LMFDB - Embedded newform 546.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,4,Mod(1,546)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.2150428631$
Analytic rank:	$1$
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)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -14.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -28.0000 q^{10} +8.00000 q^{11} +12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} -42.0000 q^{15} +16.0000 q^{16} -98.0000 q^{17} +18.0000 q^{18} -28.0000 q^{19} -56.0000 q^{20} -21.0000 q^{21} +16.0000 q^{22} -52.0000 q^{23} +24.0000 q^{24} +71.0000 q^{25} +26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -2.00000 q^{29} -84.0000 q^{30} -168.000 q^{31} +32.0000 q^{32} +24.0000 q^{33} -196.000 q^{34} +98.0000 q^{35} +36.0000 q^{36} -146.000 q^{37} -56.0000 q^{38} +39.0000 q^{39} -112.000 q^{40} -514.000 q^{41} -42.0000 q^{42} -236.000 q^{43} +32.0000 q^{44} -126.000 q^{45} -104.000 q^{46} -216.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} +142.000 q^{50} -294.000 q^{51} +52.0000 q^{52} -66.0000 q^{53} +54.0000 q^{54} -112.000 q^{55} -56.0000 q^{56} -84.0000 q^{57} -4.00000 q^{58} -84.0000 q^{59} -168.000 q^{60} +446.000 q^{61} -336.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -182.000 q^{65} +48.0000 q^{66} +292.000 q^{67} -392.000 q^{68} -156.000 q^{69} +196.000 q^{70} +100.000 q^{71} +72.0000 q^{72} +450.000 q^{73} -292.000 q^{74} +213.000 q^{75} -112.000 q^{76} -56.0000 q^{77} +78.0000 q^{78} +392.000 q^{79} -224.000 q^{80} +81.0000 q^{81} -1028.00 q^{82} -292.000 q^{83} -84.0000 q^{84} +1372.00 q^{85} -472.000 q^{86} -6.00000 q^{87} +64.0000 q^{88} -402.000 q^{89} -252.000 q^{90} -91.0000 q^{91} -208.000 q^{92} -504.000 q^{93} -432.000 q^{94} +392.000 q^{95} +96.0000 q^{96} +314.000 q^{97} +98.0000 q^{98} +72.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−14.0000	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$5$	−14.0000	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−28.0000	−0.885438
$11$	8.00000	0.219281	0.109640	−	0.993971i	$-0.465030\pi$
$11$	8.00000	0.219281	0.109640	+	0.993971i	$0.465030\pi$
$12$	12.0000	0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	−42.0000	−0.722957
$16$	16.0000	0.250000
$17$	−98.0000	−1.39815	−0.699073	−	0.715050i	$-0.746404\pi$
$17$	−98.0000	−1.39815	−0.699073	+	0.715050i	$0.746404\pi$
$18$	18.0000	0.235702
$19$	−28.0000	−0.338086	−0.169043	−	0.985609i	$-0.554068\pi$
$19$	−28.0000	−0.338086	−0.169043	+	0.985609i	$0.554068\pi$
$20$	−56.0000	−0.626099
$21$	−21.0000	−0.218218
$22$	16.0000	0.155055
$23$	−52.0000	−0.471424	−0.235712	−	0.971823i	$-0.575742\pi$
$23$	−52.0000	−0.471424	−0.235712	+	0.971823i	$0.575742\pi$
$24$	24.0000	0.204124
$25$	71.0000	0.568000
$26$	26.0000	0.196116
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−2.00000	−0.0128066	−0.00640329	−	0.999979i	$-0.502038\pi$
$29$	−2.00000	−0.0128066	−0.00640329	+	0.999979i	$0.502038\pi$
$30$	−84.0000	−0.511208
$31$	−168.000	−0.973345	−0.486672	−	0.873585i	$-0.661789\pi$
$31$	−168.000	−0.973345	−0.486672	+	0.873585i	$0.661789\pi$
$32$	32.0000	0.176777
$33$	24.0000	0.126602
$34$	−196.000	−0.988639
$35$	98.0000	0.473286
$36$	36.0000	0.166667
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	−56.0000	−0.239063
$39$	39.0000	0.160128
$40$	−112.000	−0.442719
$41$	−514.000	−1.95789	−0.978943	−	0.204135i	$-0.934562\pi$
$41$	−514.000	−1.95789	−0.978943	+	0.204135i	$0.934562\pi$
$42$	−42.0000	−0.154303
$43$	−236.000	−0.836969	−0.418484	−	0.908224i	$-0.637439\pi$
$43$	−236.000	−0.836969	−0.418484	+	0.908224i	$0.637439\pi$
$44$	32.0000	0.109640
$45$	−126.000	−0.417399
$46$	−104.000	−0.333347
$47$	−216.000	−0.670358	−0.335179	−	0.942154i	$-0.608797\pi$
$47$	−216.000	−0.670358	−0.335179	+	0.942154i	$0.608797\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	142.000	0.401637
$51$	−294.000	−0.807220
$52$	52.0000	0.138675
$53$	−66.0000	−0.171053	−0.0855264	−	0.996336i	$-0.527257\pi$
$53$	−66.0000	−0.171053	−0.0855264	+	0.996336i	$0.527257\pi$
$54$	54.0000	0.136083
$55$	−112.000	−0.274583
$56$	−56.0000	−0.133631
$57$	−84.0000	−0.195194
$58$	−4.00000	−0.00905562
$59$	−84.0000	−0.185354	−0.0926769	−	0.995696i	$-0.529542\pi$
$59$	−84.0000	−0.185354	−0.0926769	+	0.995696i	$0.529542\pi$
$60$	−168.000	−0.361478
$61$	446.000	0.936138	0.468069	−	0.883692i	$-0.344950\pi$
$61$	446.000	0.936138	0.468069	+	0.883692i	$0.344950\pi$
$62$	−336.000	−0.688259
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−182.000	−0.347297
$66$	48.0000	0.0895211
$67$	292.000	0.532440	0.266220	−	0.963912i	$-0.414225\pi$
$67$	292.000	0.532440	0.266220	+	0.963912i	$0.414225\pi$
$68$	−392.000	−0.699073
$69$	−156.000	−0.272177
$70$	196.000	0.334664
$71$	100.000	0.167152	0.0835762	−	0.996501i	$-0.473366\pi$
$71$	100.000	0.167152	0.0835762	+	0.996501i	$0.473366\pi$
$72$	72.0000	0.117851
$73$	450.000	0.721487	0.360743	−	0.932665i	$-0.382523\pi$
$73$	450.000	0.721487	0.360743	+	0.932665i	$0.382523\pi$
$74$	−292.000	−0.458707
$75$	213.000	0.327935
$76$	−112.000	−0.169043
$77$	−56.0000	−0.0828804
$78$	78.0000	0.113228
$79$	392.000	0.558271	0.279136	−	0.960252i	$-0.409952\pi$
$79$	392.000	0.558271	0.279136	+	0.960252i	$0.409952\pi$
$80$	−224.000	−0.313050
$81$	81.0000	0.111111
