LMFDB - Embedded newform 390.4.a.a.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)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -14.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +6.00000 q^{6} -14.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} -36.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} +28.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} +68.0000 q^{17} -18.0000 q^{18} -158.000 q^{19} -20.0000 q^{20} +42.0000 q^{21} +72.0000 q^{22} +46.0000 q^{23} +24.0000 q^{24} +25.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} -56.0000 q^{28} -8.00000 q^{29} -30.0000 q^{30} -176.000 q^{31} -32.0000 q^{32} +108.000 q^{33} -136.000 q^{34} +70.0000 q^{35} +36.0000 q^{36} +62.0000 q^{37} +316.000 q^{38} +39.0000 q^{39} +40.0000 q^{40} +30.0000 q^{41} -84.0000 q^{42} +252.000 q^{43} -144.000 q^{44} -45.0000 q^{45} -92.0000 q^{46} -120.000 q^{47} -48.0000 q^{48} -147.000 q^{49} -50.0000 q^{50} -204.000 q^{51} -52.0000 q^{52} +758.000 q^{53} +54.0000 q^{54} +180.000 q^{55} +112.000 q^{56} +474.000 q^{57} +16.0000 q^{58} +252.000 q^{59} +60.0000 q^{60} +398.000 q^{61} +352.000 q^{62} -126.000 q^{63} +64.0000 q^{64} +65.0000 q^{65} -216.000 q^{66} +884.000 q^{67} +272.000 q^{68} -138.000 q^{69} -140.000 q^{70} -80.0000 q^{71} -72.0000 q^{72} -660.000 q^{73} -124.000 q^{74} -75.0000 q^{75} -632.000 q^{76} +504.000 q^{77} -78.0000 q^{78} +568.000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -60.0000 q^{82} +1084.00 q^{83} +168.000 q^{84} -340.000 q^{85} -504.000 q^{86} +24.0000 q^{87} +288.000 q^{88} +1250.00 q^{89} +90.0000 q^{90} +182.000 q^{91} +184.000 q^{92} +528.000 q^{93} +240.000 q^{94} +790.000 q^{95} +96.0000 q^{96} +84.0000 q^{97} +294.000 q^{98} -324.000 q^{99} +O(q^{100})\)

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
$7$	−14.0000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−14.0000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	28.0000	0.534522
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	68.0000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	68.0000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−18.0000	−0.235702
$19$	−158.000	−1.90777	−0.953886	−	0.300168i	$-0.902957\pi$
$19$	−158.000	−1.90777	−0.953886	+	0.300168i	$0.902957\pi$
$20$	−20.0000	−0.223607
$21$	42.0000	0.436436
$22$	72.0000	0.697748
$23$	46.0000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	46.0000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	−56.0000	−0.377964
$29$	−8.00000	−0.0512263	−0.0256132	−	0.999672i	$-0.508154\pi$
$29$	−8.00000	−0.0512263	−0.0256132	+	0.999672i	$0.508154\pi$
$30$	−30.0000	−0.182574
$31$	−176.000	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$31$	−176.000	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$32$	−32.0000	−0.176777
$33$	108.000	0.569709
$34$	−136.000	−0.685994
$35$	70.0000	0.338062
$36$	36.0000	0.166667
$37$	62.0000	0.275479	0.137740	−	0.990468i	$-0.456016\pi$
$37$	62.0000	0.275479	0.137740	+	0.990468i	$0.456016\pi$
$38$	316.000	1.34900
$39$	39.0000	0.160128
$40$	40.0000	0.158114
$41$	30.0000	0.114273	0.0571367	−	0.998366i	$-0.481803\pi$
$41$	30.0000	0.114273	0.0571367	+	0.998366i	$0.481803\pi$
$42$	−84.0000	−0.308607
$43$	252.000	0.893713	0.446856	−	0.894606i	$-0.352544\pi$
$43$	252.000	0.893713	0.446856	+	0.894606i	$0.352544\pi$
$44$	−144.000	−0.493382
$45$	−45.0000	−0.149071
$46$	−92.0000	−0.294884
$47$	−120.000	−0.372421	−0.186211	−	0.982510i	$-0.559621\pi$
$47$	−120.000	−0.372421	−0.186211	+	0.982510i	$0.559621\pi$
$48$	−48.0000	−0.144338
$49$	−147.000	−0.428571
$50$	−50.0000	−0.141421
$51$	−204.000	−0.560112
$52$	−52.0000	−0.138675
$53$	758.000	1.96452	0.982258	−	0.187537i	$-0.0600503\pi$
$53$	758.000	1.96452	0.982258	+	0.187537i	$0.0600503\pi$
$54$	54.0000	0.136083
$55$	180.000	0.441294
$56$	112.000	0.267261
$57$	474.000	1.10145
$58$	16.0000	0.0362225
$59$	252.000	0.556061	0.278031	−	0.960572i	$-0.410318\pi$
$59$	252.000	0.556061	0.278031	+	0.960572i	$0.410318\pi$
$60$	60.0000	0.129099
$61$	398.000	0.835388	0.417694	−	0.908588i	$-0.362838\pi$
$61$	398.000	0.835388	0.417694	+	0.908588i	$0.362838\pi$
$62$	352.000	0.721033
$63$	−126.000	−0.251976
$64$	64.0000	0.125000
$65$	65.0000	0.124035
$66$	−216.000	−0.402845
$67$	884.000	1.61191	0.805954	−	0.591979i	$-0.201653\pi$
$67$	884.000	1.61191	0.805954	+	0.591979i	$0.201653\pi$
$68$	272.000	0.485071
$69$	−138.000	−0.240772
$70$	−140.000	−0.239046
$71$	−80.0000	−0.133722	−0.0668609	−	0.997762i	$-0.521298\pi$
$71$	−80.0000	−0.133722	−0.0668609	+	0.997762i	$0.521298\pi$
$72$	−72.0000	−0.117851
$73$	−660.000	−1.05818	−0.529090	−	0.848566i	$-0.677467\pi$
$73$	−660.000	−1.05818	−0.529090	+	0.848566i	$0.677467\pi$
$74$	−124.000	−0.194793
$75$	−75.0000	−0.115470
$76$	−632.000	−0.953886
$77$	504.000	0.745924
$78$	−78.0000	−0.113228
$79$	568.000	0.808924	0.404462	−	0.914555i	$-0.367459\pi$
$79$	568.000	0.808924	0.404462	+	0.914555i	$0.367459\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−60.0000	−0.0808036
$83$	1084.00	1.43355	0.716774	−	0.697306i	$-0.245618\pi$
$83$	1084.00	1.43355	0.716774	+	0.697306i	$0.245618\pi$
$84$	168.000	0.218218
$85$	−340.000	−0.433861
$86$	−504.000	−0.631950
