LMFDB - Embedded newform 507.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +7.00000 q^{5} -3.00000 q^{6} -10.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +7.00000 q^{5} -3.00000 q^{6} -10.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} -7.00000 q^{10} -22.0000 q^{11} -21.0000 q^{12} +10.0000 q^{14} +21.0000 q^{15} +41.0000 q^{16} +37.0000 q^{17} -9.00000 q^{18} +30.0000 q^{19} -49.0000 q^{20} -30.0000 q^{21} +22.0000 q^{22} -162.000 q^{23} +45.0000 q^{24} -76.0000 q^{25} +27.0000 q^{27} +70.0000 q^{28} -113.000 q^{29} -21.0000 q^{30} +196.000 q^{31} -161.000 q^{32} -66.0000 q^{33} -37.0000 q^{34} -70.0000 q^{35} -63.0000 q^{36} +13.0000 q^{37} -30.0000 q^{38} +105.000 q^{40} +285.000 q^{41} +30.0000 q^{42} -246.000 q^{43} +154.000 q^{44} +63.0000 q^{45} +162.000 q^{46} -462.000 q^{47} +123.000 q^{48} -243.000 q^{49} +76.0000 q^{50} +111.000 q^{51} -537.000 q^{53} -27.0000 q^{54} -154.000 q^{55} -150.000 q^{56} +90.0000 q^{57} +113.000 q^{58} +576.000 q^{59} -147.000 q^{60} -635.000 q^{61} -196.000 q^{62} -90.0000 q^{63} -167.000 q^{64} +66.0000 q^{66} +202.000 q^{67} -259.000 q^{68} -486.000 q^{69} +70.0000 q^{70} -1086.00 q^{71} +135.000 q^{72} -805.000 q^{73} -13.0000 q^{74} -228.000 q^{75} -210.000 q^{76} +220.000 q^{77} +884.000 q^{79} +287.000 q^{80} +81.0000 q^{81} -285.000 q^{82} +518.000 q^{83} +210.000 q^{84} +259.000 q^{85} +246.000 q^{86} -339.000 q^{87} -330.000 q^{88} +194.000 q^{89} -63.0000 q^{90} +1134.00 q^{92} +588.000 q^{93} +462.000 q^{94} +210.000 q^{95} -483.000 q^{96} -1202.00 q^{97} +243.000 q^{98} -198.000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.00000	−0.875000
$5$	7.00000	0.626099	0.313050	−	0.949737i	$-0.398649\pi$
$5$	7.00000	0.626099	0.313050	+	0.949737i	$0.398649\pi$
$6$	−3.00000	−0.204124
$7$	−10.0000	−0.539949	−0.269975	−	0.962867i	$-0.587015\pi$
$7$	−10.0000	−0.539949	−0.269975	+	0.962867i	$0.587015\pi$
$8$	15.0000	0.662913
$9$	9.00000	0.333333
$10$	−7.00000	−0.221359
$11$	−22.0000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−22.0000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−21.0000	−0.505181
$13$	0	0
$13$	0	0
$14$	10.0000	0.190901
$15$	21.0000	0.361478
$16$	41.0000	0.640625
$17$	37.0000	0.527872	0.263936	−	0.964540i	$-0.414979\pi$
$17$	37.0000	0.527872	0.263936	+	0.964540i	$0.414979\pi$
$18$	−9.00000	−0.117851
$19$	30.0000	0.362235	0.181118	−	0.983461i	$-0.442029\pi$
$19$	30.0000	0.362235	0.181118	+	0.983461i	$0.442029\pi$
$20$	−49.0000	−0.547837
$21$	−30.0000	−0.311740
$22$	22.0000	0.213201
$23$	−162.000	−1.46867	−0.734333	−	0.678789i	$-0.762505\pi$
$23$	−162.000	−1.46867	−0.734333	+	0.678789i	$0.762505\pi$
$24$	45.0000	0.382733
$25$	−76.0000	−0.608000
$26$	0	0
$27$	27.0000	0.192450
$28$	70.0000	0.472456
$29$	−113.000	−0.723571	−0.361786	−	0.932261i	$-0.617833\pi$
$29$	−113.000	−0.723571	−0.361786	+	0.932261i	$0.617833\pi$
$30$	−21.0000	−0.127802
$31$	196.000	1.13557	0.567785	−	0.823177i	$-0.307801\pi$
$31$	196.000	1.13557	0.567785	+	0.823177i	$0.307801\pi$
$32$	−161.000	−0.889408
$33$	−66.0000	−0.348155
$34$	−37.0000	−0.186631
$35$	−70.0000	−0.338062
$36$	−63.0000	−0.291667
$37$	13.0000	0.0577618	0.0288809	−	0.999583i	$-0.490806\pi$
$37$	13.0000	0.0577618	0.0288809	+	0.999583i	$0.490806\pi$
$38$	−30.0000	−0.128070
$39$	0	0
$40$	105.000	0.415049
$41$	285.000	1.08560	0.542799	−	0.839863i	$-0.317365\pi$
$41$	285.000	1.08560	0.542799	+	0.839863i	$0.317365\pi$
$42$	30.0000	0.110217
$43$	−246.000	−0.872434	−0.436217	−	0.899842i	$-0.643682\pi$
$43$	−246.000	−0.872434	−0.436217	+	0.899842i	$0.643682\pi$
$44$	154.000	0.527645
$45$	63.0000	0.208700
$46$	162.000	0.519252
$47$	−462.000	−1.43382	−0.716911	−	0.697165i	$-0.754445\pi$
$47$	−462.000	−1.43382	−0.716911	+	0.697165i	$0.754445\pi$
$48$	123.000	0.369865
$49$	−243.000	−0.708455
$50$	76.0000	0.214960
$51$	111.000	0.304767
$52$	0	0
$53$	−537.000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$53$	−537.000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$54$	−27.0000	−0.0680414
$55$	−154.000	−0.377552
$56$	−150.000	−0.357939
$57$	90.0000	0.209137
$58$	113.000	0.255821
$59$	576.000	1.27100	0.635498	−	0.772102i	$-0.280795\pi$
$59$	576.000	1.27100	0.635498	+	0.772102i	$0.280795\pi$
$60$	−147.000	−0.316294
$61$	−635.000	−1.33284	−0.666421	−	0.745575i	$-0.732175\pi$
$61$	−635.000	−1.33284	−0.666421	+	0.745575i	$0.732175\pi$
$62$	−196.000	−0.401484
$63$	−90.0000	−0.179983
$64$	−167.000	−0.326172
$65$	0	0
$66$	66.0000	0.123091
$67$	202.000	0.368332	0.184166	−	0.982895i	$-0.441042\pi$
$67$	202.000	0.368332	0.184166	+	0.982895i	$0.441042\pi$
$68$	−259.000	−0.461888
$69$	−486.000	−0.847935
$70$	70.0000	0.119523
$71$	−1086.00	−1.81527	−0.907637	−	0.419755i	$-0.862116\pi$
$71$	−1086.00	−1.81527	−0.907637	+	0.419755i	$0.862116\pi$
$72$	135.000	0.220971
$73$	−805.000	−1.29066	−0.645330	−	0.763904i	$-0.723280\pi$
$73$	−805.000	−1.29066	−0.645330	+	0.763904i	$0.723280\pi$
$74$	−13.0000	−0.0204219
$75$	−228.000	−0.351029
$76$	−210.000	−0.316956
$77$	220.000	0.325602
$78$	0	0
$79$	884.000	1.25896	0.629480	−	0.777017i	$-0.283268\pi$
$79$	884.000	1.25896	0.629480	+	0.777017i	$0.283268\pi$
$80$	287.000	0.401095
$81$	81.0000	0.111111
