LMFDB - Embedded newform 49.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,4,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	49.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:	$2.89109359028$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -7.00000 q^{3} -4.00000 q^{4} -7.00000 q^{5} -14.0000 q^{6} -24.0000 q^{8} +22.0000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -7.00000 q^{3} -4.00000 q^{4} -7.00000 q^{5} -14.0000 q^{6} -24.0000 q^{8} +22.0000 q^{9} -14.0000 q^{10} -5.00000 q^{11} +28.0000 q^{12} +14.0000 q^{13} +49.0000 q^{15} -16.0000 q^{16} +21.0000 q^{17} +44.0000 q^{18} -49.0000 q^{19} +28.0000 q^{20} -10.0000 q^{22} -159.000 q^{23} +168.000 q^{24} -76.0000 q^{25} +28.0000 q^{26} +35.0000 q^{27} +58.0000 q^{29} +98.0000 q^{30} -147.000 q^{31} +160.000 q^{32} +35.0000 q^{33} +42.0000 q^{34} -88.0000 q^{36} +219.000 q^{37} -98.0000 q^{38} -98.0000 q^{39} +168.000 q^{40} -350.000 q^{41} -124.000 q^{43} +20.0000 q^{44} -154.000 q^{45} -318.000 q^{46} -525.000 q^{47} +112.000 q^{48} -152.000 q^{50} -147.000 q^{51} -56.0000 q^{52} +303.000 q^{53} +70.0000 q^{54} +35.0000 q^{55} +343.000 q^{57} +116.000 q^{58} +105.000 q^{59} -196.000 q^{60} +413.000 q^{61} -294.000 q^{62} +448.000 q^{64} -98.0000 q^{65} +70.0000 q^{66} +415.000 q^{67} -84.0000 q^{68} +1113.00 q^{69} -432.000 q^{71} -528.000 q^{72} +1113.00 q^{73} +438.000 q^{74} +532.000 q^{75} +196.000 q^{76} -196.000 q^{78} -103.000 q^{79} +112.000 q^{80} -839.000 q^{81} -700.000 q^{82} -1092.00 q^{83} -147.000 q^{85} -248.000 q^{86} -406.000 q^{87} +120.000 q^{88} +329.000 q^{89} -308.000 q^{90} +636.000 q^{92} +1029.00 q^{93} -1050.00 q^{94} +343.000 q^{95} -1120.00 q^{96} +882.000 q^{97} -110.000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	2.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	−4.00000	−0.500000
$5$	−7.00000	−0.626099	−0.313050	−	0.949737i	$-0.601351\pi$
$5$	−7.00000	−0.626099	−0.313050	+	0.949737i	$0.601351\pi$
$6$	−14.0000	−0.952579
$7$	0	0
$7$	0	0
$8$	−24.0000	−1.06066
$9$	22.0000	0.814815
$10$	−14.0000	−0.442719
$11$	−5.00000	−0.137051	−0.0685253	−	0.997649i	$-0.521829\pi$
$11$	−5.00000	−0.137051	−0.0685253	+	0.997649i	$0.521829\pi$
$12$	28.0000	0.673575
$13$	14.0000	0.298685	0.149342	−	0.988786i	$-0.452284\pi$
$13$	14.0000	0.298685	0.149342	+	0.988786i	$0.452284\pi$
$14$	0	0
$15$	49.0000	0.843450
$16$	−16.0000	−0.250000
$17$	21.0000	0.299603	0.149801	−	0.988716i	$-0.452137\pi$
$17$	21.0000	0.299603	0.149801	+	0.988716i	$0.452137\pi$
$18$	44.0000	0.576161
$19$	−49.0000	−0.591651	−0.295826	−	0.955242i	$-0.595595\pi$
$19$	−49.0000	−0.591651	−0.295826	+	0.955242i	$0.595595\pi$
$20$	28.0000	0.313050
$21$	0	0
$22$	−10.0000	−0.0969094
$23$	−159.000	−1.44147	−0.720735	−	0.693211i	$-0.756195\pi$
$23$	−159.000	−1.44147	−0.720735	+	0.693211i	$0.756195\pi$
$24$	168.000	1.42887
$25$	−76.0000	−0.608000
$26$	28.0000	0.211202
$27$	35.0000	0.249472
$28$	0	0
$29$	58.0000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	58.0000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	98.0000	0.596409
$31$	−147.000	−0.851677	−0.425838	−	0.904799i	$-0.640021\pi$
$31$	−147.000	−0.851677	−0.425838	+	0.904799i	$0.640021\pi$
$32$	160.000	0.883883
$33$	35.0000	0.184628
$34$	42.0000	0.211851
$35$	0	0
$36$	−88.0000	−0.407407
$37$	219.000	0.973064	0.486532	−	0.873663i	$-0.338262\pi$
$37$	219.000	0.973064	0.486532	+	0.873663i	$0.338262\pi$
$38$	−98.0000	−0.418361
$39$	−98.0000	−0.402373
$40$	168.000	0.664078
$41$	−350.000	−1.33319	−0.666595	−	0.745420i	$-0.732249\pi$
$41$	−350.000	−1.33319	−0.666595	+	0.745420i	$0.732249\pi$
$42$	0	0
$43$	−124.000	−0.439763	−0.219882	−	0.975527i	$-0.570567\pi$
$43$	−124.000	−0.439763	−0.219882	+	0.975527i	$0.570567\pi$
$44$	20.0000	0.0685253
$45$	−154.000	−0.510155
$46$	−318.000	−1.01927
$47$	−525.000	−1.62934	−0.814671	−	0.579923i	$-0.803083\pi$
$47$	−525.000	−1.62934	−0.814671	+	0.579923i	$0.803083\pi$
$48$	112.000	0.336788
$49$	0	0
$50$	−152.000	−0.429921
$51$	−147.000	−0.403610
$52$	−56.0000	−0.149342
$53$	303.000	0.785288	0.392644	−	0.919691i	$-0.371561\pi$
$53$	303.000	0.785288	0.392644	+	0.919691i	$0.371561\pi$
$54$	70.0000	0.176404
$55$	35.0000	0.0858073
$56$	0	0
$57$	343.000	0.797043
$58$	116.000	0.262613
$59$	105.000	0.231692	0.115846	−	0.993267i	$-0.463042\pi$
$59$	105.000	0.231692	0.115846	+	0.993267i	$0.463042\pi$
$60$	−196.000	−0.421725
$61$	413.000	0.866873	0.433436	−	0.901184i	$-0.357301\pi$
$61$	413.000	0.866873	0.433436	+	0.901184i	$0.357301\pi$
$62$	−294.000	−0.602226
$63$	0	0
$64$	448.000	0.875000
$65$	−98.0000	−0.187006
$66$	70.0000	0.130552
$67$	415.000	0.756721	0.378361	−	0.925658i	$-0.376488\pi$
$67$	415.000	0.756721	0.378361	+	0.925658i	$0.376488\pi$
$68$	−84.0000	−0.149801
$69$	1113.00	1.94188
$70$	0	0
$71$	−432.000	−0.722098	−0.361049	−	0.932547i	$-0.617581\pi$
$71$	−432.000	−0.722098	−0.361049	+	0.932547i	$0.617581\pi$
$72$	−528.000	−0.864242
$73$	1113.00	1.78448	0.892238	−	0.451565i	$-0.149134\pi$
$73$	1113.00	1.78448	0.892238	+	0.451565i	$0.149134\pi$
$74$	438.000	0.688060
$75$	532.000	0.819068
$76$	196.000	0.295826
$77$	0	0
$78$	−196.000	−0.284521
$79$	−103.000	−0.146689	−0.0733443	−	0.997307i	$-0.523367\pi$