$87$	24.0000	0.0295755
$88$	288.000	0.348874
$89$	1250.00	1.48876	0.744381	−	0.667756i	$-0.232745\pi$
$89$	1250.00	1.48876	0.744381	+	0.667756i	$0.232745\pi$
$90$	90.0000	0.105409
$91$	182.000	0.209657
$92$	184.000	0.208514
$93$	528.000	0.588721
$94$	240.000	0.263342
$95$	790.000	0.853182
$96$	96.0000	0.102062
$97$	84.0000	0.0879269	0.0439634	−	0.999033i	$-0.486001\pi$
$97$	84.0000	0.0879269	0.0439634	+	0.999033i	$0.486001\pi$
$98$	294.000	0.303046
$99$	−324.000	−0.328921
$100$	100.000	0.100000
$101$	980.000	0.965482	0.482741	−	0.875763i	$-0.339641\pi$
$101$	980.000	0.965482	0.482741	+	0.875763i	$0.339641\pi$
$102$	408.000	0.396059
$103$	−1708.00	−1.63392	−0.816962	−	0.576691i	$-0.804344\pi$
$103$	−1708.00	−1.63392	−0.816962	+	0.576691i	$0.804344\pi$
$104$	104.000	0.0980581
$105$	−210.000	−0.195180
$106$	−1516.00	−1.38912
$107$	1908.00	1.72386	0.861931	−	0.507025i	$-0.169255\pi$
$107$	1908.00	1.72386	0.861931	+	0.507025i	$0.169255\pi$
$108$	−108.000	−0.0962250
$109$	656.000	0.576453	0.288227	−	0.957562i	$-0.406934\pi$
$109$	656.000	0.576453	0.288227	+	0.957562i	$0.406934\pi$
$110$	−360.000	−0.312042
$111$	−186.000	−0.159048
$112$	−224.000	−0.188982
$113$	−488.000	−0.406258	−0.203129	−	0.979152i	$-0.565111\pi$
$113$	−488.000	−0.406258	−0.203129	+	0.979152i	$0.565111\pi$
$114$	−948.000	−0.778845
$115$	−230.000	−0.186501
$116$	−32.0000	−0.0256132
$117$	−117.000	−0.0924500
$118$	−504.000	−0.393195
$119$	−952.000	−0.733359
$120$	−120.000	−0.0912871
$121$	−35.0000	−0.0262960
$122$	−796.000	−0.590709
$123$	−90.0000	−0.0659758
$124$	−704.000	−0.509847
$125$	−125.000	−0.0894427
$126$	252.000	0.178174
$127$	−1740.00	−1.21575	−0.607874	−	0.794033i	$-0.707977\pi$
$127$	−1740.00	−1.21575	−0.607874	+	0.794033i	$0.707977\pi$
$128$	−128.000	−0.0883883
$129$	−756.000	−0.515985
$130$	−130.000	−0.0877058
$131$	−2486.00	−1.65804	−0.829018	−	0.559221i	$-0.811100\pi$
$131$	−2486.00	−1.65804	−0.829018	+	0.559221i	$0.811100\pi$
$132$	432.000	0.284854
$133$	2212.00	1.44214
$134$	−1768.00	−1.13979
$135$	135.000	0.0860663
$136$	−544.000	−0.342997
$137$	−1614.00	−1.00652	−0.503260	−	0.864135i	$-0.667866\pi$
$137$	−1614.00	−1.00652	−0.503260	+	0.864135i	$0.667866\pi$
$138$	276.000	0.170251
$139$	376.000	0.229438	0.114719	−	0.993398i	$-0.463403\pi$
$139$	376.000	0.229438	0.114719	+	0.993398i	$0.463403\pi$
$140$	280.000	0.169031
$141$	360.000	0.215018
$142$	160.000	0.0945556
$143$	468.000	0.273679
$144$	144.000	0.0833333
$145$	40.0000	0.0229091
$146$	1320.00	0.748246
$147$	441.000	0.247436
$148$	248.000	0.137740
$149$	−58.0000	−0.0318896	−0.0159448	−	0.999873i	$-0.505076\pi$
$149$	−58.0000	−0.0318896	−0.0159448	+	0.999873i	$0.505076\pi$
$150$	150.000	0.0816497
$151$	−224.000	−0.120721	−0.0603605	−	0.998177i	$-0.519225\pi$
$151$	−224.000	−0.120721	−0.0603605	+	0.998177i	$0.519225\pi$
$152$	1264.00	0.674500
$153$	612.000	0.323381
$154$	−1008.00	−0.527448
$155$	880.000	0.456021
$156$	156.000	0.0800641
$157$	−1890.00	−0.960754	−0.480377	−	0.877062i	$-0.659500\pi$
$157$	−1890.00	−0.960754	−0.480377	+	0.877062i	$0.659500\pi$
$158$	−1136.00	−0.571996
$159$	−2274.00	−1.13421
$160$	160.000	0.0790569
$161$	−644.000	−0.315244
$162$	−162.000	−0.0785674
$163$	2056.00	0.987965	0.493983	−	0.869472i	$-0.335541\pi$
$163$	2056.00	0.987965	0.493983	+	0.869472i	$0.335541\pi$
$164$	120.000	0.0571367
$165$	−540.000	−0.254781
$166$	−2168.00	−1.01367
$167$	1556.00	0.720999	0.360500	−	0.932759i	$-0.382606\pi$
$167$	1556.00	0.720999	0.360500	+	0.932759i	$0.382606\pi$
$168$	−336.000	−0.154303
$169$	169.000	0.0769231
$170$	680.000	0.306786
$171$	−1422.00	−0.635924
$172$	1008.00	0.446856
$173$	22.0000	0.00966838	0.00483419	−	0.999988i	$-0.498461\pi$
$173$	22.0000	0.00966838	0.00483419	+	0.999988i	$0.498461\pi$
$174$	−48.0000	−0.0209130
$175$	−350.000	−0.151186
$176$	−576.000	−0.246691
$177$	−756.000	−0.321042
$178$	−2500.00	−1.05271
$179$	998.000	0.416726	0.208363	−	0.978052i	$-0.433186\pi$
$179$	998.000	0.416726	0.208363	+	0.978052i	$0.433186\pi$
$180$	−180.000	−0.0745356
$181$	−422.000	−0.173298	−0.0866492	−	0.996239i	$-0.527616\pi$
$181$	−422.000	−0.173298	−0.0866492	+	0.996239i	$0.527616\pi$
$182$	−364.000	−0.148250
$183$	−1194.00	−0.482312
$184$	−368.000	−0.147442
$185$	−310.000	−0.123198
$186$	−1056.00	−0.416289
$187$	−2448.00	−0.957302
$188$	−480.000	−0.186211
$189$	378.000	0.145479
$190$	−1580.00	−0.603291
$191$	−4928.00	−1.86690	−0.933449	−	0.358710i	$-0.883217\pi$
$191$	−4928.00	−1.86690	−0.933449	+	0.358710i	$0.883217\pi$
$192$	−192.000	−0.0721688
$193$	1416.00	0.528114	0.264057	−	0.964507i	$-0.414939\pi$
$193$	1416.00	0.528114	0.264057	+	0.964507i	$0.414939\pi$
$194$	−168.000	−0.0621737
$195$	−195.000	−0.0716115
$196$	−588.000	−0.214286
$197$	1110.00	0.401443	0.200721	−	0.979648i	$-0.435671\pi$
$197$	1110.00	0.401443	0.200721	+	0.979648i	$0.435671\pi$
$198$	648.000	0.232583
$199$	3856.00	1.37359	0.686795	−	0.726851i	$-0.259017\pi$
$199$	3856.00	1.37359	0.686795	+	0.726851i	$0.259017\pi$
$200$	−200.000	−0.0707107
$201$	−2652.00	−0.930635
$202$	−1960.00	−0.682699
$203$	112.000	0.0387234
$204$	−816.000	−0.280056
$205$	−150.000	−0.0511047
$206$	3416.00	1.15536
$207$	414.000	0.139010
$208$	−208.000	−0.0693375
$209$	5688.00	1.88252