$82$	−285.000	−0.383817
$83$	518.000	0.685035	0.342517	−	0.939511i	$-0.388720\pi$
$83$	518.000	0.685035	0.342517	+	0.939511i	$0.388720\pi$
$84$	210.000	0.272772
$85$	259.000	0.330500
$86$	246.000	0.308452
$87$	−339.000	−0.417754
$88$	−330.000	−0.399751
$89$	194.000	0.231056	0.115528	−	0.993304i	$-0.463144\pi$
$89$	194.000	0.231056	0.115528	+	0.993304i	$0.463144\pi$
$90$	−63.0000	−0.0737865
$91$	0	0
$92$	1134.00	1.28508
$93$	588.000	0.655621
$94$	462.000	0.506933
$95$	210.000	0.226795
$96$	−483.000	−0.513500
$97$	−1202.00	−1.25819	−0.629096	−	0.777328i	$-0.716575\pi$
$97$	−1202.00	−1.25819	−0.629096	+	0.777328i	$0.716575\pi$
$98$	243.000	0.250477
$99$	−198.000	−0.201008
$100$	532.000	0.532000
$101$	−429.000	−0.422645	−0.211322	−	0.977416i	$-0.567777\pi$
$101$	−429.000	−0.422645	−0.211322	+	0.977416i	$0.567777\pi$
$102$	−111.000	−0.107751
$103$	−1302.00	−1.24553	−0.622766	−	0.782408i	$-0.713991\pi$
$103$	−1302.00	−1.24553	−0.622766	+	0.782408i	$0.713991\pi$
$104$	0	0
$105$	−210.000	−0.195180
$106$	537.000	0.492057
$107$	−1338.00	−1.20887	−0.604436	−	0.796654i	$-0.706602\pi$
$107$	−1338.00	−1.20887	−0.604436	+	0.796654i	$0.706602\pi$
$108$	−189.000	−0.168394
$109$	−1034.00	−0.908617	−0.454308	−	0.890844i	$-0.650114\pi$
$109$	−1034.00	−0.908617	−0.454308	+	0.890844i	$0.650114\pi$
$110$	154.000	0.133485
$111$	39.0000	0.0333488
$112$	−410.000	−0.345905
$113$	1077.00	0.896599	0.448299	−	0.893884i	$-0.352030\pi$
$113$	1077.00	0.896599	0.448299	+	0.893884i	$0.352030\pi$
$114$	−90.0000	−0.0739410
$115$	−1134.00	−0.919531
$116$	791.000	0.633125
$117$	0	0
$118$	−576.000	−0.449365
$119$	−370.000	−0.285024
$120$	315.000	0.239629
$121$	−847.000	−0.636364
$122$	635.000	0.471231
$123$	855.000	0.626770
$124$	−1372.00	−0.993623
$125$	−1407.00	−1.00677
$126$	90.0000	0.0636336
$127$	−988.000	−0.690321	−0.345161	−	0.938544i	$-0.612176\pi$
$127$	−988.000	−0.690321	−0.345161	+	0.938544i	$0.612176\pi$
$128$	1455.00	1.00473
$129$	−738.000	−0.503700
$130$	0	0
$131$	560.000	0.373492	0.186746	−	0.982408i	$-0.440206\pi$
$131$	560.000	0.373492	0.186746	+	0.982408i	$0.440206\pi$
$132$	462.000	0.304636
$133$	−300.000	−0.195589
$134$	−202.000	−0.130225
$135$	189.000	0.120493
$136$	555.000	0.349933
$137$	−519.000	−0.323658	−0.161829	−	0.986819i	$-0.551739\pi$
$137$	−519.000	−0.323658	−0.161829	+	0.986819i	$0.551739\pi$
$138$	486.000	0.299790
$139$	−348.000	−0.212352	−0.106176	−	0.994347i	$-0.533861\pi$
$139$	−348.000	−0.212352	−0.106176	+	0.994347i	$0.533861\pi$
$140$	490.000	0.295804
$141$	−1386.00	−0.827817
$142$	1086.00	0.641796
$143$	0	0
$144$	369.000	0.213542
$145$	−791.000	−0.453027
$146$	805.000	0.456317
$147$	−729.000	−0.409027
$148$	−91.0000	−0.0505416
$149$	−645.000	−0.354634	−0.177317	−	0.984154i	$-0.556742\pi$
$149$	−645.000	−0.354634	−0.177317	+	0.984154i	$0.556742\pi$
$150$	228.000	0.124107
$151$	2914.00	1.57045	0.785225	−	0.619211i	$-0.212547\pi$
$151$	2914.00	1.57045	0.785225	+	0.619211i	$0.212547\pi$
$152$	450.000	0.240130
$153$	333.000	0.175957
$154$	−220.000	−0.115118
$155$	1372.00	0.710979
$156$	0	0
$157$	−2079.00	−1.05683	−0.528415	−	0.848986i	$-0.677213\pi$
$157$	−2079.00	−1.05683	−0.528415	+	0.848986i	$0.677213\pi$
$158$	−884.000	−0.445109
$159$	−1611.00	−0.803526
$160$	−1127.00	−0.556857
$161$	1620.00	0.793006
$162$	−81.0000	−0.0392837
$163$	1700.00	0.816897	0.408449	−	0.912781i	$-0.366070\pi$
$163$	1700.00	0.816897	0.408449	+	0.912781i	$0.366070\pi$
$164$	−1995.00	−0.949898
$165$	−462.000	−0.217980
$166$	−518.000	−0.242196
$167$	3680.00	1.70519	0.852596	−	0.522571i	$-0.175027\pi$
$167$	3680.00	1.70519	0.852596	+	0.522571i	$0.175027\pi$
$168$	−450.000	−0.206656
$169$	0	0
$170$	−259.000	−0.116849
$171$	270.000	0.120745
$172$	1722.00	0.763379
$173$	4146.00	1.82205	0.911025	−	0.412352i	$-0.135293\pi$
$173$	4146.00	1.82205	0.911025	+	0.412352i	$0.135293\pi$
$174$	339.000	0.147698
$175$	760.000	0.328289
$176$	−902.000	−0.386311
$177$	1728.00	0.733810
$178$	−194.000	−0.0816905
$179$	3674.00	1.53412	0.767060	−	0.641575i	$-0.221719\pi$
$179$	3674.00	1.53412	0.767060	+	0.641575i	$0.221719\pi$
$180$	−441.000	−0.182612
$181$	−3283.00	−1.34820	−0.674098	−	0.738642i	$-0.735467\pi$
$181$	−3283.00	−1.34820	−0.674098	+	0.738642i	$0.735467\pi$
$182$	0	0
$183$	−1905.00	−0.769517
$184$	−2430.00	−0.973598
$185$	91.0000	0.0361646
$186$	−588.000	−0.231797
$187$	−814.000	−0.318319
$188$	3234.00	1.25459
$189$	−270.000	−0.103913
$190$	−210.000	−0.0801842
$191$	−596.000	−0.225786	−0.112893	−	0.993607i	$-0.536012\pi$
$191$	−596.000	−0.225786	−0.112893	+	0.993607i	$0.536012\pi$
$192$	−501.000	−0.188315
$193$	−393.000	−0.146574	−0.0732869	−	0.997311i	$-0.523349\pi$
$193$	−393.000	−0.146574	−0.0732869	+	0.997311i	$0.523349\pi$
$194$	1202.00	0.444838
$195$	0	0
$196$	1701.00	0.619898
$197$	−3522.00	−1.27377	−0.636884	−	0.770960i	$-0.719777\pi$
$197$	−3522.00	−1.27377	−0.636884	+	0.770960i	$0.719777\pi$
$198$	198.000	0.0710669
$199$	2018.00	0.718855	0.359428	−	0.933173i	$-0.382972\pi$
$199$	2018.00	0.718855	0.359428	+	0.933173i	$0.382972\pi$
$200$	−1140.00	−0.403051
$201$	606.000	0.212656
$202$	429.000	0.149427
$203$	1130.00	0.390692
$204$	−777.000	−0.266671
$205$	1995.00	0.679692
$206$	1302.00	0.440362