$82$	−1028.00	−1.38443
$83$	−292.000	−0.386159	−0.193079	−	0.981183i	$-0.561847\pi$
$83$	−292.000	−0.386159	−0.193079	+	0.981183i	$0.561847\pi$
$84$	−84.0000	−0.109109
$85$	1372.00	1.75076
$86$	−472.000	−0.591826
$87$	−6.00000	−0.00739388
$88$	64.0000	0.0775275
$89$	−402.000	−0.478786	−0.239393	−	0.970923i	$-0.576948\pi$
$89$	−402.000	−0.478786	−0.239393	+	0.970923i	$0.576948\pi$
$90$	−252.000	−0.295146
$91$	−91.0000	−0.104828
$92$	−208.000	−0.235712
$93$	−504.000	−0.561961
$94$	−432.000	−0.474015
$95$	392.000	0.423351
$96$	96.0000	0.102062
$97$	314.000	0.328679	0.164340	−	0.986404i	$-0.447451\pi$
$97$	314.000	0.328679	0.164340	+	0.986404i	$0.447451\pi$
$98$	98.0000	0.101015
$99$	72.0000	0.0730937
$100$	284.000	0.284000
$101$	858.000	0.845289	0.422645	−	0.906296i	$-0.361102\pi$
$101$	858.000	0.845289	0.422645	+	0.906296i	$0.361102\pi$
$102$	−588.000	−0.570791
$103$	−944.000	−0.903059	−0.451530	−	0.892256i	$-0.649121\pi$
$103$	−944.000	−0.903059	−0.451530	+	0.892256i	$0.649121\pi$
$104$	104.000	0.0980581
$105$	294.000	0.273252
$106$	−132.000	−0.120953
$107$	840.000	0.758933	0.379467	−	0.925205i	$-0.376107\pi$
$107$	840.000	0.758933	0.379467	+	0.925205i	$0.376107\pi$
$108$	108.000	0.0962250
$109$	1334.00	1.17224	0.586119	−	0.810225i	$-0.300655\pi$
$109$	1334.00	1.17224	0.586119	+	0.810225i	$0.300655\pi$
$110$	−224.000	−0.194160
$111$	−438.000	−0.374533
$112$	−112.000	−0.0944911
$113$	946.000	0.787542	0.393771	−	0.919209i	$-0.371170\pi$
$113$	946.000	0.787542	0.393771	+	0.919209i	$0.371170\pi$
$114$	−168.000	−0.138023
$115$	728.000	0.590316
$116$	−8.00000	−0.00640329
$117$	117.000	0.0924500
$118$	−168.000	−0.131065
$119$	686.000	0.528450
$120$	−336.000	−0.255604
$121$	−1267.00	−0.951916
$122$	892.000	0.661950
$123$	−1542.00	−1.13039
$124$	−672.000	−0.486672
$125$	756.000	0.540950
$126$	−126.000	−0.0890871
$127$	904.000	0.631630	0.315815	−	0.948821i	$-0.397722\pi$
$127$	904.000	0.631630	0.315815	+	0.948821i	$0.397722\pi$
$128$	128.000	0.0883883
$129$	−708.000	−0.483224
$130$	−364.000	−0.245576
$131$	−1476.00	−0.984418	−0.492209	−	0.870477i	$-0.663810\pi$
$131$	−1476.00	−0.984418	−0.492209	+	0.870477i	$0.663810\pi$
$132$	96.0000	0.0633010
$133$	196.000	0.127785
$134$	584.000	0.376492
$135$	−378.000	−0.240986
$136$	−784.000	−0.494319
$137$	−62.0000	−0.0386644	−0.0193322	−	0.999813i	$-0.506154\pi$
$137$	−62.0000	−0.0386644	−0.0193322	+	0.999813i	$0.506154\pi$
$138$	−312.000	−0.192458
$139$	−908.000	−0.554069	−0.277034	−	0.960860i	$-0.589352\pi$
$139$	−908.000	−0.554069	−0.277034	+	0.960860i	$0.589352\pi$
$140$	392.000	0.236643
$141$	−648.000	−0.387032
$142$	200.000	0.118195
$143$	104.000	0.0608176
$144$	144.000	0.0833333
$145$	28.0000	0.0160364
$146$	900.000	0.510168
$147$	147.000	0.0824786
$148$	−584.000	−0.324355
$149$	−2514.00	−1.38225	−0.691124	−	0.722736i	$-0.742884\pi$
$149$	−2514.00	−1.38225	−0.691124	+	0.722736i	$0.742884\pi$
$150$	426.000	0.231885
$151$	−1048.00	−0.564802	−0.282401	−	0.959297i	$-0.591131\pi$
$151$	−1048.00	−0.564802	−0.282401	+	0.959297i	$0.591131\pi$
$152$	−224.000	−0.119532
$153$	−882.000	−0.466049
$154$	−112.000	−0.0586053
$155$	2352.00	1.21882
$156$	156.000	0.0800641
$157$	3166.00	1.60939	0.804695	−	0.593688i	$-0.202329\pi$
$157$	3166.00	1.60939	0.804695	+	0.593688i	$0.202329\pi$
$158$	784.000	0.394758
$159$	−198.000	−0.0987574
$160$	−448.000	−0.221359
$161$	364.000	0.178181
$162$	162.000	0.0785674
$163$	−492.000	−0.236420	−0.118210	−	0.992989i	$-0.537716\pi$
$163$	−492.000	−0.236420	−0.118210	+	0.992989i	$0.537716\pi$
$164$	−2056.00	−0.978943
$165$	−336.000	−0.158531
$166$	−584.000	−0.273055
$167$	−1104.00	−0.511557	−0.255779	−	0.966735i	$-0.582332\pi$
$167$	−1104.00	−0.511557	−0.255779	+	0.966735i	$0.582332\pi$
$168$	−168.000	−0.0771517
$169$	169.000	0.0769231
$170$	2744.00	1.23797
$171$	−252.000	−0.112695
$172$	−944.000	−0.418484
$173$	−2686.00	−1.18042	−0.590210	−	0.807249i	$-0.700955\pi$
$173$	−2686.00	−1.18042	−0.590210	+	0.807249i	$0.700955\pi$
$174$	−12.0000	−0.00522826
$175$	−497.000	−0.214684
$176$	128.000	0.0548202
$177$	−252.000	−0.107014
$178$	−804.000	−0.338553
$179$	2088.00	0.871868	0.435934	−	0.899979i	$-0.356418\pi$
$179$	2088.00	0.871868	0.435934	+	0.899979i	$0.356418\pi$
$180$	−504.000	−0.208700
$181$	−242.000	−0.0993797	−0.0496898	−	0.998765i	$-0.515823\pi$
$181$	−242.000	−0.0993797	−0.0496898	+	0.998765i	$0.515823\pi$
$182$	−182.000	−0.0741249
$183$	1338.00	0.540480
$184$	−416.000	−0.166674
$185$	2044.00	0.812313
$186$	−1008.00	−0.397366
$187$	−784.000	−0.306587
$188$	−864.000	−0.335179
$189$	−189.000	−0.0727393
$190$	784.000	0.299354
$191$	−1284.00	−0.486424	−0.243212	−	0.969973i	$-0.578201\pi$
$191$	−1284.00	−0.486424	−0.243212	+	0.969973i	$0.578201\pi$
$192$	192.000	0.0721688
$193$	−2734.00	−1.01968	−0.509838	−	0.860270i	$-0.670295\pi$
$193$	−2734.00	−1.01968	−0.509838	+	0.860270i	$0.670295\pi$
$194$	628.000	0.232411
$195$	−546.000	−0.200512
$196$	196.000	0.0714286
$197$	5118.00	1.85098	0.925488	−	0.378776i	$-0.123655\pi$
$197$	5118.00	1.85098	0.925488	+	0.378776i	$0.123655\pi$
$198$	144.000	0.0516850
$199$	1528.00	0.544307	0.272153	−	0.962254i	$-0.412264\pi$
$199$	1528.00	0.544307	0.272153	+	0.962254i	$0.412264\pi$
$200$	568.000	0.200818
$201$	876.000	0.307404