$210$	420.000	0.138013
$211$	−3684.00	−1.20198	−0.600988	−	0.799258i	$-0.705226\pi$
$211$	−3684.00	−1.20198	−0.600988	+	0.799258i	$0.705226\pi$
$212$	3032.00	0.982258
$213$	240.000	0.0772044
$214$	−3816.00	−1.21896
$215$	−1260.00	−0.399680
$216$	216.000	0.0680414
$217$	2464.00	0.770817
$218$	−1312.00	−0.407614
$219$	1980.00	0.610941
$220$	720.000	0.220647
$221$	−884.000	−0.269069
$222$	372.000	0.112464
$223$	−4146.00	−1.24501	−0.622504	−	0.782617i	$-0.713884\pi$
$223$	−4146.00	−1.24501	−0.622504	+	0.782617i	$0.713884\pi$
$224$	448.000	0.133631
$225$	225.000	0.0666667
$226$	976.000	0.287268
$227$	−1356.00	−0.396480	−0.198240	−	0.980154i	$-0.563522\pi$
$227$	−1356.00	−0.396480	−0.198240	+	0.980154i	$0.563522\pi$
$228$	1896.00	0.550727
$229$	2072.00	0.597911	0.298955	−	0.954267i	$-0.403362\pi$
$229$	2072.00	0.597911	0.298955	+	0.954267i	$0.403362\pi$
$230$	460.000	0.131876
$231$	−1512.00	−0.430659
$232$	64.0000	0.0181112
$233$	−1516.00	−0.426251	−0.213125	−	0.977025i	$-0.568364\pi$
$233$	−1516.00	−0.426251	−0.213125	+	0.977025i	$0.568364\pi$
$234$	234.000	0.0653720
$235$	600.000	0.166552
$236$	1008.00	0.278031
$237$	−1704.00	−0.467032
$238$	1904.00	0.518563
$239$	3888.00	1.05228	0.526138	−	0.850399i	$-0.323640\pi$
$239$	3888.00	1.05228	0.526138	+	0.850399i	$0.323640\pi$
$240$	240.000	0.0645497
$241$	−5662.00	−1.51337	−0.756684	−	0.653781i	$-0.773182\pi$
$241$	−5662.00	−1.51337	−0.756684	+	0.653781i	$0.773182\pi$
$242$	70.0000	0.0185941
$243$	−243.000	−0.0641500
$244$	1592.00	0.417694
$245$	735.000	0.191663
$246$	180.000	0.0466520
$247$	2054.00	0.529121
$248$	1408.00	0.360516
$249$	−3252.00	−0.827659
$250$	250.000	0.0632456
$251$	−5370.00	−1.35040	−0.675202	−	0.737633i	$-0.735944\pi$
$251$	−5370.00	−1.35040	−0.675202	+	0.737633i	$0.735944\pi$
$252$	−504.000	−0.125988
$253$	−1656.00	−0.411509
$254$	3480.00	0.859664
$255$	1020.00	0.250490
$256$	256.000	0.0625000
$257$	6944.00	1.68543	0.842714	−	0.538362i	$-0.180957\pi$
$257$	6944.00	1.68543	0.842714	+	0.538362i	$0.180957\pi$
$258$	1512.00	0.364857
$259$	−868.000	−0.208243
$260$	260.000	0.0620174
$261$	−72.0000	−0.0170754
$262$	4972.00	1.17241
$263$	4266.00	1.00020	0.500100	−	0.865967i	$-0.333296\pi$
$263$	4266.00	1.00020	0.500100	+	0.865967i	$0.333296\pi$
$264$	−864.000	−0.201422
$265$	−3790.00	−0.878558
$266$	−4424.00	−1.01975
$267$	−3750.00	−0.859537
$268$	3536.00	0.805954
$269$	524.000	0.118769	0.0593845	−	0.998235i	$-0.481086\pi$
$269$	524.000	0.118769	0.0593845	+	0.998235i	$0.481086\pi$
$270$	−270.000	−0.0608581
$271$	3712.00	0.832059	0.416029	−	0.909351i	$-0.363421\pi$
$271$	3712.00	0.832059	0.416029	+	0.909351i	$0.363421\pi$
$272$	1088.00	0.242536
$273$	−546.000	−0.121046
$274$	3228.00	0.711718
$275$	−900.000	−0.197353
$276$	−552.000	−0.120386
$277$	7854.00	1.70361	0.851807	−	0.523856i	$-0.175507\pi$
$277$	7854.00	1.70361	0.851807	+	0.523856i	$0.175507\pi$
$278$	−752.000	−0.162237
$279$	−1584.00	−0.339898
$280$	−560.000	−0.119523
$281$	2010.00	0.426714	0.213357	−	0.976974i	$-0.431560\pi$
$281$	2010.00	0.426714	0.213357	+	0.976974i	$0.431560\pi$
$282$	−720.000	−0.152040
$283$	2444.00	0.513359	0.256680	−	0.966497i	$-0.417371\pi$
$283$	2444.00	0.513359	0.256680	+	0.966497i	$0.417371\pi$
$284$	−320.000	−0.0668609
$285$	−2370.00	−0.492585
$286$	−936.000	−0.193520
$287$	−420.000	−0.0863826
$288$	−288.000	−0.0589256
$289$	−289.000	−0.0588235
$290$	−80.0000	−0.0161992
$291$	−252.000	−0.0507646
$292$	−2640.00	−0.529090
$293$	6702.00	1.33630	0.668148	−	0.744028i	$-0.267087\pi$
$293$	6702.00	1.33630	0.668148	+	0.744028i	$0.267087\pi$
$294$	−882.000	−0.174964
$295$	−1260.00	−0.248678
$296$	−496.000	−0.0973967
$297$	972.000	0.189903
$298$	116.000	0.0225493
$299$	−598.000	−0.115663
$300$	−300.000	−0.0577350
$301$	−3528.00	−0.675583
$302$	448.000	0.0853626
$303$	−2940.00	−0.557421
$304$	−2528.00	−0.476943
$305$	−1990.00	−0.373597
$306$	−1224.00	−0.228665
$307$	−300.000	−0.0557717	−0.0278858	−	0.999611i	$-0.508877\pi$
$307$	−300.000	−0.0557717	−0.0278858	+	0.999611i	$0.508877\pi$
$308$	2016.00	0.372962
$309$	5124.00	0.943347
$310$	−1760.00	−0.322456
$311$	−3816.00	−0.695773	−0.347887	−	0.937537i	$-0.613101\pi$
$311$	−3816.00	−0.695773	−0.347887	+	0.937537i	$0.613101\pi$
$312$	−312.000	−0.0566139
$313$	2910.00	0.525505	0.262752	−	0.964863i	$-0.415370\pi$
$313$	2910.00	0.525505	0.262752	+	0.964863i	$0.415370\pi$
$314$	3780.00	0.679356
$315$	630.000	0.112687
$316$	2272.00	0.404462
$317$	26.0000	0.00460664	0.00230332	−	0.999997i	$-0.499267\pi$
$317$	26.0000	0.00460664	0.00230332	+	0.999997i	$0.499267\pi$
$318$	4548.00	0.802010
$319$	288.000	0.0505483
$320$	−320.000	−0.0559017
$321$	−5724.00	−0.995273
$322$	1288.00	0.222911
$323$	−10744.0	−1.85081
$324$	324.000	0.0555556
$325$	−325.000	−0.0554700
$326$	−4112.00	−0.698597
$327$	−1968.00	−0.332815
$328$	−240.000	−0.0404018
$329$	1680.00	0.281524
$330$	1080.00	0.180158
$331$	3250.00	0.539686	0.269843	−	0.962904i	$-0.413028\pi$
$331$	3250.00	0.539686	0.269843	+	0.962904i	$0.413028\pi$
$332$	4336.00	0.716774
$333$	558.000	0.0918265
$334$	−3112.00	−0.509824
$335$	−4420.00	−0.720867
$336$	672.000	0.109109
$337$	758.000	0.122525	0.0612624	−	0.998122i	$-0.480487\pi$