$79$	−103.000	−0.146689	−0.0733443	+	0.997307i	$0.523367\pi$
$80$	112.000	0.156525
$81$	−839.000	−1.15089
$82$	−700.000	−0.942708
$83$	−1092.00	−1.44413	−0.722064	−	0.691827i	$-0.756806\pi$
$83$	−1092.00	−1.44413	−0.722064	+	0.691827i	$0.756806\pi$
$84$	0	0
$85$	−147.000	−0.187581
$86$	−248.000	−0.310960
$87$	−406.000	−0.500319
$88$	120.000	0.145364
$89$	329.000	0.391842	0.195921	−	0.980620i	$-0.437230\pi$
$89$	329.000	0.391842	0.195921	+	0.980620i	$0.437230\pi$
$90$	−308.000	−0.360734
$91$	0	0
$92$	636.000	0.720735
$93$	1029.00	1.14734
$94$	−1050.00	−1.15212
$95$	343.000	0.370432
$96$	−1120.00	−1.19072
$97$	882.000	0.923232	0.461616	−	0.887080i	$-0.347270\pi$
$97$	882.000	0.923232	0.461616	+	0.887080i	$0.347270\pi$
$98$	0	0
$99$	−110.000	−0.111671
$100$	304.000	0.304000
$101$	−1379.00	−1.35857	−0.679285	−	0.733874i	$-0.737710\pi$
$101$	−1379.00	−1.35857	−0.679285	+	0.733874i	$0.737710\pi$
$102$	−294.000	−0.285395
$103$	679.000	0.649552	0.324776	−	0.945791i	$-0.394711\pi$
$103$	679.000	0.649552	0.324776	+	0.945791i	$0.394711\pi$
$104$	−336.000	−0.316803
$105$	0	0
$106$	606.000	0.555282
$107$	457.000	0.412896	0.206448	−	0.978458i	$-0.433810\pi$
$107$	457.000	0.412896	0.206448	+	0.978458i	$0.433810\pi$
$108$	−140.000	−0.124736
$109$	−1125.00	−0.988582	−0.494291	−	0.869296i	$-0.664572\pi$
$109$	−1125.00	−0.988582	−0.494291	+	0.869296i	$0.664572\pi$
$110$	70.0000	0.0606749
$111$	−1533.00	−1.31086
$112$	0	0
$113$	−1538.00	−1.28038	−0.640190	−	0.768217i	$-0.721144\pi$
$113$	−1538.00	−1.28038	−0.640190	+	0.768217i	$0.721144\pi$
$114$	686.000	0.563595
$115$	1113.00	0.902502
$116$	−232.000	−0.185695
$117$	308.000	0.243373
$118$	210.000	0.163831
$119$	0	0
$120$	−1176.00	−0.894614
$121$	−1306.00	−0.981217
$122$	826.000	0.612972
$123$	2450.00	1.79601
$124$	588.000	0.425838
$125$	1407.00	1.00677
$126$	0	0
$127$	72.0000	0.0503068	0.0251534	−	0.999684i	$-0.491993\pi$
$127$	72.0000	0.0503068	0.0251534	+	0.999684i	$0.491993\pi$
$128$	−384.000	−0.265165
$129$	868.000	0.592427
$130$	−196.000	−0.132233
$131$	−2149.00	−1.43327	−0.716637	−	0.697446i	$-0.754320\pi$
$131$	−2149.00	−1.43327	−0.716637	+	0.697446i	$0.754320\pi$
$132$	−140.000	−0.0923139
$133$	0	0
$134$	830.000	0.535083
$135$	−245.000	−0.156194
$136$	−504.000	−0.317777
$137$	−1125.00	−0.701571	−0.350786	−	0.936456i	$-0.614085\pi$
$137$	−1125.00	−0.701571	−0.350786	+	0.936456i	$0.614085\pi$
$138$	2226.00	1.37311
$139$	−252.000	−0.153772	−0.0768862	−	0.997040i	$-0.524498\pi$
$139$	−252.000	−0.153772	−0.0768862	+	0.997040i	$0.524498\pi$
$140$	0	0
$141$	3675.00	2.19497
$142$	−864.000	−0.510600
$143$	−70.0000	−0.0409349
$144$	−352.000	−0.203704
$145$	−406.000	−0.232527
$146$	2226.00	1.26182
$147$	0	0
$148$	−876.000	−0.486532
$149$	−201.000	−0.110514	−0.0552569	−	0.998472i	$-0.517598\pi$
$149$	−201.000	−0.110514	−0.0552569	+	0.998472i	$0.517598\pi$
$150$	1064.00	0.579168
$151$	1619.00	0.872532	0.436266	−	0.899818i	$-0.356301\pi$
$151$	1619.00	0.872532	0.436266	+	0.899818i	$0.356301\pi$
$152$	1176.00	0.627541
$153$	462.000	0.244121
$154$	0	0
$155$	1029.00	0.533234
$156$	392.000	0.201187
$157$	−679.000	−0.345160	−0.172580	−	0.984996i	$-0.555210\pi$
$157$	−679.000	−0.345160	−0.172580	+	0.984996i	$0.555210\pi$
$158$	−206.000	−0.103725
$159$	−2121.00	−1.05790
$160$	−1120.00	−0.553399
$161$	0	0
$162$	−1678.00	−0.813803
$163$	−467.000	−0.224407	−0.112203	−	0.993685i	$-0.535791\pi$
$163$	−467.000	−0.224407	−0.112203	+	0.993685i	$0.535791\pi$
$164$	1400.00	0.666595
$165$	−245.000	−0.115595
$166$	−2184.00	−1.02115
$167$	−1204.00	−0.557894	−0.278947	−	0.960306i	$-0.589985\pi$
$167$	−1204.00	−0.557894	−0.278947	+	0.960306i	$0.589985\pi$
$168$	0	0
$169$	−2001.00	−0.910787
$170$	−294.000	−0.132640
$171$	−1078.00	−0.482086
$172$	496.000	0.219882
$173$	2821.00	1.23975	0.619875	−	0.784701i	$-0.287183\pi$
$173$	2821.00	1.23975	0.619875	+	0.784701i	$0.287183\pi$
$174$	−812.000	−0.353779
$175$	0	0
$176$	80.0000	0.0342627
$177$	−735.000	−0.312124
$178$	658.000	0.277074
$179$	−3253.00	−1.35833	−0.679164	−	0.733987i	$-0.737657\pi$
$179$	−3253.00	−1.35833	−0.679164	+	0.733987i	$0.737657\pi$
$180$	616.000	0.255077
$181$	−1582.00	−0.649664	−0.324832	−	0.945772i	$-0.605308\pi$
$181$	−1582.00	−0.649664	−0.324832	+	0.945772i	$0.605308\pi$
$182$	0	0
$183$	−2891.00	−1.16781
$184$	3816.00	1.52891
$185$	−1533.00	−0.609235
$186$	2058.00	0.811290
$187$	−105.000	−0.0410608
$188$	2100.00	0.814671
$189$	0	0
$190$	686.000	0.261935
$191$	2557.00	0.968681	0.484340	−	0.874880i	$-0.339060\pi$
$191$	2557.00	0.968681	0.484340	+	0.874880i	$0.339060\pi$
$192$	−3136.00	−1.17876
$193$	−397.000	−0.148066	−0.0740329	−	0.997256i	$-0.523587\pi$
$193$	−397.000	−0.148066	−0.0740329	+	0.997256i	$0.523587\pi$
$194$	1764.00	0.652824
$195$	686.000	0.251926
$196$	0	0
$197$	2914.00	1.05388	0.526939	−	0.849903i	$-0.323340\pi$
$197$	2914.00	1.05388	0.526939	+	0.849903i	$0.323340\pi$
$198$	−220.000	−0.0789632
$199$	−3339.00	−1.18942	−0.594712	−	0.803939i	$-0.702734\pi$
$199$	−3339.00	−1.18942	−0.594712	+	0.803939i	$0.702734\pi$
$200$	1824.00	0.644881
$201$	−2905.00	−1.01942
$202$	−2758.00	−0.960654
$203$	0	0
$204$	588.000	0.201805
$205$	2450.00	0.834709
$206$	1358.00	0.459303
$207$	−3498.00	−1.17453
$208$	−224.000	−0.0746712
