LMFDB - Embedded newform 390.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,4,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	390.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:	$23.0107449022$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -25.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} -25.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -21.0000 q^{11} -12.0000 q^{12} +13.0000 q^{13} -50.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} +123.000 q^{17} +18.0000 q^{18} +146.000 q^{19} -20.0000 q^{20} +75.0000 q^{21} -42.0000 q^{22} +99.0000 q^{23} -24.0000 q^{24} +25.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} -100.000 q^{28} -246.000 q^{29} +30.0000 q^{30} +182.000 q^{31} +32.0000 q^{32} +63.0000 q^{33} +246.000 q^{34} +125.000 q^{35} +36.0000 q^{36} -295.000 q^{37} +292.000 q^{38} -39.0000 q^{39} -40.0000 q^{40} +9.00000 q^{41} +150.000 q^{42} +452.000 q^{43} -84.0000 q^{44} -45.0000 q^{45} +198.000 q^{46} +390.000 q^{47} -48.0000 q^{48} +282.000 q^{49} +50.0000 q^{50} -369.000 q^{51} +52.0000 q^{52} +315.000 q^{53} -54.0000 q^{54} +105.000 q^{55} -200.000 q^{56} -438.000 q^{57} -492.000 q^{58} -24.0000 q^{59} +60.0000 q^{60} -727.000 q^{61} +364.000 q^{62} -225.000 q^{63} +64.0000 q^{64} -65.0000 q^{65} +126.000 q^{66} +596.000 q^{67} +492.000 q^{68} -297.000 q^{69} +250.000 q^{70} +771.000 q^{71} +72.0000 q^{72} +326.000 q^{73} -590.000 q^{74} -75.0000 q^{75} +584.000 q^{76} +525.000 q^{77} -78.0000 q^{78} -889.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} +18.0000 q^{82} -96.0000 q^{83} +300.000 q^{84} -615.000 q^{85} +904.000 q^{86} +738.000 q^{87} -168.000 q^{88} +795.000 q^{89} -90.0000 q^{90} -325.000 q^{91} +396.000 q^{92} -546.000 q^{93} +780.000 q^{94} -730.000 q^{95} -96.0000 q^{96} +983.000 q^{97} +564.000 q^{98} -189.000 q^{99} +O(q^{100})\)

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
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−6.00000	−0.408248
$7$	−25.0000	−1.34987	−0.674937	−	0.737876i	$-0.735829\pi$
$7$	−25.0000	−1.34987	−0.674937	+	0.737876i	$0.735829\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−21.0000	−0.575613	−0.287806	−	0.957689i	$-0.592926\pi$
$11$	−21.0000	−0.575613	−0.287806	+	0.957689i	$0.592926\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−50.0000	−0.954504
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	123.000	1.75482	0.877408	−	0.479744i	$-0.159271\pi$
$17$	123.000	1.75482	0.877408	+	0.479744i	$0.159271\pi$
$18$	18.0000	0.235702
$19$	146.000	1.76288	0.881439	−	0.472297i	$-0.156575\pi$
$19$	146.000	1.76288	0.881439	+	0.472297i	$0.156575\pi$
$20$	−20.0000	−0.223607
$21$	75.0000	0.779350
$22$	−42.0000	−0.407020
$23$	99.0000	0.897519	0.448759	−	0.893653i	$-0.351866\pi$
$23$	99.0000	0.897519	0.448759	+	0.893653i	$0.351866\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	−100.000	−0.674937
$29$	−246.000	−1.57521	−0.787604	−	0.616181i	$-0.788679\pi$
$29$	−246.000	−1.57521	−0.787604	+	0.616181i	$0.788679\pi$
$30$	30.0000	0.182574
$31$	182.000	1.05446	0.527228	−	0.849724i	$-0.323231\pi$
$31$	182.000	1.05446	0.527228	+	0.849724i	$0.323231\pi$
$32$	32.0000	0.176777
$33$	63.0000	0.332330
$34$	246.000	1.24084
$35$	125.000	0.603682
$36$	36.0000	0.166667
$37$	−295.000	−1.31075	−0.655374	−	0.755304i	$-0.727489\pi$
$37$	−295.000	−1.31075	−0.655374	+	0.755304i	$0.727489\pi$
$38$	292.000	1.24654
$39$	−39.0000	−0.160128
$40$	−40.0000	−0.158114
$41$	9.00000	0.0342820	0.0171410	−	0.999853i	$-0.494544\pi$
$41$	9.00000	0.0342820	0.0171410	+	0.999853i	$0.494544\pi$
$42$	150.000	0.551083
$43$	452.000	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
$43$	452.000	1.60301	0.801504	+	0.597989i	$0.204033\pi$
$44$	−84.0000	−0.287806
$45$	−45.0000	−0.149071
$46$	198.000	0.634641
$47$	390.000	1.21037	0.605185	−	0.796085i	$-0.293099\pi$
$47$	390.000	1.21037	0.605185	+	0.796085i	$0.293099\pi$
$48$	−48.0000	−0.144338
$49$	282.000	0.822157
$50$	50.0000	0.141421
$51$	−369.000	−1.01314
$52$	52.0000	0.138675
$53$	315.000	0.816388	0.408194	−	0.912895i	$-0.366159\pi$
$53$	315.000	0.816388	0.408194	+	0.912895i	$0.366159\pi$
$54$	−54.0000	−0.136083
$55$	105.000	0.257422
$56$	−200.000	−0.477252
$57$	−438.000	−1.01780
$58$	−492.000	−1.11384
$59$	−24.0000	−0.0529582	−0.0264791	−	0.999649i	$-0.508430\pi$
$59$	−24.0000	−0.0529582	−0.0264791	+	0.999649i	$0.508430\pi$
$60$	60.0000	0.129099
$61$	−727.000	−1.52595	−0.762974	−	0.646429i	$-0.776262\pi$
$61$	−727.000	−1.52595	−0.762974	+	0.646429i	$0.776262\pi$
$62$	364.000	0.745614
$63$	−225.000	−0.449958
$64$	64.0000	0.125000
$65$	−65.0000	−0.124035
$66$	126.000	0.234993
$67$	596.000	1.08676	0.543381	−	0.839487i	$-0.317144\pi$
$67$	596.000	1.08676	0.543381	+	0.839487i	$0.317144\pi$
$68$	492.000	0.877408
$69$	−297.000	−0.518183
$70$	250.000	0.426867
$71$	771.000	1.28874	0.644372	−	0.764712i	$-0.277119\pi$
$71$	771.000	1.28874	0.644372	+	0.764712i	$0.277119\pi$
$72$	72.0000	0.117851
$73$	326.000	0.522677	0.261338	−	0.965247i	$-0.415836\pi$
$73$	326.000	0.522677	0.261338	+	0.965247i	$0.415836\pi$
$74$	−590.000	−0.926839
$75$	−75.0000	−0.115470
$76$	584.000	0.881439
$77$	525.000	0.777004
$78$	−78.0000	−0.113228
$79$	−889.000	−1.26608	−0.633040	−	0.774119i	$-0.718193\pi$