$337$	758.000	0.122525	0.0612624	+	0.998122i	$0.480487\pi$
$338$	−338.000	−0.0543928
$339$	1464.00	0.234553
$340$	−1360.00	−0.216930
$341$	6336.00	1.00620
$342$	2844.00	0.449666
$343$	6860.00	1.07990
$344$	−2016.00	−0.315975
$345$	690.000	0.107676
$346$	−44.0000	−0.00683657
$347$	−11168.0	−1.72775	−0.863876	−	0.503705i	$-0.831970\pi$
$347$	−11168.0	−1.72775	−0.863876	+	0.503705i	$0.831970\pi$
$348$	96.0000	0.0147878
$349$	12968.0	1.98900	0.994500	−	0.104735i	$-0.0333994\pi$
$349$	12968.0	1.98900	0.994500	+	0.104735i	$0.0333994\pi$
$350$	700.000	0.106904
$351$	351.000	0.0533761
$352$	1152.00	0.174437
$353$	6218.00	0.937538	0.468769	−	0.883321i	$-0.344698\pi$
$353$	6218.00	0.937538	0.468769	+	0.883321i	$0.344698\pi$
$354$	1512.00	0.227011
$355$	400.000	0.0598022
$356$	5000.00	0.744381
$357$	2856.00	0.423405
$358$	−1996.00	−0.294670
$359$	7296.00	1.07261	0.536307	−	0.844023i	$-0.319819\pi$
$359$	7296.00	1.07261	0.536307	+	0.844023i	$0.319819\pi$
$360$	360.000	0.0527046
$361$	18105.0	2.63960
$362$	844.000	0.122540
$363$	105.000	0.0151820
$364$	728.000	0.104828
$365$	3300.00	0.473233
$366$	2388.00	0.341046
$367$	4492.00	0.638911	0.319456	−	0.947601i	$-0.396500\pi$
$367$	4492.00	0.638911	0.319456	+	0.947601i	$0.396500\pi$
$368$	736.000	0.104257
$369$	270.000	0.0380912
$370$	620.000	0.0871142
$371$	−10612.0	−1.48503
$372$	2112.00	0.294360
$373$	−10378.0	−1.44062	−0.720312	−	0.693651i	$-0.756001\pi$
$373$	−10378.0	−1.44062	−0.720312	+	0.693651i	$0.756001\pi$
$374$	4896.00	0.676915
$375$	375.000	0.0516398
$376$	960.000	0.131671
$377$	104.000	0.0142076
$378$	−756.000	−0.102869
$379$	614.000	0.0832165	0.0416083	−	0.999134i	$-0.486752\pi$
$379$	614.000	0.0832165	0.0416083	+	0.999134i	$0.486752\pi$
$380$	3160.00	0.426591
$381$	5220.00	0.701913
$382$	9856.00	1.32010
$383$	−7460.00	−0.995269	−0.497635	−	0.867387i	$-0.665798\pi$
$383$	−7460.00	−0.995269	−0.497635	+	0.867387i	$0.665798\pi$
$384$	384.000	0.0510310
$385$	−2520.00	−0.333587
$386$	−2832.00	−0.373433
$387$	2268.00	0.297904
$388$	336.000	0.0439634
$389$	−2020.00	−0.263286	−0.131643	−	0.991297i	$-0.542025\pi$
$389$	−2020.00	−0.263286	−0.131643	+	0.991297i	$0.542025\pi$
$390$	390.000	0.0506370
$391$	3128.00	0.404577
$392$	1176.00	0.151523
$393$	7458.00	0.957268
$394$	−2220.00	−0.283863
$395$	−2840.00	−0.361762
$396$	−1296.00	−0.164461
$397$	3262.00	0.412381	0.206190	−	0.978512i	$-0.433893\pi$
$397$	3262.00	0.412381	0.206190	+	0.978512i	$0.433893\pi$
$398$	−7712.00	−0.971275
$399$	−6636.00	−0.832620
$400$	400.000	0.0500000
$401$	−12050.0	−1.50062	−0.750310	−	0.661087i	$-0.770096\pi$
$401$	−12050.0	−1.50062	−0.750310	+	0.661087i	$0.770096\pi$
$402$	5304.00	0.658058
$403$	2288.00	0.282812
$404$	3920.00	0.482741
$405$	−405.000	−0.0496904
$406$	−224.000	−0.0273816
$407$	−2232.00	−0.271833
$408$	1632.00	0.198030
$409$	1198.00	0.144834	0.0724172	−	0.997374i	$-0.476929\pi$
$409$	1198.00	0.144834	0.0724172	+	0.997374i	$0.476929\pi$
$410$	300.000	0.0361364
$411$	4842.00	0.581115
$412$	−6832.00	−0.816962
$413$	−3528.00	−0.420343
$414$	−828.000	−0.0982946
$415$	−5420.00	−0.641102
$416$	416.000	0.0490290
$417$	−1128.00	−0.132466
$418$	−11376.0	−1.33114
$419$	6906.00	0.805203	0.402602	−	0.915375i	$-0.368106\pi$
$419$	6906.00	0.805203	0.402602	+	0.915375i	$0.368106\pi$
$420$	−840.000	−0.0975900
$421$	2724.00	0.315344	0.157672	−	0.987492i	$-0.449601\pi$
$421$	2724.00	0.315344	0.157672	+	0.987492i	$0.449601\pi$
$422$	7368.00	0.849926
$423$	−1080.00	−0.124140
$424$	−6064.00	−0.694561
$425$	1700.00	0.194029
$426$	−480.000	−0.0545917
$427$	−5572.00	−0.631494
$428$	7632.00	0.861931
$429$	−1404.00	−0.158009
$430$	2520.00	0.282617
$431$	5136.00	0.573996	0.286998	−	0.957931i	$-0.407343\pi$
$431$	5136.00	0.573996	0.286998	+	0.957931i	$0.407343\pi$
$432$	−432.000	−0.0481125
$433$	−6218.00	−0.690111	−0.345055	−	0.938582i	$-0.612140\pi$
$433$	−6218.00	−0.690111	−0.345055	+	0.938582i	$0.612140\pi$
$434$	−4928.00	−0.545050
$435$	−120.000	−0.0132266
$436$	2624.00	0.288227
$437$	−7268.00	−0.795596
$438$	−3960.00	−0.432000
$439$	2632.00	0.286147	0.143073	−	0.989712i	$-0.454301\pi$
$439$	2632.00	0.286147	0.143073	+	0.989712i	$0.454301\pi$
$440$	−1440.00	−0.156021
$441$	−1323.00	−0.142857
$442$	1768.00	0.190261
$443$	−408.000	−0.0437577	−0.0218789	−	0.999761i	$-0.506965\pi$
$443$	−408.000	−0.0437577	−0.0218789	+	0.999761i	$0.506965\pi$
$444$	−744.000	−0.0795240
$445$	−6250.00	−0.665794
$446$	8292.00	0.880353
$447$	174.000	0.0184114
$448$	−896.000	−0.0944911
$449$	−4902.00	−0.515233	−0.257617	−	0.966247i	$-0.582937\pi$
$449$	−4902.00	−0.515233	−0.257617	+	0.966247i	$0.582937\pi$
$450$	−450.000	−0.0471405
$451$	−1080.00	−0.112761
$452$	−1952.00	−0.203129
$453$	672.000	0.0696983
$454$	2712.00	0.280353
$455$	−910.000	−0.0937614
$456$	−3792.00	−0.389423
$457$	1788.00	0.183018	0.0915089	−	0.995804i	$-0.470831\pi$
$457$	1788.00	0.183018	0.0915089	+	0.995804i	$0.470831\pi$
$458$	−4144.00	−0.422787
$459$	−1836.00	−0.186704
$460$	−920.000	−0.0932505
$461$	−6590.00	−0.665785	−0.332893	−	0.942965i	$-0.608025\pi$
$461$	−6590.00	−0.665785	−0.332893	+	0.942965i	$0.608025\pi$
$462$	3024.00	0.304522
$463$	10406.0	1.04451	0.522255	−	0.852790i	$-0.325091\pi$
$463$	10406.0	1.04451	0.522255	+	0.852790i	$0.325091\pi$