$207$	−1458.00	−0.489556
$208$	0	0
$209$	−660.000	−0.218436
$210$	210.000	0.0690066
$211$	160.000	0.0522031	0.0261016	−	0.999659i	$-0.491691\pi$
$211$	160.000	0.0522031	0.0261016	+	0.999659i	$0.491691\pi$
$212$	3759.00	1.21778
$213$	−3258.00	−1.04805
$214$	1338.00	0.427401
$215$	−1722.00	−0.546230
$216$	405.000	0.127578
$217$	−1960.00	−0.613150
$218$	1034.00	0.321245
$219$	−2415.00	−0.745162
$220$	1078.00	0.330358
$221$	0	0
$222$	−39.0000	−0.0117906
$223$	4072.00	1.22279	0.611393	−	0.791327i	$-0.290609\pi$
$223$	4072.00	1.22279	0.611393	+	0.791327i	$0.290609\pi$
$224$	1610.00	0.480235
$225$	−684.000	−0.202667
$226$	−1077.00	−0.316995
$227$	−5794.00	−1.69410	−0.847051	−	0.531511i	$-0.821624\pi$
$227$	−5794.00	−1.69410	−0.847051	+	0.531511i	$0.821624\pi$
$228$	−630.000	−0.182995
$229$	6482.00	1.87049	0.935246	−	0.353999i	$-0.115178\pi$
$229$	6482.00	1.87049	0.935246	+	0.353999i	$0.115178\pi$
$230$	1134.00	0.325103
$231$	660.000	0.187986
$232$	−1695.00	−0.479665
$233$	6890.00	1.93725	0.968624	−	0.248530i	$-0.0799474\pi$
$233$	6890.00	1.93725	0.968624	+	0.248530i	$0.0799474\pi$
$234$	0	0
$235$	−3234.00	−0.897714
$236$	−4032.00	−1.11212
$237$	2652.00	0.726860
$238$	370.000	0.100771
$239$	2466.00	0.667415	0.333708	−	0.942677i	$-0.391700\pi$
$239$	2466.00	0.667415	0.333708	+	0.942677i	$0.391700\pi$
$240$	861.000	0.231572
$241$	−3617.00	−0.966770	−0.483385	−	0.875408i	$-0.660593\pi$
$241$	−3617.00	−0.966770	−0.483385	+	0.875408i	$0.660593\pi$
$242$	847.000	0.224989
$243$	243.000	0.0641500
$244$	4445.00	1.16624
$245$	−1701.00	−0.443563
$246$	−855.000	−0.221597
$247$	0	0
$248$	2940.00	0.752783
$249$	1554.00	0.395505
$250$	1407.00	0.355946
$251$	4860.00	1.22215	0.611077	−	0.791571i	$-0.290737\pi$
$251$	4860.00	1.22215	0.611077	+	0.791571i	$0.290737\pi$
$252$	630.000	0.157485
$253$	3564.00	0.885639
$254$	988.000	0.244065
$255$	777.000	0.190814
$256$	−119.000	−0.0290527
$257$	565.000	0.137135	0.0685676	−	0.997646i	$-0.478157\pi$
$257$	565.000	0.137135	0.0685676	+	0.997646i	$0.478157\pi$
$258$	738.000	0.178085
$259$	−130.000	−0.0311884
$260$	0	0
$261$	−1017.00	−0.241190
$262$	−560.000	−0.132049
$263$	−498.000	−0.116760	−0.0583802	−	0.998294i	$-0.518594\pi$
$263$	−498.000	−0.116760	−0.0583802	+	0.998294i	$0.518594\pi$
$264$	−990.000	−0.230797
$265$	−3759.00	−0.871372
$266$	300.000	0.0691511
$267$	582.000	0.133400
$268$	−1414.00	−0.322290
$269$	5546.00	1.25705	0.628523	−	0.777791i	$-0.283660\pi$
$269$	5546.00	1.25705	0.628523	+	0.777791i	$0.283660\pi$
$270$	−189.000	−0.0426006
$271$	−2256.00	−0.505691	−0.252845	−	0.967507i	$-0.581366\pi$
$271$	−2256.00	−0.505691	−0.252845	+	0.967507i	$0.581366\pi$
$272$	1517.00	0.338168
$273$	0	0
$274$	519.000	0.114430
$275$	1672.00	0.366638
$276$	3402.00	0.741943
$277$	2309.00	0.500846	0.250423	−	0.968137i	$-0.419430\pi$
$277$	2309.00	0.500846	0.250423	+	0.968137i	$0.419430\pi$
$278$	348.000	0.0750779
$279$	1764.00	0.378523
$280$	−1050.00	−0.224105
$281$	5833.00	1.23832	0.619159	−	0.785265i	$-0.287473\pi$
$281$	5833.00	1.23832	0.619159	+	0.785265i	$0.287473\pi$
$282$	1386.00	0.292678
$283$	1650.00	0.346581	0.173290	−	0.984871i	$-0.444560\pi$
$283$	1650.00	0.346581	0.173290	+	0.984871i	$0.444560\pi$
$284$	7602.00	1.58837
$285$	630.000	0.130940
$286$	0	0
$287$	−2850.00	−0.586168
$288$	−1449.00	−0.296469
$289$	−3544.00	−0.721352
$290$	791.000	0.160169
$291$	−3606.00	−0.726417
$292$	5635.00	1.12933
$293$	2991.00	0.596369	0.298184	−	0.954508i	$-0.403619\pi$
$293$	2991.00	0.596369	0.298184	+	0.954508i	$0.403619\pi$
$294$	729.000	0.144613
$295$	4032.00	0.795770
$296$	195.000	0.0382910
$297$	−594.000	−0.116052
$298$	645.000	0.125382
$299$	0	0
$300$	1596.00	0.307150
$301$	2460.00	0.471070
$302$	−2914.00	−0.555238
$303$	−1287.00	−0.244014
$304$	1230.00	0.232057
$305$	−4445.00	−0.834492
$306$	−333.000	−0.0622103
$307$	−2422.00	−0.450263	−0.225132	−	0.974328i	$-0.572281\pi$
$307$	−2422.00	−0.450263	−0.225132	+	0.974328i	$0.572281\pi$
$308$	−1540.00	−0.284901
$309$	−3906.00	−0.719109
$310$	−1372.00	−0.251369
$311$	−3402.00	−0.620288	−0.310144	−	0.950690i	$-0.600377\pi$
$311$	−3402.00	−0.620288	−0.310144	+	0.950690i	$0.600377\pi$
$312$	0	0
$313$	2310.00	0.417153	0.208577	−	0.978006i	$-0.433117\pi$
$313$	2310.00	0.417153	0.208577	+	0.978006i	$0.433117\pi$
$314$	2079.00	0.373646
$315$	−630.000	−0.112687
$316$	−6188.00	−1.10159
$317$	−257.000	−0.0455349	−0.0227674	−	0.999741i	$-0.507248\pi$
$317$	−257.000	−0.0455349	−0.0227674	+	0.999741i	$0.507248\pi$
$318$	1611.00	0.284089
$319$	2486.00	0.436330
$320$	−1169.00	−0.204216
$321$	−4014.00	−0.697943
$322$	−1620.00	−0.280370
$323$	1110.00	0.191214
$324$	−567.000	−0.0972222
$325$	0	0
$326$	−1700.00	−0.288817
$327$	−3102.00	−0.524590
$328$	4275.00	0.719657
$329$	4620.00	0.774191
$330$	462.000	0.0770675
$331$	1028.00	0.170707	0.0853535	−	0.996351i	$-0.472798\pi$
$331$	1028.00	0.170707	0.0853535	+	0.996351i	$0.472798\pi$
$332$	−3626.00	−0.599405
$333$	117.000	0.0192539
$334$	−3680.00	−0.602876
$335$	1414.00	0.230612
$336$	−1230.00	−0.199708
$337$	2487.00	0.402005	0.201002	−	0.979591i	$-0.435580\pi$
$337$	2487.00	0.402005	0.201002	+	0.979591i	$0.435580\pi$
$338$	0	0
$339$	3231.00	0.517651
$340$	−1813.00	−0.289187