$79$	−889.000	−1.26608	−0.633040	+	0.774119i	$0.718193\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	18.0000	0.0242411
$83$	−96.0000	−0.126956	−0.0634781	−	0.997983i	$-0.520219\pi$
$83$	−96.0000	−0.126956	−0.0634781	+	0.997983i	$0.520219\pi$
$84$	300.000	0.389675
$85$	−615.000	−0.784778
$86$	904.000	1.13350
$87$	738.000	0.909447
$88$	−168.000	−0.203510
$89$	795.000	0.946852	0.473426	−	0.880834i	$-0.343017\pi$
$89$	795.000	0.946852	0.473426	+	0.880834i	$0.343017\pi$
$90$	−90.0000	−0.105409
$91$	−325.000	−0.374387
$92$	396.000	0.448759
$93$	−546.000	−0.608791
$94$	780.000	0.855860
$95$	−730.000	−0.788383
$96$	−96.0000	−0.102062
$97$	983.000	1.02895	0.514477	−	0.857504i	$-0.327986\pi$
$97$	983.000	1.02895	0.514477	+	0.857504i	$0.327986\pi$
$98$	564.000	0.581353
$99$	−189.000	−0.191871
$100$	100.000	0.100000
$101$	−732.000	−0.721156	−0.360578	−	0.932729i	$-0.617420\pi$
$101$	−732.000	−0.721156	−0.360578	+	0.932729i	$0.617420\pi$
$102$	−738.000	−0.716401
$103$	−1600.00	−1.53061	−0.765304	−	0.643669i	$-0.777412\pi$
$103$	−1600.00	−1.53061	−0.765304	+	0.643669i	$0.777412\pi$
$104$	104.000	0.0980581
$105$	−375.000	−0.348536
$106$	630.000	0.577274
$107$	1011.00	0.913430	0.456715	−	0.889613i	$-0.349026\pi$
$107$	1011.00	0.913430	0.456715	+	0.889613i	$0.349026\pi$
$108$	−108.000	−0.0962250
$109$	1100.00	0.966614	0.483307	−	0.875451i	$-0.339436\pi$
$109$	1100.00	0.966614	0.483307	+	0.875451i	$0.339436\pi$
$110$	210.000	0.182025
$111$	885.000	0.756761
$112$	−400.000	−0.337468
$113$	−1770.00	−1.47352	−0.736759	−	0.676155i	$-0.763645\pi$
$113$	−1770.00	−1.47352	−0.736759	+	0.676155i	$0.763645\pi$
$114$	−876.000	−0.719692
$115$	−495.000	−0.401383
$116$	−984.000	−0.787604
$117$	117.000	0.0924500
$118$	−48.0000	−0.0374471
$119$	−3075.00	−2.36878
$120$	120.000	0.0912871
$121$	−890.000	−0.668670
$122$	−1454.00	−1.07901
$123$	−27.0000	−0.0197927
$124$	728.000	0.527228
$125$	−125.000	−0.0894427
$126$	−450.000	−0.318168
$127$	2666.00	1.86275	0.931375	−	0.364061i	$-0.118610\pi$
$127$	2666.00	1.86275	0.931375	+	0.364061i	$0.118610\pi$
$128$	128.000	0.0883883
$129$	−1356.00	−0.925497
$130$	−130.000	−0.0877058
$131$	582.000	0.388165	0.194082	−	0.980985i	$-0.437827\pi$
$131$	582.000	0.388165	0.194082	+	0.980985i	$0.437827\pi$
$132$	252.000	0.166165
$133$	−3650.00	−2.37966
$134$	1192.00	0.768456
$135$	135.000	0.0860663
$136$	984.000	0.620421
$137$	270.000	0.168377	0.0841885	−	0.996450i	$-0.473170\pi$
$137$	270.000	0.168377	0.0841885	+	0.996450i	$0.473170\pi$
$138$	−594.000	−0.366410
$139$	−7.00000	−0.00427146	−0.00213573	−	0.999998i	$-0.500680\pi$
$139$	−7.00000	−0.00427146	−0.00213573	+	0.999998i	$0.500680\pi$
$140$	500.000	0.301841
$141$	−1170.00	−0.698807
$142$	1542.00	0.911280
$143$	−273.000	−0.159646
$144$	144.000	0.0833333
$145$	1230.00	0.704455
$146$	652.000	0.369588
$147$	−846.000	−0.474673
$148$	−1180.00	−0.655374
$149$	3099.00	1.70389	0.851946	−	0.523629i	$-0.175422\pi$
$149$	3099.00	1.70389	0.851946	+	0.523629i	$0.175422\pi$
$150$	−150.000	−0.0816497
$151$	−2860.00	−1.54135	−0.770674	−	0.637230i	$-0.780080\pi$
$151$	−2860.00	−1.54135	−0.770674	+	0.637230i	$0.780080\pi$
$152$	1168.00	0.623272
$153$	1107.00	0.584939
$154$	1050.00	0.549425
$155$	−910.000	−0.471567
$156$	−156.000	−0.0800641
$157$	−2734.00	−1.38979	−0.694895	−	0.719111i	$-0.744549\pi$
$157$	−2734.00	−1.38979	−0.694895	+	0.719111i	$0.744549\pi$
$158$	−1778.00	−0.895254
$159$	−945.000	−0.471342
$160$	−160.000	−0.0790569
$161$	−2475.00	−1.21154
$162$	162.000	0.0785674
$163$	−853.000	−0.409890	−0.204945	−	0.978773i	$-0.565702\pi$
$163$	−853.000	−0.409890	−0.204945	+	0.978773i	$0.565702\pi$
$164$	36.0000	0.0171410
$165$	−315.000	−0.148623
$166$	−192.000	−0.0897716
$167$	2556.00	1.18437	0.592183	−	0.805803i	$-0.298266\pi$
$167$	2556.00	1.18437	0.592183	+	0.805803i	$0.298266\pi$
$168$	600.000	0.275542
$169$	169.000	0.0769231
$170$	−1230.00	−0.554922
$171$	1314.00	0.587626
$172$	1808.00	0.801504
$173$	−1134.00	−0.498361	−0.249180	−	0.968457i	$-0.580161\pi$
$173$	−1134.00	−0.498361	−0.249180	+	0.968457i	$0.580161\pi$
$174$	1476.00	0.643076
$175$	−625.000	−0.269975
$176$	−336.000	−0.143903
$177$	72.0000	0.0305754
$178$	1590.00	0.669525
$179$	498.000	0.207946	0.103973	−	0.994580i	$-0.466845\pi$
$179$	498.000	0.207946	0.103973	+	0.994580i	$0.466845\pi$
$180$	−180.000	−0.0745356
$181$	1901.00	0.780664	0.390332	−	0.920674i	$-0.372360\pi$
$181$	1901.00	0.780664	0.390332	+	0.920674i	$0.372360\pi$
$182$	−650.000	−0.264732
$183$	2181.00	0.881006
$184$	792.000	0.317321
$185$	1475.00	0.586185
$186$	−1092.00	−0.430480
$187$	−2583.00	−1.01009
$188$	1560.00	0.605185
$189$	675.000	0.259783
$190$	−1460.00	−0.557471
$191$	2340.00	0.886474	0.443237	−	0.896405i	$-0.353830\pi$
$191$	2340.00	0.886474	0.443237	+	0.896405i	$0.353830\pi$
$192$	−192.000	−0.0721688
$193$	1685.00	0.628440	0.314220	−	0.949350i	$-0.398257\pi$
$193$	1685.00	0.628440	0.314220	+	0.949350i	$0.398257\pi$
$194$	1966.00	0.727580
$195$	195.000	0.0716115
$196$	1128.00	0.411079
$197$	1992.00	0.720427	0.360214	−	0.932870i	$-0.382704\pi$
$197$	1992.00	0.720427	0.360214	+	0.932870i	$0.382704\pi$
$198$	−378.000	−0.135673
$199$	−4732.00	−1.68564	−0.842821	−	0.538195i	$-0.819107\pi$
$199$	−4732.00	−1.68564	−0.842821	+	0.538195i	$0.819107\pi$