$202$	1716.00	0.597710
$203$	14.0000	0.00484043
$204$	−1176.00	−0.403610
$205$	7196.00	2.45166
$206$	−1888.00	−0.638559
$207$	−468.000	−0.157141
$208$	208.000	0.0693375
$209$	−224.000	−0.0741359
$210$	588.000	0.193218
$211$	268.000	0.0874402	0.0437201	−	0.999044i	$-0.486079\pi$
$211$	268.000	0.0874402	0.0437201	+	0.999044i	$0.486079\pi$
$212$	−264.000	−0.0855264
$213$	300.000	0.0965055
$214$	1680.00	0.536647
$215$	3304.00	1.04805
$216$	216.000	0.0680414
$217$	1176.00	0.367890
$218$	2668.00	0.828898
$219$	1350.00	0.416550
$220$	−448.000	−0.137292
$221$	−1274.00	−0.387776
$222$	−876.000	−0.264835
$223$	1392.00	0.418005	0.209003	−	0.977915i	$-0.432978\pi$
$223$	1392.00	0.418005	0.209003	+	0.977915i	$0.432978\pi$
$224$	−224.000	−0.0668153
$225$	639.000	0.189333
$226$	1892.00	0.556876
$227$	−6140.00	−1.79527	−0.897635	−	0.440740i	$-0.854716\pi$
$227$	−6140.00	−1.79527	−0.897635	+	0.440740i	$0.854716\pi$
$228$	−336.000	−0.0975971
$229$	1870.00	0.539620	0.269810	−	0.962914i	$-0.413039\pi$
$229$	1870.00	0.539620	0.269810	+	0.962914i	$0.413039\pi$
$230$	1456.00	0.417417
$231$	−168.000	−0.0478510
$232$	−16.0000	−0.00452781
$233$	50.0000	0.0140584	0.00702920	−	0.999975i	$-0.497763\pi$
$233$	50.0000	0.0140584	0.00702920	+	0.999975i	$0.497763\pi$
$234$	234.000	0.0653720
$235$	3024.00	0.839421
$236$	−336.000	−0.0926769
$237$	1176.00	0.322318
$238$	1372.00	0.373670
$239$	−3228.00	−0.873648	−0.436824	−	0.899547i	$-0.643897\pi$
$239$	−3228.00	−0.873648	−0.436824	+	0.899547i	$0.643897\pi$
$240$	−672.000	−0.180739
$241$	5978.00	1.59783	0.798915	−	0.601444i	$-0.205408\pi$
$241$	5978.00	1.59783	0.798915	+	0.601444i	$0.205408\pi$
$242$	−2534.00	−0.673106
$243$	243.000	0.0641500
$244$	1784.00	0.468069
$245$	−686.000	−0.178885
$246$	−3084.00	−0.799303
$247$	−364.000	−0.0937683
$248$	−1344.00	−0.344129
$249$	−876.000	−0.222949
$250$	1512.00	0.382509
$251$	−4084.00	−1.02701	−0.513506	−	0.858086i	$-0.671653\pi$
$251$	−4084.00	−1.02701	−0.513506	+	0.858086i	$0.671653\pi$
$252$	−252.000	−0.0629941
$253$	−416.000	−0.103374
$254$	1808.00	0.446630
$255$	4116.00	1.01080
$256$	256.000	0.0625000
$257$	2646.00	0.642229	0.321115	−	0.947040i	$-0.395943\pi$
$257$	2646.00	0.642229	0.321115	+	0.947040i	$0.395943\pi$
$258$	−1416.00	−0.341691
$259$	1022.00	0.245189
$260$	−728.000	−0.173649
$261$	−18.0000	−0.00426886
$262$	−2952.00	−0.696088
$263$	5164.00	1.21074	0.605372	−	0.795942i	$-0.293024\pi$
$263$	5164.00	1.21074	0.605372	+	0.795942i	$0.293024\pi$
$264$	192.000	0.0447605
$265$	924.000	0.214192
$266$	392.000	0.0903574
$267$	−1206.00	−0.276427
$268$	1168.00	0.266220
$269$	−4710.00	−1.06756	−0.533780	−	0.845623i	$-0.679229\pi$
$269$	−4710.00	−1.06756	−0.533780	+	0.845623i	$0.679229\pi$
$270$	−756.000	−0.170403
$271$	−3144.00	−0.704739	−0.352370	−	0.935861i	$-0.614624\pi$
$271$	−3144.00	−0.704739	−0.352370	+	0.935861i	$0.614624\pi$
$272$	−1568.00	−0.349537
$273$	−273.000	−0.0605228
$274$	−124.000	−0.0273398
$275$	568.000	0.124552
$276$	−624.000	−0.136088
$277$	3310.00	0.717973	0.358987	−	0.933343i	$-0.383122\pi$
$277$	3310.00	0.717973	0.358987	+	0.933343i	$0.383122\pi$
$278$	−1816.00	−0.391786
$279$	−1512.00	−0.324448
$280$	784.000	0.167332
$281$	−4910.00	−1.04237	−0.521185	−	0.853444i	$-0.674510\pi$
$281$	−4910.00	−1.04237	−0.521185	+	0.853444i	$0.674510\pi$
$282$	−1296.00	−0.273673
$283$	7436.00	1.56192	0.780962	−	0.624579i	$-0.214729\pi$
$283$	7436.00	1.56192	0.780962	+	0.624579i	$0.214729\pi$
$284$	400.000	0.0835762
$285$	1176.00	0.244422
$286$	208.000	0.0430045
$287$	3598.00	0.740011
$288$	288.000	0.0589256
$289$	4691.00	0.954814
$290$	56.0000	0.0113394
$291$	942.000	0.189763
$292$	1800.00	0.360743
$293$	−310.000	−0.0618102	−0.0309051	−	0.999522i	$-0.509839\pi$
$293$	−310.000	−0.0618102	−0.0309051	+	0.999522i	$0.509839\pi$
$294$	294.000	0.0583212
$295$	1176.00	0.232100
$296$	−1168.00	−0.229353
$297$	216.000	0.0422006
$298$	−5028.00	−0.977397
$299$	−676.000	−0.130749
$300$	852.000	0.163967
$301$	1652.00	0.316345
$302$	−2096.00	−0.399375
$303$	2574.00	0.488028
$304$	−448.000	−0.0845216
$305$	−6244.00	−1.17223
$306$	−1764.00	−0.329546
$307$	−3516.00	−0.653644	−0.326822	−	0.945086i	$-0.605978\pi$
$307$	−3516.00	−0.653644	−0.326822	+	0.945086i	$0.605978\pi$
$308$	−224.000	−0.0414402
$309$	−2832.00	−0.521381
$310$	4704.00	0.861836
$311$	3216.00	0.586375	0.293188	−	0.956055i	$-0.405284\pi$
$311$	3216.00	0.586375	0.293188	+	0.956055i	$0.405284\pi$
$312$	312.000	0.0566139
$313$	−1398.00	−0.252459	−0.126229	−	0.992001i	$-0.540288\pi$
$313$	−1398.00	−0.252459	−0.126229	+	0.992001i	$0.540288\pi$
$314$	6332.00	1.13801
$315$	882.000	0.157762
$316$	1568.00	0.279136
$317$	−9034.00	−1.60063	−0.800315	−	0.599579i	$-0.795335\pi$
$317$	−9034.00	−1.60063	−0.800315	+	0.599579i	$0.795335\pi$
$318$	−396.000	−0.0698320
$319$	−16.0000	−0.00280824
$320$	−896.000	−0.156525
$321$	2520.00	0.438170
$322$	728.000	0.125993
$323$	2744.00	0.472694
$324$	324.000	0.0555556
$325$	923.000	0.157535
$326$	−984.000	−0.167174
$327$	4002.00	0.676792
$328$	−4112.00	−0.692217
$329$	1512.00	0.253372
$330$	−672.000	−0.112098
$331$	9108.00	1.51245	0.756225	−	0.654312i	$-0.227042\pi$
$331$	9108.00	1.51245	0.756225	+	0.654312i	$0.227042\pi$
$332$	−1168.00	−0.193079
$333$	−1314.00	−0.216237
$334$	−2208.00	−0.361726