$209$	245.000	0.0810861
$210$	0	0
$211$	1780.00	0.580759	0.290380	−	0.956911i	$-0.406218\pi$
$211$	1780.00	0.580759	0.290380	+	0.956911i	$0.406218\pi$
$212$	−1212.00	−0.392644
$213$	3024.00	0.972775
$214$	914.000	0.291961
$215$	868.000	0.275335
$216$	−840.000	−0.264605
$217$	0	0
$218$	−2250.00	−0.699033
$219$	−7791.00	−2.40396
$220$	−140.000	−0.0429036
$221$	294.000	0.0894868
$222$	−3066.00	−0.926921
$223$	1400.00	0.420408	0.210204	−	0.977658i	$-0.432587\pi$
$223$	1400.00	0.420408	0.210204	+	0.977658i	$0.432587\pi$
$224$	0	0
$225$	−1672.00	−0.495407
$226$	−3076.00	−0.905365
$227$	2205.00	0.644718	0.322359	−	0.946617i	$-0.395524\pi$
$227$	2205.00	0.644718	0.322359	+	0.946617i	$0.395524\pi$
$228$	−1372.00	−0.398522
$229$	−287.000	−0.0828188	−0.0414094	−	0.999142i	$-0.513185\pi$
$229$	−287.000	−0.0828188	−0.0414094	+	0.999142i	$0.513185\pi$
$230$	2226.00	0.638166
$231$	0	0
$232$	−1392.00	−0.393919
$233$	4587.00	1.28972	0.644859	−	0.764301i	$-0.276916\pi$
$233$	4587.00	1.28972	0.644859	+	0.764301i	$0.276916\pi$
$234$	616.000	0.172091
$235$	3675.00	1.02013
$236$	−420.000	−0.115846
$237$	721.000	0.197612
$238$	0	0
$239$	1668.00	0.451439	0.225720	−	0.974192i	$-0.427527\pi$
$239$	1668.00	0.451439	0.225720	+	0.974192i	$0.427527\pi$
$240$	−784.000	−0.210862
$241$	3409.00	0.911174	0.455587	−	0.890191i	$-0.349429\pi$
$241$	3409.00	0.911174	0.455587	+	0.890191i	$0.349429\pi$
$242$	−2612.00	−0.693825
$243$	4928.00	1.30095
$244$	−1652.00	−0.433436
$245$	0	0
$246$	4900.00	1.26997
$247$	−686.000	−0.176717
$248$	3528.00	0.903340
$249$	7644.00	1.94546
$250$	2814.00	0.711892
$251$	4760.00	1.19701	0.598503	−	0.801121i	$-0.295762\pi$
$251$	4760.00	1.19701	0.598503	+	0.801121i	$0.295762\pi$
$252$	0	0
$253$	795.000	0.197554
$254$	144.000	0.0355723
$255$	1029.00	0.252700
$256$	−4352.00	−1.06250
$257$	805.000	0.195387	0.0976936	−	0.995217i	$-0.468853\pi$
$257$	805.000	0.195387	0.0976936	+	0.995217i	$0.468853\pi$
$258$	1736.00	0.418909
$259$	0	0
$260$	392.000	0.0935031
$261$	1276.00	0.302615
$262$	−4298.00	−1.01348
$263$	−257.000	−0.0602559	−0.0301279	−	0.999546i	$-0.509591\pi$
$263$	−257.000	−0.0602559	−0.0301279	+	0.999546i	$0.509591\pi$
$264$	−840.000	−0.195827
$265$	−2121.00	−0.491668
$266$	0	0
$267$	−2303.00	−0.527870
$268$	−1660.00	−0.378361
$269$	−3591.00	−0.813930	−0.406965	−	0.913444i	$-0.633413\pi$
$269$	−3591.00	−0.813930	−0.406965	+	0.913444i	$0.633413\pi$
$270$	−490.000	−0.110446
$271$	−1393.00	−0.312246	−0.156123	−	0.987738i	$-0.549900\pi$
$271$	−1393.00	−0.312246	−0.156123	+	0.987738i	$0.549900\pi$
$272$	−336.000	−0.0749007
$273$	0	0
$274$	−2250.00	−0.496086
$275$	380.000	0.0833268
$276$	−4452.00	−0.970938
$277$	415.000	0.0900178	0.0450089	−	0.998987i	$-0.485668\pi$
$277$	415.000	0.0900178	0.0450089	+	0.998987i	$0.485668\pi$
$278$	−504.000	−0.108733
$279$	−3234.00	−0.693959
$280$	0	0
$281$	−4954.00	−1.05171	−0.525856	−	0.850574i	$-0.676255\pi$
$281$	−4954.00	−1.05171	−0.525856	+	0.850574i	$0.676255\pi$
$282$	7350.00	1.55208
$283$	4277.00	0.898379	0.449190	−	0.893437i	$-0.351713\pi$
$283$	4277.00	0.898379	0.449190	+	0.893437i	$0.351713\pi$
$284$	1728.00	0.361049
$285$	−2401.00	−0.499028
$286$	−140.000	−0.0289454
$287$	0	0
$288$	3520.00	0.720201
$289$	−4472.00	−0.910238
$290$	−812.000	−0.164422
$291$	−6174.00	−1.24373
$292$	−4452.00	−0.892238
$293$	−7742.00	−1.54366	−0.771830	−	0.635829i	$-0.780658\pi$
$293$	−7742.00	−1.54366	−0.771830	+	0.635829i	$0.780658\pi$
$294$	0	0
$295$	−735.000	−0.145062
$296$	−5256.00	−1.03209
$297$	−175.000	−0.0341903
$298$	−402.000	−0.0781451
$299$	−2226.00	−0.430545
$300$	−2128.00	−0.409534
$301$	0	0
$302$	3238.00	0.616973
$303$	9653.00	1.83020
$304$	784.000	0.147913
$305$	−2891.00	−0.542748
$306$	924.000	0.172619
$307$	7364.00	1.36901	0.684504	−	0.729009i	$-0.260019\pi$
$307$	7364.00	1.36901	0.684504	+	0.729009i	$0.260019\pi$
$308$	0	0
$309$	−4753.00	−0.875044
$310$	2058.00	0.377053
$311$	−9975.00	−1.81875	−0.909374	−	0.415980i	$-0.863438\pi$
$311$	−9975.00	−1.81875	−0.909374	+	0.415980i	$0.863438\pi$
$312$	2352.00	0.426781
$313$	4753.00	0.858324	0.429162	−	0.903228i	$-0.358809\pi$
$313$	4753.00	0.858324	0.429162	+	0.903228i	$0.358809\pi$
$314$	−1358.00	−0.244065
$315$	0	0
$316$	412.000	0.0733443
$317$	−3477.00	−0.616050	−0.308025	−	0.951378i	$-0.599668\pi$
$317$	−3477.00	−0.616050	−0.308025	+	0.951378i	$0.599668\pi$
$318$	−4242.00	−0.748049
$319$	−290.000	−0.0508993
$320$	−3136.00	−0.547837
$321$	−3199.00	−0.556233
$322$	0	0
$323$	−1029.00	−0.177260
$324$	3356.00	0.575446
$325$	−1064.00	−0.181600
$326$	−934.000	−0.158679
$327$	7875.00	1.33177
$328$	8400.00	1.41406
$329$	0	0
$330$	−490.000	−0.0817382
$331$	3341.00	0.554797	0.277399	−	0.960755i	$-0.410528\pi$
$331$	3341.00	0.554797	0.277399	+	0.960755i	$0.410528\pi$
$332$	4368.00	0.722064
$333$	4818.00	0.792867
$334$	−2408.00	−0.394491
$335$	−2905.00	−0.473782
$336$	0	0
$337$	7366.00	1.19066	0.595329	−	0.803482i	$-0.297022\pi$
$337$	7366.00	1.19066	0.595329	+	0.803482i	$0.297022\pi$
$338$	−4002.00	−0.644024
$339$	10766.0	1.72486
$340$	588.000	0.0937905
$341$	735.000	0.116723
$342$	−2156.00	−0.340886
$343$	0	0
$344$	2976.00	0.466439
$345$	−7791.00	−1.21581
$346$	5642.00	0.876635
$347$	7415.00	1.14714	0.573571	−	0.819156i	$-0.305558\pi$