$200$	200.000	0.0707107
$201$	−1788.00	−0.627442
$202$	−1464.00	−0.509934
$203$	6150.00	2.12633
$204$	−1476.00	−0.506572
$205$	−45.0000	−0.0153314
$206$	−3200.00	−1.08230
$207$	891.000	0.299173
$208$	208.000	0.0693375
$209$	−3066.00	−1.01474
$210$	−750.000	−0.246452
$211$	−4012.00	−1.30899	−0.654496	−	0.756065i	$-0.727119\pi$
$211$	−4012.00	−1.30899	−0.654496	+	0.756065i	$0.727119\pi$
$212$	1260.00	0.408194
$213$	−2313.00	−0.744057
$214$	2022.00	0.645893
$215$	−2260.00	−0.716887
$216$	−216.000	−0.0680414
$217$	−4550.00	−1.42338
$218$	2200.00	0.683499
$219$	−978.000	−0.301768
$220$	420.000	0.128711
$221$	1599.00	0.486699
$222$	1770.00	0.535111
$223$	−1096.00	−0.329119	−0.164560	−	0.986367i	$-0.552620\pi$
$223$	−1096.00	−0.329119	−0.164560	+	0.986367i	$0.552620\pi$
$224$	−800.000	−0.238626
$225$	225.000	0.0666667
$226$	−3540.00	−1.04193
$227$	6138.00	1.79468	0.897342	−	0.441335i	$-0.145495\pi$
$227$	6138.00	1.79468	0.897342	+	0.441335i	$0.145495\pi$
$228$	−1752.00	−0.508899
$229$	−1294.00	−0.373406	−0.186703	−	0.982416i	$-0.559780\pi$
$229$	−1294.00	−0.373406	−0.186703	+	0.982416i	$0.559780\pi$
$230$	−990.000	−0.283820
$231$	−1575.00	−0.448603
$232$	−1968.00	−0.556920
$233$	873.000	0.245460	0.122730	−	0.992440i	$-0.460835\pi$
$233$	873.000	0.245460	0.122730	+	0.992440i	$0.460835\pi$
$234$	234.000	0.0653720
$235$	−1950.00	−0.541294
$236$	−96.0000	−0.0264791
$237$	2667.00	0.730972
$238$	−6150.00	−1.67498
$239$	5163.00	1.39735	0.698675	−	0.715439i	$-0.253773\pi$
$239$	5163.00	1.39735	0.698675	+	0.715439i	$0.253773\pi$
$240$	240.000	0.0645497
$241$	−2482.00	−0.663401	−0.331701	−	0.943385i	$-0.607622\pi$
$241$	−2482.00	−0.663401	−0.331701	+	0.943385i	$0.607622\pi$
$242$	−1780.00	−0.472821
$243$	−243.000	−0.0641500
$244$	−2908.00	−0.762974
$245$	−1410.00	−0.367680
$246$	−54.0000	−0.0139956
$247$	1898.00	0.488935
$248$	1456.00	0.372807
$249$	288.000	0.0732982
$250$	−250.000	−0.0632456
$251$	5172.00	1.30061	0.650306	−	0.759672i	$-0.274641\pi$
$251$	5172.00	1.30061	0.650306	+	0.759672i	$0.274641\pi$
$252$	−900.000	−0.224979
$253$	−2079.00	−0.516623
$254$	5332.00	1.31716
$255$	1845.00	0.453092
$256$	256.000	0.0625000
$257$	198.000	0.0480580	0.0240290	−	0.999711i	$-0.492351\pi$
$257$	198.000	0.0480580	0.0240290	+	0.999711i	$0.492351\pi$
$258$	−2712.00	−0.654425
$259$	7375.00	1.76934
$260$	−260.000	−0.0620174
$261$	−2214.00	−0.525070
$262$	1164.00	0.274474
$263$	−2568.00	−0.602090	−0.301045	−	0.953610i	$-0.597335\pi$
$263$	−2568.00	−0.602090	−0.301045	+	0.953610i	$0.597335\pi$
$264$	504.000	0.117496
$265$	−1575.00	−0.365100
$266$	−7300.00	−1.68268
$267$	−2385.00	−0.546665
$268$	2384.00	0.543381
$269$	−6924.00	−1.56938	−0.784691	−	0.619887i	$-0.787178\pi$
$269$	−6924.00	−1.56938	−0.784691	+	0.619887i	$0.787178\pi$
$270$	270.000	0.0608581
$271$	254.000	0.0569351	0.0284675	−	0.999595i	$-0.490937\pi$
$271$	254.000	0.0569351	0.0284675	+	0.999595i	$0.490937\pi$
$272$	1968.00	0.438704
$273$	975.000	0.216153
$274$	540.000	0.119061
$275$	−525.000	−0.115123
$276$	−1188.00	−0.259091
$277$	6590.00	1.42944	0.714720	−	0.699411i	$-0.246554\pi$
$277$	6590.00	1.42944	0.714720	+	0.699411i	$0.246554\pi$
$278$	−14.0000	−0.00302037
$279$	1638.00	0.351486
$280$	1000.00	0.213434
$281$	2370.00	0.503140	0.251570	−	0.967839i	$-0.419053\pi$
$281$	2370.00	0.503140	0.251570	+	0.967839i	$0.419053\pi$
$282$	−2340.00	−0.494131
$283$	−6280.00	−1.31911	−0.659553	−	0.751658i	$-0.729255\pi$
$283$	−6280.00	−1.31911	−0.659553	+	0.751658i	$0.729255\pi$
$284$	3084.00	0.644372
$285$	2190.00	0.455173
$286$	−546.000	−0.112887
$287$	−225.000	−0.0462764
$288$	288.000	0.0589256
$289$	10216.0	2.07938
$290$	2460.00	0.498125
$291$	−2949.00	−0.594067
$292$	1304.00	0.261338
$293$	−3588.00	−0.715403	−0.357702	−	0.933836i	$-0.616440\pi$
$293$	−3588.00	−0.715403	−0.357702	+	0.933836i	$0.616440\pi$
$294$	−1692.00	−0.335644
$295$	120.000	0.0236836
$296$	−2360.00	−0.463420
$297$	567.000	0.110777
$298$	6198.00	1.20483
$299$	1287.00	0.248927
$300$	−300.000	−0.0577350
$301$	−11300.0	−2.16386
$302$	−5720.00	−1.08990
$303$	2196.00	0.416359
$304$	2336.00	0.440720
$305$	3635.00	0.682425
$306$	2214.00	0.413614
$307$	−6469.00	−1.20262	−0.601312	−	0.799015i	$-0.705355\pi$
$307$	−6469.00	−1.20262	−0.601312	+	0.799015i	$0.705355\pi$
$308$	2100.00	0.388502
$309$	4800.00	0.883697
$310$	−1820.00	−0.333449
$311$	1536.00	0.280060	0.140030	−	0.990147i	$-0.455280\pi$
$311$	1536.00	0.280060	0.140030	+	0.990147i	$0.455280\pi$
$312$	−312.000	−0.0566139
$313$	1514.00	0.273407	0.136703	−	0.990612i	$-0.456349\pi$
$313$	1514.00	0.273407	0.136703	+	0.990612i	$0.456349\pi$
$314$	−5468.00	−0.982730
$315$	1125.00	0.201227
$316$	−3556.00	−0.633040
$317$	−2724.00	−0.482634	−0.241317	−	0.970446i	$-0.577579\pi$
$317$	−2724.00	−0.482634	−0.241317	+	0.970446i	$0.577579\pi$
$318$	−1890.00	−0.333289
$319$	5166.00	0.906710
$320$	−320.000	−0.0559017
$321$	−3033.00	−0.527369
$322$	−4950.00	−0.856685
$323$	17958.0	3.09353
$324$	324.000	0.0555556
$325$	325.000	0.0554700
$326$	−1706.00	−0.289836
$327$	−3300.00	−0.558075
$328$	72.0000	0.0121205
$329$	−9750.00	−1.63384
$330$	−630.000	−0.105092
$331$	1100.00	0.182663	0.0913315	−	0.995821i	$-0.470888\pi$
$331$	1100.00	0.182663	0.0913315	+	0.995821i	$0.470888\pi$
$332$	−384.000	−0.0634781