$335$	−4088.00	−0.666720
$336$	−336.000	−0.0545545
$337$	−1566.00	−0.253132	−0.126566	−	0.991958i	$-0.540396\pi$
$337$	−1566.00	−0.253132	−0.126566	+	0.991958i	$0.540396\pi$
$338$	338.000	0.0543928
$339$	2838.00	0.454687
$340$	5488.00	0.875378
$341$	−1344.00	−0.213436
$342$	−504.000	−0.0796877
$343$	−343.000	−0.0539949
$344$	−1888.00	−0.295913
$345$	2184.00	0.340819
$346$	−5372.00	−0.834684
$347$	4528.00	0.700507	0.350253	−	0.936655i	$-0.386096\pi$
$347$	4528.00	0.700507	0.350253	+	0.936655i	$0.386096\pi$
$348$	−24.0000	−0.00369694
$349$	−1090.00	−0.167182	−0.0835908	−	0.996500i	$-0.526639\pi$
$349$	−1090.00	−0.167182	−0.0835908	+	0.996500i	$0.526639\pi$
$350$	−994.000	−0.151804
$351$	351.000	0.0533761
$352$	256.000	0.0387638
$353$	−11234.0	−1.69384	−0.846920	−	0.531720i	$-0.821546\pi$
$353$	−11234.0	−1.69384	−0.846920	+	0.531720i	$0.821546\pi$
$354$	−504.000	−0.0756703
$355$	−1400.00	−0.209308
$356$	−1608.00	−0.239393
$357$	2058.00	0.305101
$358$	4176.00	0.616504
$359$	3444.00	0.506316	0.253158	−	0.967425i	$-0.418531\pi$
$359$	3444.00	0.506316	0.253158	+	0.967425i	$0.418531\pi$
$360$	−1008.00	−0.147573
$361$	−6075.00	−0.885698
$362$	−484.000	−0.0702720
$363$	−3801.00	−0.549589
$364$	−364.000	−0.0524142
$365$	−6300.00	−0.903444
$366$	2676.00	0.382177
$367$	11712.0	1.66583	0.832917	−	0.553397i	$-0.186669\pi$
$367$	11712.0	1.66583	0.832917	+	0.553397i	$0.186669\pi$
$368$	−832.000	−0.117856
$369$	−4626.00	−0.652629
$370$	4088.00	0.574392
$371$	462.000	0.0646519
$372$	−2016.00	−0.280980
$373$	10990.0	1.52558	0.762789	−	0.646647i	$-0.223829\pi$
$373$	10990.0	1.52558	0.762789	+	0.646647i	$0.223829\pi$
$374$	−1568.00	−0.216790
$375$	2268.00	0.312317
$376$	−1728.00	−0.237007
$377$	−26.0000	−0.00355190
$378$	−378.000	−0.0514344
$379$	−7004.00	−0.949265	−0.474632	−	0.880184i	$-0.657419\pi$
$379$	−7004.00	−0.949265	−0.474632	+	0.880184i	$0.657419\pi$
$380$	1568.00	0.211676
$381$	2712.00	0.364672
$382$	−2568.00	−0.343954
$383$	−5472.00	−0.730042	−0.365021	−	0.930999i	$-0.618938\pi$
$383$	−5472.00	−0.730042	−0.365021	+	0.930999i	$0.618938\pi$
$384$	384.000	0.0510310
$385$	784.000	0.103783
$386$	−5468.00	−0.721020
$387$	−2124.00	−0.278990
$388$	1256.00	0.164340
$389$	1910.00	0.248948	0.124474	−	0.992223i	$-0.460276\pi$
$389$	1910.00	0.248948	0.124474	+	0.992223i	$0.460276\pi$
$390$	−1092.00	−0.141784
$391$	5096.00	0.659120
$392$	392.000	0.0505076
$393$	−4428.00	−0.568354
$394$	10236.0	1.30884
$395$	−5488.00	−0.699066
$396$	288.000	0.0365468
$397$	−13234.0	−1.67304	−0.836518	−	0.547939i	$-0.815413\pi$
$397$	−13234.0	−1.67304	−0.836518	+	0.547939i	$0.815413\pi$
$398$	3056.00	0.384883
$399$	588.000	0.0737765
$400$	1136.00	0.142000
$401$	4586.00	0.571107	0.285554	−	0.958363i	$-0.407823\pi$
$401$	4586.00	0.571107	0.285554	+	0.958363i	$0.407823\pi$
$402$	1752.00	0.217368
$403$	−2184.00	−0.269957
$404$	3432.00	0.422645
$405$	−1134.00	−0.139133
$406$	28.0000	0.00342270
$407$	−1168.00	−0.142250
$408$	−2352.00	−0.285395
$409$	−4174.00	−0.504624	−0.252312	−	0.967646i	$-0.581191\pi$
$409$	−4174.00	−0.504624	−0.252312	+	0.967646i	$0.581191\pi$
$410$	14392.0	1.73359
$411$	−186.000	−0.0223229
$412$	−3776.00	−0.451530
$413$	588.000	0.0700571
$414$	−936.000	−0.111116
$415$	4088.00	0.483547
$416$	416.000	0.0490290
$417$	−2724.00	−0.319892
$418$	−448.000	−0.0524220
$419$	−10012.0	−1.16735	−0.583673	−	0.811989i	$-0.698385\pi$
$419$	−10012.0	−1.16735	−0.583673	+	0.811989i	$0.698385\pi$
$420$	1176.00	0.136626
$421$	−1250.00	−0.144706	−0.0723531	−	0.997379i	$-0.523051\pi$
$421$	−1250.00	−0.144706	−0.0723531	+	0.997379i	$0.523051\pi$
$422$	536.000	0.0618296
$423$	−1944.00	−0.223453
$424$	−528.000	−0.0604763
$425$	−6958.00	−0.794147
$426$	600.000	0.0682397
$427$	−3122.00	−0.353827
$428$	3360.00	0.379467
$429$	312.000	0.0351131
$430$	6608.00	0.741084
$431$	6236.00	0.696932	0.348466	−	0.937321i	$-0.386703\pi$
$431$	6236.00	0.696932	0.348466	+	0.937321i	$0.386703\pi$
$432$	432.000	0.0481125
$433$	−3966.00	−0.440170	−0.220085	−	0.975481i	$-0.570634\pi$
$433$	−3966.00	−0.440170	−0.220085	+	0.975481i	$0.570634\pi$
$434$	2352.00	0.260137
$435$	84.0000	0.00925860
$436$	5336.00	0.586119
$437$	1456.00	0.159382
$438$	2700.00	0.294546
$439$	8776.00	0.954113	0.477057	−	0.878873i	$-0.341704\pi$
$439$	8776.00	0.954113	0.477057	+	0.878873i	$0.341704\pi$
$440$	−896.000	−0.0970798
$441$	441.000	0.0476190
$442$	−2548.00	−0.274199
$443$	6096.00	0.653792	0.326896	−	0.945060i	$-0.393997\pi$
$443$	6096.00	0.653792	0.326896	+	0.945060i	$0.393997\pi$
$444$	−1752.00	−0.187266
$445$	5628.00	0.599534
$446$	2784.00	0.295574
$447$	−7542.00	−0.798041
$448$	−448.000	−0.0472456
$449$	−9390.00	−0.986952	−0.493476	−	0.869759i	$-0.664274\pi$
$449$	−9390.00	−0.986952	−0.493476	+	0.869759i	$0.664274\pi$
$450$	1278.00	0.133879
$451$	−4112.00	−0.429327
$452$	3784.00	0.393771
$453$	−3144.00	−0.326088
$454$	−12280.0	−1.26945
$455$	1274.00	0.131266
$456$	−672.000	−0.0690116
$457$	−6.00000	−0.000614154 0	−0.000307077	−	1.00000i	$-0.500098\pi$
$457$	−6.00000	−0.000614154 0	−0.000307077		1.00000i	$0.500098\pi$
$458$	3740.00	0.381569
$459$	−2646.00	−0.269073
$460$	2912.00	0.295158
$461$	−9374.00	−0.947051	−0.473526	−	0.880780i	$-0.657019\pi$
$461$	−9374.00	−0.947051	−0.473526	+	0.880780i	$0.657019\pi$
