$
$149$	−442.000	−0.243020	−0.121510	+	0.992590i	$0.538774\pi$
$150$	−264.000	−0.143703
$151$	−1016.00	−0.547556	−0.273778	−	0.961793i	$-0.588273\pi$
$151$	−1016.00	−0.547556	−0.273778	+	0.961793i	$0.588273\pi$
$152$	632.000	0.337250
$153$	−144.000	−0.0760896
$154$	−868.000	−0.454191
$155$	2187.00	1.13332
$156$	−156.000	−0.0800641
$157$	−3880.00	−1.97234	−0.986171	−	0.165731i	$-0.947002\pi$
$157$	−3880.00	−1.97234	−0.986171	+	0.165731i	$0.947002\pi$
$158$	−1934.00	−0.973802
$159$	−507.000	−0.252879
$160$	288.000	0.142302
$161$	1085.00	0.531118
$162$	162.000	0.0785674
$163$	−1604.00	−0.770767	−0.385383	−	0.922757i	$-0.625931\pi$
$163$	−1604.00	−0.770767	−0.385383	+	0.922757i	$0.625931\pi$
$164$	−1624.00	−0.773251
$165$	1674.00	0.789823
$166$	−1358.00	−0.634947
$167$	959.000	0.444369	0.222185	−	0.975005i	$-0.428681\pi$
$167$	959.000	0.444369	0.222185	+	0.975005i	$0.428681\pi$
$168$	−168.000	−0.0771517
$169$	169.000	0.0769231
$170$	−288.000	−0.129933
$171$	711.000	0.317962
$172$	−412.000	−0.182644
$173$	−488.000	−0.214462	−0.107231	−	0.994234i	$-0.534198\pi$
$173$	−488.000	−0.214462	−0.107231	+	0.994234i	$0.534198\pi$
$174$	306.000	0.133321
$175$	308.000	0.133043
$176$	992.000	0.424857
$177$	960.000	0.407672
$178$	2118.00	0.891858
$179$	2533.00	1.05768	0.528842	−	0.848721i	$-0.322627\pi$
$179$	2533.00	1.05768	0.528842	+	0.848721i	$0.322627\pi$
$180$	324.000	0.134164
$181$	−3626.00	−1.48905	−0.744526	−	0.667593i	$-0.767325\pi$
$181$	−3626.00	−1.48905	−0.744526	+	0.667593i	$0.767325\pi$
$182$	182.000	0.0741249
$183$	−1842.00	−0.744069
$184$	−1240.00	−0.496815
$185$	3708.00	1.47361
$186$	1458.00	0.574762
$187$	−992.000	−0.387926
$188$	1716.00	0.665703
$189$	−189.000	−0.0727393
$190$	1422.00	0.542962
$191$	−4992.00	−1.89114	−0.945572	−	0.325413i	$-0.894497\pi$
$191$	−4992.00	−1.89114	−0.945572	+	0.325413i	$0.894497\pi$
$192$	192.000	0.0721688
$193$	−470.000	−0.175292	−0.0876460	−	0.996152i	$-0.527934\pi$
$193$	−470.000	−0.175292	−0.0876460	+	0.996152i	$0.527934\pi$
$194$	−42.0000	−0.0155434
$195$	−351.000	−0.128901
$196$	196.000	0.0714286
$197$	−3190.00	−1.15370	−0.576848	−	0.816852i	$-0.695717\pi$
$197$	−3190.00	−1.15370	−0.576848	+	0.816852i	$0.695717\pi$
$198$	1116.00	0.400559
$199$	1944.00	0.692495	0.346248	−	0.938143i	$-0.387456\pi$
$199$	1944.00	0.692495	0.346248	+	0.938143i	$0.387456\pi$
$200$	−352.000	−0.124451
$201$	774.000	0.271611
$202$	1788.00	0.622788
$203$	−357.000	−0.123431
$204$	−192.000	−0.0658955
$205$	−3654.00	−1.24491
$206$	256.000	0.0865843
$207$	−1395.00	−0.468402
$208$	−208.000	−0.0693375
$209$	4898.00	1.62106
$210$	−378.000	−0.124212
$211$	−1139.00	−0.371621	−0.185810	−	0.982586i	$-0.559491\pi$
$211$	−1139.00	−0.371621	−0.185810	+	0.982586i	$0.559491\pi$
$212$	−676.000	−0.218999
$213$	−792.000	−0.254774
$214$	−1416.00	−0.452317
$215$	−927.000	−0.294051
$216$	216.000	0.0680414
$217$	−1701.00	−0.532126
$218$	2476.00	0.769247
$219$	−363.000	−0.112006
$220$	2232.00	0.684006
$221$	208.000	0.0633104
$222$	2472.00	0.747341
$223$	833.000	0.250143	0.125071	−	0.992148i	$-0.460084\pi$
$223$	833.000	0.250143	0.125071	+	0.992148i	$0.460084\pi$
$224$	−224.000	−0.0668153
$225$	−396.000	−0.117333
$226$	2146.00	0.631636
$227$	−3516.00	−1.02804	−0.514020	−	0.857778i	$-0.671844\pi$
$227$	−3516.00	−1.02804	−0.514020	+	0.857778i	$0.671844\pi$
$228$	948.000	0.275363
$229$	3222.00	0.929763	0.464882	−	0.885373i	$-0.346097\pi$
$229$	3222.00	0.929763	0.464882	+	0.885373i	$0.346097\pi$
$230$	−2790.00	−0.799857
$231$	−1302.00	−0.370846
$232$	408.000	0.115459
$233$	−477.000	−0.134117	−0.0670586	−	0.997749i	$-0.521361\pi$
$233$	−477.000	−0.134117	−0.0670586	+	0.997749i	$0.521361\pi$
$234$	−234.000	−0.0653720
$235$	3861.00	1.07176
$236$	1280.00	0.353055
$237$	−2901.00	−0.795106
$238$	224.000	0.0610074
$239$	−6368.00	−1.72348	−0.861740	−	0.507350i	$-0.830625\pi$
$239$	−6368.00	−1.72348	−0.861740	+	0.507350i	$0.830625\pi$
$240$	432.000	0.116190
$241$	747.000	0.199662	0.0998309	−	0.995004i	$-0.468170\pi$
$241$	747.000	0.199662	0.0998309	+	0.995004i	$0.468170\pi$
$242$	5026.00	1.33506
$243$	243.000	0.0641500
$244$	−2456.00	−0.644382
$245$	441.000	0.114998
$246$	−2436.00	−0.631356
$247$	−1027.00	−0.264561
$248$	1944.00	0.497759
$249$	−2037.00	−0.518432