$347$	7415.00	1.14714	0.573571	+	0.819156i	$0.305558\pi$
$348$	1624.00	0.250160
$349$	3878.00	0.594798	0.297399	−	0.954753i	$-0.403881\pi$
$349$	3878.00	0.594798	0.297399	+	0.954753i	$0.403881\pi$
$350$	0	0
$351$	490.000	0.0745136
$352$	−800.000	−0.121137
$353$	−1267.00	−0.191036	−0.0955179	−	0.995428i	$-0.530451\pi$
$353$	−1267.00	−0.191036	−0.0955179	+	0.995428i	$0.530451\pi$
$354$	−1470.00	−0.220705
$355$	3024.00	0.452105
$356$	−1316.00	−0.195921
$357$	0	0
$358$	−6506.00	−0.960483
$359$	4685.00	0.688760	0.344380	−	0.938830i	$-0.388089\pi$
$359$	4685.00	0.688760	0.344380	+	0.938830i	$0.388089\pi$
$360$	3696.00	0.541101
$361$	−4458.00	−0.649949
$362$	−3164.00	−0.459382
$363$	9142.00	1.32185
$364$	0	0
$365$	−7791.00	−1.11726
$366$	−5782.00	−0.825765
$367$	4641.00	0.660104	0.330052	−	0.943963i	$-0.392934\pi$
$367$	4641.00	0.660104	0.330052	+	0.943963i	$0.392934\pi$
$368$	2544.00	0.360367
$369$	−7700.00	−1.08630
$370$	−3066.00	−0.430794
$371$	0	0
$372$	−4116.00	−0.573668
$373$	−8797.00	−1.22116	−0.610578	−	0.791956i	$-0.709063\pi$
$373$	−8797.00	−1.22116	−0.610578	+	0.791956i	$0.709063\pi$
$374$	−210.000	−0.0290343
$375$	−9849.00	−1.35627
$376$	12600.0	1.72818
$377$	812.000	0.110929
$378$	0	0
$379$	13680.0	1.85407	0.927037	−	0.374969i	$-0.122347\pi$
$379$	13680.0	1.85407	0.927037	+	0.374969i	$0.122347\pi$
$380$	−1372.00	−0.185216
$381$	−504.000	−0.0677709
$382$	5114.00	0.684961
$383$	−9765.00	−1.30279	−0.651395	−	0.758739i	$-0.725816\pi$
$383$	−9765.00	−1.30279	−0.651395	+	0.758739i	$0.725816\pi$
$384$	2688.00	0.357217
$385$	0	0
$386$	−794.000	−0.104698
$387$	−2728.00	−0.358326
$388$	−3528.00	−0.461616
$389$	1731.00	0.225617	0.112809	−	0.993617i	$-0.464015\pi$
$389$	1731.00	0.225617	0.112809	+	0.993617i	$0.464015\pi$
$390$	1372.00	0.178138
$391$	−3339.00	−0.431868
$392$	0	0
$393$	15043.0	1.93084
$394$	5828.00	0.745204
$395$	721.000	0.0918416
$396$	440.000	0.0558354
$397$	−10983.0	−1.38847	−0.694233	−	0.719750i	$-0.744256\pi$
$397$	−10983.0	−1.38847	−0.694233	+	0.719750i	$0.744256\pi$
$398$	−6678.00	−0.841050
$399$	0	0
$400$	1216.00	0.152000
$401$	6603.00	0.822289	0.411145	−	0.911570i	$-0.365129\pi$
$401$	6603.00	0.822289	0.411145	+	0.911570i	$0.365129\pi$
$402$	−5810.00	−0.720837
$403$	−2058.00	−0.254383
$404$	5516.00	0.679285
$405$	5873.00	0.720572
$406$	0	0
$407$	−1095.00	−0.133359
$408$	3528.00	0.428093
$409$	−10955.0	−1.32443	−0.662213	−	0.749316i	$-0.730382\pi$
$409$	−10955.0	−1.32443	−0.662213	+	0.749316i	$0.730382\pi$
$410$	4900.00	0.590229
$411$	7875.00	0.945122
$412$	−2716.00	−0.324776
$413$	0	0
$414$	−6996.00	−0.830518
$415$	7644.00	0.904167
$416$	2240.00	0.264002
$417$	1764.00	0.207155
$418$	490.000	0.0573366
$419$	−6636.00	−0.773723	−0.386861	−	0.922138i	$-0.626441\pi$
$419$	−6636.00	−0.773723	−0.386861	+	0.922138i	$0.626441\pi$
$420$	0	0
$421$	−16630.0	−1.92517	−0.962585	−	0.270980i	$-0.912652\pi$
$421$	−16630.0	−1.92517	−0.962585	+	0.270980i	$0.912652\pi$
$422$	3560.00	0.410659
$423$	−11550.0	−1.32761
$424$	−7272.00	−0.832923
$425$	−1596.00	−0.182159
$426$	6048.00	0.687856
$427$	0	0
$428$	−1828.00	−0.206448
$429$	490.000	0.0551455
$430$	1736.00	0.194692
$431$	4923.00	0.550192	0.275096	−	0.961417i	$-0.411290\pi$
$431$	4923.00	0.550192	0.275096	+	0.961417i	$0.411290\pi$
$432$	−560.000	−0.0623681
$433$	−8974.00	−0.995988	−0.497994	−	0.867180i	$-0.665930\pi$
$433$	−8974.00	−0.995988	−0.497994	+	0.867180i	$0.665930\pi$
$434$	0	0
$435$	2842.00	0.313249
$436$	4500.00	0.494291
$437$	7791.00	0.852847
$438$	−15582.0	−1.69986
$439$	4179.00	0.454334	0.227167	−	0.973856i	$-0.427054\pi$
$439$	4179.00	0.454334	0.227167	+	0.973856i	$0.427054\pi$
$440$	−840.000	−0.0910123
$441$	0	0
$442$	588.000	0.0632767
$443$	−12927.0	−1.38641	−0.693206	−	0.720740i	$-0.743802\pi$
$443$	−12927.0	−1.38641	−0.693206	+	0.720740i	$0.743802\pi$
$444$	6132.00	0.655432
$445$	−2303.00	−0.245332
$446$	2800.00	0.297273
$447$	1407.00	0.148879
$448$	0	0
$449$	−2826.00	−0.297032	−0.148516	−	0.988910i	$-0.547450\pi$
$449$	−2826.00	−0.297032	−0.148516	+	0.988910i	$0.547450\pi$
$450$	−3344.00	−0.350306
$451$	1750.00	0.182715
$452$	6152.00	0.640190
$453$	−11333.0	−1.17543
$454$	4410.00	0.455884
$455$	0	0
$456$	−8232.00	−0.845392
$457$	8479.00	0.867901	0.433951	−	0.900937i	$-0.357119\pi$
$457$	8479.00	0.867901	0.433951	+	0.900937i	$0.357119\pi$
$458$	−574.000	−0.0585617
$459$	735.000	0.0747426
$460$	−4452.00	−0.451251
$461$	−9338.00	−0.943414	−0.471707	−	0.881755i	$-0.656362\pi$
$461$	−9338.00	−0.943414	−0.471707	+	0.881755i	$0.656362\pi$
$462$	0	0
$463$	−4016.00	−0.403109	−0.201554	−	0.979477i	$-0.564599\pi$
$463$	−4016.00	−0.403109	−0.201554	+	0.979477i	$0.564599\pi$
$464$	−928.000	−0.0928477
$465$	−7203.00	−0.718347
$466$	9174.00	0.911969
$467$	5859.00	0.580561	0.290281	−	0.956942i	$-0.406251\pi$
$467$	5859.00	0.580561	0.290281	+	0.956942i	$0.406251\pi$
$468$	−1232.00	−0.121686
$469$	0	0
$470$	7350.00	0.721341
$471$	4753.00	0.464982
$472$	−2520.00	−0.245747
$473$	620.000	0.0602698
$474$	1442.00	0.139733
$475$	3724.00	0.359724
$476$	0	0
$477$	6666.00	0.639864
$478$	3336.00	0.319216
$479$	−6503.00	−0.620312	−0.310156	−	0.950686i	$-0.600381\pi$
$479$	−6503.00	−0.620312	−0.310156	+	0.950686i	$0.600381\pi$