$462$	−336.000	−0.0338358
$463$	4008.00	0.402306	0.201153	−	0.979560i	$-0.435531\pi$
$463$	4008.00	0.402306	0.201153	+	0.979560i	$0.435531\pi$
$464$	−32.0000	−0.00320164
$465$	7056.00	0.703686
$466$	100.000	0.00994080
$467$	−3260.00	−0.323030	−0.161515	−	0.986870i	$-0.551638\pi$
$467$	−3260.00	−0.323030	−0.161515	+	0.986870i	$0.551638\pi$
$468$	468.000	0.0462250
$469$	−2044.00	−0.201243
$470$	6048.00	0.593561
$471$	9498.00	0.929182
$472$	−672.000	−0.0655324
$473$	−1888.00	−0.183531
$474$	2352.00	0.227913
$475$	−1988.00	−0.192033
$476$	2744.00	0.264225
$477$	−594.000	−0.0570176
$478$	−6456.00	−0.617763
$479$	12696.0	1.21105	0.605527	−	0.795825i	$-0.292962\pi$
$479$	12696.0	1.21105	0.605527	+	0.795825i	$0.292962\pi$
$480$	−1344.00	−0.127802
$481$	−1898.00	−0.179920
$482$	11956.0	1.12984
$483$	1092.00	0.102873
$484$	−5068.00	−0.475958
$485$	−4396.00	−0.411571
$486$	486.000	0.0453609
$487$	10760.0	1.00120	0.500598	−	0.865680i	$-0.333114\pi$
$487$	10760.0	1.00120	0.500598	+	0.865680i	$0.333114\pi$
$488$	3568.00	0.330975
$489$	−1476.00	−0.136497
$490$	−1372.00	−0.126491
$491$	−11600.0	−1.06619	−0.533096	−	0.846055i	$-0.678972\pi$
$491$	−11600.0	−1.06619	−0.533096	+	0.846055i	$0.678972\pi$
$492$	−6168.00	−0.565193
$493$	196.000	0.0179055
$494$	−728.000	−0.0663042
$495$	−1008.00	−0.0915277
$496$	−2688.00	−0.243336
$497$	−700.000	−0.0631776
$498$	−1752.00	−0.157649
$499$	−15908.0	−1.42713	−0.713567	−	0.700587i	$-0.752922\pi$
$499$	−15908.0	−1.42713	−0.713567	+	0.700587i	$0.752922\pi$
$500$	3024.00	0.270475
$501$	−3312.00	−0.295348
$502$	−8168.00	−0.726207
$503$	−12232.0	−1.08429	−0.542145	−	0.840285i	$-0.682388\pi$
$503$	−12232.0	−1.08429	−0.542145	+	0.840285i	$0.682388\pi$
$504$	−504.000	−0.0445435
$505$	−12012.0	−1.05847
$506$	−832.000	−0.0730967
$507$	507.000	0.0444116
$508$	3616.00	0.315815
$509$	−13350.0	−1.16253	−0.581266	−	0.813714i	$-0.697442\pi$
$509$	−13350.0	−1.16253	−0.581266	+	0.813714i	$0.697442\pi$
$510$	8232.00	0.714743
$511$	−3150.00	−0.272696
$512$	512.000	0.0441942
$513$	−756.000	−0.0650647
$514$	5292.00	0.454125
$515$	13216.0	1.13081
$516$	−2832.00	−0.241612
$517$	−1728.00	−0.146997
$518$	2044.00	0.173375
$519$	−8058.00	−0.681516
$520$	−1456.00	−0.122788
$521$	−18282.0	−1.53733	−0.768665	−	0.639652i	$-0.779079\pi$
$521$	−18282.0	−1.53733	−0.768665	+	0.639652i	$0.779079\pi$
$522$	−36.0000	−0.00301854
$523$	−6108.00	−0.510677	−0.255339	−	0.966852i	$-0.582187\pi$
$523$	−6108.00	−0.510677	−0.255339	+	0.966852i	$0.582187\pi$
$524$	−5904.00	−0.492209
$525$	−1491.00	−0.123948
$526$	10328.0	0.856126
$527$	16464.0	1.36088
$528$	384.000	0.0316505
$529$	−9463.00	−0.777760
$530$	1848.00	0.151457
$531$	−756.000	−0.0617846
$532$	784.000	0.0638923
$533$	−6682.00	−0.543020
$534$	−2412.00	−0.195463
$535$	−11760.0	−0.950335
$536$	2336.00	0.188246
$537$	6264.00	0.503373
$538$	−9420.00	−0.754879
$539$	392.000	0.0313259
$540$	−1512.00	−0.120493
$541$	−14906.0	−1.18458	−0.592291	−	0.805724i	$-0.701776\pi$
$541$	−14906.0	−1.18458	−0.592291	+	0.805724i	$0.701776\pi$
$542$	−6288.00	−0.498326
$543$	−726.000	−0.0573769
$544$	−3136.00	−0.247160
$545$	−18676.0	−1.46788
$546$	−546.000	−0.0427960
$547$	5492.00	0.429289	0.214644	−	0.976692i	$-0.431141\pi$
$547$	5492.00	0.429289	0.214644	+	0.976692i	$0.431141\pi$
$548$	−248.000	−0.0193322
$549$	4014.00	0.312046
$550$	1136.00	0.0880713
$551$	56.0000	0.00432973
$552$	−1248.00	−0.0962290
$553$	−2744.00	−0.211007
$554$	6620.00	0.507684
$555$	6132.00	0.468989
$556$	−3632.00	−0.277034
$557$	−6746.00	−0.513173	−0.256586	−	0.966521i	$-0.582598\pi$
$557$	−6746.00	−0.513173	−0.256586	+	0.966521i	$0.582598\pi$
$558$	−3024.00	−0.229420
$559$	−3068.00	−0.232133
$560$	1568.00	0.118322
$561$	−2352.00	−0.177008
$562$	−9820.00	−0.737067
$563$	596.000	0.0446153	0.0223076	−	0.999751i	$-0.492899\pi$
$563$	596.000	0.0446153	0.0223076	+	0.999751i	$0.492899\pi$
$564$	−2592.00	−0.193516
$565$	−13244.0	−0.986158
$566$	14872.0	1.10445
$567$	−567.000	−0.0419961
$568$	800.000	0.0590973
$569$	−2638.00	−0.194360	−0.0971799	−	0.995267i	$-0.530982\pi$
$569$	−2638.00	−0.194360	−0.0971799	+	0.995267i	$0.530982\pi$
$570$	2352.00	0.172832
$571$	−22612.0	−1.65724	−0.828619	−	0.559813i	$-0.810873\pi$
$571$	−22612.0	−1.65724	−0.828619	+	0.559813i	$0.810873\pi$
$572$	416.000	0.0304088
$573$	−3852.00	−0.280837
$574$	7196.00	0.523267
$575$	−3692.00	−0.267769
$576$	576.000	0.0416667
$577$	−1806.00	−0.130303	−0.0651514	−	0.997875i	$-0.520753\pi$
$577$	−1806.00	−0.130303	−0.0651514	+	0.997875i	$0.520753\pi$
$578$	9382.00	0.675155
$579$	−8202.00	−0.588711
$580$	112.000	0.00801818
$581$	2044.00	0.145954
$582$	1884.00	0.134183
$583$	−528.000	−0.0375086
$584$	3600.00	0.255084
$585$	−1638.00	−0.115766
$586$	−620.000	−0.0437064
$587$	−27156.0	−1.90945	−0.954726	−	0.297487i	$-0.903851\pi$
$587$	−27156.0	−1.90945	−0.954726	+	0.297487i	$0.903851\pi$
$588$	588.000	0.0412393
$589$	4704.00	0.329075
$590$	2352.00	0.164119
$591$	15354.0	1.06866
$592$	−2336.00	−0.162177
$593$	−6034.00	−0.417853	−0.208926	−	0.977931i	$-0.566997\pi$
$593$	−6034.00	−0.417853	−0.208926	+	0.977931i	$0.566997\pi$
$594$	432.000	0.0298404
$595$	−9604.00	−0.661724
$596$	−10056.0	−0.691124
$597$	4584.00	0.314256
$598$	−1352.00	−0.0924538
$599$	19284.0	1.31540	0.657699	−	0.753281i	$-0.271530\pi$