$333$	−2655.00	−0.436916
$334$	5112.00	0.837474
$335$	−2980.00	−0.486014
$336$	1200.00	0.194837
$337$	−196.000	−0.0316819	−0.0158410	−	0.999875i	$-0.505043\pi$
$337$	−196.000	−0.0316819	−0.0158410	+	0.999875i	$0.505043\pi$
$338$	338.000	0.0543928
$339$	5310.00	0.850736
$340$	−2460.00	−0.392389
$341$	−3822.00	−0.606959
$342$	2628.00	0.415515
$343$	1525.00	0.240065
$344$	3616.00	0.566749
$345$	1485.00	0.231738
$346$	−2268.00	−0.352394
$347$	11055.0	1.71027	0.855135	−	0.518406i	$-0.173474\pi$
$347$	11055.0	1.71027	0.855135	+	0.518406i	$0.173474\pi$
$348$	2952.00	0.454724
$349$	−11176.0	−1.71415	−0.857074	−	0.515194i	$-0.827720\pi$
$349$	−11176.0	−1.71415	−0.857074	+	0.515194i	$0.827720\pi$
$350$	−1250.00	−0.190901
$351$	−351.000	−0.0533761
$352$	−672.000	−0.101755
$353$	2130.00	0.321157	0.160579	−	0.987023i	$-0.448664\pi$
$353$	2130.00	0.321157	0.160579	+	0.987023i	$0.448664\pi$
$354$	144.000	0.0216201
$355$	−3855.00	−0.576344
$356$	3180.00	0.473426
$357$	9225.00	1.36762
$358$	996.000	0.147040
$359$	2304.00	0.338720	0.169360	−	0.985554i	$-0.445830\pi$
$359$	2304.00	0.338720	0.169360	+	0.985554i	$0.445830\pi$
$360$	−360.000	−0.0527046
$361$	14457.0	2.10774
$362$	3802.00	0.552013
$363$	2670.00	0.386057
$364$	−1300.00	−0.187194
$365$	−1630.00	−0.233748
$366$	4362.00	0.622966
$367$	13412.0	1.90763	0.953816	−	0.300393i	$-0.0971177\pi$
$367$	13412.0	1.90763	0.953816	+	0.300393i	$0.0971177\pi$
$368$	1584.00	0.224380
$369$	81.0000	0.0114273
$370$	2950.00	0.414495
$371$	−7875.00	−1.10202
$372$	−2184.00	−0.304395
$373$	9056.00	1.25711	0.628555	−	0.777765i	$-0.283647\pi$
$373$	9056.00	1.25711	0.628555	+	0.777765i	$0.283647\pi$
$374$	−5166.00	−0.714245
$375$	375.000	0.0516398
$376$	3120.00	0.427930
$377$	−3198.00	−0.436884
$378$	1350.00	0.183694
$379$	−4066.00	−0.551072	−0.275536	−	0.961291i	$-0.588855\pi$
$379$	−4066.00	−0.551072	−0.275536	+	0.961291i	$0.588855\pi$
$380$	−2920.00	−0.394192
$381$	−7998.00	−1.07546
$382$	4680.00	0.626831
$383$	786.000	0.104864	0.0524318	−	0.998625i	$-0.483303\pi$
$383$	786.000	0.104864	0.0524318	+	0.998625i	$0.483303\pi$
$384$	−384.000	−0.0510310
$385$	−2625.00	−0.347487
$386$	3370.00	0.444374
$387$	4068.00	0.534336
$388$	3932.00	0.514477
$389$	8364.00	1.09016	0.545079	−	0.838385i	$-0.316500\pi$
$389$	8364.00	1.09016	0.545079	+	0.838385i	$0.316500\pi$
$390$	390.000	0.0506370
$391$	12177.0	1.57498
$392$	2256.00	0.290677
$393$	−1746.00	−0.224107
$394$	3984.00	0.509419
$395$	4445.00	0.566208
$396$	−756.000	−0.0959354
$397$	7625.00	0.963949	0.481975	−	0.876185i	$-0.339920\pi$
$397$	7625.00	0.963949	0.481975	+	0.876185i	$0.339920\pi$
$398$	−9464.00	−1.19193
$399$	10950.0	1.37390
$400$	400.000	0.0500000
$401$	954.000	0.118804	0.0594021	−	0.998234i	$-0.481081\pi$
$401$	954.000	0.118804	0.0594021	+	0.998234i	$0.481081\pi$
$402$	−3576.00	−0.443668
$403$	2366.00	0.292454
$404$	−2928.00	−0.360578
$405$	−405.000	−0.0496904
$406$	12300.0	1.50354
$407$	6195.00	0.754483
$408$	−2952.00	−0.358200
$409$	182.000	0.0220032	0.0110016	−	0.999939i	$-0.496498\pi$
$409$	182.000	0.0220032	0.0110016	+	0.999939i	$0.496498\pi$
$410$	−90.0000	−0.0108409
$411$	−810.000	−0.0972125
$412$	−6400.00	−0.765304
$413$	600.000	0.0714869
$414$	1782.00	0.211547
$415$	480.000	0.0567766
$416$	416.000	0.0490290
$417$	21.0000	0.00246613
$418$	−6132.00	−0.717526
$419$	−4806.00	−0.560354	−0.280177	−	0.959948i	$-0.590393\pi$
$419$	−4806.00	−0.560354	−0.280177	+	0.959948i	$0.590393\pi$
$420$	−1500.00	−0.174268
$421$	−10024.0	−1.16043	−0.580214	−	0.814464i	$-0.697031\pi$
$421$	−10024.0	−1.16043	−0.580214	+	0.814464i	$0.697031\pi$
$422$	−8024.00	−0.925598
$423$	3510.00	0.403456
$424$	2520.00	0.288637
$425$	3075.00	0.350963
$426$	−4626.00	−0.526128
$427$	18175.0	2.05984
$428$	4044.00	0.456715
$429$	819.000	0.0921718
$430$	−4520.00	−0.506916
$431$	−11544.0	−1.29015	−0.645075	−	0.764119i	$-0.723174\pi$
$431$	−11544.0	−1.29015	−0.645075	+	0.764119i	$0.723174\pi$
$432$	−432.000	−0.0481125
$433$	−268.000	−0.0297442	−0.0148721	−	0.999889i	$-0.504734\pi$
$433$	−268.000	−0.0297442	−0.0148721	+	0.999889i	$0.504734\pi$
$434$	−9100.00	−1.00648
$435$	−3690.00	−0.406717
$436$	4400.00	0.483307
$437$	14454.0	1.58222
$438$	−1956.00	−0.213382
$439$	−6235.00	−0.677859	−0.338930	−	0.940812i	$-0.610065\pi$
$439$	−6235.00	−0.677859	−0.338930	+	0.940812i	$0.610065\pi$
$440$	840.000	0.0910123
$441$	2538.00	0.274052
$442$	3198.00	0.344148
$443$	9363.00	1.00418	0.502088	−	0.864817i	$-0.332566\pi$
$443$	9363.00	1.00418	0.502088	+	0.864817i	$0.332566\pi$
$444$	3540.00	0.378381
$445$	−3975.00	−0.423445
$446$	−2192.00	−0.232722
$447$	−9297.00	−0.983743
$448$	−1600.00	−0.168734
$449$	−7023.00	−0.738165	−0.369082	−	0.929397i	$-0.620328\pi$
$449$	−7023.00	−0.738165	−0.369082	+	0.929397i	$0.620328\pi$
$450$	450.000	0.0471405
$451$	−189.000	−0.0197332
$452$	−7080.00	−0.736759
$453$	8580.00	0.889897
$454$	12276.0	1.26903
$455$	1625.00	0.167431
$456$	−3504.00	−0.359846
$457$	2603.00	0.266440	0.133220	−	0.991086i	$-0.457468\pi$
$457$	2603.00	0.266440	0.133220	+	0.991086i	$0.457468\pi$
$458$	−2588.00	−0.264038
$459$	−3321.00	−0.337715
$460$	−1980.00	−0.200691
$461$	10377.0	1.04838	0.524192	−	0.851600i	$-0.324367\pi$
$461$	10377.0	1.04838	0.524192	+	0.851600i	$0.324367\pi$
$462$	−3150.00	−0.317211