$464$	−128.000	−0.0128066
$465$	−2640.00	−0.263284
$466$	3032.00	0.301405
$467$	7284.00	0.721763	0.360882	−	0.932612i	$-0.382476\pi$
$467$	7284.00	0.721763	0.360882	+	0.932612i	$0.382476\pi$
$468$	−468.000	−0.0462250
$469$	−12376.0	−1.21849
$470$	−1200.00	−0.117770
$471$	5670.00	0.554692
$472$	−2016.00	−0.196597
$473$	−9072.00	−0.881884
$474$	3408.00	0.330242
$475$	−3950.00	−0.381555
$476$	−3808.00	−0.366679
$477$	6822.00	0.654838
$478$	−7776.00	−0.744071
$479$	−8384.00	−0.799738	−0.399869	−	0.916572i	$-0.630944\pi$
$479$	−8384.00	−0.799738	−0.399869	+	0.916572i	$0.630944\pi$
$480$	−480.000	−0.0456435
$481$	−806.000	−0.0764042
$482$	11324.0	1.07011
$483$	1932.00	0.182006
$484$	−140.000	−0.0131480
$485$	−420.000	−0.0393221
$486$	486.000	0.0453609
$487$	−20294.0	−1.88831	−0.944157	−	0.329496i	$-0.893121\pi$
$487$	−20294.0	−1.88831	−0.944157	+	0.329496i	$0.893121\pi$
$488$	−3184.00	−0.295354
$489$	−6168.00	−0.570402
$490$	−1470.00	−0.135526
$491$	−19302.0	−1.77411	−0.887054	−	0.461666i	$-0.847252\pi$
$491$	−19302.0	−1.77411	−0.887054	+	0.461666i	$0.847252\pi$
$492$	−360.000	−0.0329879
$493$	−544.000	−0.0496968
$494$	−4108.00	−0.374145
$495$	1620.00	0.147098
$496$	−2816.00	−0.254924
$497$	1120.00	0.101084
$498$	6504.00	0.585243
$499$	−5186.00	−0.465245	−0.232622	−	0.972567i	$-0.574731\pi$
$499$	−5186.00	−0.465245	−0.232622	+	0.972567i	$0.574731\pi$
$500$	−500.000	−0.0447214
$501$	−4668.00	−0.416269
$502$	10740.0	0.954880
$503$	−11806.0	−1.04653	−0.523264	−	0.852171i	$-0.675286\pi$
$503$	−11806.0	−1.04653	−0.523264	+	0.852171i	$0.675286\pi$
$504$	1008.00	0.0890871
$505$	−4900.00	−0.431777
$506$	3312.00	0.290981
$507$	−507.000	−0.0444116
$508$	−6960.00	−0.607874
$509$	3002.00	0.261417	0.130709	−	0.991421i	$-0.458275\pi$
$509$	3002.00	0.261417	0.130709	+	0.991421i	$0.458275\pi$
$510$	−2040.00	−0.177123
$511$	9240.00	0.799909
$512$	−512.000	−0.0441942
$513$	4266.00	0.367151
$514$	−13888.0	−1.19178
$515$	8540.00	0.730713
$516$	−3024.00	−0.257993
$517$	4320.00	0.367492
$518$	1736.00	0.147250
$519$	−66.0000	−0.00558204
$520$	−520.000	−0.0438529
$521$	−194.000	−0.0163134	−0.00815671	−	0.999967i	$-0.502596\pi$
$521$	−194.000	−0.0163134	−0.00815671	+	0.999967i	$0.502596\pi$
$522$	144.000	0.0120742
$523$	10268.0	0.858486	0.429243	−	0.903189i	$-0.358780\pi$
$523$	10268.0	0.858486	0.429243	+	0.903189i	$0.358780\pi$
$524$	−9944.00	−0.829018
$525$	1050.00	0.0872872
$526$	−8532.00	−0.707249
$527$	−11968.0	−0.989249
$528$	1728.00	0.142427
$529$	−10051.0	−0.826087
$530$	7580.00	0.621234
$531$	2268.00	0.185354
$532$	8848.00	0.721070
$533$	−390.000	−0.0316938
$534$	7500.00	0.607784
$535$	−9540.00	−0.770935
$536$	−7072.00	−0.569895
$537$	−2994.00	−0.240597
$538$	−1048.00	−0.0839823
$539$	5292.00	0.422899
$540$	540.000	0.0430331
$541$	10900.0	0.866225	0.433112	−	0.901340i	$-0.357415\pi$
$541$	10900.0	0.866225	0.433112	+	0.901340i	$0.357415\pi$
$542$	−7424.00	−0.588354
$543$	1266.00	0.100054
$544$	−2176.00	−0.171499
$545$	−3280.00	−0.257798
$546$	1092.00	0.0855921
$547$	23556.0	1.84128	0.920642	−	0.390409i	$-0.127666\pi$
$547$	23556.0	1.84128	0.920642	+	0.390409i	$0.127666\pi$
$548$	−6456.00	−0.503260
$549$	3582.00	0.278463
$550$	1800.00	0.139550
$551$	1264.00	0.0977281
$552$	1104.00	0.0851257
$553$	−7952.00	−0.611489
$554$	−15708.0	−1.20464
$555$	930.000	0.0711285
$556$	1504.00	0.114719
$557$	−10470.0	−0.796460	−0.398230	−	0.917286i	$-0.630375\pi$
$557$	−10470.0	−0.796460	−0.398230	+	0.917286i	$0.630375\pi$
$558$	3168.00	0.240344
$559$	−3276.00	−0.247871
$560$	1120.00	0.0845154
$561$	7344.00	0.552699
$562$	−4020.00	−0.301732
$563$	22608.0	1.69239	0.846193	−	0.532876i	$-0.178889\pi$
$563$	22608.0	1.69239	0.846193	+	0.532876i	$0.178889\pi$
$564$	1440.00	0.107509
$565$	2440.00	0.181684
$566$	−4888.00	−0.363000
$567$	−1134.00	−0.0839921
$568$	640.000	0.0472778
$569$	23550.0	1.73509	0.867546	−	0.497357i	$-0.165696\pi$
$569$	23550.0	1.73509	0.867546	+	0.497357i	$0.165696\pi$
$570$	4740.00	0.348310
$571$	26424.0	1.93662	0.968310	−	0.249751i	$-0.0803489\pi$
$571$	26424.0	1.93662	0.968310	+	0.249751i	$0.0803489\pi$
$572$	1872.00	0.136840
$573$	14784.0	1.07785
$574$	840.000	0.0610817
$575$	1150.00	0.0834058
$576$	576.000	0.0416667
$577$	17912.0	1.29235	0.646175	−	0.763189i	$-0.276367\pi$
$577$	17912.0	1.29235	0.646175	+	0.763189i	$0.276367\pi$
$578$	578.000	0.0415945
$579$	−4248.00	−0.304906
$580$	160.000	0.0114545
$581$	−15176.0	−1.08366
$582$	504.000	0.0358960
$583$	−27288.0	−1.93851
$584$	5280.00	0.374123
$585$	585.000	0.0413449
$586$	−13404.0	−0.944905
$587$	−23076.0	−1.62257	−0.811285	−	0.584651i	$-0.801231\pi$
$587$	−23076.0	−1.62257	−0.811285	+	0.584651i	$0.801231\pi$
$588$	1764.00	0.123718
$589$	27808.0	1.94535
$590$	2520.00	0.175842
$591$	−3330.00	−0.231773
$592$	992.000	0.0688698
$593$	3382.00	0.234203	0.117101	−	0.993120i	$-0.462640\pi$
$593$	3382.00	0.234203	0.117101	+	0.993120i	$0.462640\pi$
$594$	−1944.00	−0.134282
$595$	4760.00	0.327968
$596$	−232.000	−0.0159448
$597$	−11568.0	−0.793043
$598$	1196.00	0.0817861
$599$	−8296.00	−0.565885	−0.282943	−	0.959137i	$-0.591311\pi$
$599$	−8296.00	−0.565885	−0.282943	+	0.959137i	$0.591311\pi$
$600$	600.000	0.0408248