$250$	−3042.00	−0.769572
$251$	7518.00	1.89057	0.945283	−	0.326252i	$-0.105786\pi$
$251$	7518.00	1.89057	0.945283	+	0.326252i	$0.105786\pi$
$252$	−252.000	−0.0629941
$253$	−9610.00	−2.38805
$254$	−3496.00	−0.863616
$255$	−432.000	−0.106090
$256$	256.000	0.0625000
$257$	1094.00	0.265532	0.132766	−	0.991147i	$-0.457614\pi$
$257$	1094.00	0.265532	0.132766	+	0.991147i	$0.457614\pi$
$258$	−618.000	−0.149128
$259$	−2884.00	−0.691904
$260$	−468.000	−0.111631
$261$	459.000	0.108856
$262$	−1872.00	−0.441422
$263$	−1289.00	−0.302217	−0.151109	−	0.988517i	$-0.548284\pi$
$263$	−1289.00	−0.302217	−0.151109	+	0.988517i	$0.548284\pi$
$264$	1488.00	0.346894
$265$	−1521.00	−0.352582
$266$	−1106.00	−0.254937
$267$	3177.00	0.728199
$268$	1032.00	0.235222
$269$	−2060.00	−0.466916	−0.233458	−	0.972367i	$-0.575004\pi$
$269$	−2060.00	−0.466916	−0.233458	+	0.972367i	$0.575004\pi$
$270$	486.000	0.109545
$271$	576.000	0.129113	0.0645563	−	0.997914i	$-0.479437\pi$
$271$	576.000	0.129113	0.0645563	+	0.997914i	$0.479437\pi$
$272$	−256.000	−0.0570672
$273$	273.000	0.0605228
$274$	−2256.00	−0.497409
$275$	−2728.00	−0.598199
$276$	−1860.00	−0.405648
$277$	−5699.00	−1.23617	−0.618086	−	0.786110i	$-0.712092\pi$
$277$	−5699.00	−1.23617	−0.618086	+	0.786110i	$0.712092\pi$
$278$	−1708.00	−0.368486
$279$	2187.00	0.469291
$280$	−504.000	−0.107571
$281$	4062.00	0.862344	0.431172	−	0.902270i	$-0.358100\pi$
$281$	4062.00	0.862344	0.431172	+	0.902270i	$0.358100\pi$
$282$	2574.00	0.543544
$283$	−772.000	−0.162158	−0.0810789	−	0.996708i	$-0.525837\pi$
$283$	−772.000	−0.162158	−0.0810789	+	0.996708i	$0.525837\pi$
$284$	−1056.00	−0.220641
$285$	2133.00	0.443326
$286$	−1612.00	−0.333285
$287$	2842.00	0.584522
$288$	288.000	0.0589256
$289$	−4657.00	−0.947893
$290$	918.000	0.185886
$291$	−63.0000	−0.0126912
$292$	−484.000	−0.0969999
$293$	7833.00	1.56180	0.780902	−	0.624653i	$-0.214760\pi$
$293$	7833.00	1.56180	0.780902	+	0.624653i	$0.214760\pi$
$294$	294.000	0.0583212
$295$	2880.00	0.568407
$296$	3296.00	0.647217
$297$	1674.00	0.327055
$298$	−884.000	−0.171841
$299$	2015.00	0.389734
$300$	−528.000	−0.101614
$301$	721.000	0.138066
$302$	−2032.00	−0.387180
$303$	2682.00	0.508505
$304$	1264.00	0.238472
$305$	−5526.00	−1.03744
$306$	−288.000	−0.0538035
$307$	−2891.00	−0.537453	−0.268727	−	0.963217i	$-0.586603\pi$
$307$	−2891.00	−0.537453	−0.268727	+	0.963217i	$0.586603\pi$
$308$	−1736.00	−0.321162
$309$	384.000	0.0706958
$310$	4374.00	0.801376
$311$	−3530.00	−0.643627	−0.321813	−	0.946803i	$-0.604292\pi$
$311$	−3530.00	−0.643627	−0.321813	+	0.946803i	$0.604292\pi$
$312$	−312.000	−0.0566139
$313$	5490.00	0.991416	0.495708	−	0.868489i	$-0.334909\pi$
$313$	5490.00	0.991416	0.495708	+	0.868489i	$0.334909\pi$
$314$	−7760.00	−1.39466
$315$	−567.000	−0.101419
$316$	−3868.00	−0.688582
$317$	−5636.00	−0.998578	−0.499289	−	0.866435i	$-0.666405\pi$
$317$	−5636.00	−0.998578	−0.499289	+	0.866435i	$0.666405\pi$
$318$	−1014.00	−0.178812
$319$	3162.00	0.554978
$320$	576.000	0.100623
$321$	−2124.00	−0.369315
$322$	2170.00	0.375557
$323$	−1264.00	−0.217743
$324$	324.000	0.0555556
$325$	572.000	0.0976272
$326$	−3208.00	−0.545014
$327$	3714.00	0.628088
$328$	−3248.00	−0.546771
$329$	−3003.00	−0.503224
$330$	3348.00	0.558489
$331$	−9962.00	−1.65426	−0.827131	−	0.562008i	$-0.810029\pi$
$331$	−9962.00	−1.65426	−0.827131	+	0.562008i	$0.810029\pi$
$332$	−2716.00	−0.448975
$333$	3708.00	0.610202
$334$	1918.00	0.314216
$335$	2322.00	0.378700
$336$	−336.000	−0.0545545
$337$	−11415.0	−1.84515	−0.922574	−	0.385821i	$-0.873918\pi$
$337$	−11415.0	−1.84515	−0.922574	+	0.385821i	$0.873918\pi$
$338$	338.000	0.0543928
$339$	3219.00	0.515729
$340$	−576.000	−0.0918764
$341$	15066.0	2.39258
$342$	1422.00	0.224833
$343$	−343.000	−0.0539949
$344$	−824.000	−0.129149
$345$	−4185.00	−0.653081
$346$	−976.000	−0.151648
$347$	4864.00	0.752488	0.376244	−	0.926521i	$-0.377216\pi$
$347$	4864.00	0.752488	0.376244	+	0.926521i	$0.377216\pi$
$348$	612.000	0.0942720
$349$	−469.000	−0.0719341	−0.0359670	−	0.999353i	$-0.511451\pi$
$349$	−469.000	−0.0719341	−0.0359670	+	0.999353i	$0.511451\pi$
$350$	616.000	0.0940760
$351$	−351.000	−0.0533761
$352$	1984.00	0.300419
$353$	4198.00	0.632966	0.316483	−	0.948598i	$-0.397498\pi$