$341$	−4312.00	−0.684774
$342$	−270.000	−0.0426898
$343$	5860.00	0.922479
$344$	−3690.00	−0.578347
$345$	−3402.00	−0.530891
$346$	−4146.00	−0.644192
$347$	−2850.00	−0.440911	−0.220455	−	0.975397i	$-0.570754\pi$
$347$	−2850.00	−0.440911	−0.220455	+	0.975397i	$0.570754\pi$
$348$	2373.00	0.365535
$349$	−2018.00	−0.309516	−0.154758	−	0.987952i	$-0.549460\pi$
$349$	−2018.00	−0.309516	−0.154758	+	0.987952i	$0.549460\pi$
$350$	−760.000	−0.116068
$351$	0	0
$352$	3542.00	0.536333
$353$	−5287.00	−0.797163	−0.398582	−	0.917133i	$-0.630497\pi$
$353$	−5287.00	−0.797163	−0.398582	+	0.917133i	$0.630497\pi$
$354$	−1728.00	−0.259441
$355$	−7602.00	−1.13654
$356$	−1358.00	−0.202174
$357$	−1110.00	−0.164559
$358$	−3674.00	−0.542394
$359$	−7278.00	−1.06997	−0.534983	−	0.844863i	$-0.679682\pi$
$359$	−7278.00	−1.06997	−0.534983	+	0.844863i	$0.679682\pi$
$360$	945.000	0.138350
$361$	−5959.00	−0.868786
$362$	3283.00	0.476659
$363$	−2541.00	−0.367405
$364$	0	0
$365$	−5635.00	−0.808080
$366$	1905.00	0.272065
$367$	−4202.00	−0.597664	−0.298832	−	0.954306i	$-0.596597\pi$
$367$	−4202.00	−0.597664	−0.298832	+	0.954306i	$0.596597\pi$
$368$	−6642.00	−0.940865
$369$	2565.00	0.361866
$370$	−91.0000	−0.0127861
$371$	5370.00	0.751473
$372$	−4116.00	−0.573668
$373$	−1583.00	−0.219744	−0.109872	−	0.993946i	$-0.535044\pi$
$373$	−1583.00	−0.219744	−0.109872	+	0.993946i	$0.535044\pi$
$374$	814.000	0.112543
$375$	−4221.00	−0.581257
$376$	−6930.00	−0.950499
$377$	0	0
$378$	270.000	0.0367389
$379$	−2052.00	−0.278111	−0.139056	−	0.990285i	$-0.544407\pi$
$379$	−2052.00	−0.278111	−0.139056	+	0.990285i	$0.544407\pi$
$380$	−1470.00	−0.198446
$381$	−2964.00	−0.398557
$382$	596.000	0.0798273
$383$	−6872.00	−0.916822	−0.458411	−	0.888740i	$-0.651581\pi$
$383$	−6872.00	−0.916822	−0.458411	+	0.888740i	$0.651581\pi$
$384$	4365.00	0.580079
$385$	1540.00	0.203859
$386$	393.000	0.0518217
$387$	−2214.00	−0.290811
$388$	8414.00	1.10092
$389$	−11653.0	−1.51884	−0.759422	−	0.650598i	$-0.774518\pi$
$389$	−11653.0	−1.51884	−0.759422	+	0.650598i	$0.774518\pi$
$390$	0	0
$391$	−5994.00	−0.775268
$392$	−3645.00	−0.469644
$393$	1680.00	0.215636
$394$	3522.00	0.450345
$395$	6188.00	0.788233
$396$	1386.00	0.175882
$397$	6134.00	0.775458	0.387729	−	0.921774i	$-0.373260\pi$
$397$	6134.00	0.775458	0.387729	+	0.921774i	$0.373260\pi$
$398$	−2018.00	−0.254154
$399$	−900.000	−0.112923
$400$	−3116.00	−0.389500
$401$	−10795.0	−1.34433	−0.672165	−	0.740401i	$-0.734636\pi$
$401$	−10795.0	−1.34433	−0.672165	+	0.740401i	$0.734636\pi$
$402$	−606.000	−0.0751854
$403$	0	0
$404$	3003.00	0.369814
$405$	567.000	0.0695666
$406$	−1130.00	−0.138130
$407$	−286.000	−0.0348317
$408$	1665.00	0.202034
$409$	−8489.00	−1.02629	−0.513147	−	0.858301i	$-0.671520\pi$
$409$	−8489.00	−1.02629	−0.513147	+	0.858301i	$0.671520\pi$
$410$	−1995.00	−0.240307
$411$	−1557.00	−0.186864
$412$	9114.00	1.08984
$413$	−5760.00	−0.686274
$414$	1458.00	0.173084
$415$	3626.00	0.428900
$416$	0	0
$417$	−1044.00	−0.122602
$418$	660.000	0.0772288
$419$	1496.00	0.174426	0.0872129	−	0.996190i	$-0.472204\pi$
$419$	1496.00	0.174426	0.0872129	+	0.996190i	$0.472204\pi$
$420$	1470.00	0.170783
$421$	−11695.0	−1.35387	−0.676935	−	0.736043i	$-0.736692\pi$
$421$	−11695.0	−1.35387	−0.676935	+	0.736043i	$0.736692\pi$
$422$	−160.000	−0.0184566
$423$	−4158.00	−0.477941
$424$	−8055.00	−0.922607
$425$	−2812.00	−0.320946
$426$	3258.00	0.370541
$427$	6350.00	0.719668
$428$	9366.00	1.05776
$429$	0	0
$430$	1722.00	0.193121
$431$	10590.0	1.18353	0.591766	−	0.806110i	$-0.298431\pi$
$431$	10590.0	1.18353	0.591766	+	0.806110i	$0.298431\pi$
$432$	1107.00	0.123288
$433$	−13949.0	−1.54814	−0.774072	−	0.633098i	$-0.781783\pi$
$433$	−13949.0	−1.54814	−0.774072	+	0.633098i	$0.781783\pi$
$434$	1960.00	0.216781
$435$	−2373.00	−0.261555
$436$	7238.00	0.795040
$437$	−4860.00	−0.532003
$438$	2415.00	0.263455
$439$	−10726.0	−1.16611	−0.583057	−	0.812431i	$-0.698144\pi$
$439$	−10726.0	−1.16611	−0.583057	+	0.812431i	$0.698144\pi$
$440$	−2310.00	−0.250284
$441$	−2187.00	−0.236152
$442$	0	0
$443$	16228.0	1.74044	0.870221	−	0.492662i	$-0.163976\pi$
$443$	16228.0	1.74044	0.870221	+	0.492662i	$0.163976\pi$
$444$	−273.000	−0.0291802
$445$	1358.00	0.144664
$446$	−4072.00	−0.432320
$447$	−1935.00	−0.204748
$448$	1670.00	0.176116
$449$	7538.00	0.792294	0.396147	−	0.918187i	$-0.370347\pi$
$449$	7538.00	0.792294	0.396147	+	0.918187i	$0.370347\pi$
$450$	684.000	0.0716535
$451$	−6270.00	−0.654640
$452$	−7539.00	−0.784524
$453$	8742.00	0.906700
$454$	5794.00	0.598956
$455$	0	0
$456$	1350.00	0.138639
$457$	15539.0	1.59056	0.795278	−	0.606245i	$-0.207325\pi$
$457$	15539.0	1.59056	0.795278	+	0.606245i	$0.207325\pi$
$458$	−6482.00	−0.661319
$459$	999.000	0.101589
$460$	7938.00	0.804589
$461$	4811.00	0.486053	0.243027	−	0.970020i	$-0.421860\pi$
$461$	4811.00	0.486053	0.243027	+	0.970020i	$0.421860\pi$
$462$	−660.000	−0.0664632
$463$	562.000	0.0564111	0.0282056	−	0.999602i	$-0.491021\pi$
$463$	562.000	0.0564111	0.0282056	+	0.999602i	$0.491021\pi$
$464$	−4633.00	−0.463538
$465$	4116.00	0.410484
$466$	−6890.00	−0.684921
$467$	4914.00	0.486922	0.243461	−	0.969911i	$-0.421717\pi$
$467$	4914.00	0.486922	0.243461	+	0.969911i	$0.421717\pi$
$468$	0	0