$480$	7840.00	0.745511
$481$	3066.00	0.290639
$482$	6818.00	0.644297
$483$	0	0
$484$	5224.00	0.490609
$485$	−6174.00	−0.578035
$486$	9856.00	0.919912
$487$	−16049.0	−1.49333	−0.746663	−	0.665203i	$-0.768345\pi$
$487$	−16049.0	−1.49333	−0.746663	+	0.665203i	$0.768345\pi$
$488$	−9912.00	−0.919457
$489$	3269.00	0.302309
$490$	0	0
$491$	8864.00	0.814718	0.407359	−	0.913268i	$-0.366450\pi$
$491$	8864.00	0.814718	0.407359	+	0.913268i	$0.366450\pi$
$492$	−9800.00	−0.898004
$493$	1218.00	0.111270
$494$	−1372.00	−0.124958
$495$	770.000	0.0699170
$496$	2352.00	0.212919
$497$	0	0
$498$	15288.0	1.37565
$499$	−10211.0	−0.916046	−0.458023	−	0.888940i	$-0.651442\pi$
$499$	−10211.0	−0.916046	−0.458023	+	0.888940i	$0.651442\pi$
$500$	−5628.00	−0.503384
$501$	8428.00	0.751567
$502$	9520.00	0.846411
$503$	1680.00	0.148921	0.0744607	−	0.997224i	$-0.476276\pi$
$503$	1680.00	0.148921	0.0744607	+	0.997224i	$0.476276\pi$
$504$	0	0
$505$	9653.00	0.850600
$506$	1590.00	0.139692
$507$	14007.0	1.22697
$508$	−288.000	−0.0251534
$509$	9457.00	0.823525	0.411762	−	0.911291i	$-0.364913\pi$
$509$	9457.00	0.823525	0.411762	+	0.911291i	$0.364913\pi$
$510$	2058.00	0.178686
$511$	0	0
$512$	−5632.00	−0.486136
$513$	−1715.00	−0.147601
$514$	1610.00	0.138160
$515$	−4753.00	−0.406684
$516$	−3472.00	−0.296214
$517$	2625.00	0.223302
$518$	0	0
$519$	−19747.0	−1.67013
$520$	2352.00	0.198350
$521$	18081.0	1.52043	0.760214	−	0.649673i	$-0.225094\pi$
$521$	18081.0	1.52043	0.760214	+	0.649673i	$0.225094\pi$
$522$	2552.00	0.213981
$523$	−20377.0	−1.70368	−0.851839	−	0.523803i	$-0.824513\pi$
$523$	−20377.0	−1.70368	−0.851839	+	0.523803i	$0.824513\pi$
$524$	8596.00	0.716637
$525$	0	0
$526$	−514.000	−0.0426073
$527$	−3087.00	−0.255165
$528$	−560.000	−0.0461570
$529$	13114.0	1.07783
$530$	−4242.00	−0.347662
$531$	2310.00	0.188786
$532$	0	0
$533$	−4900.00	−0.398204
$534$	−4606.00	−0.373261
$535$	−3199.00	−0.258514
$536$	−9960.00	−0.802624
$537$	22771.0	1.82987
$538$	−7182.00	−0.575535
$539$	0	0
$540$	980.000	0.0780972
$541$	−6193.00	−0.492159	−0.246079	−	0.969250i	$-0.579142\pi$
$541$	−6193.00	−0.492159	−0.246079	+	0.969250i	$0.579142\pi$
$542$	−2786.00	−0.220791
$543$	11074.0	0.875195
$544$	3360.00	0.264814
$545$	7875.00	0.618950
$546$	0	0
$547$	−18464.0	−1.44326	−0.721630	−	0.692279i	$-0.756607\pi$
$547$	−18464.0	−1.44326	−0.721630	+	0.692279i	$0.756607\pi$
$548$	4500.00	0.350786
$549$	9086.00	0.706341
$550$	760.000	0.0589209
$551$	−2842.00	−0.219734
$552$	−26712.0	−2.05967
$553$	0	0
$554$	830.000	0.0636522
$555$	10731.0	0.820731
$556$	1008.00	0.0768862
$557$	−9413.00	−0.716053	−0.358027	−	0.933711i	$-0.616550\pi$
$557$	−9413.00	−0.716053	−0.358027	+	0.933711i	$0.616550\pi$
$558$	−6468.00	−0.490703
$559$	−1736.00	−0.131351
$560$	0	0
$561$	735.000	0.0553150
$562$	−9908.00	−0.743672
$563$	−3199.00	−0.239470	−0.119735	−	0.992806i	$-0.538205\pi$
$563$	−3199.00	−0.239470	−0.119735	+	0.992806i	$0.538205\pi$
$564$	−14700.0	−1.09749
$565$	10766.0	0.801644
$566$	8554.00	0.635250
$567$	0	0
$568$	10368.0	0.765901
$569$	21583.0	1.59017	0.795085	−	0.606498i	$-0.207426\pi$
$569$	21583.0	1.59017	0.795085	+	0.606498i	$0.207426\pi$
$570$	−4802.00	−0.352866
$571$	20267.0	1.48537	0.742686	−	0.669640i	$-0.233551\pi$
$571$	20267.0	1.48537	0.742686	+	0.669640i	$0.233551\pi$
$572$	280.000	0.0204675
$573$	−17899.0	−1.30496
$574$	0	0
$575$	12084.0	0.876413
$576$	9856.00	0.712963
$577$	−13951.0	−1.00656	−0.503282	−	0.864122i	$-0.667874\pi$
$577$	−13951.0	−1.00656	−0.503282	+	0.864122i	$0.667874\pi$
$578$	−8944.00	−0.643636
$579$	2779.00	0.199467
$580$	1624.00	0.116264
$581$	0	0
$582$	−12348.0	−0.879452
$583$	−1515.00	−0.107624
$584$	−26712.0	−1.89272
$585$	−2156.00	−0.152375
$586$	−15484.0	−1.09153
$587$	20972.0	1.47463	0.737314	−	0.675550i	$-0.236094\pi$
$587$	20972.0	1.47463	0.737314	+	0.675550i	$0.236094\pi$
$588$	0	0
$589$	7203.00	0.503895
$590$	−1470.00	−0.102574
$591$	−20398.0	−1.41973
$592$	−3504.00	−0.243266
$593$	189.000	0.0130882	0.00654410	−	0.999979i	$-0.497917\pi$
$593$	189.000	0.0130882	0.00654410	+	0.999979i	$0.497917\pi$
$594$	−350.000	−0.0241762
$595$	0	0
$596$	804.000	0.0552569
$597$	23373.0	1.60233
$598$	−4452.00	−0.304441
$599$	−10281.0	−0.701286	−0.350643	−	0.936509i	$-0.614037\pi$
$599$	−10281.0	−0.701286	−0.350643	+	0.936509i	$0.614037\pi$
$600$	−12768.0	−0.868752
$601$	6090.00	0.413338	0.206669	−	0.978411i	$-0.433738\pi$
$601$	6090.00	0.413338	0.206669	+	0.978411i	$0.433738\pi$
$602$	0	0
$603$	9130.00	0.616588
$604$	−6476.00	−0.436266
$605$	9142.00	0.614339
$606$	19306.0	1.29415
$607$	−4949.00	−0.330929	−0.165464	−	0.986216i	$-0.552912\pi$
$607$	−4949.00	−0.330929	−0.165464	+	0.986216i	$0.552912\pi$
$608$	−7840.00	−0.522951
$609$	0	0
$610$	−5782.00	−0.383781
$611$	−7350.00	−0.486660
$612$	−1848.00	−0.122060
$613$	−15797.0	−1.04084	−0.520420	−	0.853910i	$-0.674225\pi$
$613$	−15797.0	−1.04084	−0.520420	+	0.853910i	$0.674225\pi$
$614$	14728.0	0.968035
$615$	−17150.0	−1.12448
$616$	0	0
$617$	−9378.00	−0.611903	−0.305951	−	0.952047i	$-0.598975\pi$
$617$	−9378.00	−0.611903	−0.305951	+	0.952047i	$0.598975\pi$
$618$	−9506.00	−0.618750
$619$	24353.0	1.58131	0.790654	−	0.612263i	$-0.209741\pi$
$619$	24353.0	1.58131	0.790654	+	0.612263i	$0.209741\pi$