$463$	−11401.0	−1.14438	−0.572192	−	0.820120i	$-0.693907\pi$
$463$	−11401.0	−1.14438	−0.572192	+	0.820120i	$0.693907\pi$
$464$	−3936.00	−0.393802
$465$	2730.00	0.272260
$466$	1746.00	0.173566
$467$	−15381.0	−1.52409	−0.762043	−	0.647527i	$-0.775803\pi$
$467$	−15381.0	−1.52409	−0.762043	+	0.647527i	$0.775803\pi$
$468$	468.000	0.0462250
$469$	−14900.0	−1.46699
$470$	−3900.00	−0.382752
$471$	8202.00	0.802395
$472$	−192.000	−0.0187236
$473$	−9492.00	−0.922712
$474$	5334.00	0.516875
$475$	3650.00	0.352576
$476$	−12300.0	−1.18439
$477$	2835.00	0.272129
$478$	10326.0	0.988076
$479$	−1767.00	−0.168552	−0.0842759	−	0.996442i	$-0.526858\pi$
$479$	−1767.00	−0.168552	−0.0842759	+	0.996442i	$0.526858\pi$
$480$	480.000	0.0456435
$481$	−3835.00	−0.363536
$482$	−4964.00	−0.469095
$483$	7425.00	0.699481
$484$	−3560.00	−0.334335
$485$	−4915.00	−0.460162
$486$	−486.000	−0.0453609
$487$	9893.00	0.920523	0.460261	−	0.887783i	$-0.347756\pi$
$487$	9893.00	0.920523	0.460261	+	0.887783i	$0.347756\pi$
$488$	−5816.00	−0.539504
$489$	2559.00	0.236650
$490$	−2820.00	−0.259989
$491$	−15336.0	−1.40958	−0.704790	−	0.709416i	$-0.748959\pi$
$491$	−15336.0	−1.40958	−0.704790	+	0.709416i	$0.748959\pi$
$492$	−108.000	−0.00989637
$493$	−30258.0	−2.76420
$494$	3796.00	0.345729
$495$	945.000	0.0858073
$496$	2912.00	0.263614
$497$	−19275.0	−1.73964
$498$	576.000	0.0518297
$499$	−16450.0	−1.47576	−0.737879	−	0.674933i	$-0.764172\pi$
$499$	−16450.0	−1.47576	−0.737879	+	0.674933i	$0.764172\pi$
$500$	−500.000	−0.0447214
$501$	−7668.00	−0.683794
$502$	10344.0	0.919672
$503$	3444.00	0.305289	0.152645	−	0.988281i	$-0.451221\pi$
$503$	3444.00	0.305289	0.152645	+	0.988281i	$0.451221\pi$
$504$	−1800.00	−0.159084
$505$	3660.00	0.322511
$506$	−4158.00	−0.365308
$507$	−507.000	−0.0444116
$508$	10664.0	0.931375
$509$	5535.00	0.481993	0.240997	−	0.970526i	$-0.422526\pi$
$509$	5535.00	0.481993	0.240997	+	0.970526i	$0.422526\pi$
$510$	3690.00	0.320384
$511$	−8150.00	−0.705548
$512$	512.000	0.0441942
$513$	−3942.00	−0.339266
$514$	396.000	0.0339821
$515$	8000.00	0.684509
$516$	−5424.00	−0.462749
$517$	−8190.00	−0.696704
$518$	14750.0	1.25112
$519$	3402.00	0.287729
$520$	−520.000	−0.0438529
$521$	−6786.00	−0.570634	−0.285317	−	0.958433i	$-0.592099\pi$
$521$	−6786.00	−0.570634	−0.285317	+	0.958433i	$0.592099\pi$
$522$	−4428.00	−0.371280
$523$	1928.00	0.161196	0.0805980	−	0.996747i	$-0.474317\pi$
$523$	1928.00	0.161196	0.0805980	+	0.996747i	$0.474317\pi$
$524$	2328.00	0.194082
$525$	1875.00	0.155870
$526$	−5136.00	−0.425742
$527$	22386.0	1.85038
$528$	1008.00	0.0830825
$529$	−2366.00	−0.194460
$530$	−3150.00	−0.258165
$531$	−216.000	−0.0176527
$532$	−14600.0	−1.18983
$533$	117.000	0.00950813
$534$	−4770.00	−0.386551
$535$	−5055.00	−0.408499
$536$	4768.00	0.384228
$537$	−1494.00	−0.120057
$538$	−13848.0	−1.10972
$539$	−5922.00	−0.473244
$540$	540.000	0.0430331
$541$	−15118.0	−1.20143	−0.600715	−	0.799463i	$-0.705117\pi$
$541$	−15118.0	−1.20143	−0.600715	+	0.799463i	$0.705117\pi$
$542$	508.000	0.0402592
$543$	−5703.00	−0.450717
$544$	3936.00	0.310211
$545$	−5500.00	−0.432283
$546$	1950.00	0.152843
$547$	3620.00	0.282962	0.141481	−	0.989941i	$-0.454814\pi$
$547$	3620.00	0.282962	0.141481	+	0.989941i	$0.454814\pi$
$548$	1080.00	0.0841885
$549$	−6543.00	−0.508649
$550$	−1050.00	−0.0814039
$551$	−35916.0	−2.77690
$552$	−2376.00	−0.183205
$553$	22225.0	1.70905
$554$	13180.0	1.01077
$555$	−4425.00	−0.338434
$556$	−28.0000	−0.00213573
$557$	−11886.0	−0.904176	−0.452088	−	0.891973i	$-0.649321\pi$
$557$	−11886.0	−0.904176	−0.452088	+	0.891973i	$0.649321\pi$
$558$	3276.00	0.248538
$559$	5876.00	0.444594
$560$	2000.00	0.150920
$561$	7749.00	0.583178
$562$	4740.00	0.355774
$563$	−11409.0	−0.854053	−0.427027	−	0.904239i	$-0.640439\pi$
$563$	−11409.0	−0.854053	−0.427027	+	0.904239i	$0.640439\pi$
$564$	−4680.00	−0.349403
$565$	8850.00	0.658978
$566$	−12560.0	−0.932749
$567$	−2025.00	−0.149986
$568$	6168.00	0.455640
$569$	11760.0	0.866441	0.433220	−	0.901288i	$-0.357377\pi$
$569$	11760.0	0.866441	0.433220	+	0.901288i	$0.357377\pi$
$570$	4380.00	0.321856
$571$	−4057.00	−0.297338	−0.148669	−	0.988887i	$-0.547499\pi$
$571$	−4057.00	−0.297338	−0.148669	+	0.988887i	$0.547499\pi$
$572$	−1092.00	−0.0798231
$573$	−7020.00	−0.511806
$574$	−450.000	−0.0327224
$575$	2475.00	0.179504
$576$	576.000	0.0416667
$577$	−5101.00	−0.368037	−0.184019	−	0.982923i	$-0.558911\pi$
$577$	−5101.00	−0.368037	−0.184019	+	0.982923i	$0.558911\pi$
$578$	20432.0	1.47034
$579$	−5055.00	−0.362830
$580$	4920.00	0.352227
$581$	2400.00	0.171375
$582$	−5898.00	−0.420069
$583$	−6615.00	−0.469923
$584$	2608.00	0.184794
$585$	−585.000	−0.0413449
$586$	−7176.00	−0.505867
$587$	−24186.0	−1.70062	−0.850309	−	0.526283i	$-0.823585\pi$
$587$	−24186.0	−1.70062	−0.850309	+	0.526283i	$0.823585\pi$
$588$	−3384.00	−0.237336
$589$	26572.0	1.85888
$590$	240.000	0.0167469
$591$	−5976.00	−0.415939
$592$	−4720.00	−0.327687
$593$	72.0000	0.00498598	0.00249299	−	0.999997i	$-0.499206\pi$
$593$	72.0000	0.00498598	0.00249299	+	0.999997i	$0.499206\pi$
$594$	1134.00	0.0783309
$595$	15375.0	1.05935
$596$	12396.0	0.851946
$597$	14196.0	0.973205
$598$	2574.00	0.176018
$599$	−14016.0	−0.956057	−0.478029	−	0.878344i	$-0.658648\pi$
$599$	−14016.0	−0.956057	−0.478029	+	0.878344i	$0.658648\pi$