$469$	−2020.00	−0.198880
$470$	3234.00	0.317390
$471$	−6237.00	−0.610161
$472$	8640.00	0.842560
$473$	5412.00	0.526097
$474$	−2652.00	−0.256984
$475$	−2280.00	−0.220239
$476$	2590.00	0.249396
$477$	−4833.00	−0.463916
$478$	−2466.00	−0.235967
$479$	−3600.00	−0.343399	−0.171700	−	0.985149i	$-0.554926\pi$
$479$	−3600.00	−0.343399	−0.171700	+	0.985149i	$0.554926\pi$
$480$	−3381.00	−0.321502
$481$	0	0
$482$	3617.00	0.341805
$483$	4860.00	0.457842
$484$	5929.00	0.556818
$485$	−8414.00	−0.787753
$486$	−243.000	−0.0226805
$487$	−17130.0	−1.59391	−0.796955	−	0.604038i	$-0.793557\pi$
$487$	−17130.0	−1.59391	−0.796955	+	0.604038i	$0.793557\pi$
$488$	−9525.00	−0.883558
$489$	5100.00	0.471636
$490$	1701.00	0.156823
$491$	−11838.0	−1.08807	−0.544034	−	0.839063i	$-0.683104\pi$
$491$	−11838.0	−1.08807	−0.544034	+	0.839063i	$0.683104\pi$
$492$	−5985.00	−0.548424
$493$	−4181.00	−0.381953
$494$	0	0
$495$	−1386.00	−0.125851
$496$	8036.00	0.727474
$497$	10860.0	0.980156
$498$	−1554.00	−0.139832
$499$	8976.00	0.805252	0.402626	−	0.915364i	$-0.368097\pi$
$499$	8976.00	0.805252	0.402626	+	0.915364i	$0.368097\pi$
$500$	9849.00	0.880921
$501$	11040.0	0.984493
$502$	−4860.00	−0.432096
$503$	1682.00	0.149099	0.0745494	−	0.997217i	$-0.476248\pi$
$503$	1682.00	0.149099	0.0745494	+	0.997217i	$0.476248\pi$
$504$	−1350.00	−0.119313
$505$	−3003.00	−0.264617
$506$	−3564.00	−0.313121
$507$	0	0
$508$	6916.00	0.604031
$509$	15167.0	1.32076	0.660379	−	0.750933i	$-0.270396\pi$
$509$	15167.0	1.32076	0.660379	+	0.750933i	$0.270396\pi$
$510$	−777.000	−0.0674630
$511$	8050.00	0.696890
$512$	−11521.0	−0.994455
$513$	810.000	0.0697122
$514$	−565.000	−0.0484846
$515$	−9114.00	−0.779827
$516$	5166.00	0.440737
$517$	10164.0	0.864627
$518$	130.000	0.0110268
$519$	12438.0	1.05196
$520$	0	0
$521$	−6783.00	−0.570381	−0.285191	−	0.958471i	$-0.592057\pi$
$521$	−6783.00	−0.570381	−0.285191	+	0.958471i	$0.592057\pi$
$522$	1017.00	0.0852737
$523$	−13918.0	−1.16366	−0.581828	−	0.813312i	$-0.697662\pi$
$523$	−13918.0	−1.16366	−0.581828	+	0.813312i	$0.697662\pi$
$524$	−3920.00	−0.326805
$525$	2280.00	0.189538
$526$	498.000	0.0412810
$527$	7252.00	0.599435
$528$	−2706.00	−0.223037
$529$	14077.0	1.15698
$530$	3759.00	0.308076
$531$	5184.00	0.423666
$532$	2100.00	0.171140
$533$	0	0
$534$	−582.000	−0.0471641
$535$	−9366.00	−0.756874
$536$	3030.00	0.244172
$537$	11022.0	0.885725
$538$	−5546.00	−0.444433
$539$	5346.00	0.427214
$540$	−1323.00	−0.105431
$541$	−1335.00	−0.106093	−0.0530463	−	0.998592i	$-0.516893\pi$
$541$	−1335.00	−0.106093	−0.0530463	+	0.998592i	$0.516893\pi$
$542$	2256.00	0.178789
$543$	−9849.00	−0.778381
$544$	−5957.00	−0.469493
$545$	−7238.00	−0.568884
$546$	0	0
$547$	−3806.00	−0.297501	−0.148750	−	0.988875i	$-0.547525\pi$
$547$	−3806.00	−0.297501	−0.148750	+	0.988875i	$0.547525\pi$
$548$	3633.00	0.283201
$549$	−5715.00	−0.444281
$550$	−1672.00	−0.129626
$551$	−3390.00	−0.262103
$552$	−7290.00	−0.562107
$553$	−8840.00	−0.679774
$554$	−2309.00	−0.177076
$555$	273.000	0.0208796
$556$	2436.00	0.185808
$557$	−1905.00	−0.144915	−0.0724573	−	0.997372i	$-0.523084\pi$
$557$	−1905.00	−0.144915	−0.0724573	+	0.997372i	$0.523084\pi$
$558$	−1764.00	−0.133828
$559$	0	0
$560$	−2870.00	−0.216571
$561$	−2442.00	−0.183781
$562$	−5833.00	−0.437812
$563$	−4800.00	−0.359318	−0.179659	−	0.983729i	$-0.557499\pi$
$563$	−4800.00	−0.359318	−0.179659	+	0.983729i	$0.557499\pi$
$564$	9702.00	0.724340
$565$	7539.00	0.561359
$566$	−1650.00	−0.122535
$567$	−810.000	−0.0599944
$568$	−16290.0	−1.20337
$569$	14678.0	1.08143	0.540715	−	0.841206i	$-0.318154\pi$
$569$	14678.0	1.08143	0.540715	+	0.841206i	$0.318154\pi$
$570$	−630.000	−0.0462944
$571$	−586.000	−0.0429481	−0.0214740	−	0.999769i	$-0.506836\pi$
$571$	−586.000	−0.0429481	−0.0214740	+	0.999769i	$0.506836\pi$
$572$	0	0
$573$	−1788.00	−0.130357
$574$	2850.00	0.207242
$575$	12312.0	0.892949
$576$	−1503.00	−0.108724
$577$	8939.00	0.644949	0.322474	−	0.946578i	$-0.395485\pi$
$577$	8939.00	0.644949	0.322474	+	0.946578i	$0.395485\pi$
$578$	3544.00	0.255036
$579$	−1179.00	−0.0846245
$580$	5537.00	0.396399
$581$	−5180.00	−0.369884
$582$	3606.00	0.256827
$583$	11814.0	0.839255
$584$	−12075.0	−0.855594
$585$	0	0
$586$	−2991.00	−0.210848
$587$	−13792.0	−0.969773	−0.484887	−	0.874577i	$-0.661139\pi$
$587$	−13792.0	−0.969773	−0.484887	+	0.874577i	$0.661139\pi$
$588$	5103.00	0.357898
$589$	5880.00	0.411343
$590$	−4032.00	−0.281347
$591$	−10566.0	−0.735410
$592$	533.000	0.0370037
$593$	9569.00	0.662650	0.331325	−	0.943517i	$-0.392504\pi$
$593$	9569.00	0.662650	0.331325	+	0.943517i	$0.392504\pi$
$594$	594.000	0.0410305
$595$	−2590.00	−0.178453
$596$	4515.00	0.310305
$597$	6054.00	0.415031
$598$	0	0
$599$	−5192.00	−0.354156	−0.177078	−	0.984197i	$-0.556664\pi$
$599$	−5192.00	−0.354156	−0.177078	+	0.984197i	$0.556664\pi$
$600$	−3420.00	−0.232702
$601$	−3677.00	−0.249564	−0.124782	−	0.992184i	$-0.539823\pi$
$601$	−3677.00	−0.249564	−0.124782	+	0.992184i	$0.539823\pi$
$602$	−2460.00	−0.166548
$603$	1818.00	0.122777
$604$	−20398.0	−1.37414
$605$	−5929.00	−0.398427
$606$	1287.00	0.0862719
$607$	−10960.0	−0.732871	−0.366435	−	0.930443i	$-0.619422\pi$
$607$	−10960.0	−0.732871	−0.366435	+	0.930443i	$0.619422\pi$