$601$	−20042.0	−1.36028	−0.680142	−	0.733081i	$-0.738082\pi$
$601$	−20042.0	−1.36028	−0.680142	+	0.733081i	$0.738082\pi$
$602$	7056.00	0.477709
$603$	7956.00	0.537302
$604$	−896.000	−0.0603605
$605$	175.000	0.0117599
$606$	5880.00	0.394156
$607$	−6524.00	−0.436245	−0.218123	−	0.975921i	$-0.569993\pi$
$607$	−6524.00	−0.436245	−0.218123	+	0.975921i	$0.569993\pi$
$608$	5056.00	0.337250
$609$	−336.000	−0.0223570
$610$	3980.00	0.264173
$611$	1560.00	0.103291
$612$	2448.00	0.161690
$613$	−1226.00	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−1226.00	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	600.000	0.0394365
$615$	450.000	0.0295053
$616$	−4032.00	−0.263724
$617$	4226.00	0.275741	0.137871	−	0.990450i	$-0.455974\pi$
$617$	4226.00	0.275741	0.137871	+	0.990450i	$0.455974\pi$
$618$	−10248.0	−0.667047
$619$	−14866.0	−0.965291	−0.482645	−	0.875816i	$-0.660324\pi$
$619$	−14866.0	−0.965291	−0.482645	+	0.875816i	$0.660324\pi$
$620$	3520.00	0.228011
$621$	−1242.00	−0.0802572
$622$	7632.00	0.491986
$623$	−17500.0	−1.12540
$624$	624.000	0.0400320
$625$	625.000	0.0400000
$626$	−5820.00	−0.371588
$627$	−17064.0	−1.08687
$628$	−7560.00	−0.480377
$629$	4216.00	0.267254
$630$	−1260.00	−0.0796819
$631$	11580.0	0.730575	0.365287	−	0.930895i	$-0.380971\pi$
$631$	11580.0	0.730575	0.365287	+	0.930895i	$0.380971\pi$
$632$	−4544.00	−0.285998
$633$	11052.0	0.693961
$634$	−52.0000	−0.00325739
$635$	8700.00	0.543699
$636$	−9096.00	−0.567107
$637$	1911.00	0.118864
$638$	−576.000	−0.0357430
$639$	−720.000	−0.0445740
$640$	640.000	0.0395285
$641$	12738.0	0.784900	0.392450	−	0.919773i	$-0.371628\pi$
$641$	12738.0	0.784900	0.392450	+	0.919773i	$0.371628\pi$
$642$	11448.0	0.703764
$643$	−22220.0	−1.36279	−0.681393	−	0.731918i	$-0.738625\pi$
$643$	−22220.0	−1.36279	−0.681393	+	0.731918i	$0.738625\pi$
$644$	−2576.00	−0.157622
$645$	3780.00	0.230756
$646$	21488.0	1.30872
$647$	17994.0	1.09338	0.546690	−	0.837335i	$-0.315888\pi$
$647$	17994.0	1.09338	0.546690	+	0.837335i	$0.315888\pi$
$648$	−648.000	−0.0392837
$649$	−9072.00	−0.548701
$650$	650.000	0.0392232
$651$	−7392.00	−0.445031
$652$	8224.00	0.493983
$653$	10146.0	0.608030	0.304015	−	0.952667i	$-0.401673\pi$
$653$	10146.0	0.608030	0.304015	+	0.952667i	$0.401673\pi$
$654$	3936.00	0.235336
$655$	12430.0	0.741497
$656$	480.000	0.0285684
$657$	−5940.00	−0.352727
$658$	−3360.00	−0.199068
$659$	8370.00	0.494763	0.247382	−	0.968918i	$-0.420430\pi$
$659$	8370.00	0.494763	0.247382	+	0.968918i	$0.420430\pi$
$660$	−2160.00	−0.127391
$661$	3404.00	0.200303	0.100151	−	0.994972i	$-0.468067\pi$
$661$	3404.00	0.200303	0.100151	+	0.994972i	$0.468067\pi$
$662$	−6500.00	−0.381616
$663$	2652.00	0.155347
$664$	−8672.00	−0.506836
$665$	−11060.0	−0.644945
$666$	−1116.00	−0.0649311
$667$	−368.000	−0.0213628
$668$	6224.00	0.360500
$669$	12438.0	0.718805
$670$	8840.00	0.509730
$671$	−14328.0	−0.824331
$672$	−1344.00	−0.0771517
$673$	−5078.00	−0.290851	−0.145425	−	0.989369i	$-0.546455\pi$
$673$	−5078.00	−0.290851	−0.145425	+	0.989369i	$0.546455\pi$
$674$	−1516.00	−0.0866382
$675$	−675.000	−0.0384900
$676$	676.000	0.0384615
$677$	−1094.00	−0.0621061	−0.0310531	−	0.999518i	$-0.509886\pi$
$677$	−1094.00	−0.0621061	−0.0310531	+	0.999518i	$0.509886\pi$
$678$	−2928.00	−0.165854
$679$	−1176.00	−0.0664665
$680$	2720.00	0.153393
$681$	4068.00	0.228908
$682$	−12672.0	−0.711490
$683$	3108.00	0.174121	0.0870603	−	0.996203i	$-0.472253\pi$
$683$	3108.00	0.174121	0.0870603	+	0.996203i	$0.472253\pi$
$684$	−5688.00	−0.317962
$685$	8070.00	0.450130
$686$	−13720.0	−0.763604
$687$	−6216.00	−0.345204
$688$	4032.00	0.223428
$689$	−9854.00	−0.544858
$690$	−1380.00	−0.0761387
$691$	−23638.0	−1.30135	−0.650674	−	0.759357i	$-0.725514\pi$
$691$	−23638.0	−1.30135	−0.650674	+	0.759357i	$0.725514\pi$
$692$	88.0000	0.00483419
$693$	4536.00	0.248641
$694$	22336.0	1.22170
$695$	−1880.00	−0.102608
$696$	−192.000	−0.0104565
$697$	2040.00	0.110862
$698$	−25936.0	−1.40644
$699$	4548.00	0.246096
$700$	−1400.00	−0.0755929
$701$	27060.0	1.45798	0.728989	−	0.684526i	$-0.239991\pi$
$701$	27060.0	1.45798	0.728989	+	0.684526i	$0.239991\pi$
$702$	−702.000	−0.0377426
$703$	−9796.00	−0.525552
$704$	−2304.00	−0.123346
$705$	−1800.00	−0.0961588
$706$	−12436.0	−0.662939
$707$	−13720.0	−0.729836
$708$	−3024.00	−0.160521
$709$	3536.00	0.187302	0.0936511	−	0.995605i	$-0.470146\pi$
$709$	3536.00	0.187302	0.0936511	+	0.995605i	$0.470146\pi$
$710$	−800.000	−0.0422866
$711$	5112.00	0.269641
$712$	−10000.0	−0.526357
$713$	−8096.00	−0.425242
$714$	−5712.00	−0.299392
$715$	−2340.00	−0.122393
$716$	3992.00	0.208363
$717$	−11664.0	−0.607531
$718$	−14592.0	−0.758452
$719$	17204.0	0.892352	0.446176	−	0.894945i	$-0.352786\pi$
$719$	17204.0	0.892352	0.446176	+	0.894945i	$0.352786\pi$
$720$	−720.000	−0.0372678
$721$	23912.0	1.23513
$722$	−36210.0	−1.86648
$723$	16986.0	0.873743
$724$	−1688.00	−0.0866492
$725$	−200.000	−0.0102453
$726$	−210.000	−0.0107353
$727$	−24224.0	−1.23579	−0.617894	−	0.786261i	$-0.712014\pi$
$727$	−24224.0	−1.23579	−0.617894	+	0.786261i	$0.712014\pi$
$728$	−1456.00	−0.0741249
$729$	729.000	0.0370370
$730$	−6600.00	−0.334626
$731$	17136.0	0.867029
$732$	−4776.00	−0.241156
$733$	25958.0	1.30802	0.654011	−	0.756485i	$-0.273085\pi$