$353$	4198.00	0.632966	0.316483	+	0.948598i	$0.397498\pi$
$354$	1920.00	0.288268
$355$	−2376.00	−0.355225
$356$	4236.00	0.630639
$357$	336.000	0.0498123
$358$	5066.00	0.747895
$359$	3408.00	0.501023	0.250512	−	0.968114i	$-0.419401\pi$
$359$	3408.00	0.501023	0.250512	+	0.968114i	$0.419401\pi$
$360$	648.000	0.0948683
$361$	−618.000	−0.0901006
$362$	−7252.00	−1.05292
$363$	7539.00	1.09007
$364$	364.000	0.0524142
$365$	−1089.00	−0.156167
$366$	−3684.00	−0.526136
$367$	−8290.00	−1.17911	−0.589557	−	0.807727i	$-0.700697\pi$
$367$	−8290.00	−1.17911	−0.589557	+	0.807727i	$0.700697\pi$
$368$	−2480.00	−0.351301
$369$	−3654.00	−0.515500
$370$	7416.00	1.04200
$371$	1183.00	0.165548
$372$	2916.00	0.406418
$373$	−9906.00	−1.37510	−0.687551	−	0.726136i	$-0.741314\pi$
$373$	−9906.00	−1.37510	−0.687551	+	0.726136i	$0.741314\pi$
$374$	−1984.00	−0.274305
$375$	−4563.00	−0.628353
$376$	3432.00	0.470723
$377$	−663.000	−0.0905736
$378$	−378.000	−0.0514344
$379$	−9238.00	−1.25204	−0.626021	−	0.779806i	$-0.715318\pi$
$379$	−9238.00	−1.25204	−0.626021	+	0.779806i	$0.715318\pi$
$380$	2844.00	0.383932
$381$	−5244.00	−0.705140
$382$	−9984.00	−1.33724
$383$	4972.00	0.663335	0.331668	−	0.943396i	$-0.392389\pi$
$383$	4972.00	0.663335	0.331668	+	0.943396i	$0.392389\pi$
$384$	384.000	0.0510310
$385$	−3906.00	−0.517060
$386$	−940.000	−0.123950
$387$	−927.000	−0.121762
$388$	−84.0000	−0.0109909
$389$	7454.00	0.971550	0.485775	−	0.874084i	$-0.338538\pi$
$389$	7454.00	0.971550	0.485775	+	0.874084i	$0.338538\pi$
$390$	−702.000	−0.0911465
$391$	2480.00	0.320765
$392$	392.000	0.0505076
$393$	−2808.00	−0.360419
$394$	−6380.00	−0.815786
$395$	−8703.00	−1.10860
$396$	2232.00	0.283238
$397$	7219.00	0.912623	0.456311	−	0.889820i	$-0.349170\pi$
$397$	7219.00	0.912623	0.456311	+	0.889820i	$0.349170\pi$
$398$	3888.00	0.489668
$399$	−1659.00	−0.208155
$400$	−704.000	−0.0880000
$401$	10772.0	1.34147	0.670733	−	0.741699i	$-0.265980\pi$
$401$	10772.0	1.34147	0.670733	+	0.741699i	$0.265980\pi$
$402$	1548.00	0.192058
$403$	−3159.00	−0.390474
$404$	3576.00	0.440378
$405$	729.000	0.0894427
$406$	−714.000	−0.0872789
$407$	25544.0	3.11098
$408$	−384.000	−0.0465952
$409$	13151.0	1.58991	0.794957	−	0.606665i	$-0.207493\pi$
$409$	13151.0	1.58991	0.794957	+	0.606665i	$0.207493\pi$
$410$	−7308.00	−0.880284
$411$	−3384.00	−0.406132
$412$	512.000	0.0612243
$413$	−2240.00	−0.266884
$414$	−2790.00	−0.331210
$415$	−6111.00	−0.722837
$416$	−416.000	−0.0490290
$417$	−2562.00	−0.300867
$418$	9796.00	1.14626
$419$	6034.00	0.703533	0.351766	−	0.936088i	$-0.385581\pi$
$419$	6034.00	0.703533	0.351766	+	0.936088i	$0.385581\pi$
$420$	−756.000	−0.0878310
$421$	8864.00	1.02614	0.513070	−	0.858347i	$-0.328508\pi$
$421$	8864.00	1.02614	0.513070	+	0.858347i	$0.328508\pi$
$422$	−2278.00	−0.262776
$423$	3861.00	0.443802
$424$	−1352.00	−0.154856
$425$	704.000	0.0803506
$426$	−1584.00	−0.180153
$427$	4298.00	0.487107
$428$	−2832.00	−0.319836
$429$	−2418.00	−0.272126
$430$	−1854.00	−0.207925
$431$	−134.000	−0.0149758	−0.00748788	−	0.999972i	$-0.502383\pi$
$431$	−134.000	−0.0149758	−0.00748788	+	0.999972i	$0.502383\pi$
$432$	432.000	0.0481125
$433$	12376.0	1.37356	0.686781	−	0.726864i	$-0.259023\pi$
$433$	12376.0	1.37356	0.686781	+	0.726864i	$0.259023\pi$
$434$	−3402.00	−0.376270
$435$	1377.00	0.151775
$436$	4952.00	0.543940
$437$	−12245.0	−1.34041
$438$	−726.000	−0.0792000
$439$	14198.0	1.54358	0.771792	−	0.635875i	$-0.219361\pi$
$439$	14198.0	1.54358	0.771792	+	0.635875i	$0.219361\pi$
$440$	4464.00	0.483666
$441$	441.000	0.0476190
$442$	416.000	0.0447672
$443$	7657.00	0.821208	0.410604	−	0.911814i	$-0.365318\pi$
$443$	7657.00	0.821208	0.410604	+	0.911814i	$0.365318\pi$
$444$	4944.00	0.528450
$445$	9531.00	1.01531
$446$	1666.00	0.176878
$447$	−1326.00	−0.140308
$448$	−448.000	−0.0472456
$449$	15500.0	1.62915	0.814577	−	0.580055i	$-0.196969\pi$
$449$	15500.0	1.62915	0.814577	+	0.580055i	$0.196969\pi$
$450$	−792.000	−0.0829672
$451$	−25172.0	−2.62817
$452$	4292.00	0.446634
$453$	−3048.00	−0.316131
$454$	−7032.00	−0.726934
$455$	819.000	0.0843853
$456$	1896.00	0.194711
$457$	10184.0	1.04242	0.521212	−	0.853427i	$-0.325480\pi$
$457$	10184.0	1.04242	0.521212	+	0.853427i	$0.325480\pi$
$458$	6444.00	0.657442
$459$	−432.000	−0.0439304