$600$	−600.000	−0.0408248
$601$	10739.0	0.728873	0.364437	−	0.931228i	$-0.381262\pi$
$601$	10739.0	0.728873	0.364437	+	0.931228i	$0.381262\pi$
$602$	−22600.0	−1.53008
$603$	5364.00	0.362254
$604$	−11440.0	−0.770674
$605$	4450.00	0.299038
$606$	4392.00	0.294411
$607$	18884.0	1.26273	0.631366	−	0.775485i	$-0.282495\pi$
$607$	18884.0	1.26273	0.631366	+	0.775485i	$0.282495\pi$
$608$	4672.00	0.311636
$609$	−18450.0	−1.22764
$610$	7270.00	0.482547
$611$	5070.00	0.335696
$612$	4428.00	0.292469
$613$	−3985.00	−0.262565	−0.131283	−	0.991345i	$-0.541910\pi$
$613$	−3985.00	−0.262565	−0.131283	+	0.991345i	$0.541910\pi$
$614$	−12938.0	−0.850383
$615$	135.000	0.00885159
$616$	4200.00	0.274712
$617$	5034.00	0.328462	0.164231	−	0.986422i	$-0.447486\pi$
$617$	5034.00	0.328462	0.164231	+	0.986422i	$0.447486\pi$
$618$	9600.00	0.624868
$619$	20954.0	1.36060	0.680301	−	0.732933i	$-0.261849\pi$
$619$	20954.0	1.36060	0.680301	+	0.732933i	$0.261849\pi$
$620$	−3640.00	−0.235784
$621$	−2673.00	−0.172728
$622$	3072.00	0.198032
$623$	−19875.0	−1.27813
$624$	−624.000	−0.0400320
$625$	625.000	0.0400000
$626$	3028.00	0.193328
$627$	9198.00	0.585858
$628$	−10936.0	−0.694895
$629$	−36285.0	−2.30012
$630$	2250.00	0.142289
$631$	16724.0	1.05511	0.527553	−	0.849522i	$-0.323110\pi$
$631$	16724.0	1.05511	0.527553	+	0.849522i	$0.323110\pi$
$632$	−7112.00	−0.447627
$633$	12036.0	0.755747
$634$	−5448.00	−0.341274
$635$	−13330.0	−0.833047
$636$	−3780.00	−0.235671
$637$	3666.00	0.228025
$638$	10332.0	0.641141
$639$	6939.00	0.429582
$640$	−640.000	−0.0395285
$641$	−20520.0	−1.26442	−0.632208	−	0.774798i	$-0.717851\pi$
$641$	−20520.0	−1.26442	−0.632208	+	0.774798i	$0.717851\pi$
$642$	−6066.00	−0.372906
$643$	−8953.00	−0.549101	−0.274550	−	0.961573i	$-0.588529\pi$
$643$	−8953.00	−0.549101	−0.274550	+	0.961573i	$0.588529\pi$
$644$	−9900.00	−0.605768
$645$	6780.00	0.413895
$646$	35916.0	2.18746
$647$	17913.0	1.08846	0.544229	−	0.838937i	$-0.316822\pi$
$647$	17913.0	1.08846	0.544229	+	0.838937i	$0.316822\pi$
$648$	648.000	0.0392837
$649$	504.000	0.0304834
$650$	650.000	0.0392232
$651$	13650.0	0.821791
$652$	−3412.00	−0.204945
$653$	20718.0	1.24159	0.620795	−	0.783973i	$-0.286810\pi$
$653$	20718.0	1.24159	0.620795	+	0.783973i	$0.286810\pi$
$654$	−6600.00	−0.394618
$655$	−2910.00	−0.173593
$656$	144.000	0.00857051
$657$	2934.00	0.174226
$658$	−19500.0	−1.15530
$659$	−32040.0	−1.89393	−0.946966	−	0.321334i	$-0.895869\pi$
$659$	−32040.0	−1.89393	−0.946966	+	0.321334i	$0.895869\pi$
$660$	−1260.00	−0.0743113
$661$	−2176.00	−0.128043	−0.0640216	−	0.997949i	$-0.520393\pi$
$661$	−2176.00	−0.128043	−0.0640216	+	0.997949i	$0.520393\pi$
$662$	2200.00	0.129162
$663$	−4797.00	−0.280996
$664$	−768.000	−0.0448858
$665$	18250.0	1.06422
$666$	−5310.00	−0.308946
$667$	−24354.0	−1.41378
$668$	10224.0	0.592183
$669$	3288.00	0.190017
$670$	−5960.00	−0.343664
$671$	15267.0	0.878355
$672$	2400.00	0.137771
$673$	−30238.0	−1.73193	−0.865965	−	0.500104i	$-0.833295\pi$
$673$	−30238.0	−1.73193	−0.865965	+	0.500104i	$0.833295\pi$
$674$	−392.000	−0.0224025
$675$	−675.000	−0.0384900
$676$	676.000	0.0384615
$677$	−25341.0	−1.43860	−0.719301	−	0.694698i	$-0.755538\pi$
$677$	−25341.0	−1.43860	−0.719301	+	0.694698i	$0.755538\pi$
$678$	10620.0	0.601561
$679$	−24575.0	−1.38896
$680$	−4920.00	−0.277461
$681$	−18414.0	−1.03616
$682$	−7644.00	−0.429185
$683$	6504.00	0.364376	0.182188	−	0.983264i	$-0.441682\pi$
$683$	6504.00	0.364376	0.182188	+	0.983264i	$0.441682\pi$
$684$	5256.00	0.293813
$685$	−1350.00	−0.0753005
$686$	3050.00	0.169752
$687$	3882.00	0.215586
$688$	7232.00	0.400752
$689$	4095.00	0.226425
$690$	2970.00	0.163864
$691$	22358.0	1.23088	0.615440	−	0.788184i	$-0.288978\pi$
$691$	22358.0	1.23088	0.615440	+	0.788184i	$0.288978\pi$
$692$	−4536.00	−0.249180
$693$	4725.00	0.259001
$694$	22110.0	1.20934
$695$	35.0000	0.00191025
$696$	5904.00	0.321538
$697$	1107.00	0.0601587
$698$	−22352.0	−1.21209
$699$	−2619.00	−0.141716
$700$	−2500.00	−0.134987
$701$	−2268.00	−0.122199	−0.0610993	−	0.998132i	$-0.519461\pi$
$701$	−2268.00	−0.122199	−0.0610993	+	0.998132i	$0.519461\pi$
$702$	−702.000	−0.0377426
$703$	−43070.0	−2.31069
$704$	−1344.00	−0.0719516
$705$	5850.00	0.312516
$706$	4260.00	0.227092
$707$	18300.0	0.973469
$708$	288.000	0.0152877
$709$	13700.0	0.725690	0.362845	−	0.931849i	$-0.381805\pi$
$709$	13700.0	0.725690	0.362845	+	0.931849i	$0.381805\pi$
$710$	−7710.00	−0.407537
$711$	−8001.00	−0.422027
$712$	6360.00	0.334763
$713$	18018.0	0.946395
$714$	18450.0	0.967050
$715$	1365.00	0.0713960
$716$	1992.00	0.103973
$717$	−15489.0	−0.806761
$718$	4608.00	0.239511
$719$	6540.00	0.339222	0.169611	−	0.985511i	$-0.445749\pi$
$719$	6540.00	0.339222	0.169611	+	0.985511i	$0.445749\pi$
$720$	−720.000	−0.0372678
$721$	40000.0	2.06613
$722$	28914.0	1.49040
$723$	7446.00	0.383015
$724$	7604.00	0.390332
$725$	−6150.00	−0.315042
$726$	5340.00	0.272983
$727$	−13822.0	−0.705130	−0.352565	−	0.935787i	$-0.614690\pi$
$727$	−13822.0	−0.705130	−0.352565	+	0.935787i	$0.614690\pi$
$728$	−2600.00	−0.132366
$729$	729.000	0.0370370
$730$	−3260.00	−0.165285
$731$	55596.0	2.81299
$732$	8724.00	0.440503
$733$	−21715.0	−1.09422	−0.547109	−	0.837061i	$-0.684272\pi$
$733$	−21715.0	−1.09422	−0.547109	+	0.837061i	$0.684272\pi$