$733$	25958.0	1.30802	0.654011	+	0.756485i	$0.273085\pi$
$734$	−8984.00	−0.451779
$735$	−2205.00	−0.110657
$736$	−1472.00	−0.0737210
$737$	−31824.0	−1.59057
$738$	−540.000	−0.0269345
$739$	12790.0	0.636655	0.318327	−	0.947981i	$-0.396879\pi$
$739$	12790.0	0.636655	0.318327	+	0.947981i	$0.396879\pi$
$740$	−1240.00	−0.0615991
$741$	−6162.00	−0.305488
$742$	21224.0	1.05008
$743$	5428.00	0.268013	0.134007	−	0.990980i	$-0.457216\pi$
$743$	5428.00	0.268013	0.134007	+	0.990980i	$0.457216\pi$
$744$	−4224.00	−0.208144
$745$	290.000	0.0142614
$746$	20756.0	1.01867
$747$	9756.00	0.477849
$748$	−9792.00	−0.478651
$749$	−26712.0	−1.30312
$750$	−750.000	−0.0365148
$751$	37720.0	1.83279	0.916393	−	0.400280i	$-0.131087\pi$
$751$	37720.0	1.83279	0.916393	+	0.400280i	$0.131087\pi$
$752$	−1920.00	−0.0931053
$753$	16110.0	0.779656
$754$	−208.000	−0.0100463
$755$	1120.00	0.0539880
$756$	1512.00	0.0727393
$757$	10034.0	0.481759	0.240880	−	0.970555i	$-0.422564\pi$
$757$	10034.0	0.481759	0.240880	+	0.970555i	$0.422564\pi$
$758$	−1228.00	−0.0588430
$759$	4968.00	0.237585
$760$	−6320.00	−0.301645
$761$	−2586.00	−0.123183	−0.0615916	−	0.998101i	$-0.519618\pi$
$761$	−2586.00	−0.123183	−0.0615916	+	0.998101i	$0.519618\pi$
$762$	−10440.0	−0.496327
$763$	−9184.00	−0.435758
$764$	−19712.0	−0.933449
$765$	−3060.00	−0.144620
$766$	14920.0	0.703762
$767$	−3276.00	−0.154224
$768$	−768.000	−0.0360844
$769$	2810.00	0.131770	0.0658850	−	0.997827i	$-0.479013\pi$
$769$	2810.00	0.131770	0.0658850	+	0.997827i	$0.479013\pi$
$770$	5040.00	0.235882
$771$	−20832.0	−0.973082
$772$	5664.00	0.264057
$773$	28958.0	1.34741	0.673704	−	0.739001i	$-0.264702\pi$
$773$	28958.0	1.34741	0.673704	+	0.739001i	$0.264702\pi$
$774$	−4536.00	−0.210650
$775$	−4400.00	−0.203939
$776$	−672.000	−0.0310868
$777$	2604.00	0.120229
$778$	4040.00	0.186171
$779$	−4740.00	−0.218008
$780$	−780.000	−0.0358057
$781$	2880.00	0.131952
$782$	−6256.00	−0.286079
$783$	216.000	0.00985851
$784$	−2352.00	−0.107143
$785$	9450.00	0.429662
$786$	−14916.0	−0.676891
$787$	35384.0	1.60267	0.801336	−	0.598214i	$-0.204123\pi$
$787$	35384.0	1.60267	0.801336	+	0.598214i	$0.204123\pi$
$788$	4440.00	0.200721
$789$	−12798.0	−0.577466
$790$	5680.00	0.255804
$791$	6832.00	0.307102
$792$	2592.00	0.116291
$793$	−5174.00	−0.231695
$794$	−6524.00	−0.291597
$795$	11370.0	0.507236
$796$	15424.0	0.686795
$797$	3526.00	0.156709	0.0783547	−	0.996926i	$-0.475033\pi$
$797$	3526.00	0.156709	0.0783547	+	0.996926i	$0.475033\pi$
$798$	13272.0	0.588752
$799$	−8160.00	−0.361302
$800$	−800.000	−0.0353553
$801$	11250.0	0.496254
$802$	24100.0	1.06110
$803$	23760.0	1.04417
$804$	−10608.0	−0.465318
$805$	3220.00	0.140981
$806$	−4576.00	−0.199979
$807$	−1572.00	−0.0685713
$808$	−7840.00	−0.341349
$809$	29666.0	1.28925	0.644624	−	0.764500i	$-0.277014\pi$
$809$	29666.0	1.28925	0.644624	+	0.764500i	$0.277014\pi$
$810$	810.000	0.0351364
$811$	19666.0	0.851500	0.425750	−	0.904841i	$-0.360010\pi$
$811$	19666.0	0.851500	0.425750	+	0.904841i	$0.360010\pi$
$812$	448.000	0.0193617
$813$	−11136.0	−0.480389
$814$	4464.00	0.192215
$815$	−10280.0	−0.441832
$816$	−3264.00	−0.140028
$817$	−39816.0	−1.70500
$818$	−2396.00	−0.102413
$819$	1638.00	0.0698857
$820$	−600.000	−0.0255523
$821$	19702.0	0.837521	0.418760	−	0.908097i	$-0.362465\pi$
$821$	19702.0	0.837521	0.418760	+	0.908097i	$0.362465\pi$
$822$	−9684.00	−0.410910
$823$	32860.0	1.39177	0.695886	−	0.718153i	$-0.255012\pi$
$823$	32860.0	1.39177	0.695886	+	0.718153i	$0.255012\pi$
$824$	13664.0	0.577680
$825$	2700.00	0.113942
$826$	7056.00	0.297227
$827$	17412.0	0.732134	0.366067	−	0.930589i	$-0.380704\pi$
$827$	17412.0	0.732134	0.366067	+	0.930589i	$0.380704\pi$
$828$	1656.00	0.0695048
$829$	46910.0	1.96532	0.982661	−	0.185412i	$-0.0593621\pi$
$829$	46910.0	1.96532	0.982661	+	0.185412i	$0.0593621\pi$
$830$	10840.0	0.453328
$831$	−23562.0	−0.983582
$832$	−832.000	−0.0346688
$833$	−9996.00	−0.415775
$834$	2256.00	0.0936677
$835$	−7780.00	−0.322441
$836$	22752.0	0.941261
$837$	4752.00	0.196240
$838$	−13812.0	−0.569365
$839$	−13720.0	−0.564561	−0.282281	−	0.959332i	$-0.591091\pi$
$839$	−13720.0	−0.564561	−0.282281	+	0.959332i	$0.591091\pi$
$840$	1680.00	0.0690066
$841$	−24325.0	−0.997376
$842$	−5448.00	−0.222982
$843$	−6030.00	−0.246363
$844$	−14736.0	−0.600988
$845$	−845.000	−0.0344010
$846$	2160.00	0.0877805
$847$	490.000	0.0198779
$848$	12128.0	0.491129
$849$	−7332.00	−0.296388
$850$	−3400.00	−0.137199
$851$	2852.00	0.114883
$852$	960.000	0.0386022
$853$	−10610.0	−0.425885	−0.212942	−	0.977065i	$-0.568305\pi$
$853$	−10610.0	−0.425885	−0.212942	+	0.977065i	$0.568305\pi$
$854$	11144.0	0.446534
$855$	7110.00	0.284394
$856$	−15264.0	−0.609478
$857$	44488.0	1.77326	0.886628	−	0.462482i	$-0.153041\pi$
$857$	44488.0	1.77326	0.886628	+	0.462482i	$0.153041\pi$
$858$	2808.00	0.111729
$859$	7764.00	0.308387	0.154193	−	0.988041i	$-0.450722\pi$
$859$	7764.00	0.308387	0.154193	+	0.988041i	$0.450722\pi$
$860$	−5040.00	−0.199840
$861$	1260.00	0.0498730
$862$	−10272.0	−0.405877
$863$	40572.0	1.60033	0.800166	−	0.599778i	$-0.204745\pi$
$863$	40572.0	1.60033	0.800166	+	0.599778i	$0.204745\pi$
$864$	864.000	0.0340207
$865$	−110.000	−0.00432383
$866$	12436.0	0.487982