$460$	−5580.00	−0.565584
$461$	19250.0	1.94482	0.972410	−	0.233279i	$-0.0749455\pi$
$461$	19250.0	1.94482	0.972410	+	0.233279i	$0.0749455\pi$
$462$	−2604.00	−0.262227
$463$	−15622.0	−1.56807	−0.784034	−	0.620717i	$-0.786841\pi$
$463$	−15622.0	−1.56807	−0.784034	+	0.620717i	$0.786841\pi$
$464$	816.000	0.0816419
$465$	6561.00	0.654321
$466$	−954.000	−0.0948352
$467$	−15594.0	−1.54519	−0.772596	−	0.634898i	$-0.781042\pi$
$467$	−15594.0	−1.54519	−0.772596	+	0.634898i	$0.781042\pi$
$468$	−468.000	−0.0462250
$469$	−1806.00	−0.177811
$470$	7722.00	0.757850
$471$	−11640.0	−1.13873
$472$	2560.00	0.249647
$473$	−6386.00	−0.620779
$474$	−5802.00	−0.562225
$475$	−3476.00	−0.335768
$476$	448.000	0.0431388
$477$	−1521.00	−0.146000
$478$	−12736.0	−1.21868
$479$	3239.00	0.308964	0.154482	−	0.987996i	$-0.450629\pi$
$479$	3239.00	0.308964	0.154482	+	0.987996i	$0.450629\pi$
$480$	864.000	0.0821584
$481$	−5356.00	−0.507718
$482$	1494.00	0.141182
$483$	3255.00	0.306641
$484$	10052.0	0.944027
$485$	−189.000	−0.0176949
$486$	486.000	0.0453609
$487$	1798.00	0.167300	0.0836501	−	0.996495i	$-0.473342\pi$
$487$	1798.00	0.167300	0.0836501	+	0.996495i	$0.473342\pi$
$488$	−4912.00	−0.455647
$489$	−4812.00	−0.445002
$490$	882.000	0.0813157
$491$	13708.0	1.25995	0.629973	−	0.776617i	$-0.283066\pi$
$491$	13708.0	1.25995	0.629973	+	0.776617i	$0.283066\pi$
$492$	−4872.00	−0.446436
$493$	−816.000	−0.0745452
$494$	−2054.00	−0.187073
$495$	5022.00	0.456004
$496$	3888.00	0.351968
$497$	1848.00	0.166789
$498$	−4074.00	−0.366587
$499$	−11936.0	−1.07080	−0.535400	−	0.844599i	$-0.679839\pi$
$499$	−11936.0	−1.07080	−0.535400	+	0.844599i	$0.679839\pi$
$500$	−6084.00	−0.544170
$501$	2877.00	0.256557
$502$	15036.0	1.33683
$503$	2562.00	0.227105	0.113553	−	0.993532i	$-0.463777\pi$
$503$	2562.00	0.227105	0.113553	+	0.993532i	$0.463777\pi$
$504$	−504.000	−0.0445435
$505$	8046.00	0.708995
$506$	−19220.0	−1.68860
$507$	507.000	0.0444116
$508$	−6992.00	−0.610669
$509$	−8055.00	−0.701437	−0.350719	−	0.936481i	$-0.614063\pi$
$509$	−8055.00	−0.701437	−0.350719	+	0.936481i	$0.614063\pi$
$510$	−864.000	−0.0750168
$511$	847.000	0.0733250
$512$	512.000	0.0441942
$513$	2133.00	0.183576
$514$	2188.00	0.187760
$515$	1152.00	0.0985693
$516$	−1236.00	−0.105449
$517$	26598.0	2.26263
$518$	−5768.00	−0.489250
$519$	−1464.00	−0.123820
$520$	−936.000	−0.0789352
$521$	−5980.00	−0.502857	−0.251429	−	0.967876i	$-0.580900\pi$
$521$	−5980.00	−0.502857	−0.251429	+	0.967876i	$0.580900\pi$
$522$	918.000	0.0769727
$523$	19238.0	1.60845	0.804225	−	0.594325i	$-0.202581\pi$
$523$	19238.0	1.60845	0.804225	+	0.594325i	$0.202581\pi$
$524$	−3744.00	−0.312132
$525$	924.000	0.0768127
$526$	−2578.00	−0.213700
$527$	−3888.00	−0.321374
$528$	2976.00	0.245291
$529$	11858.0	0.974603
$530$	−3042.00	−0.249313
$531$	2880.00	0.235370
$532$	−2212.00	−0.180268
$533$	5278.00	0.428922
$534$	6354.00	0.514915
$535$	−6372.00	−0.514926
$536$	2064.00	0.166327
$537$	7599.00	0.610654
$538$	−4120.00	−0.330160
$539$	3038.00	0.242775
$540$	972.000	0.0774597
$541$	13012.0	1.03407	0.517033	−	0.855966i	$-0.327036\pi$
$541$	13012.0	1.03407	0.517033	+	0.855966i	$0.327036\pi$
$542$	1152.00	0.0912964
$543$	−10878.0	−0.859705
$544$	−512.000	−0.0403526
$545$	11142.0	0.875726
$546$	546.000	0.0427960
$547$	18055.0	1.41129	0.705645	−	0.708565i	$-0.250657\pi$
$547$	18055.0	1.41129	0.705645	+	0.708565i	$0.250657\pi$
$548$	−4512.00	−0.351721
$549$	−5526.00	−0.429588
$550$	−5456.00	−0.422990
$551$	4029.00	0.311508
$552$	−3720.00	−0.286836
$553$	6769.00	0.520519
$554$	−11398.0	−0.874106
$555$	11124.0	0.850788
$556$	−3416.00	−0.260559
$557$	−19608.0	−1.49159	−0.745797	−	0.666174i	$-0.767931\pi$
$557$	−19608.0	−1.49159	−0.745797	+	0.666174i	$0.767931\pi$
$558$	4374.00	0.331839
$559$	1339.00	0.101312
$560$	−1008.00	−0.0760639
$561$	−2976.00	−0.223969
$562$	8124.00	0.609769
$563$	−13512.0	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−13512.0	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	5148.00	0.384344
$565$	9657.00	0.719067
$566$	−1544.00	−0.114663
$567$	−567.000	−0.0419961
$568$	−2112.00	−0.156017
$569$	−4061.00	−0.299202	−0.149601	−	0.988746i	$-0.547799\pi$
$569$	−4061.00	−0.299202	−0.149601	+	0.988746i	$0.547799\pi$
$570$	4266.00	0.313479