$599$	19284.0	1.31540	0.657699	+	0.753281i	$0.271530\pi$
$600$	1704.00	0.115943
$601$	10490.0	0.711973	0.355987	−	0.934491i	$-0.384145\pi$
$601$	10490.0	0.711973	0.355987	+	0.934491i	$0.384145\pi$
$602$	3304.00	0.223689
$603$	2628.00	0.177480
$604$	−4192.00	−0.282401
$605$	17738.0	1.19199
$606$	5148.00	0.345088
$607$	−13216.0	−0.883725	−0.441862	−	0.897083i	$-0.645682\pi$
$607$	−13216.0	−0.883725	−0.441862	+	0.897083i	$0.645682\pi$
$608$	−896.000	−0.0597658
$609$	42.0000	0.00279462
$610$	−12488.0	−0.828892
$611$	−2808.00	−0.185924
$612$	−3528.00	−0.233024
$613$	−8242.00	−0.543053	−0.271526	−	0.962431i	$-0.587528\pi$
$613$	−8242.00	−0.543053	−0.271526	+	0.962431i	$0.587528\pi$
$614$	−7032.00	−0.462196
$615$	21588.0	1.41547
$616$	−448.000	−0.0293027
$617$	−18206.0	−1.18792	−0.593959	−	0.804495i	$-0.702436\pi$
$617$	−18206.0	−1.18792	−0.593959	+	0.804495i	$0.702436\pi$
$618$	−5664.00	−0.368672
$619$	16580.0	1.07659	0.538293	−	0.842758i	$-0.319069\pi$
$619$	16580.0	1.07659	0.538293	+	0.842758i	$0.319069\pi$
$620$	9408.00	0.609410
$621$	−1404.00	−0.0907256
$622$	6432.00	0.414630
$623$	2814.00	0.180964
$624$	624.000	0.0400320
$625$	−19459.0	−1.24538
$626$	−2796.00	−0.178515
$627$	−672.000	−0.0428024
$628$	12664.0	0.804695
$629$	14308.0	0.906991
$630$	1764.00	0.111555
$631$	21040.0	1.32740	0.663700	−	0.747999i	$-0.268985\pi$
$631$	21040.0	1.32740	0.663700	+	0.747999i	$0.268985\pi$
$632$	3136.00	0.197379
$633$	804.000	0.0504836
$634$	−18068.0	−1.13182
$635$	−12656.0	−0.790926
$636$	−792.000	−0.0493787
$637$	637.000	0.0396214
$638$	−32.0000	−0.00198572
$639$	900.000	0.0557174
$640$	−1792.00	−0.110680
$641$	170.000	0.0104752	0.00523759	−	0.999986i	$-0.498333\pi$
$641$	170.000	0.0104752	0.00523759	+	0.999986i	$0.498333\pi$
$642$	5040.00	0.309833
$643$	23924.0	1.46729	0.733647	−	0.679530i	$-0.237816\pi$
$643$	23924.0	1.46729	0.733647	+	0.679530i	$0.237816\pi$
$644$	1456.00	0.0890907
$645$	9912.00	0.605092
$646$	5488.00	0.334245
$647$	22992.0	1.39708	0.698538	−	0.715572i	$-0.253834\pi$
$647$	22992.0	1.39708	0.698538	+	0.715572i	$0.253834\pi$
$648$	648.000	0.0392837
$649$	−672.000	−0.0406445
$650$	1846.00	0.111394
$651$	3528.00	0.212401
$652$	−1968.00	−0.118210
$653$	12990.0	0.778466	0.389233	−	0.921139i	$-0.372740\pi$
$653$	12990.0	0.778466	0.389233	+	0.921139i	$0.372740\pi$
$654$	8004.00	0.478564
$655$	20664.0	1.23269
$656$	−8224.00	−0.489471
$657$	4050.00	0.240496
$658$	3024.00	0.179161
$659$	11088.0	0.655428	0.327714	−	0.944777i	$-0.393722\pi$
$659$	11088.0	0.655428	0.327714	+	0.944777i	$0.393722\pi$
$660$	−1344.00	−0.0792653
$661$	6886.00	0.405196	0.202598	−	0.979262i	$-0.435062\pi$
$661$	6886.00	0.405196	0.202598	+	0.979262i	$0.435062\pi$
$662$	18216.0	1.06946
$663$	−3822.00	−0.223883
$664$	−2336.00	−0.136528
$665$	−2744.00	−0.160012
$666$	−2628.00	−0.152902
$667$	104.000	0.00603733
$668$	−4416.00	−0.255779
$669$	4176.00	0.241336
$670$	−8176.00	−0.471442
$671$	3568.00	0.205277
$672$	−672.000	−0.0385758
$673$	−8238.00	−0.471845	−0.235922	−	0.971772i	$-0.575811\pi$
$673$	−8238.00	−0.471845	−0.235922	+	0.971772i	$0.575811\pi$
$674$	−3132.00	−0.178991
$675$	1917.00	0.109312
$676$	676.000	0.0384615
$677$	778.000	0.0441669	0.0220834	−	0.999756i	$-0.492970\pi$
$677$	778.000	0.0441669	0.0220834	+	0.999756i	$0.492970\pi$
$678$	5676.00	0.321512
$679$	−2198.00	−0.124229
$680$	10976.0	0.618986
$681$	−18420.0	−1.03650
$682$	−2688.00	−0.150922
$683$	19552.0	1.09537	0.547684	−	0.836685i	$-0.315510\pi$
$683$	19552.0	1.09537	0.547684	+	0.836685i	$0.315510\pi$
$684$	−1008.00	−0.0563477
$685$	868.000	0.0484154
$686$	−686.000	−0.0381802
$687$	5610.00	0.311550
$688$	−3776.00	−0.209242
$689$	−858.000	−0.0474415
$690$	4368.00	0.240996
$691$	−31700.0	−1.74519	−0.872594	−	0.488446i	$-0.837564\pi$
$691$	−31700.0	−1.74519	−0.872594	+	0.488446i	$0.837564\pi$
$692$	−10744.0	−0.590210
$693$	−504.000	−0.0276268
$694$	9056.00	0.495333
$695$	12712.0	0.693804
$696$	−48.0000	−0.00261413
$697$	50372.0	2.73741
$698$	−2180.00	−0.118215
$699$	150.000	0.00811663
$700$	−1988.00	−0.107342
$701$	1470.00	0.0792028	0.0396014	−	0.999216i	$-0.487391\pi$
$701$	1470.00	0.0792028	0.0396014	+	0.999216i	$0.487391\pi$
$702$	702.000	0.0377426
$703$	4088.00	0.219320
$704$	512.000	0.0274101
$705$	9072.00	0.484640
$706$	−22468.0	−1.19773
$707$	−6006.00	−0.319489
$708$	−1008.00	−0.0535070
$709$	26718.0	1.41525	0.707627	−	0.706586i	$-0.249766\pi$
$709$	26718.0	1.41525	0.707627	+	0.706586i	$0.249766\pi$
$710$	−2800.00	−0.148003
$711$	3528.00	0.186090
$712$	−3216.00	−0.169276
$713$	8736.00	0.458858
$714$	4116.00	0.215739
$715$	−1456.00	−0.0761557
$716$	8352.00	0.435934
$717$	−9684.00	−0.504401
$718$	6888.00	0.358019
$719$	−7408.00	−0.384244	−0.192122	−	0.981371i	$-0.561537\pi$
$719$	−7408.00	−0.384244	−0.192122	+	0.981371i	$0.561537\pi$
$720$	−2016.00	−0.104350
$721$	6608.00	0.341324
$722$	−12150.0	−0.626283
$723$	17934.0	0.922507
$724$	−968.000	−0.0496898
$725$	−142.000	−0.00727413
$726$	−7602.00	−0.388618
$727$	−21120.0	−1.07744	−0.538719	−	0.842486i	$-0.681092\pi$
$727$	−21120.0	−1.07744	−0.538719	+	0.842486i	$0.681092\pi$
$728$	−728.000	−0.0370625
$729$	729.000	0.0370370
$730$	−12600.0	−0.638831
$731$	23128.0	1.17021
$732$	5352.00	0.270240