$608$	−4830.00	−0.322175
$609$	3390.00	0.225566
$610$	4445.00	0.295037
$611$	0	0
$612$	−2331.00	−0.153963
$613$	−26027.0	−1.71488	−0.857439	−	0.514585i	$-0.827946\pi$
$613$	−26027.0	−1.71488	−0.857439	+	0.514585i	$0.827946\pi$
$614$	2422.00	0.159192
$615$	5985.00	0.392420
$616$	3300.00	0.215845
$617$	17681.0	1.15366	0.576832	−	0.816863i	$-0.304289\pi$
$617$	17681.0	1.15366	0.576832	+	0.816863i	$0.304289\pi$
$618$	3906.00	0.254243
$619$	3192.00	0.207265	0.103633	−	0.994616i	$-0.466953\pi$
$619$	3192.00	0.207265	0.103633	+	0.994616i	$0.466953\pi$
$620$	−9604.00	−0.622106
$621$	−4374.00	−0.282645
$622$	3402.00	0.219305
$623$	−1940.00	−0.124758
$624$	0	0
$625$	−349.000	−0.0223360
$626$	−2310.00	−0.147486
$627$	−1980.00	−0.126114
$628$	14553.0	0.924726
$629$	481.000	0.0304908
$630$	630.000	0.0398410
$631$	7580.00	0.478217	0.239109	−	0.970993i	$-0.423145\pi$
$631$	7580.00	0.478217	0.239109	+	0.970993i	$0.423145\pi$
$632$	13260.0	0.834580
$633$	480.000	0.0301395
$634$	257.000	0.0160990
$635$	−6916.00	−0.432210
$636$	11277.0	0.703085
$637$	0	0
$638$	−2486.00	−0.154266
$639$	−9774.00	−0.605091
$640$	10185.0	0.629059
$641$	−27707.0	−1.70727	−0.853635	−	0.520871i	$-0.825607\pi$
$641$	−27707.0	−1.70727	−0.853635	+	0.520871i	$0.825607\pi$
$642$	4014.00	0.246760
$643$	11216.0	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$643$	11216.0	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$644$	−11340.0	−0.693880
$645$	−5166.00	−0.315366
$646$	−1110.00	−0.0676043
$647$	−2536.00	−0.154097	−0.0770483	−	0.997027i	$-0.524550\pi$
$647$	−2536.00	−0.154097	−0.0770483	+	0.997027i	$0.524550\pi$
$648$	1215.00	0.0736570
$649$	−12672.0	−0.766440
$650$	0	0
$651$	−5880.00	−0.354002
$652$	−11900.0	−0.714785
$653$	17730.0	1.06252	0.531262	−	0.847207i	$-0.321718\pi$
$653$	17730.0	1.06252	0.531262	+	0.847207i	$0.321718\pi$
$654$	3102.00	0.185471
$655$	3920.00	0.233843
$656$	11685.0	0.695461
$657$	−7245.00	−0.430220
$658$	−4620.00	−0.273718
$659$	18920.0	1.11839	0.559195	−	0.829036i	$-0.311110\pi$
$659$	18920.0	1.11839	0.559195	+	0.829036i	$0.311110\pi$
$660$	3234.00	0.190732
$661$	5241.00	0.308398	0.154199	−	0.988040i	$-0.450720\pi$
$661$	5241.00	0.308398	0.154199	+	0.988040i	$0.450720\pi$
$662$	−1028.00	−0.0603540
$663$	0	0
$664$	7770.00	0.454118
$665$	−2100.00	−0.122458
$666$	−117.000	−0.00680729
$667$	18306.0	1.06269
$668$	−25760.0	−1.49204
$669$	12216.0	0.705976
$670$	−1414.00	−0.0815337
$671$	13970.0	0.803735
$672$	4830.00	0.277264
$673$	20467.0	1.17228	0.586140	−	0.810210i	$-0.300647\pi$
$673$	20467.0	1.17228	0.586140	+	0.810210i	$0.300647\pi$
$674$	−2487.00	−0.142130
$675$	−2052.00	−0.117010
$676$	0	0
$677$	−70.0000	−0.00397388	−0.00198694	−	0.999998i	$-0.500632\pi$
$677$	−70.0000	−0.00397388	−0.00198694	+	0.999998i	$0.500632\pi$
$678$	−3231.00	−0.183017
$679$	12020.0	0.679360
$680$	3885.00	0.219093
$681$	−17382.0	−0.978091
$682$	4312.00	0.242104
$683$	6432.00	0.360342	0.180171	−	0.983635i	$-0.442335\pi$
$683$	6432.00	0.360342	0.180171	+	0.983635i	$0.442335\pi$
$684$	−1890.00	−0.105652
$685$	−3633.00	−0.202642
$686$	−5860.00	−0.326146
$687$	19446.0	1.07993
$688$	−10086.0	−0.558903
$689$	0	0
$690$	3402.00	0.187698
$691$	−6666.00	−0.366985	−0.183492	−	0.983021i	$-0.558740\pi$
$691$	−6666.00	−0.366985	−0.183492	+	0.983021i	$0.558740\pi$
$692$	−29022.0	−1.59429
$693$	1980.00	0.108534
$694$	2850.00	0.155885
$695$	−2436.00	−0.132954
$696$	−5085.00	−0.276935
$697$	10545.0	0.573056
$698$	2018.00	0.109430
$699$	20670.0	1.11847
$700$	−5320.00	−0.287253
$701$	−14054.0	−0.757221	−0.378611	−	0.925556i	$-0.623598\pi$
$701$	−14054.0	−0.757221	−0.378611	+	0.925556i	$0.623598\pi$
$702$	0	0
$703$	390.000	0.0209234
$704$	3674.00	0.196689
$705$	−9702.00	−0.518296
$706$	5287.00	0.281840
$707$	4290.00	0.228207
$708$	−12096.0	−0.642084
$709$	−71.0000	−0.00376088	−0.00188044	−	0.999998i	$-0.500599\pi$
$709$	−71.0000	−0.00376088	−0.00188044	+	0.999998i	$0.500599\pi$
$710$	7602.00	0.401828
$711$	7956.00	0.419653
$712$	2910.00	0.153170
$713$	−31752.0	−1.66777
$714$	1110.00	0.0581803
$715$	0	0
$716$	−25718.0	−1.34236
$717$	7398.00	0.385332
$718$	7278.00	0.378290
$719$	3936.00	0.204156	0.102078	−	0.994776i	$-0.467451\pi$
$719$	3936.00	0.204156	0.102078	+	0.994776i	$0.467451\pi$
$720$	2583.00	0.133698
$721$	13020.0	0.672524
$722$	5959.00	0.307162
$723$	−10851.0	−0.558165
$724$	22981.0	1.17967
$725$	8588.00	0.439931
$726$	2541.00	0.129897
$727$	34202.0	1.74482	0.872409	−	0.488777i	$-0.162557\pi$
$727$	34202.0	1.74482	0.872409	+	0.488777i	$0.162557\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	5635.00	0.285700
$731$	−9102.00	−0.460533
$732$	13335.0	0.673328
$733$	−27363.0	−1.37882	−0.689410	−	0.724371i	$-0.742130\pi$
$733$	−27363.0	−1.37882	−0.689410	+	0.724371i	$0.742130\pi$
$734$	4202.00	0.211306
$735$	−5103.00	−0.256091
$736$	26082.0	1.30624
$737$	−4444.00	−0.222112
$738$	−2565.00	−0.127939
$739$	21776.0	1.08396	0.541978	−	0.840393i	$-0.317676\pi$
$739$	21776.0	1.08396	0.541978	+	0.840393i	$0.317676\pi$
$740$	−637.000	−0.0316440
$741$	0	0
$742$	−5370.00	−0.265686
$743$	2484.00	0.122650	0.0613251	−	0.998118i	$-0.480467\pi$
$743$	2484.00	0.122650	0.0613251	+	0.998118i	$0.480467\pi$