$571$	−2255.00	−0.165269	−0.0826347	−	0.996580i	$-0.526333\pi$
$571$	−2255.00	−0.165269	−0.0826347	+	0.996580i	$0.526333\pi$
$572$	−3224.00	−0.235668
$573$	−14976.0	−1.09185
$574$	5684.00	0.413320
$575$	6820.00	0.494632
$576$	576.000	0.0416667
$577$	−254.000	−0.0183261	−0.00916305	−	0.999958i	$-0.502917\pi$
$577$	−254.000	−0.0183261	−0.00916305	+	0.999958i	$0.502917\pi$
$578$	−9314.00	−0.670262
$579$	−1410.00	−0.101205
$580$	1836.00	0.131441
$581$	4753.00	0.339394
$582$	−126.000	−0.00897400
$583$	−10478.0	−0.744347
$584$	−968.000	−0.0685893
$585$	−1053.00	−0.0744208
$586$	15666.0	1.10436
$587$	15057.0	1.05872	0.529360	−	0.848397i	$-0.322432\pi$
$587$	15057.0	1.05872	0.529360	+	0.848397i	$0.322432\pi$
$588$	588.000	0.0412393
$589$	19197.0	1.34295
$590$	5760.00	0.401924
$591$	−9570.00	−0.666087
$592$	6592.00	0.457651
$593$	19917.0	1.37925	0.689623	−	0.724168i	$-0.257776\pi$
$593$	19917.0	1.37925	0.689623	+	0.724168i	$0.257776\pi$
$594$	3348.00	0.231263
$595$	1008.00	0.0694521
$596$	−1768.00	−0.121510
$597$	5832.00	0.399812
$598$	4030.00	0.275584
$599$	−15107.0	−1.03048	−0.515238	−	0.857047i	$-0.672297\pi$
$599$	−15107.0	−1.03048	−0.515238	+	0.857047i	$0.672297\pi$
$600$	−1056.00	−0.0718517
$601$	23674.0	1.60679	0.803397	−	0.595444i	$-0.203024\pi$
$601$	23674.0	1.60679	0.803397	+	0.595444i	$0.203024\pi$
$602$	1442.00	0.0976271
$603$	2322.00	0.156815
$604$	−4064.00	−0.273778
$605$	22617.0	1.51985
$606$	5364.00	0.359567
$607$	23746.0	1.58784	0.793921	−	0.608021i	$-0.208036\pi$
$607$	23746.0	1.58784	0.793921	+	0.608021i	$0.208036\pi$
$608$	2528.00	0.168625
$609$	−1071.00	−0.0712629
$610$	−11052.0	−0.733578
$611$	−5577.00	−0.369266
$612$	−576.000	−0.0380448
$613$	−19216.0	−1.26611	−0.633056	−	0.774106i	$-0.718200\pi$
$613$	−19216.0	−1.26611	−0.633056	+	0.774106i	$0.718200\pi$
$614$	−5782.00	−0.380037
$615$	−10962.0	−0.718749
$616$	−3472.00	−0.227096
$617$	20862.0	1.36122	0.680610	−	0.732646i	$-0.261715\pi$
$617$	20862.0	1.36122	0.680610	+	0.732646i	$0.261715\pi$
$618$	768.000	0.0499895
$619$	124.000	0.00805167	0.00402583	−	0.999992i	$-0.498719\pi$
$619$	124.000	0.00805167	0.00402583	+	0.999992i	$0.498719\pi$
$620$	8748.00	0.566658
$621$	−4185.00	−0.270432
$622$	−7060.00	−0.455113
$623$	−7413.00	−0.476718
$624$	−624.000	−0.0400320
$625$	−8189.00	−0.524096
$626$	10980.0	0.701037
$627$	14694.0	0.935920
$628$	−15520.0	−0.986171
$629$	−6592.00	−0.417870
$630$	−1134.00	−0.0717137
$631$	23942.0	1.51048	0.755242	−	0.655446i	$-0.227519\pi$
$631$	23942.0	1.51048	0.755242	+	0.655446i	$0.227519\pi$
$632$	−7736.00	−0.486901
$633$	−3417.00	−0.214555
$634$	−11272.0	−0.706101
$635$	−15732.0	−0.983158
$636$	−2028.00	−0.126439
$637$	−637.000	−0.0396214
$638$	6324.00	0.392429
$639$	−2376.00	−0.147094
$640$	1152.00	0.0711512
$641$	−15723.0	−0.968832	−0.484416	−	0.874838i	$-0.660968\pi$
$641$	−15723.0	−0.968832	−0.484416	+	0.874838i	$0.660968\pi$
$642$	−4248.00	−0.261145
$643$	−5544.00	−0.340022	−0.170011	−	0.985442i	$-0.554380\pi$
$643$	−5544.00	−0.340022	−0.170011	+	0.985442i	$0.554380\pi$
$644$	4340.00	0.265559
$645$	−2781.00	−0.169770
$646$	−2528.00	−0.153967
$647$	−6078.00	−0.369321	−0.184661	−	0.982802i	$-0.559119\pi$
$647$	−6078.00	−0.369321	−0.184661	+	0.982802i	$0.559119\pi$
$648$	648.000	0.0392837
$649$	19840.0	1.19998
$650$	1144.00	0.0690329
$651$	−5103.00	−0.307223
$652$	−6416.00	−0.385383
$653$	30778.0	1.84447	0.922233	−	0.386635i	$-0.126363\pi$
$653$	30778.0	1.84447	0.922233	+	0.386635i	$0.126363\pi$
$654$	7428.00	0.444125
$655$	−8424.00	−0.502524
$656$	−6496.00	−0.386625
$657$	−1089.00	−0.0646666
$658$	−6006.00	−0.355833
$659$	−11205.0	−0.662344	−0.331172	−	0.943570i	$-0.607444\pi$
$659$	−11205.0	−0.662344	−0.331172	+	0.943570i	$0.607444\pi$
$660$	6696.00	0.394911
$661$	−20785.0	−1.22306	−0.611530	−	0.791221i	$-0.709446\pi$
$661$	−20785.0	−1.22306	−0.611530	+	0.791221i	$0.709446\pi$
$662$	−19924.0	−1.16974
$663$	624.000	0.0365523
$664$	−5432.00	−0.317474
$665$	−4977.00	−0.290225
$666$	7416.00	0.431478
$667$	−7905.00	−0.458895
$668$	3836.00	0.222185
$669$	2499.00	0.144420
$670$	4644.00	0.267781
$671$	−38068.0	−2.19016
$672$	−672.000	−0.0385758
$673$	1297.00	0.0742878	0.0371439	−	0.999310i	$-0.488174\pi$
$673$	1297.00	0.0742878	0.0371439	+	0.999310i	$0.488174\pi$