$620$	−4116.00	−0.266617
$621$	−5565.00	−0.359607
$622$	−19950.0	−1.28605
$623$	0	0
$624$	1568.00	0.100593
$625$	−349.000	−0.0223360
$626$	9506.00	0.606927
$627$	−1715.00	−0.109235
$628$	2716.00	0.172580
$629$	4599.00	0.291533
$630$	0	0
$631$	−12640.0	−0.797449	−0.398725	−	0.917071i	$-0.630547\pi$
$631$	−12640.0	−0.797449	−0.398725	+	0.917071i	$0.630547\pi$
$632$	2472.00	0.155587
$633$	−12460.0	−0.782371
$634$	−6954.00	−0.435613
$635$	−504.000	−0.0314971
$636$	8484.00	0.528950
$637$	0	0
$638$	−580.000	−0.0359913
$639$	−9504.00	−0.588376
$640$	2688.00	0.166020
$641$	−1041.00	−0.0641451	−0.0320726	−	0.999486i	$-0.510211\pi$
$641$	−1041.00	−0.0641451	−0.0320726	+	0.999486i	$0.510211\pi$
$642$	−6398.00	−0.393316
$643$	−9548.00	−0.585593	−0.292797	−	0.956175i	$-0.594586\pi$
$643$	−9548.00	−0.585593	−0.292797	+	0.956175i	$0.594586\pi$
$644$	0	0
$645$	−6076.00	−0.370918
$646$	−2058.00	−0.125342
$647$	3241.00	0.196935	0.0984674	−	0.995140i	$-0.468606\pi$
$647$	3241.00	0.196935	0.0984674	+	0.995140i	$0.468606\pi$
$648$	20136.0	1.22070
$649$	−525.000	−0.0317535
$650$	−2128.00	−0.128411
$651$	0	0
$652$	1868.00	0.112203
$653$	−8853.00	−0.530543	−0.265272	−	0.964174i	$-0.585462\pi$
$653$	−8853.00	−0.530543	−0.265272	+	0.964174i	$0.585462\pi$
$654$	15750.0	0.941703
$655$	15043.0	0.897372
$656$	5600.00	0.333298
$657$	24486.0	1.45402
$658$	0	0
$659$	7044.00	0.416381	0.208191	−	0.978088i	$-0.433243\pi$
$659$	7044.00	0.416381	0.208191	+	0.978088i	$0.433243\pi$
$660$	980.000	0.0577976
$661$	12089.0	0.711358	0.355679	−	0.934608i	$-0.384250\pi$
$661$	12089.0	0.711358	0.355679	+	0.934608i	$0.384250\pi$
$662$	6682.00	0.392301
$663$	−2058.00	−0.120552
$664$	26208.0	1.53173
$665$	0	0
$666$	9636.00	0.560642
$667$	−9222.00	−0.535348
$668$	4816.00	0.278947
$669$	−9800.00	−0.566353
$670$	−5810.00	−0.335015
$671$	−2065.00	−0.118805
$672$	0	0
$673$	982.000	0.0562456	0.0281228	−	0.999604i	$-0.491047\pi$
$673$	982.000	0.0562456	0.0281228	+	0.999604i	$0.491047\pi$
$674$	14732.0	0.841922
$675$	−2660.00	−0.151679
$676$	8004.00	0.455394
$677$	30513.0	1.73222	0.866108	−	0.499857i	$-0.166614\pi$
$677$	30513.0	1.73222	0.866108	+	0.499857i	$0.166614\pi$
$678$	21532.0	1.21966
$679$	0	0
$680$	3528.00	0.198960
$681$	−15435.0	−0.868532
$682$	1470.00	0.0825355
$683$	11475.0	0.642868	0.321434	−	0.946932i	$-0.395835\pi$
$683$	11475.0	0.642868	0.321434	+	0.946932i	$0.395835\pi$
$684$	4312.00	0.241043
$685$	7875.00	0.439253
$686$	0	0
$687$	2009.00	0.111569
$688$	1984.00	0.109941
$689$	4242.00	0.234553
$690$	−15582.0	−0.859705
$691$	28315.0	1.55883	0.779416	−	0.626506i	$-0.215516\pi$
$691$	28315.0	1.55883	0.779416	+	0.626506i	$0.215516\pi$
$692$	−11284.0	−0.619875
$693$	0	0
$694$	14830.0	0.811151
$695$	1764.00	0.0962767
$696$	9744.00	0.530669
$697$	−7350.00	−0.399428
$698$	7756.00	0.420586
$699$	−32109.0	−1.73744
$700$	0	0
$701$	10614.0	0.571876	0.285938	−	0.958248i	$-0.407695\pi$
$701$	10614.0	0.571876	0.285938	+	0.958248i	$0.407695\pi$
$702$	980.000	0.0526891
$703$	−10731.0	−0.575715
$704$	−2240.00	−0.119919
$705$	−25725.0	−1.37427
$706$	−2534.00	−0.135083
$707$	0	0
$708$	2940.00	0.156062
$709$	10299.0	0.545539	0.272769	−	0.962079i	$-0.412060\pi$
$709$	10299.0	0.545539	0.272769	+	0.962079i	$0.412060\pi$
$710$	6048.00	0.319686
$711$	−2266.00	−0.119524
$712$	−7896.00	−0.415611
$713$	23373.0	1.22767
$714$	0	0
$715$	490.000	0.0256293
$716$	13012.0	0.679164
$717$	−11676.0	−0.608156
$718$	9370.00	0.487027
$719$	−32529.0	−1.68724	−0.843621	−	0.536939i	$-0.819580\pi$
$719$	−32529.0	−1.68724	−0.843621	+	0.536939i	$0.819580\pi$
$720$	2464.00	0.127539
$721$	0	0
$722$	−8916.00	−0.459583
$723$	−23863.0	−1.22749
$724$	6328.00	0.324832
$725$	−4408.00	−0.225806
$726$	18284.0	0.934687
$727$	−29456.0	−1.50270	−0.751350	−	0.659904i	$-0.770597\pi$
$727$	−29456.0	−1.50270	−0.751350	+	0.659904i	$0.770597\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	−15582.0	−0.790021
$731$	−2604.00	−0.131754
$732$	11564.0	0.583904
$733$	−27867.0	−1.40422	−0.702109	−	0.712070i	$-0.747758\pi$
$733$	−27867.0	−1.40422	−0.702109	+	0.712070i	$0.747758\pi$
$734$	9282.00	0.466764
$735$	0	0
$736$	−25440.0	−1.27409
$737$	−2075.00	−0.103709
$738$	−15400.0	−0.768133
$739$	19539.0	0.972603	0.486302	−	0.873791i	$-0.338346\pi$
$739$	19539.0	0.972603	0.486302	+	0.873791i	$0.338346\pi$
$740$	6132.00	0.304617
$741$	4802.00	0.238065
$742$	0	0
$743$	1248.00	0.0616214	0.0308107	−	0.999525i	$-0.490191\pi$
$743$	1248.00	0.0616214	0.0308107	+	0.999525i	$0.490191\pi$
$744$	−24696.0	−1.21693
$745$	1407.00	0.0691926
$746$	−17594.0	−0.863488
$747$	−24024.0	−1.17670
$748$	420.000	0.0205304
$749$	0	0
$750$	−19698.0	−0.959026
$751$	28093.0	1.36502	0.682509	−	0.730877i	$-0.260889\pi$
$751$	28093.0	1.36502	0.682509	+	0.730877i	$0.260889\pi$
$752$	8400.00	0.407336
$753$	−33320.0	−1.61255
$754$	1624.00	0.0784385
$755$	−11333.0	−0.546292
$756$	0	0
$757$	35954.0	1.72625	0.863124	−	0.504991i	$-0.168504\pi$
$757$	35954.0	1.72625	0.863124	+	0.504991i	$0.168504\pi$
$758$	27360.0	1.31103
$759$	−5565.00	−0.266135
$760$	−8232.00	−0.392903
$761$	861.000	0.0410134	0.0205067	−	0.999790i	$-0.493472\pi$
$761$	861.000	0.0410134	0.0205067	+	0.999790i	$0.493472\pi$
$762$	−1008.00	−0.0479212