$744$	8820.00	0.434619
$745$	−4515.00	−0.222036
$746$	1583.00	0.0776914
$747$	4662.00	0.228345
$748$	5698.00	0.278529
$749$	13380.0	0.652730
$750$	4221.00	0.205506
$751$	32906.0	1.59888	0.799439	−	0.600748i	$-0.205130\pi$
$751$	32906.0	1.59888	0.799439	+	0.600748i	$0.205130\pi$
$752$	−18942.0	−0.918542
$753$	14580.0	0.705611
$754$	0	0
$755$	20398.0	0.983257
$756$	1890.00	0.0909241
$757$	−3914.00	−0.187922	−0.0939609	−	0.995576i	$-0.529953\pi$
$757$	−3914.00	−0.187922	−0.0939609	+	0.995576i	$0.529953\pi$
$758$	2052.00	0.0983272
$759$	10692.0	0.511324
$760$	3150.00	0.150345
$761$	−33038.0	−1.57375	−0.786877	−	0.617110i	$-0.788303\pi$
$761$	−33038.0	−1.57375	−0.786877	+	0.617110i	$0.788303\pi$
$762$	2964.00	0.140911
$763$	10340.0	0.490607
$764$	4172.00	0.197562
$765$	2331.00	0.110167
$766$	6872.00	0.324145
$767$	0	0
$768$	−357.000	−0.0167736
$769$	−17586.0	−0.824665	−0.412332	−	0.911033i	$-0.635286\pi$
$769$	−17586.0	−0.824665	−0.412332	+	0.911033i	$0.635286\pi$
$770$	−1540.00	−0.0720750
$771$	1695.00	0.0791750
$772$	2751.00	0.128252
$773$	18314.0	0.852146	0.426073	−	0.904689i	$-0.359897\pi$
$773$	18314.0	0.852146	0.426073	+	0.904689i	$0.359897\pi$
$774$	2214.00	0.102817
$775$	−14896.0	−0.690426
$776$	−18030.0	−0.834071
$777$	−390.000	−0.0180067
$778$	11653.0	0.536993
$779$	8550.00	0.393242
$780$	0	0
$781$	23892.0	1.09465
$782$	5994.00	0.274098
$783$	−3051.00	−0.139251
$784$	−9963.00	−0.453854
$785$	−14553.0	−0.661680
$786$	−1680.00	−0.0762387
$787$	−42068.0	−1.90542	−0.952708	−	0.303888i	$-0.901715\pi$
$787$	−42068.0	−1.90542	−0.952708	+	0.303888i	$0.901715\pi$
$788$	24654.0	1.11455
$789$	−1494.00	−0.0674117
$790$	−6188.00	−0.278682
$791$	−10770.0	−0.484118
$792$	−2970.00	−0.133250
$793$	0	0
$794$	−6134.00	−0.274166
$795$	−11277.0	−0.503087
$796$	−14126.0	−0.628998
$797$	−4282.00	−0.190309	−0.0951545	−	0.995463i	$-0.530335\pi$
$797$	−4282.00	−0.190309	−0.0951545	+	0.995463i	$0.530335\pi$
$798$	900.000	0.0399244
$799$	−17094.0	−0.756874
$800$	12236.0	0.540760
$801$	1746.00	0.0770186
$802$	10795.0	0.475293
$803$	17710.0	0.778297
$804$	−4242.00	−0.186074
$805$	11340.0	0.496500
$806$	0	0
$807$	16638.0	0.725756
$808$	−6435.00	−0.280176
$809$	40221.0	1.74795	0.873977	−	0.485967i	$-0.161532\pi$
$809$	40221.0	1.74795	0.873977	+	0.485967i	$0.161532\pi$
$810$	−567.000	−0.0245955
$811$	−7084.00	−0.306724	−0.153362	−	0.988170i	$-0.549010\pi$
$811$	−7084.00	−0.306724	−0.153362	+	0.988170i	$0.549010\pi$
$812$	−7910.00	−0.341855
$813$	−6768.00	−0.291961
$814$	286.000	0.0123149
$815$	11900.0	0.511459
$816$	4551.00	0.195241
$817$	−7380.00	−0.316026
$818$	8489.00	0.362850
$819$	0	0
$820$	−13965.0	−0.594730
$821$	17338.0	0.737028	0.368514	−	0.929622i	$-0.379867\pi$
$821$	17338.0	0.737028	0.368514	+	0.929622i	$0.379867\pi$
$822$	1557.00	0.0660664
$823$	35496.0	1.50342	0.751709	−	0.659495i	$-0.229230\pi$
$823$	35496.0	1.50342	0.751709	+	0.659495i	$0.229230\pi$
$824$	−19530.0	−0.825679
$825$	5016.00	0.211678
$826$	5760.00	0.242634
$827$	−14992.0	−0.630378	−0.315189	−	0.949029i	$-0.602068\pi$
$827$	−14992.0	−0.630378	−0.315189	+	0.949029i	$0.602068\pi$
$828$	10206.0	0.428361
$829$	−20659.0	−0.865521	−0.432760	−	0.901509i	$-0.642460\pi$
$829$	−20659.0	−0.865521	−0.432760	+	0.901509i	$0.642460\pi$
$830$	−3626.00	−0.151639
$831$	6927.00	0.289164
$832$	0	0
$833$	−8991.00	−0.373973
$834$	1044.00	0.0433462
$835$	25760.0	1.06762
$836$	4620.00	0.191132
$837$	5292.00	0.218540
$838$	−1496.00	−0.0616688
$839$	28716.0	1.18163	0.590814	−	0.806808i	$-0.298807\pi$
$839$	28716.0	1.18163	0.590814	+	0.806808i	$0.298807\pi$
$840$	−3150.00	−0.129387
$841$	−11620.0	−0.476444
$842$	11695.0	0.478665
$843$	17499.0	0.714944
$844$	−1120.00	−0.0456777
$845$	0	0
$846$	4158.00	0.168978
$847$	8470.00	0.343604
$848$	−22017.0	−0.891588
$849$	4950.00	0.200098
$850$	2812.00	0.113472
$851$	−2106.00	−0.0848328
$852$	22806.0	0.917043
$853$	13377.0	0.536952	0.268476	−	0.963286i	$-0.413480\pi$
$853$	13377.0	0.536952	0.268476	+	0.963286i	$0.413480\pi$
$854$	−6350.00	−0.254441
$855$	1890.00	0.0755984
$856$	−20070.0	−0.801377
$857$	−27419.0	−1.09290	−0.546450	−	0.837492i	$-0.684021\pi$
$857$	−27419.0	−1.09290	−0.546450	+	0.837492i	$0.684021\pi$
$858$	0	0
$859$	2422.00	0.0962021	0.0481010	−	0.998842i	$-0.484683\pi$
$859$	2422.00	0.0962021	0.0481010	+	0.998842i	$0.484683\pi$
$860$	12054.0	0.477951
$861$	−8550.00	−0.338424
$862$	−10590.0	−0.418442
$863$	−34522.0	−1.36169	−0.680847	−	0.732425i	$-0.738388\pi$
$863$	−34522.0	−1.36169	−0.680847	+	0.732425i	$0.738388\pi$
$864$	−4347.00	−0.171167
$865$	29022.0	1.14078
$866$	13949.0	0.547351
$867$	−10632.0	−0.416472
$868$	13720.0	0.536506
$869$	−19448.0	−0.759181
$870$	2373.00	0.0924738
$871$	0	0
$872$	−15510.0	−0.602334
$873$	−10818.0	−0.419397
$874$	4860.00	0.188091
$875$	14070.0	0.543603
$876$	16905.0	0.652017
$877$	13733.0	0.528769	0.264385	−	0.964417i	$-0.414831\pi$
$877$	13733.0	0.528769	0.264385	+	0.964417i	$0.414831\pi$
$878$	10726.0	0.412284
$879$	8973.00	0.344314
$880$	−6314.00	−0.241869
$881$	−22759.0	−0.870341	−0.435170	−	0.900348i	$-0.643312\pi$
$881$	−22759.0	−0.870341	−0.435170	+	0.900348i	$0.643312\pi$
$882$	2187.00	0.0834922
$883$	−2168.00	−0.0826263	−0.0413131	−	0.999146i	$-0.513154\pi$