$674$	−22830.0	−1.30472
$675$	−1188.00	−0.0677424
$676$	676.000	0.0384615
$677$	7310.00	0.414987	0.207493	−	0.978236i	$-0.433469\pi$
$677$	7310.00	0.414987	0.207493	+	0.978236i	$0.433469\pi$
$678$	6438.00	0.364675
$679$	147.000	0.00830831
$680$	−1152.00	−0.0649664
$681$	−10548.0	−0.593539
$682$	30132.0	1.69181
$683$	−26384.0	−1.47812	−0.739060	−	0.673640i	$-0.764730\pi$
$683$	−26384.0	−1.47812	−0.739060	+	0.673640i	$0.764730\pi$
$684$	2844.00	0.158981
$685$	−10152.0	−0.566260
$686$	−686.000	−0.0381802
$687$	9666.00	0.536799
$688$	−1648.00	−0.0913218
$689$	2197.00	0.121479
$690$	−8370.00	−0.461798
$691$	−27823.0	−1.53175	−0.765873	−	0.642992i	$-0.777693\pi$
$691$	−27823.0	−1.53175	−0.765873	+	0.642992i	$0.777693\pi$
$692$	−1952.00	−0.107231
$693$	−3906.00	−0.214108
$694$	9728.00	0.532089
$695$	−7686.00	−0.419492
$696$	1224.00	0.0666603
$697$	6496.00	0.353018
$698$	−938.000	−0.0508651
$699$	−1431.00	−0.0774326
$700$	1232.00	0.0665217
$701$	24117.0	1.29941	0.649705	−	0.760186i	$-0.274892\pi$
$701$	24117.0	1.29941	0.649705	+	0.760186i	$0.274892\pi$
$702$	−702.000	−0.0377426
$703$	32548.0	1.74619
$704$	3968.00	0.212428
$705$	11583.0	0.618782
$706$	8396.00	0.447575
$707$	−6258.00	−0.332894
$708$	3840.00	0.203836
$709$	28286.0	1.49831	0.749156	−	0.662394i	$-0.230459\pi$
$709$	28286.0	1.49831	0.749156	+	0.662394i	$0.230459\pi$
$710$	−4752.00	−0.251182
$711$	−8703.00	−0.459055
$712$	8472.00	0.445929
$713$	−37665.0	−1.97835
$714$	672.000	0.0352226
$715$	−7254.00	−0.379418
$716$	10132.0	0.528842
$717$	−19104.0	−0.995052
$718$	6816.00	0.354277
$719$	−26346.0	−1.36654	−0.683268	−	0.730167i	$-0.739442\pi$
$719$	−26346.0	−1.36654	−0.683268	+	0.730167i	$0.739442\pi$
$720$	1296.00	0.0670820
$721$	−896.000	−0.0462813
$722$	−1236.00	−0.0637107
$723$	2241.00	0.115275
$724$	−14504.0	−0.744526
$725$	−2244.00	−0.114952
$726$	15078.0	0.770795
$727$	−22862.0	−1.16631	−0.583153	−	0.812362i	$-0.698181\pi$
$727$	−22862.0	−1.16631	−0.583153	+	0.812362i	$0.698181\pi$
$728$	728.000	0.0370625
$729$	729.000	0.0370370
$730$	−2178.00	−0.110427
$731$	1648.00	0.0833837
$732$	−7368.00	−0.372034
$733$	18587.0	0.936598	0.468299	−	0.883570i	$-0.344867\pi$
$733$	18587.0	0.936598	0.468299	+	0.883570i	$0.344867\pi$
$734$	−16580.0	−0.833759
$735$	1323.00	0.0663940
$736$	−4960.00	−0.248408
$737$	15996.0	0.799485
$738$	−7308.00	−0.364514
$739$	5102.00	0.253965	0.126982	−	0.991905i	$-0.459471\pi$
$739$	5102.00	0.253965	0.126982	+	0.991905i	$0.459471\pi$
$740$	14832.0	0.736804
$741$	−3081.00	−0.152744
$742$	2366.00	0.117060
$743$	27428.0	1.35429	0.677144	−	0.735851i	$-0.263217\pi$
$743$	27428.0	1.35429	0.677144	+	0.735851i	$0.263217\pi$
$744$	5832.00	0.287381
$745$	−3978.00	−0.195628
$746$	−19812.0	−0.972344
$747$	−6111.00	−0.299317
$748$	−3968.00	−0.193963
$749$	4956.00	0.241773
$750$	−9126.00	−0.444313
$751$	125.000	0.00607365	0.00303683	−	0.999995i	$-0.499033\pi$
$751$	125.000	0.00607365	0.00303683	+	0.999995i	$0.499033\pi$
$752$	6864.00	0.332851
$753$	22554.0	1.09152
$754$	−1326.00	−0.0640452
$755$	−9144.00	−0.440774
$756$	−756.000	−0.0363696
$757$	−27907.0	−1.33989	−0.669945	−	0.742410i	$-0.733682\pi$
$757$	−27907.0	−1.33989	−0.669945	+	0.742410i	$0.733682\pi$
$758$	−18476.0	−0.885328
$759$	−28830.0	−1.37874
$760$	5688.00	0.271481
$761$	7793.00	0.371217	0.185608	−	0.982624i	$-0.440574\pi$
$761$	7793.00	0.371217	0.185608	+	0.982624i	$0.440574\pi$
$762$	−10488.0	−0.498609
$763$	−8666.00	−0.411180
$764$	−19968.0	−0.945572
$765$	−1296.00	−0.0612510
$766$	9944.00	0.469049
$767$	−4160.00	−0.195839
$768$	768.000	0.0360844
$769$	29393.0	1.37833	0.689167	−	0.724603i	$-0.257977\pi$
$769$	29393.0	1.37833	0.689167	+	0.724603i	$0.257977\pi$
$770$	−7812.00	−0.365617
$771$	3282.00	0.153305
$772$	−1880.00	−0.0876460
$773$	20326.0	0.945764	0.472882	−	0.881126i	$-0.343214\pi$
$773$	20326.0	0.945764	0.472882	+	0.881126i	$0.343214\pi$
$774$	−1854.00	−0.0860990
$775$	−10692.0	−0.495572
$776$	−168.000	−0.00777171
$777$	−8652.00	−0.399471
$778$	14908.0	0.686989
$779$	−32074.0	−1.47519
$780$	−1404.00	−0.0644503
$781$	−16368.0	−0.749927
$782$	4960.00	0.226815
$783$	1377.00	0.0628480
$784$	784.000	0.0357143
$785$	−34920.0	−1.58770
$786$	−5616.00	−0.254855