$867$	867.000	0.0339618
$868$	9856.00	0.385408
$869$	−20448.0	−0.798217
$870$	240.000	0.00935260
$871$	−11492.0	−0.447063
$872$	−5248.00	−0.203807
$873$	756.000	0.0293090
$874$	14536.0	0.562572
$875$	1750.00	0.0676123
$876$	7920.00	0.305470
$877$	−49734.0	−1.91493	−0.957467	−	0.288541i	$-0.906830\pi$
$877$	−49734.0	−1.91493	−0.957467	+	0.288541i	$0.906830\pi$
$878$	−5264.00	−0.202336
$879$	−20106.0	−0.771511
$880$	2880.00	0.110324
$881$	−27498.0	−1.05157	−0.525784	−	0.850618i	$-0.676228\pi$
$881$	−27498.0	−1.05157	−0.525784	+	0.850618i	$0.676228\pi$
$882$	2646.00	0.101015
$883$	−31908.0	−1.21607	−0.608035	−	0.793910i	$-0.708042\pi$
$883$	−31908.0	−1.21607	−0.608035	+	0.793910i	$0.708042\pi$
$884$	−3536.00	−0.134535
$885$	3780.00	0.143574
$886$	816.000	0.0309414
$887$	−36842.0	−1.39463	−0.697313	−	0.716767i	$-0.745621\pi$
$887$	−36842.0	−1.39463	−0.697313	+	0.716767i	$0.745621\pi$
$888$	1488.00	0.0562320
$889$	24360.0	0.919019
$890$	12500.0	0.470788
$891$	−2916.00	−0.109640
$892$	−16584.0	−0.622504
$893$	18960.0	0.710495
$894$	−348.000	−0.0130189
$895$	−4990.00	−0.186366
$896$	1792.00	0.0668153
$897$	1794.00	0.0667781
$898$	9804.00	0.364325
$899$	1408.00	0.0522352
$900$	900.000	0.0333333
$901$	51544.0	1.90586
$902$	2160.00	0.0797341
$903$	10584.0	0.390048
$904$	3904.00	0.143634
$905$	2110.00	0.0775014
$906$	−1344.00	−0.0492841
$907$	10852.0	0.397282	0.198641	−	0.980072i	$-0.436347\pi$
$907$	10852.0	0.397282	0.198641	+	0.980072i	$0.436347\pi$
$908$	−5424.00	−0.198240
$909$	8820.00	0.321827
$910$	1820.00	0.0662994
$911$	7484.00	0.272180	0.136090	−	0.990696i	$-0.456546\pi$
$911$	7484.00	0.272180	0.136090	+	0.990696i	$0.456546\pi$
$912$	7584.00	0.275363
$913$	−39024.0	−1.41457
$914$	−3576.00	−0.129413
$915$	5970.00	0.215696
$916$	8288.00	0.298955
$917$	34804.0	1.25336
$918$	3672.00	0.132020
$919$	4768.00	0.171145	0.0855723	−	0.996332i	$-0.472728\pi$
$919$	4768.00	0.171145	0.0855723	+	0.996332i	$0.472728\pi$
$920$	1840.00	0.0659380
$921$	900.000	0.0321998
$922$	13180.0	0.470781
$923$	1040.00	0.0370878
$924$	−6048.00	−0.215330
$925$	1550.00	0.0550959
$926$	−20812.0	−0.738580
$927$	−15372.0	−0.544642
$928$	256.000	0.00905562
$929$	6994.00	0.247003	0.123501	−	0.992344i	$-0.460588\pi$
$929$	6994.00	0.247003	0.123501	+	0.992344i	$0.460588\pi$
$930$	5280.00	0.186170
$931$	23226.0	0.817617
$932$	−6064.00	−0.213125
$933$	11448.0	0.401705
$934$	−14568.0	−0.510364
$935$	12240.0	0.428119
$936$	936.000	0.0326860
$937$	−48270.0	−1.68294	−0.841469	−	0.540306i	$-0.818308\pi$
$937$	−48270.0	−1.68294	−0.841469	+	0.540306i	$0.818308\pi$
$938$	24752.0	0.861601
$939$	−8730.00	−0.303400
$940$	2400.00	0.0832759
$941$	−19458.0	−0.674084	−0.337042	−	0.941490i	$-0.609426\pi$
$941$	−19458.0	−0.674084	−0.337042	+	0.941490i	$0.609426\pi$
$942$	−11340.0	−0.392226
$943$	1380.00	0.0476553
$944$	4032.00	0.139015
$945$	−1890.00	−0.0650600
$946$	18144.0	0.623586
$947$	−11548.0	−0.396261	−0.198131	−	0.980176i	$-0.563487\pi$
$947$	−11548.0	−0.396261	−0.198131	+	0.980176i	$0.563487\pi$
$948$	−6816.00	−0.233516
$949$	8580.00	0.293486
$950$	7900.00	0.269800
$951$	−78.0000	−0.00265965
$952$	7616.00	0.259281
$953$	35172.0	1.19552	0.597761	−	0.801674i	$-0.296057\pi$
$953$	35172.0	1.19552	0.597761	+	0.801674i	$0.296057\pi$
$954$	−13644.0	−0.463041
$955$	24640.0	0.834902
$956$	15552.0	0.526138
$957$	−864.000	−0.0291841
$958$	16768.0	0.565501
$959$	22596.0	0.760858
$960$	960.000	0.0322749
$961$	1185.00	0.0397771
$962$	1612.00	0.0540260
$963$	17172.0	0.574621
$964$	−22648.0	−0.756684
$965$	−7080.00	−0.236180
$966$	−3864.00	−0.128698
$967$	−56062.0	−1.86436	−0.932178	−	0.362000i	$-0.882094\pi$
$967$	−56062.0	−1.86436	−0.932178	+	0.362000i	$0.882094\pi$
$968$	280.000	0.00929705
$969$	32232.0	1.06857
$970$	840.000	0.0278049
$971$	−20058.0	−0.662916	−0.331458	−	0.943470i	$-0.607541\pi$
$971$	−20058.0	−0.662916	−0.331458	+	0.943470i	$0.607541\pi$
$972$	−972.000	−0.0320750
$973$	−5264.00	−0.173439
$974$	40588.0	1.33524
$975$	975.000	0.0320256
$976$	6368.00	0.208847
$977$	53134.0	1.73993	0.869963	−	0.493117i	$-0.164143\pi$
$977$	53134.0	1.73993	0.869963	+	0.493117i	$0.164143\pi$
$978$	12336.0	0.403335
$979$	−45000.0	−1.46906
$980$	2940.00	0.0958315
$981$	5904.00	0.192151
$982$	38604.0	1.25448
$983$	6244.00	0.202597	0.101298	−	0.994856i	$-0.467700\pi$
$983$	6244.00	0.202597	0.101298	+	0.994856i	$0.467700\pi$
$984$	720.000	0.0233260
$985$	−5550.00	−0.179531
$986$	1088.00	0.0351410
$987$	−5040.00	−0.162538
$988$	8216.00	0.264561
$989$	11592.0	0.372704
$990$	−3240.00	−0.104014
$991$	55840.0	1.78993	0.894963	−	0.446141i	$-0.147202\pi$
$991$	55840.0	1.78993	0.894963	+	0.446141i	$0.147202\pi$
$992$	5632.00	0.180258
$993$	−9750.00	−0.311588
$994$	−2240.00	−0.0714773
$995$	−19280.0	−0.614289
$996$	−13008.0	−0.413830
$997$	−1282.00	−0.0407235	−0.0203618	−	0.999793i	$-0.506482\pi$
$997$	−1282.00	−0.0407235	−0.0203618	+	0.999793i	$0.506482\pi$
$998$	10372.0	0.328978
$999$	−1674.00	−0.0530160

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.4.a.a.1.1	✓	1	1.1	even	1	trivial
1170.4.a.n.1.1		1	3.2	odd	2
1950.4.a.q.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

Related objects

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