$883$	−2168.00	−0.0826263	−0.0413131	+	0.999146i	$0.513154\pi$
$884$	0	0
$885$	12096.0	0.459438
$886$	−16228.0	−0.615339
$887$	15888.0	0.601428	0.300714	−	0.953714i	$-0.402775\pi$
$887$	15888.0	0.601428	0.300714	+	0.953714i	$0.402775\pi$
$888$	585.000	0.0221073
$889$	9880.00	0.372739
$890$	−1358.00	−0.0511464
$891$	−1782.00	−0.0670025
$892$	−28504.0	−1.06994
$893$	−13860.0	−0.519381
$894$	1935.00	0.0723894
$895$	25718.0	0.960512
$896$	−14550.0	−0.542502
$897$	0	0
$898$	−7538.00	−0.280118
$899$	−22148.0	−0.821665
$900$	4788.00	0.177333
$901$	−19869.0	−0.734664
$902$	6270.00	0.231450
$903$	7380.00	0.271972
$904$	16155.0	0.594366
$905$	−22981.0	−0.844104
$906$	−8742.00	−0.320567
$907$	−11628.0	−0.425691	−0.212845	−	0.977086i	$-0.568273\pi$
$907$	−11628.0	−0.425691	−0.212845	+	0.977086i	$0.568273\pi$
$908$	40558.0	1.48234
$909$	−3861.00	−0.140882
$910$	0	0
$911$	−12584.0	−0.457658	−0.228829	−	0.973467i	$-0.573490\pi$
$911$	−12584.0	−0.457658	−0.228829	+	0.973467i	$0.573490\pi$
$912$	3690.00	0.133978
$913$	−11396.0	−0.413092
$914$	−15539.0	−0.562346
$915$	−13335.0	−0.481794
$916$	−45374.0	−1.63668
$917$	−5600.00	−0.201667
$918$	−999.000	−0.0359171
$919$	17184.0	0.616809	0.308405	−	0.951255i	$-0.400205\pi$
$919$	17184.0	0.616809	0.308405	+	0.951255i	$0.400205\pi$
$920$	−17010.0	−0.609569
$921$	−7266.00	−0.259960
$922$	−4811.00	−0.171846
$923$	0	0
$924$	−4620.00	−0.164488
$925$	−988.000	−0.0351192
$926$	−562.000	−0.0199443
$927$	−11718.0	−0.415178
$928$	18193.0	0.643550
$929$	12777.0	0.451238	0.225619	−	0.974216i	$-0.427560\pi$
$929$	12777.0	0.451238	0.225619	+	0.974216i	$0.427560\pi$
$930$	−4116.00	−0.145128
$931$	−7290.00	−0.256627
$932$	−48230.0	−1.69509
$933$	−10206.0	−0.358124
$934$	−4914.00	−0.172153
$935$	−5698.00	−0.199299
$936$	0	0
$937$	9191.00	0.320445	0.160222	−	0.987081i	$-0.448779\pi$
$937$	9191.00	0.320445	0.160222	+	0.987081i	$0.448779\pi$
$938$	2020.00	0.0703149
$939$	6930.00	0.240843
$940$	22638.0	0.785500
$941$	50498.0	1.74940	0.874701	−	0.484662i	$-0.161058\pi$
$941$	50498.0	1.74940	0.874701	+	0.484662i	$0.161058\pi$
$942$	6237.00	0.215724
$943$	−46170.0	−1.59438
$944$	23616.0	0.814232
$945$	−1890.00	−0.0650600
$946$	−5412.00	−0.186003
$947$	1560.00	0.0535303	0.0267651	−	0.999642i	$-0.491479\pi$
$947$	1560.00	0.0535303	0.0267651	+	0.999642i	$0.491479\pi$
$948$	−18564.0	−0.636003
$949$	0	0
$950$	2280.00	0.0778663
$951$	−771.000	−0.0262896
$952$	−5550.00	−0.188946
$953$	−21498.0	−0.730733	−0.365366	−	0.930864i	$-0.619056\pi$
$953$	−21498.0	−0.730733	−0.365366	+	0.930864i	$0.619056\pi$
$954$	4833.00	0.164019
$955$	−4172.00	−0.141364
$956$	−17262.0	−0.583988
$957$	7458.00	0.251915
$958$	3600.00	0.121410
$959$	5190.00	0.174759
$960$	−3507.00	−0.117904
$961$	8625.00	0.289517
$962$	0	0
$963$	−12042.0	−0.402957
$964$	25319.0	0.845923
$965$	−2751.00	−0.0917698
$966$	−4860.00	−0.161872
$967$	−418.000	−0.0139007	−0.00695035	−	0.999976i	$-0.502212\pi$
$967$	−418.000	−0.0139007	−0.00695035	+	0.999976i	$0.502212\pi$
$968$	−12705.0	−0.421853
$969$	3330.00	0.110397
$970$	8414.00	0.278513
$971$	18132.0	0.599262	0.299631	−	0.954055i	$-0.403136\pi$
$971$	18132.0	0.599262	0.299631	+	0.954055i	$0.403136\pi$
$972$	−1701.00	−0.0561313
$973$	3480.00	0.114659
$974$	17130.0	0.563532
$975$	0	0
$976$	−26035.0	−0.853853
$977$	12501.0	0.409358	0.204679	−	0.978829i	$-0.434385\pi$
$977$	12501.0	0.409358	0.204679	+	0.978829i	$0.434385\pi$
$978$	−5100.00	−0.166748
$979$	−4268.00	−0.139332
$980$	11907.0	0.388118
$981$	−9306.00	−0.302872
$982$	11838.0	0.384690
$983$	−43708.0	−1.41818	−0.709089	−	0.705119i	$-0.750894\pi$
$983$	−43708.0	−1.41818	−0.709089	+	0.705119i	$0.750894\pi$
$984$	12825.0	0.415494
$985$	−24654.0	−0.797504
$986$	4181.00	0.135041
$987$	13860.0	0.446979
$988$	0	0
$989$	39852.0	1.28131
$990$	1386.00	0.0444949
$991$	−39614.0	−1.26981	−0.634904	−	0.772591i	$-0.718960\pi$
$991$	−39614.0	−1.26981	−0.634904	+	0.772591i	$0.718960\pi$
$992$	−31556.0	−1.00998
$993$	3084.00	0.0985577
$994$	−10860.0	−0.346538
$995$	14126.0	0.450075
$996$	−10878.0	−0.346067
$997$	−36503.0	−1.15954	−0.579770	−	0.814780i	$-0.696858\pi$
$997$	−36503.0	−1.15954	−0.579770	+	0.814780i	$0.696858\pi$
$998$	−8976.00	−0.284700
$999$	351.000	0.0111163

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.b.1.1		1
3.2	odd	2		1521.4.a.h.1.1		1
13.3	even	3		39.4.e.b.22.1	yes	2
13.5	odd	4		507.4.b.d.337.2		2
13.8	odd	4		507.4.b.d.337.1		2
13.9	even	3		39.4.e.b.16.1	✓	2
13.12	even	2		507.4.a.d.1.1		1
39.29	odd	6		117.4.g.a.100.1		2
39.35	odd	6		117.4.g.a.55.1		2
39.38	odd	2		1521.4.a.e.1.1		1
52.3	odd	6		624.4.q.c.529.1		2
52.35	odd	6		624.4.q.c.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.b.16.1	✓	2	13.9	even	3
39.4.e.b.22.1	yes	2	13.3	even	3
117.4.g.a.55.1		2	39.35	odd	6
117.4.g.a.100.1		2	39.29	odd	6
507.4.a.b.1.1		1	1.1	even	1	trivial
507.4.a.d.1.1		1	13.12	even	2
507.4.b.d.337.1		2	13.8	odd	4
507.4.b.d.337.2		2	13.5	odd	4
624.4.q.c.289.1		2	52.35	odd	6
624.4.q.c.529.1		2	52.3	odd	6
1521.4.a.e.1.1		1	39.38	odd	2
1521.4.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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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