$787$	16679.0	0.755454	0.377727	−	0.925917i	$-0.376706\pi$
$787$	16679.0	0.755454	0.377727	+	0.925917i	$0.376706\pi$
$788$	−12760.0	−0.576848
$789$	−3867.00	−0.174485
$790$	−17406.0	−0.783896
$791$	−7511.00	−0.337624
$792$	4464.00	0.200279
$793$	7982.00	0.357439
$794$	14438.0	0.645322
$795$	−4563.00	−0.203563
$796$	7776.00	0.346248
$797$	−9282.00	−0.412529	−0.206264	−	0.978496i	$-0.566131\pi$
$797$	−9282.00	−0.412529	−0.206264	+	0.978496i	$0.566131\pi$
$798$	−3318.00	−0.147188
$799$	−6864.00	−0.303918
$800$	−1408.00	−0.0622254
$801$	9531.00	0.420426
$802$	21544.0	0.948560
$803$	−7502.00	−0.329688
$804$	3096.00	0.135805
$805$	9765.00	0.427542
$806$	−6318.00	−0.276107
$807$	−6180.00	−0.269574
$808$	7152.00	0.311394
$809$	−883.000	−0.0383741	−0.0191870	−	0.999816i	$-0.506108\pi$
$809$	−883.000	−0.0383741	−0.0191870	+	0.999816i	$0.506108\pi$
$810$	1458.00	0.0632456
$811$	−12684.0	−0.549193	−0.274596	−	0.961560i	$-0.588544\pi$
$811$	−12684.0	−0.549193	−0.274596	+	0.961560i	$0.588544\pi$
$812$	−1428.00	−0.0617155
$813$	1728.00	0.0745432
$814$	51088.0	2.19980
$815$	−14436.0	−0.620455
$816$	−768.000	−0.0329478
$817$	−8137.00	−0.348443
$818$	26302.0	1.12424
$819$	819.000	0.0349428
$820$	−14616.0	−0.622455
$821$	−34126.0	−1.45068	−0.725338	−	0.688393i	$-0.758317\pi$
$821$	−34126.0	−1.45068	−0.725338	+	0.688393i	$0.758317\pi$
$822$	−6768.00	−0.287179
$823$	−7568.00	−0.320539	−0.160270	−	0.987073i	$-0.551236\pi$
$823$	−7568.00	−0.320539	−0.160270	+	0.987073i	$0.551236\pi$
$824$	1024.00	0.0432921
$825$	−8184.00	−0.345370
$826$	−4480.00	−0.188716
$827$	32076.0	1.34872	0.674360	−	0.738403i	$-0.264420\pi$
$827$	32076.0	1.34872	0.674360	+	0.738403i	$0.264420\pi$
$828$	−5580.00	−0.234201
$829$	−22598.0	−0.946756	−0.473378	−	0.880859i	$-0.656966\pi$
$829$	−22598.0	−0.946756	−0.473378	+	0.880859i	$0.656966\pi$
$830$	−12222.0	−0.511123
$831$	−17097.0	−0.713704
$832$	−832.000	−0.0346688
$833$	−784.000	−0.0326098
$834$	−5124.00	−0.212745
$835$	8631.00	0.357710
$836$	19592.0	0.810530
$837$	6561.00	0.270945
$838$	12068.0	0.497473
$839$	25480.0	1.04847	0.524236	−	0.851573i	$-0.324351\pi$
$839$	25480.0	1.04847	0.524236	+	0.851573i	$0.324351\pi$
$840$	−1512.00	−0.0621059
$841$	−21788.0	−0.893354
$842$	17728.0	0.725591
$843$	12186.0	0.497874
$844$	−4556.00	−0.185810
$845$	1521.00	0.0619219
$846$	7722.00	0.313815
$847$	−17591.0	−0.713617
$848$	−2704.00	−0.109500
$849$	−2316.00	−0.0936218
$850$	1408.00	0.0568165
$851$	−63860.0	−2.57238
$852$	−3168.00	−0.127387
$853$	−22939.0	−0.920770	−0.460385	−	0.887719i	$-0.652289\pi$
$853$	−22939.0	−0.920770	−0.460385	+	0.887719i	$0.652289\pi$
$854$	8596.00	0.344437
$855$	6399.00	0.255955
$856$	−5664.00	−0.226158
$857$	−38782.0	−1.54582	−0.772910	−	0.634516i	$-0.781200\pi$
$857$	−38782.0	−1.54582	−0.772910	+	0.634516i	$0.781200\pi$
$858$	−4836.00	−0.192422
$859$	−13970.0	−0.554890	−0.277445	−	0.960742i	$-0.589488\pi$
$859$	−13970.0	−0.554890	−0.277445	+	0.960742i	$0.589488\pi$
$860$	−3708.00	−0.147025
$861$	8526.00	0.337474
$862$	−268.000	−0.0105895
$863$	−22212.0	−0.876136	−0.438068	−	0.898942i	$-0.644337\pi$
$863$	−22212.0	−0.876136	−0.438068	+	0.898942i	$0.644337\pi$
$864$	864.000	0.0340207
$865$	−4392.00	−0.172639
$866$	24752.0	0.971255
$867$	−13971.0	−0.547266
$868$	−6804.00	−0.266063
$869$	−59954.0	−2.34039
$870$	2754.00	0.107321
$871$	−3354.00	−0.130478
$872$	9904.00	0.384624
$873$	−189.000	−0.00732724
$874$	−24490.0	−0.947811
$875$	10647.0	0.411353
$876$	−1452.00	−0.0560029
$877$	−24816.0	−0.955504	−0.477752	−	0.878495i	$-0.658548\pi$
$877$	−24816.0	−0.955504	−0.477752	+	0.878495i	$0.658548\pi$
$878$	28396.0	1.09148
$879$	23499.0	0.901708
$880$	8928.00	0.342003
$881$	7490.00	0.286430	0.143215	−	0.989692i	$-0.454256\pi$
$881$	7490.00	0.286430	0.143215	+	0.989692i	$0.454256\pi$
$882$	882.000	0.0336718
$883$	6092.00	0.232177	0.116088	−	0.993239i	$-0.462964\pi$
$883$	6092.00	0.232177	0.116088	+	0.993239i	$0.462964\pi$
$884$	832.000	0.0316552
$885$	8640.00	0.328170
$886$	15314.0	0.580682
$887$	−6480.00	−0.245295	−0.122648	−	0.992450i	$-0.539139\pi$
$887$	−6480.00	−0.245295	−0.122648	+	0.992450i	$0.539139\pi$
$888$	9888.00	0.373671
$889$	12236.0	0.461622
$890$	19062.0	0.717932
$891$	5022.00	0.188825
$892$	3332.00	0.125071