$733$	−39354.0	−1.98305	−0.991523	−	0.129929i	$-0.958525\pi$
$733$	−39354.0	−1.98305	−0.991523	+	0.129929i	$0.958525\pi$
$734$	23424.0	1.17792
$735$	−2058.00	−0.103280
$736$	−1664.00	−0.0833368
$737$	2336.00	0.116754
$738$	−9252.00	−0.461478
$739$	−15596.0	−0.776330	−0.388165	−	0.921590i	$-0.626891\pi$
$739$	−15596.0	−0.776330	−0.388165	+	0.921590i	$0.626891\pi$
$740$	8176.00	0.406156
$741$	−1092.00	−0.0541371
$742$	924.000	0.0457158
$743$	4468.00	0.220612	0.110306	−	0.993898i	$-0.464817\pi$
$743$	4468.00	0.220612	0.110306	+	0.993898i	$0.464817\pi$
$744$	−4032.00	−0.198683
$745$	35196.0	1.73085
$746$	21980.0	1.07875
$747$	−2628.00	−0.128720
$748$	−3136.00	−0.153293
$749$	−5880.00	−0.286850
$750$	4536.00	0.220842
$751$	5440.00	0.264325	0.132163	−	0.991228i	$-0.457808\pi$
$751$	5440.00	0.264325	0.132163	+	0.991228i	$0.457808\pi$
$752$	−3456.00	−0.167590
$753$	−12252.0	−0.592945
$754$	−52.0000	−0.00251158
$755$	14672.0	0.707243
$756$	−756.000	−0.0363696
$757$	14638.0	0.702810	0.351405	−	0.936224i	$-0.385704\pi$
$757$	14638.0	0.702810	0.351405	+	0.936224i	$0.385704\pi$
$758$	−14008.0	−0.671231
$759$	−1248.00	−0.0596832
$760$	3136.00	0.149677
$761$	35950.0	1.71247	0.856233	−	0.516590i	$-0.172799\pi$
$761$	35950.0	1.71247	0.856233	+	0.516590i	$0.172799\pi$
$762$	5424.00	0.257862
$763$	−9338.00	−0.443065
$764$	−5136.00	−0.243212
$765$	12348.0	0.583585
$766$	−10944.0	−0.516218
$767$	−1092.00	−0.0514079
$768$	768.000	0.0360844
$769$	−24974.0	−1.17111	−0.585556	−	0.810632i	$-0.699124\pi$
$769$	−24974.0	−1.17111	−0.585556	+	0.810632i	$0.699124\pi$
$770$	1568.00	0.0733855
$771$	7938.00	0.370791
$772$	−10936.0	−0.509838
$773$	−41254.0	−1.91954	−0.959769	−	0.280790i	$-0.909404\pi$
$773$	−41254.0	−1.91954	−0.959769	+	0.280790i	$0.909404\pi$
$774$	−4248.00	−0.197275
$775$	−11928.0	−0.552860
$776$	2512.00	0.116206
$777$	3066.00	0.141560
$778$	3820.00	0.176033
$779$	14392.0	0.661934
$780$	−2184.00	−0.100256
$781$	800.000	0.0366533
$782$	10192.0	0.466068
$783$	−54.0000	−0.00246463
$784$	784.000	0.0357143
$785$	−44324.0	−2.01528
$786$	−8856.00	−0.401887
$787$	26780.0	1.21297	0.606483	−	0.795097i	$-0.292580\pi$
$787$	26780.0	1.21297	0.606483	+	0.795097i	$0.292580\pi$
$788$	20472.0	0.925488
$789$	15492.0	0.699024
$790$	−10976.0	−0.494315
$791$	−6622.00	−0.297663
$792$	576.000	0.0258425
$793$	5798.00	0.259638
$794$	−26468.0	−1.18302
$795$	2772.00	0.123664
$796$	6112.00	0.272153
$797$	−18222.0	−0.809857	−0.404929	−	0.914348i	$-0.632704\pi$
$797$	−18222.0	−0.809857	−0.404929	+	0.914348i	$0.632704\pi$
$798$	1176.00	0.0521679
$799$	21168.0	0.937259
$800$	2272.00	0.100409
$801$	−3618.00	−0.159595
$802$	9172.00	0.403834
$803$	3600.00	0.158208
$804$	3504.00	0.153702
$805$	−5096.00	−0.223119
$806$	−4368.00	−0.190889
$807$	−14130.0	−0.616356
$808$	6864.00	0.298855
$809$	7338.00	0.318900	0.159450	−	0.987206i	$-0.449028\pi$
$809$	7338.00	0.318900	0.159450	+	0.987206i	$0.449028\pi$
$810$	−2268.00	−0.0983820
$811$	12500.0	0.541226	0.270613	−	0.962688i	$-0.412774\pi$
$811$	12500.0	0.541226	0.270613	+	0.962688i	$0.412774\pi$
$812$	56.0000	0.00242022
$813$	−9432.00	−0.406882
$814$	−2336.00	−0.100586
$815$	6888.00	0.296044
$816$	−4704.00	−0.201805
$817$	6608.00	0.282968
$818$	−8348.00	−0.356823
$819$	−819.000	−0.0349428
$820$	28784.0	1.22583
$821$	44830.0	1.90570	0.952849	−	0.303445i	$-0.0981370\pi$
$821$	44830.0	1.90570	0.952849	+	0.303445i	$0.0981370\pi$
$822$	−372.000	−0.0157847
$823$	11152.0	0.472338	0.236169	−	0.971712i	$-0.424108\pi$
$823$	11152.0	0.472338	0.236169	+	0.971712i	$0.424108\pi$
$824$	−7552.00	−0.319280
$825$	1704.00	0.0719099
$826$	1176.00	0.0495379
$827$	−2408.00	−0.101251	−0.0506254	−	0.998718i	$-0.516121\pi$
$827$	−2408.00	−0.101251	−0.0506254	+	0.998718i	$0.516121\pi$
$828$	−1872.00	−0.0785706
$829$	−44922.0	−1.88203	−0.941017	−	0.338360i	$-0.890128\pi$
$829$	−44922.0	−1.88203	−0.941017	+	0.338360i	$0.890128\pi$
$830$	8176.00	0.341919
$831$	9930.00	0.414522
$832$	832.000	0.0346688
$833$	−4802.00	−0.199735
$834$	−5448.00	−0.226198
$835$	15456.0	0.640571
$836$	−896.000	−0.0370680
$837$	−4536.00	−0.187320
$838$	−20024.0	−0.825439
$839$	36720.0	1.51098	0.755492	−	0.655158i	$-0.227398\pi$
$839$	36720.0	1.51098	0.755492	+	0.655158i	$0.227398\pi$
$840$	2352.00	0.0966092
$841$	−24385.0	−0.999836
$842$	−2500.00	−0.102323
$843$	−14730.0	−0.601813
$844$	1072.00	0.0437201
$845$	−2366.00	−0.0963229
$846$	−3888.00	−0.158005
$847$	8869.00	0.359790
$848$	−1056.00	−0.0427632
$849$	22308.0	0.901777
$850$	−13916.0	−0.561547
$851$	7592.00	0.305817
$852$	1200.00	0.0482527
$853$	16006.0	0.642479	0.321240	−	0.946998i	$-0.395900\pi$
$853$	16006.0	0.642479	0.321240	+	0.946998i	$0.395900\pi$
$854$	−6244.00	−0.250194
$855$	3528.00	0.141117
$856$	6720.00	0.268323
$857$	37278.0	1.48587	0.742936	−	0.669363i	$-0.233433\pi$
$857$	37278.0	1.48587	0.742936	+	0.669363i	$0.233433\pi$
$858$	624.000	0.0248287
$859$	−45236.0	−1.79678	−0.898389	−	0.439201i	$-0.855262\pi$
$859$	−45236.0	−1.79678	−0.898389	+	0.439201i	$0.855262\pi$
$860$	13216.0	0.524025
$861$	10794.0	0.427246
$862$	12472.0	0.492805
$863$	−23916.0	−0.943349	−0.471674	−	0.881773i	$-0.656350\pi$
$863$	−23916.0	−0.943349	−0.471674	+	0.881773i	$0.656350\pi$
$864$	864.000	0.0340207
$865$	37604.0	1.47812
$866$	−7932.00	−0.311247