$763$	0	0
$764$	−10228.0	−0.484340
$765$	−3234.00	−0.152844
$766$	−19530.0	−0.921211
$767$	1470.00	0.0692029
$768$	30464.0	1.43135
$769$	−24710.0	−1.15873	−0.579366	−	0.815067i	$-0.696700\pi$
$769$	−24710.0	−1.15873	−0.579366	+	0.815067i	$0.696700\pi$
$770$	0	0
$771$	−5635.00	−0.263216
$772$	1588.00	0.0740329
$773$	−16499.0	−0.767694	−0.383847	−	0.923397i	$-0.625401\pi$
$773$	−16499.0	−0.767694	−0.383847	+	0.923397i	$0.625401\pi$
$774$	−5456.00	−0.253375
$775$	11172.0	0.517819
$776$	−21168.0	−0.979236
$777$	0	0
$778$	3462.00	0.159536
$779$	17150.0	0.788784
$780$	−2744.00	−0.125963
$781$	2160.00	0.0989640
$782$	−6678.00	−0.305377
$783$	2030.00	0.0926517
$784$	0	0
$785$	4753.00	0.216104
$786$	30086.0	1.36531
$787$	−16471.0	−0.746033	−0.373016	−	0.927825i	$-0.621676\pi$
$787$	−16471.0	−0.746033	−0.373016	+	0.927825i	$0.621676\pi$
$788$	−11656.0	−0.526939
$789$	1799.00	0.0811738
$790$	1442.00	0.0649418
$791$	0	0
$792$	2640.00	0.118445
$793$	5782.00	0.258922
$794$	−21966.0	−0.981794
$795$	14847.0	0.662351
$796$	13356.0	0.594712
$797$	36470.0	1.62087	0.810435	−	0.585828i	$-0.199231\pi$
$797$	36470.0	1.62087	0.810435	+	0.585828i	$0.199231\pi$
$798$	0	0
$799$	−11025.0	−0.488156
$800$	−12160.0	−0.537401
$801$	7238.00	0.319279
$802$	13206.0	0.581446
$803$	−5565.00	−0.244564
$804$	11620.0	0.509709
$805$	0	0
$806$	−4116.00	−0.179876
$807$	25137.0	1.09649
$808$	33096.0	1.44098
$809$	35751.0	1.55369	0.776847	−	0.629690i	$-0.216818\pi$
$809$	35751.0	1.55369	0.776847	+	0.629690i	$0.216818\pi$
$810$	11746.0	0.509521
$811$	16492.0	0.714072	0.357036	−	0.934091i	$-0.383787\pi$
$811$	16492.0	0.714072	0.357036	+	0.934091i	$0.383787\pi$
$812$	0	0
$813$	9751.00	0.420643
$814$	−2190.00	−0.0942991
$815$	3269.00	0.140501
$816$	2352.00	0.100903
$817$	6076.00	0.260186
$818$	−21910.0	−0.936510
$819$	0	0
$820$	−9800.00	−0.417355
$821$	−41473.0	−1.76299	−0.881497	−	0.472190i	$-0.843464\pi$
$821$	−41473.0	−1.76299	−0.881497	+	0.472190i	$0.843464\pi$
$822$	15750.0	0.668302
$823$	−25065.0	−1.06162	−0.530809	−	0.847492i	$-0.678112\pi$
$823$	−25065.0	−1.06162	−0.530809	+	0.847492i	$0.678112\pi$
$824$	−16296.0	−0.688954
$825$	−2660.00	−0.112254
$826$	0	0
$827$	9732.00	0.409208	0.204604	−	0.978845i	$-0.434409\pi$
$827$	9732.00	0.409208	0.204604	+	0.978845i	$0.434409\pi$
$828$	13992.0	0.587265
$829$	−27755.0	−1.16281	−0.581406	−	0.813614i	$-0.697497\pi$
$829$	−27755.0	−1.16281	−0.581406	+	0.813614i	$0.697497\pi$
$830$	15288.0	0.639342
$831$	−2905.00	−0.121268
$832$	6272.00	0.261349
$833$	0	0
$834$	3528.00	0.146480
$835$	8428.00	0.349297
$836$	−980.000	−0.0405431
$837$	−5145.00	−0.212470
$838$	−13272.0	−0.547105
$839$	−21112.0	−0.868733	−0.434367	−	0.900736i	$-0.643028\pi$
$839$	−21112.0	−0.868733	−0.434367	+	0.900736i	$0.643028\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	−33260.0	−1.36130
$843$	34678.0	1.41681
$844$	−7120.00	−0.290380
$845$	14007.0	0.570243
$846$	−23100.0	−0.938764
$847$	0	0
$848$	−4848.00	−0.196322
$849$	−29939.0	−1.21025
$850$	−3192.00	−0.128806
$851$	−34821.0	−1.40264
$852$	−12096.0	−0.486387
$853$	21238.0	0.852492	0.426246	−	0.904607i	$-0.359836\pi$
$853$	21238.0	0.852492	0.426246	+	0.904607i	$0.359836\pi$
$854$	0	0
$855$	7546.00	0.301834
$856$	−10968.0	−0.437942
$857$	35609.0	1.41935	0.709673	−	0.704531i	$-0.248842\pi$
$857$	35609.0	1.41935	0.709673	+	0.704531i	$0.248842\pi$
$858$	980.000	0.0389938
$859$	−2177.00	−0.0864706	−0.0432353	−	0.999065i	$-0.513767\pi$
$859$	−2177.00	−0.0864706	−0.0432353	+	0.999065i	$0.513767\pi$
$860$	−3472.00	−0.137668
$861$	0	0
$862$	9846.00	0.389044
$863$	−32247.0	−1.27196	−0.635980	−	0.771706i	$-0.719404\pi$
$863$	−32247.0	−1.27196	−0.635980	+	0.771706i	$0.719404\pi$
$864$	5600.00	0.220504
$865$	−19747.0	−0.776206
$866$	−17948.0	−0.704270
$867$	31304.0	1.22623
$868$	0	0
$869$	515.000	0.0201038
$870$	5684.00	0.221501
$871$	5810.00	0.226021
$872$	27000.0	1.04855
$873$	19404.0	0.752263
$874$	15582.0	0.603054
$875$	0	0
$876$	31164.0	1.20198
$877$	27631.0	1.06389	0.531946	−	0.846779i	$-0.321461\pi$
$877$	27631.0	1.06389	0.531946	+	0.846779i	$0.321461\pi$
$878$	8358.00	0.321263
$879$	54194.0	2.07954
$880$	−560.000	−0.0214518
$881$	−24402.0	−0.933172	−0.466586	−	0.884476i	$-0.654516\pi$
$881$	−24402.0	−0.933172	−0.466586	+	0.884476i	$0.654516\pi$
$882$	0	0
$883$	−19612.0	−0.747448	−0.373724	−	0.927540i	$-0.621919\pi$
$883$	−19612.0	−0.747448	−0.373724	+	0.927540i	$0.621919\pi$
$884$	−1176.00	−0.0447434
$885$	5145.00	0.195421
$886$	−25854.0	−0.980341
$887$	−2261.00	−0.0855884	−0.0427942	−	0.999084i	$-0.513626\pi$
$887$	−2261.00	−0.0855884	−0.0427942	+	0.999084i	$0.513626\pi$
$888$	36792.0	1.39038
$889$	0	0
$890$	−4606.00	−0.173476
$891$	4195.00	0.157730
$892$	−5600.00	−0.210204
$893$	25725.0	0.964003
$894$	2814.00	0.105273
$895$	22771.0	0.850448
$896$	0	0
$897$	15582.0	0.580009
$898$	−5652.00	−0.210033
$899$	−8526.00	−0.316305
$900$	6688.00	0.247704
$901$	6363.00	0.235274
$902$	3500.00	0.129199
$903$	0	0
$904$	36912.0	1.35805
$905$	11074.0	0.406754
$906$	−22666.0	−0.831156
$907$	−23833.0	−0.872505	−0.436252	−	0.899824i	$-0.643695\pi$
$907$	−23833.0	−0.872505	−0.436252	+	0.899824i	$0.643695\pi$
$908$	−8820.00	−0.322359
$909$	−30338.0	−1.10698
$910$	0	0