$893$	33891.0	1.27001
$894$	−2652.00	−0.0992127
$895$	22797.0	0.851419
$896$	−896.000	−0.0334077
$897$	6045.00	0.225013
$898$	31000.0	1.15199
$899$	12393.0	0.459766
$900$	−1584.00	−0.0586667
$901$	2704.00	0.0999815
$902$	−50344.0	−1.85839
$903$	2163.00	0.0797122
$904$	8584.00	0.315818
$905$	−32634.0	−1.19866
$906$	−6096.00	−0.223539
$907$	−12111.0	−0.443373	−0.221686	−	0.975118i	$-0.571156\pi$
$907$	−12111.0	−0.443373	−0.221686	+	0.975118i	$0.571156\pi$
$908$	−14064.0	−0.514020
$909$	8046.00	0.293585
$910$	1638.00	0.0596694
$911$	7191.00	0.261524	0.130762	−	0.991414i	$-0.458258\pi$
$911$	7191.00	0.261524	0.130762	+	0.991414i	$0.458258\pi$
$912$	3792.00	0.137682
$913$	−42098.0	−1.52600
$914$	20368.0	0.737105
$915$	−16578.0	−0.598964
$916$	12888.0	0.464882
$917$	6552.00	0.235950
$918$	−864.000	−0.0310635
$919$	−12272.0	−0.440496	−0.220248	−	0.975444i	$-0.570687\pi$
$919$	−12272.0	−0.440496	−0.220248	+	0.975444i	$0.570687\pi$
$920$	−11160.0	−0.399929
$921$	−8673.00	−0.310299
$922$	38500.0	1.37520
$923$	3432.00	0.122390
$924$	−5208.00	−0.185423
$925$	−18128.0	−0.644373
$926$	−31244.0	−1.10879
$927$	1152.00	0.0408162
$928$	1632.00	0.0577296
$929$	3285.00	0.116014	0.0580072	−	0.998316i	$-0.481525\pi$
$929$	3285.00	0.116014	0.0580072	+	0.998316i	$0.481525\pi$
$930$	13122.0	0.462675
$931$	3871.00	0.136269
$932$	−1908.00	−0.0670586
$933$	−10590.0	−0.371598
$934$	−31188.0	−1.09262
$935$	−8928.00	−0.312275
$936$	−936.000	−0.0326860
$937$	−5068.00	−0.176696	−0.0883481	−	0.996090i	$-0.528159\pi$
$937$	−5068.00	−0.176696	−0.0883481	+	0.996090i	$0.528159\pi$
$938$	−3612.00	−0.125731
$939$	16470.0	0.572394
$940$	15444.0	0.535881
$941$	13571.0	0.470140	0.235070	−	0.971978i	$-0.424468\pi$
$941$	13571.0	0.470140	0.235070	+	0.971978i	$0.424468\pi$
$942$	−23280.0	−0.805205
$943$	62930.0	2.17315
$944$	5120.00	0.176527
$945$	−1701.00	−0.0585540
$946$	−12772.0	−0.438957
$947$	−17662.0	−0.606059	−0.303030	−	0.952981i	$-0.597998\pi$
$947$	−17662.0	−0.606059	−0.303030	+	0.952981i	$0.597998\pi$
$948$	−11604.0	−0.397553
$949$	1573.00	0.0538058
$950$	−6952.00	−0.237424
$951$	−16908.0	−0.576529
$952$	896.000	0.0305037
$953$	37893.0	1.28801	0.644006	−	0.765021i	$-0.277271\pi$
$953$	37893.0	1.28801	0.644006	+	0.765021i	$0.277271\pi$
$954$	−3042.00	−0.103237
$955$	−44928.0	−1.52234
$956$	−25472.0	−0.861740
$957$	9486.00	0.320417
$958$	6478.00	0.218470
$959$	7896.00	0.265876
$960$	1728.00	0.0580948
$961$	29258.0	0.982109
$962$	−10712.0	−0.359011
$963$	−6372.00	−0.213224
$964$	2988.00	0.0998309
$965$	−4230.00	−0.141107
$966$	6510.00	0.216828
$967$	−48914.0	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$967$	−48914.0	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$968$	20104.0	0.667528
$969$	−3792.00	−0.125714
$970$	−378.000	−0.0125122
$971$	31110.0	1.02818	0.514092	−	0.857735i	$-0.328129\pi$
$971$	31110.0	1.02818	0.514092	+	0.857735i	$0.328129\pi$
$972$	972.000	0.0320750
$973$	5978.00	0.196964
$974$	3596.00	0.118299
$975$	1716.00	0.0563651
$976$	−9824.00	−0.322191
$977$	22294.0	0.730039	0.365020	−	0.931000i	$-0.381062\pi$
$977$	22294.0	0.730039	0.365020	+	0.931000i	$0.381062\pi$
$978$	−9624.00	−0.314664
$979$	65658.0	2.14345
$980$	1764.00	0.0574989
$981$	11142.0	0.362627
$982$	27416.0	0.890916
$983$	−317.000	−0.0102856	−0.00514279	−	0.999987i	$-0.501637\pi$
$983$	−317.000	−0.0102856	−0.00514279	+	0.999987i	$0.501637\pi$
$984$	−9744.00	−0.315678
$985$	−28710.0	−0.928707
$986$	−1632.00	−0.0527114
$987$	−9009.00	−0.290537
$988$	−4108.00	−0.132280
$989$	15965.0	0.513304
$990$	10044.0	0.322444
$991$	40564.0	1.30026	0.650130	−	0.759823i	$-0.274714\pi$
$991$	40564.0	1.30026	0.650130	+	0.759823i	$0.274714\pi$
$992$	7776.00	0.248879
$993$	−29886.0	−0.955089
$994$	3696.00	0.117938
$995$	17496.0	0.557448
$996$	−8148.00	−0.259216
$997$	−57528.0	−1.82741	−0.913706	−	0.406376i	$-0.866792\pi$
$997$	−57528.0	−1.82741	−0.913706	+	0.406376i	$0.866792\pi$
$998$	−23872.0	−0.757169
$999$	11124.0	0.352300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.4.a.e.1.1	✓	1
3.2	odd	2		1638.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.e.1.1	✓	1	1.1	even	1	trivial
1638.4.a.b.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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