$867$	14073.0	0.551262
$868$	4704.00	0.183945
$869$	3136.00	0.122418
$870$	168.000	0.00654682
$871$	3796.00	0.147672
$872$	10672.0	0.414449
$873$	2826.00	0.109560
$874$	2912.00	0.112700
$875$	−5292.00	−0.204460
$876$	5400.00	0.208275
$877$	−24538.0	−0.944800	−0.472400	−	0.881384i	$-0.656612\pi$
$877$	−24538.0	−0.944800	−0.472400	+	0.881384i	$0.656612\pi$
$878$	17552.0	0.674660
$879$	−930.000	−0.0356861
$880$	−1792.00	−0.0686458
$881$	910.000	0.0347999	0.0173999	−	0.999849i	$-0.494461\pi$
$881$	910.000	0.0347999	0.0173999	+	0.999849i	$0.494461\pi$
$882$	882.000	0.0336718
$883$	10460.0	0.398649	0.199324	−	0.979934i	$-0.436125\pi$
$883$	10460.0	0.398649	0.199324	+	0.979934i	$0.436125\pi$
$884$	−5096.00	−0.193888
$885$	3528.00	0.134003
$886$	12192.0	0.462301
$887$	20048.0	0.758902	0.379451	−	0.925212i	$-0.376113\pi$
$887$	20048.0	0.758902	0.379451	+	0.925212i	$0.376113\pi$
$888$	−3504.00	−0.132417
$889$	−6328.00	−0.238734
$890$	11256.0	0.423935
$891$	648.000	0.0243646
$892$	5568.00	0.209003
$893$	6048.00	0.226639
$894$	−15084.0	−0.564300
$895$	−29232.0	−1.09175
$896$	−896.000	−0.0334077
$897$	−2028.00	−0.0754882
$898$	−18780.0	−0.697881
$899$	336.000	0.0124652
$900$	2556.00	0.0946667
$901$	6468.00	0.239157
$902$	−8224.00	−0.303580
$903$	4956.00	0.182642
$904$	7568.00	0.278438
$905$	3388.00	0.124443
$906$	−6288.00	−0.230579
$907$	−876.000	−0.0320696	−0.0160348	−	0.999871i	$-0.505104\pi$
$907$	−876.000	−0.0320696	−0.0160348	+	0.999871i	$0.505104\pi$
$908$	−24560.0	−0.897635
$909$	7722.00	0.281763
$910$	2548.00	0.0928191
$911$	14796.0	0.538105	0.269052	−	0.963126i	$-0.413290\pi$
$911$	14796.0	0.538105	0.269052	+	0.963126i	$0.413290\pi$
$912$	−1344.00	−0.0487986
$913$	−2336.00	−0.0846772
$914$	−12.0000	−0.000434272 0
$915$	−18732.0	−0.676788
$916$	7480.00	0.269810
$917$	10332.0	0.372075
$918$	−5292.00	−0.190264
$919$	728.000	0.0261311	0.0130656	−	0.999915i	$-0.495841\pi$
$919$	728.000	0.0261311	0.0130656	+	0.999915i	$0.495841\pi$
$920$	5824.00	0.208708
$921$	−10548.0	−0.377382
$922$	−18748.0	−0.669666
$923$	1300.00	0.0463597
$924$	−672.000	−0.0239255
$925$	−10366.0	−0.368467
$926$	8016.00	0.284473
$927$	−8496.00	−0.301020
$928$	−64.0000	−0.00226390
$929$	39894.0	1.40891	0.704456	−	0.709747i	$-0.251191\pi$
$929$	39894.0	1.40891	0.704456	+	0.709747i	$0.251191\pi$
$930$	14112.0	0.497581
$931$	−1372.00	−0.0482980
$932$	200.000	0.00702920
$933$	9648.00	0.338544
$934$	−6520.00	−0.228416
$935$	10976.0	0.383908
$936$	936.000	0.0326860
$937$	18906.0	0.659159	0.329580	−	0.944128i	$-0.393093\pi$
$937$	18906.0	0.659159	0.329580	+	0.944128i	$0.393093\pi$
$938$	−4088.00	−0.142301
$939$	−4194.00	−0.145757
$940$	12096.0	0.419711
$941$	49514.0	1.71531	0.857657	−	0.514222i	$-0.171919\pi$
$941$	49514.0	1.71531	0.857657	+	0.514222i	$0.171919\pi$
$942$	18996.0	0.657031
$943$	26728.0	0.922994
$944$	−1344.00	−0.0463384
$945$	2646.00	0.0910840
$946$	−3776.00	−0.129776
$947$	16384.0	0.562205	0.281103	−	0.959678i	$-0.409300\pi$
$947$	16384.0	0.562205	0.281103	+	0.959678i	$0.409300\pi$
$948$	4704.00	0.161159
$949$	5850.00	0.200104
$950$	−3976.00	−0.135788
$951$	−27102.0	−0.924125
$952$	5488.00	0.186835
$953$	−30742.0	−1.04494	−0.522472	−	0.852657i	$-0.674990\pi$
$953$	−30742.0	−1.04494	−0.522472	+	0.852657i	$0.674990\pi$
$954$	−1188.00	−0.0403175
$955$	17976.0	0.609099
$956$	−12912.0	−0.436824
$957$	−48.0000	−0.00162134
$958$	25392.0	0.856345
$959$	434.000	0.0146138
$960$	−2688.00	−0.0903696
$961$	−1567.00	−0.0525998
$962$	−3796.00	−0.127222
$963$	7560.00	0.252978
$964$	23912.0	0.798915
$965$	38276.0	1.27684
$966$	2184.00	0.0727423
$967$	−41376.0	−1.37597	−0.687985	−	0.725725i	$-0.741504\pi$
$967$	−41376.0	−1.37597	−0.687985	+	0.725725i	$0.741504\pi$
$968$	−10136.0	−0.336553
$969$	8232.00	0.272910
$970$	−8792.00	−0.291025
$971$	57300.0	1.89376	0.946882	−	0.321582i	$-0.104214\pi$
$971$	57300.0	1.89376	0.946882	+	0.321582i	$0.104214\pi$
$972$	972.000	0.0320750
$973$	6356.00	0.209418
$974$	21520.0	0.707952
$975$	2769.00	0.0909528
$976$	7136.00	0.234035
$977$	−7734.00	−0.253258	−0.126629	−	0.991950i	$-0.540416\pi$
$977$	−7734.00	−0.253258	−0.126629	+	0.991950i	$0.540416\pi$
$978$	−2952.00	−0.0965179
$979$	−3216.00	−0.104989
$980$	−2744.00	−0.0894427
$981$	12006.0	0.390746
$982$	−23200.0	−0.753912
$983$	−31432.0	−1.01986	−0.509931	−	0.860215i	$-0.670329\pi$
$983$	−31432.0	−1.01986	−0.509931	+	0.860215i	$0.670329\pi$
$984$	−12336.0	−0.399652
$985$	−71652.0	−2.31779
$986$	392.000	0.0126611
$987$	4536.00	0.146284
$988$	−1456.00	−0.0468841
$989$	12272.0	0.394567
$990$	−2016.00	−0.0647199
$991$	−24432.0	−0.783156	−0.391578	−	0.920145i	$-0.628071\pi$
$991$	−24432.0	−0.783156	−0.391578	+	0.920145i	$0.628071\pi$
$992$	−5376.00	−0.172065
$993$	27324.0	0.873213
$994$	−1400.00	−0.0446733
$995$	−21392.0	−0.681580
$996$	−3504.00	−0.111474
$997$	3094.00	0.0982828	0.0491414	−	0.998792i	$-0.484352\pi$
$997$	3094.00	0.0982828	0.0491414	+	0.998792i	$0.484352\pi$
$998$	−31816.0	−1.00914
$999$	−3942.00	−0.124844

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.c.1.1	✓	1	1.1	even	1	trivial
1638.4.a.h.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 $\F_{q}$

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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