$911$	31824.0	1.15738	0.578692	−	0.815546i	$-0.303563\pi$
$911$	31824.0	1.15738	0.578692	+	0.815546i	$0.303563\pi$
$912$	−5488.00	−0.199261
$913$	5460.00	0.197919
$914$	16958.0	0.613699
$915$	20237.0	0.731163
$916$	1148.00	0.0414094
$917$	0	0
$918$	1470.00	0.0528510
$919$	−16819.0	−0.603708	−0.301854	−	0.953354i	$-0.597606\pi$
$919$	−16819.0	−0.603708	−0.301854	+	0.953354i	$0.597606\pi$
$920$	−26712.0	−0.957248
$921$	−51548.0	−1.84426
$922$	−18676.0	−0.667095
$923$	−6048.00	−0.215680
$924$	0	0
$925$	−16644.0	−0.591623
$926$	−8032.00	−0.285041
$927$	14938.0	0.529265
$928$	9280.00	0.328266
$929$	−1799.00	−0.0635342	−0.0317671	−	0.999495i	$-0.510113\pi$
$929$	−1799.00	−0.0635342	−0.0317671	+	0.999495i	$0.510113\pi$
$930$	−14406.0	−0.507948
$931$	0	0
$932$	−18348.0	−0.644859
$933$	69825.0	2.45013
$934$	11718.0	0.410519
$935$	735.000	0.0257081
$936$	−7392.00	−0.258136
$937$	−14154.0	−0.493480	−0.246740	−	0.969082i	$-0.579359\pi$
$937$	−14154.0	−0.493480	−0.246740	+	0.969082i	$0.579359\pi$
$938$	0	0
$939$	−33271.0	−1.15629
$940$	−14700.0	−0.510065
$941$	−12047.0	−0.417344	−0.208672	−	0.977986i	$-0.566914\pi$
$941$	−12047.0	−0.417344	−0.208672	+	0.977986i	$0.566914\pi$
$942$	9506.00	0.328792
$943$	55650.0	1.92175
$944$	−1680.00	−0.0579230
$945$	0	0
$946$	1240.00	0.0426172
$947$	−24379.0	−0.836548	−0.418274	−	0.908321i	$-0.637365\pi$
$947$	−24379.0	−0.836548	−0.418274	+	0.908321i	$0.637365\pi$
$948$	−2884.00	−0.0988059
$949$	15582.0	0.532996
$950$	7448.00	0.254363
$951$	24339.0	0.829912
$952$	0	0
$953$	−52330.0	−1.77874	−0.889368	−	0.457192i	$-0.848855\pi$
$953$	−52330.0	−1.77874	−0.889368	+	0.457192i	$0.848855\pi$
$954$	13332.0	0.452452
$955$	−17899.0	−0.606490
$956$	−6672.00	−0.225720
$957$	2030.00	0.0685690
$958$	−13006.0	−0.438627
$959$	0	0
$960$	21952.0	0.738018
$961$	−8182.00	−0.274647
$962$	6132.00	0.205513
$963$	10054.0	0.336434
$964$	−13636.0	−0.455587
$965$	2779.00	0.0927038
$966$	0	0
$967$	−12416.0	−0.412897	−0.206449	−	0.978457i	$-0.566191\pi$
$967$	−12416.0	−0.412897	−0.206449	+	0.978457i	$0.566191\pi$
$968$	31344.0	1.04074
$969$	7203.00	0.238796
$970$	−12348.0	−0.408732
$971$	−36813.0	−1.21667	−0.608334	−	0.793681i	$-0.708162\pi$
$971$	−36813.0	−1.21667	−0.608334	+	0.793681i	$0.708162\pi$
$972$	−19712.0	−0.650476
$973$	0	0
$974$	−32098.0	−1.05594
$975$	7448.00	0.244643
$976$	−6608.00	−0.216718
$977$	34995.0	1.14595	0.572973	−	0.819574i	$-0.305790\pi$
$977$	34995.0	1.14595	0.572973	+	0.819574i	$0.305790\pi$
$978$	6538.00	0.213765
$979$	−1645.00	−0.0537022
$980$	0	0
$981$	−24750.0	−0.805511
$982$	17728.0	0.576093
$983$	14301.0	0.464019	0.232010	−	0.972713i	$-0.425470\pi$
$983$	14301.0	0.464019	0.232010	+	0.972713i	$0.425470\pi$
$984$	−58800.0	−1.90495
$985$	−20398.0	−0.659832
$986$	2436.00	0.0786796
$987$	0	0
$988$	2744.00	0.0883586
$989$	19716.0	0.633905
$990$	1540.00	0.0494388
$991$	−2665.00	−0.0854253	−0.0427127	−	0.999087i	$-0.513600\pi$
$991$	−2665.00	−0.0854253	−0.0427127	+	0.999087i	$0.513600\pi$
$992$	−23520.0	−0.752783
$993$	−23387.0	−0.747396
$994$	0	0
$995$	23373.0	0.744697
$996$	−30576.0	−0.972729
$997$	−24871.0	−0.790043	−0.395021	−	0.918672i	$-0.629263\pi$
$997$	−24871.0	−0.790043	−0.395021	+	0.918672i	$0.629263\pi$
$998$	−20422.0	−0.647743
$999$	7665.00	0.242753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.a.c.1.1		1
3.2	odd	2		441.4.a.e.1.1		1
4.3	odd	2		784.4.a.r.1.1		1
5.4	even	2		1225.4.a.d.1.1		1
7.2	even	3		49.4.c.a.18.1		2
7.3	odd	6		7.4.c.a.2.1	✓	2
7.4	even	3		49.4.c.a.30.1		2
7.5	odd	6		7.4.c.a.4.1	yes	2
7.6	odd	2		49.4.a.d.1.1		1
21.2	odd	6		441.4.e.k.361.1		2
21.5	even	6		63.4.e.b.46.1		2
21.11	odd	6		441.4.e.k.226.1		2
21.17	even	6		63.4.e.b.37.1		2
21.20	even	2		441.4.a.d.1.1		1
28.3	even	6		112.4.i.c.65.1		2
28.19	even	6		112.4.i.c.81.1		2
28.27	even	2		784.4.a.b.1.1		1
35.3	even	12		175.4.k.a.149.2		4
35.12	even	12		175.4.k.a.74.2		4
35.17	even	12		175.4.k.a.149.1		4
35.19	odd	6		175.4.e.a.151.1		2
35.24	odd	6		175.4.e.a.51.1		2
35.33	even	12		175.4.k.a.74.1		4
35.34	odd	2		1225.4.a.c.1.1		1
56.3	even	6		448.4.i.a.65.1		2
56.5	odd	6		448.4.i.f.193.1		2
56.19	even	6		448.4.i.a.193.1		2
56.45	odd	6		448.4.i.f.65.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.4.c.a.2.1	✓	2	7.3	odd	6
7.4.c.a.4.1	yes	2	7.5	odd	6
49.4.a.c.1.1		1	1.1	even	1	trivial
49.4.a.d.1.1		1	7.6	odd	2
49.4.c.a.18.1		2	7.2	even	3
49.4.c.a.30.1		2	7.4	even	3
63.4.e.b.37.1		2	21.17	even	6
63.4.e.b.46.1		2	21.5	even	6
112.4.i.c.65.1		2	28.3	even	6
112.4.i.c.81.1		2	28.19	even	6
175.4.e.a.51.1		2	35.24	odd	6
175.4.e.a.151.1		2	35.19	odd	6
175.4.k.a.74.1		4	35.33	even	12
175.4.k.a.74.2		4	35.12	even	12
175.4.k.a.149.1		4	35.17	even	12
175.4.k.a.149.2		4	35.3	even	12
441.4.a.d.1.1		1	21.20	even	2
441.4.a.e.1.1		1	3.2	odd	2
441.4.e.k.226.1		2	21.11	odd	6
441.4.e.k.361.1		2	21.2	odd	6
448.4.i.a.65.1		2	56.3	even	6
448.4.i.a.193.1		2	56.19	even	6
448.4.i.f.65.1		2	56.45	odd	6
448.4.i.f.193.1		2	56.5	odd	6
784.4.a.b.1.1		1	28.27	even	2
784.4.a.r.1.1		1	4.3	odd	2
1225.4.a.c.1.1		1	35.34	odd	2
1225.4.a.d.1.1		1	5.4	even	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:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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