$734$	26824.0	1.34890
$735$	4230.00	0.212280
$736$	3168.00	0.158660
$737$	−12516.0	−0.625553
$738$	162.000	0.00808036
$739$	−3310.00	−0.164764	−0.0823818	−	0.996601i	$-0.526253\pi$
$739$	−3310.00	−0.164764	−0.0823818	+	0.996601i	$0.526253\pi$
$740$	5900.00	0.293092
$741$	−5694.00	−0.282287
$742$	−15750.0	−0.779246
$743$	11178.0	0.551926	0.275963	−	0.961168i	$-0.411003\pi$
$743$	11178.0	0.551926	0.275963	+	0.961168i	$0.411003\pi$
$744$	−4368.00	−0.215240
$745$	−15495.0	−0.762004
$746$	18112.0	0.888911
$747$	−864.000	−0.0423188
$748$	−10332.0	−0.505047
$749$	−25275.0	−1.23302
$750$	750.000	0.0365148
$751$	16337.0	0.793802	0.396901	−	0.917861i	$-0.370086\pi$
$751$	16337.0	0.793802	0.396901	+	0.917861i	$0.370086\pi$
$752$	6240.00	0.302592
$753$	−15516.0	−0.750909
$754$	−6396.00	−0.308924
$755$	14300.0	0.689312
$756$	2700.00	0.129892
$757$	23096.0	1.10890	0.554451	−	0.832217i	$-0.312928\pi$
$757$	23096.0	1.10890	0.554451	+	0.832217i	$0.312928\pi$
$758$	−8132.00	−0.389667
$759$	6237.00	0.298272
$760$	−5840.00	−0.278736
$761$	−4578.00	−0.218071	−0.109036	−	0.994038i	$-0.534776\pi$
$761$	−4578.00	−0.218071	−0.109036	+	0.994038i	$0.534776\pi$
$762$	−15996.0	−0.760464
$763$	−27500.0	−1.30481
$764$	9360.00	0.443237
$765$	−5535.00	−0.261593
$766$	1572.00	0.0741497
$767$	−312.000	−0.0146880
$768$	−768.000	−0.0360844
$769$	29036.0	1.36159	0.680796	−	0.732473i	$-0.261634\pi$
$769$	29036.0	1.36159	0.680796	+	0.732473i	$0.261634\pi$
$770$	−5250.00	−0.245710
$771$	−594.000	−0.0277463
$772$	6740.00	0.314220
$773$	−37056.0	−1.72421	−0.862103	−	0.506733i	$-0.830853\pi$
$773$	−37056.0	−1.72421	−0.862103	+	0.506733i	$0.830853\pi$
$774$	8136.00	0.377833
$775$	4550.00	0.210891
$776$	7864.00	0.363790
$777$	−22125.0	−1.02153
$778$	16728.0	0.770858
$779$	1314.00	0.0604351
$780$	780.000	0.0358057
$781$	−16191.0	−0.741818
$782$	24354.0	1.11368
$783$	6642.00	0.303149
$784$	4512.00	0.205539
$785$	13670.0	0.621533
$786$	−3492.00	−0.158468
$787$	−17332.0	−0.785031	−0.392515	−	0.919745i	$-0.628395\pi$
$787$	−17332.0	−0.785031	−0.392515	+	0.919745i	$0.628395\pi$
$788$	7968.00	0.360214
$789$	7704.00	0.347617
$790$	8890.00	0.400370
$791$	44250.0	1.98906
$792$	−1512.00	−0.0678366
$793$	−9451.00	−0.423222
$794$	15250.0	0.681615
$795$	4725.00	0.210791
$796$	−18928.0	−0.842821
$797$	−30759.0	−1.36705	−0.683526	−	0.729927i	$-0.739554\pi$
$797$	−30759.0	−1.36705	−0.683526	+	0.729927i	$0.739554\pi$
$798$	21900.0	0.971493
$799$	47970.0	2.12398
$800$	800.000	0.0353553
$801$	7155.00	0.315617
$802$	1908.00	0.0840073
$803$	−6846.00	−0.300859
$804$	−7152.00	−0.313721
$805$	12375.0	0.541815
$806$	4732.00	0.206796
$807$	20772.0	0.906083
$808$	−5856.00	−0.254967
$809$	−30102.0	−1.30820	−0.654098	−	0.756410i	$-0.726951\pi$
$809$	−30102.0	−1.30820	−0.654098	+	0.756410i	$0.726951\pi$
$810$	−810.000	−0.0351364
$811$	−14308.0	−0.619509	−0.309755	−	0.950817i	$-0.600247\pi$
$811$	−14308.0	−0.619509	−0.309755	+	0.950817i	$0.600247\pi$
$812$	24600.0	1.06317
$813$	−762.000	−0.0328715
$814$	12390.0	0.533500
$815$	4265.00	0.183309
$816$	−5904.00	−0.253286
$817$	65992.0	2.82591
$818$	364.000	0.0155586
$819$	−2925.00	−0.124796
$820$	−180.000	−0.00766570
$821$	27501.0	1.16905	0.584526	−	0.811375i	$-0.301281\pi$
$821$	27501.0	1.16905	0.584526	+	0.811375i	$0.301281\pi$
$822$	−1620.00	−0.0687396
$823$	2288.00	0.0969072	0.0484536	−	0.998825i	$-0.484571\pi$
$823$	2288.00	0.0969072	0.0484536	+	0.998825i	$0.484571\pi$
$824$	−12800.0	−0.541152
$825$	1575.00	0.0664660
$826$	1200.00	0.0505488
$827$	−1422.00	−0.0597918	−0.0298959	−	0.999553i	$-0.509518\pi$
$827$	−1422.00	−0.0597918	−0.0298959	+	0.999553i	$0.509518\pi$
$828$	3564.00	0.149586
$829$	−42298.0	−1.77210	−0.886050	−	0.463590i	$-0.846561\pi$
$829$	−42298.0	−1.77210	−0.886050	+	0.463590i	$0.846561\pi$
$830$	960.000	0.0401471
$831$	−19770.0	−0.825287
$832$	832.000	0.0346688
$833$	34686.0	1.44274
$834$	42.0000	0.00174381
$835$	−12780.0	−0.529665
$836$	−12264.0	−0.507368
$837$	−4914.00	−0.202930
$838$	−9612.00	−0.396230
$839$	17595.0	0.724013	0.362006	−	0.932176i	$-0.382092\pi$
$839$	17595.0	0.724013	0.362006	+	0.932176i	$0.382092\pi$
$840$	−3000.00	−0.123226
$841$	36127.0	1.48128
$842$	−20048.0	−0.820546
$843$	−7110.00	−0.290488
$844$	−16048.0	−0.654496
$845$	−845.000	−0.0344010
$846$	7020.00	0.285287
$847$	22250.0	0.902620
$848$	5040.00	0.204097
$849$	18840.0	0.761587
$850$	6150.00	0.248169
$851$	−29205.0	−1.17642
$852$	−9252.00	−0.372029
$853$	−13633.0	−0.547227	−0.273614	−	0.961840i	$-0.588219\pi$
$853$	−13633.0	−0.547227	−0.273614	+	0.961840i	$0.588219\pi$
$854$	36350.0	1.45652
$855$	−6570.00	−0.262794
$856$	8088.00	0.322946
$857$	−29319.0	−1.16863	−0.584316	−	0.811526i	$-0.698637\pi$
$857$	−29319.0	−1.16863	−0.584316	+	0.811526i	$0.698637\pi$
$858$	1638.00	0.0651753
$859$	−24235.0	−0.962616	−0.481308	−	0.876551i	$-0.659838\pi$
$859$	−24235.0	−0.962616	−0.481308	+	0.876551i	$0.659838\pi$
$860$	−9040.00	−0.358444
$861$	675.000	0.0267177
$862$	−23088.0	−0.912274
$863$	−7470.00	−0.294649	−0.147324	−	0.989088i	$-0.547066\pi$
$863$	−7470.00	−0.294649	−0.147324	+	0.989088i	$0.547066\pi$
$864$	−864.000	−0.0340207
$865$	5670.00	0.222874
$866$	−536.000	−0.0210324
$867$	−30648.0	−1.20053
$868$	−18200.0	−0.711692
$869$	18669.0	0.728772
$870$	−7380.00	−0.287592
$871$	7748.00	0.301413
$872$	8800.00	0.341750
$873$	8847.00	0.342985
$874$	28908.0	1.11880
$875$	3125.00	0.120736
$876$	−3912.00	−0.150884
$877$	−28762.0	−1.10744	−0.553719	−	0.832703i	$-0.686792\pi$
$877$	−28762.0	−1.10744	−0.553719	+	0.832703i	$0.686792\pi$
$878$	−12470.0	−0.479319
$879$	10764.0	0.413038
$880$	1680.00	0.0643554
$881$	11442.0	0.437560	0.218780	−	0.975774i	$-0.429792\pi$
$881$	11442.0	0.437560	0.218780	+	0.975774i	$0.429792\pi$
$882$	5076.00	0.193784
$883$	−15208.0	−0.579604	−0.289802	−	0.957087i	$-0.593589\pi$
$883$	−15208.0	−0.579604	−0.289802	+	0.957087i	$0.593589\pi$
$884$	6396.00	0.243349
$885$	−360.000	−0.0136737
$886$	18726.0	0.710059
$887$	25461.0	0.963807	0.481903	−	0.876224i	$-0.339946\pi$
$887$	25461.0	0.963807	0.481903	+	0.876224i	$0.339946\pi$
$888$	7080.00	0.267555
$889$	−66650.0	−2.51448
$890$	−7950.00	−0.299421
$891$	−1701.00	−0.0639570
$892$	−4384.00	−0.164560
$893$	56940.0	2.13373
$894$	−18594.0	−0.695611
$895$	−2490.00	−0.0929961
$896$	−3200.00	−0.119313
$897$	−3861.00	−0.143718
$898$	−14046.0	−0.521961
$899$	−44772.0	−1.66099
$900$	900.000	0.0333333
$901$	38745.0	1.43261
$902$	−378.000	−0.0139535
$903$	33900.0	1.24930
$904$	−14160.0	−0.520967
$905$	−9505.00	−0.349124
$906$	17160.0	0.629253
$907$	27326.0	1.00038	0.500190	−	0.865916i	$-0.333263\pi$
$907$	27326.0	1.00038	0.500190	+	0.865916i	$0.333263\pi$
$908$	24552.0	0.897342
$909$	−6588.00	−0.240385
$910$	3250.00	0.118392
$911$	24420.0	0.888113	0.444056	−	0.895999i	$-0.353539\pi$
$911$	24420.0	0.888113	0.444056	+	0.895999i	$0.353539\pi$
$912$	−7008.00	−0.254450
$913$	2016.00	0.0730776
$914$	5206.00	0.188402
$915$	−10905.0	−0.393998
$916$	−5176.00	−0.186703
$917$	−14550.0	−0.523973
$918$	−6642.00	−0.238800
$919$	32951.0	1.18276	0.591378	−	0.806394i	$-0.298584\pi$
$919$	32951.0	1.18276	0.591378	+	0.806394i	$0.298584\pi$
$920$	−3960.00	−0.141910
$921$	19407.0	0.694335
$922$	20754.0	0.741320
$923$	10023.0	0.357433
$924$	−6300.00	−0.224302
$925$	−7375.00	−0.262150
$926$	−22802.0	−0.809201
$927$	−14400.0	−0.510203
$928$	−7872.00	−0.278460
$929$	49779.0	1.75802	0.879008	−	0.476808i	$-0.158206\pi$
$929$	49779.0	1.75802	0.879008	+	0.476808i	$0.158206\pi$
$930$	5460.00	0.192517
$931$	41172.0	1.44936
$932$	3492.00	0.122730
$933$	−4608.00	−0.161693
$934$	−30762.0	−1.07769
$935$	12915.0	0.451728
$936$	936.000	0.0326860
$937$	−44494.0	−1.55129	−0.775643	−	0.631171i	$-0.782574\pi$
$937$	−44494.0	−1.55129	−0.775643	+	0.631171i	$0.782574\pi$
$938$	−29800.0	−1.03732
$939$	−4542.00	−0.157852
$940$	−7800.00	−0.270647
$941$	41235.0	1.42850	0.714252	−	0.699888i	$-0.246767\pi$
$941$	41235.0	1.42850	0.714252	+	0.699888i	$0.246767\pi$
$942$	16404.0	0.567379
$943$	891.000	0.0307688
$944$	−384.000	−0.0132396
$945$	−3375.00	−0.116179
$946$	−18984.0	−0.652456
$947$	−34536.0	−1.18508	−0.592539	−	0.805542i	$-0.701874\pi$
$947$	−34536.0	−1.18508	−0.592539	+	0.805542i	$0.701874\pi$
$948$	10668.0	0.365486
$949$	4238.00	0.144964
$950$	7300.00	0.249309
$951$	8172.00	0.278649
$952$	−24600.0	−0.837490
$953$	−48693.0	−1.65511	−0.827556	−	0.561384i	$-0.810269\pi$
$953$	−48693.0	−1.65511	−0.827556	+	0.561384i	$0.810269\pi$
$954$	5670.00	0.192425
$955$	−11700.0	−0.396443
$956$	20652.0	0.698675
$957$	−15498.0	−0.523489
$958$	−3534.00	−0.119184
$959$	−6750.00	−0.227288
$960$	960.000	0.0322749
$961$	3333.00	0.111879
$962$	−7670.00	−0.257059
$963$	9099.00	0.304477
$964$	−9928.00	−0.331701
$965$	−8425.00	−0.281047
$966$	14850.0	0.494608
$967$	−18088.0	−0.601521	−0.300761	−	0.953700i	$-0.597240\pi$
$967$	−18088.0	−0.601521	−0.300761	+	0.953700i	$0.597240\pi$
$968$	−7120.00	−0.236411
$969$	−53874.0	−1.78605
$970$	−9830.00	−0.325384
$971$	15396.0	0.508837	0.254419	−	0.967094i	$-0.418116\pi$
$971$	15396.0	0.508837	0.254419	+	0.967094i	$0.418116\pi$
$972$	−972.000	−0.0320750
$973$	175.000	0.00576592
$974$	19786.0	0.650908
$975$	−975.000	−0.0320256
$976$	−11632.0	−0.381487
$977$	29700.0	0.972556	0.486278	−	0.873804i	$-0.338354\pi$
$977$	29700.0	0.972556	0.486278	+	0.873804i	$0.338354\pi$
$978$	5118.00	0.167337
$979$	−16695.0	−0.545020
$980$	−5640.00	−0.183840
$981$	9900.00	0.322205
$982$	−30672.0	−0.996724
$983$	54444.0	1.76652	0.883262	−	0.468879i	$-0.155342\pi$
$983$	54444.0	1.76652	0.883262	+	0.468879i	$0.155342\pi$
$984$	−216.000	−0.00699779
$985$	−9960.00	−0.322185
$986$	−60516.0	−1.95459
$987$	29250.0	0.943301
$988$	7592.00	0.244467
$989$	44748.0	1.43873
$990$	1890.00	0.0606749
$991$	−12571.0	−0.402958	−0.201479	−	0.979493i	$-0.564575\pi$
$991$	−12571.0	−0.402958	−0.201479	+	0.979493i	$0.564575\pi$
$992$	5824.00	0.186403
$993$	−3300.00	−0.105461
$994$	−38550.0	−1.23011
$995$	23660.0	0.753842
$996$	1152.00	0.0366491
$997$	−3796.00	−0.120582	−0.0602911	−	0.998181i	$-0.519203\pi$
$997$	−3796.00	−0.120582	−0.0602911	+	0.998181i	$0.519203\pi$
$998$	−32900.0	−1.04352
$999$	7965.00	0.252254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.4.a.g.1.1	✓	1
3.2	odd	2		1170.4.a.e.1.1		1
5.4	even	2		1950.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.g.1.1	✓	1	1.1	even	1	trivial
1170.4.a.e.1.1		1	3.2	odd	2
1950.4.a.g.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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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