LMFDB - Embedded newform 390.4.a.b.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} +5.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -35.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} -10.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} +23.0000 q^{17} -18.0000 q^{18} -30.0000 q^{19} +20.0000 q^{20} -15.0000 q^{21} +70.0000 q^{22} +63.0000 q^{23} +24.0000 q^{24} +25.0000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +20.0000 q^{28} -190.000 q^{29} +30.0000 q^{30} +330.000 q^{31} -32.0000 q^{32} +105.000 q^{33} -46.0000 q^{34} +25.0000 q^{35} +36.0000 q^{36} +43.0000 q^{37} +60.0000 q^{38} +39.0000 q^{39} -40.0000 q^{40} -473.000 q^{41} +30.0000 q^{42} -232.000 q^{43} -140.000 q^{44} +45.0000 q^{45} -126.000 q^{46} +270.000 q^{47} -48.0000 q^{48} -318.000 q^{49} -50.0000 q^{50} -69.0000 q^{51} -52.0000 q^{52} -193.000 q^{53} +54.0000 q^{54} -175.000 q^{55} -40.0000 q^{56} +90.0000 q^{57} +380.000 q^{58} -200.000 q^{59} -60.0000 q^{60} -679.000 q^{61} -660.000 q^{62} +45.0000 q^{63} +64.0000 q^{64} -65.0000 q^{65} -210.000 q^{66} -12.0000 q^{67} +92.0000 q^{68} -189.000 q^{69} -50.0000 q^{70} -899.000 q^{71} -72.0000 q^{72} +154.000 q^{73} -86.0000 q^{74} -75.0000 q^{75} -120.000 q^{76} -175.000 q^{77} -78.0000 q^{78} +215.000 q^{79} +80.0000 q^{80} +81.0000 q^{81} +946.000 q^{82} -1308.00 q^{83} -60.0000 q^{84} +115.000 q^{85} +464.000 q^{86} +570.000 q^{87} +280.000 q^{88} -1019.00 q^{89} -90.0000 q^{90} -65.0000 q^{91} +252.000 q^{92} -990.000 q^{93} -540.000 q^{94} -150.000 q^{95} +96.0000 q^{96} -427.000 q^{97} +636.000 q^{98} -315.000 q^{99} +O(q^{100})\)

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
$7$	5.00000	0.269975	0.134987	−	0.990847i	$-0.456901\pi$
$7$	5.00000	0.269975	0.134987	+	0.990847i	$0.456901\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−35.0000	−0.959354	−0.479677	−	0.877445i	$-0.659246\pi$
$11$	−35.0000	−0.959354	−0.479677	+	0.877445i	$0.659246\pi$
$12$	−12.0000	−0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−10.0000	−0.190901
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	23.0000	0.328136	0.164068	−	0.986449i	$-0.447538\pi$
$17$	23.0000	0.328136	0.164068	+	0.986449i	$0.447538\pi$
$18$	−18.0000	−0.235702
$19$	−30.0000	−0.362235	−0.181118	−	0.983461i	$-0.557971\pi$
$19$	−30.0000	−0.362235	−0.181118	+	0.983461i	$0.557971\pi$
$20$	20.0000	0.223607
$21$	−15.0000	−0.155870
$22$	70.0000	0.678366
$23$	63.0000	0.571148	0.285574	−	0.958357i	$-0.407816\pi$
$23$	63.0000	0.571148	0.285574	+	0.958357i	$0.407816\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	20.0000	0.134987
$29$	−190.000	−1.21662	−0.608312	−	0.793698i	$-0.708153\pi$
$29$	−190.000	−1.21662	−0.608312	+	0.793698i	$0.708153\pi$
$30$	30.0000	0.182574
$31$	330.000	1.91193	0.955964	−	0.293485i	$-0.0948150\pi$
$31$	330.000	1.91193	0.955964	+	0.293485i	$0.0948150\pi$
$32$	−32.0000	−0.176777
$33$	105.000	0.553883
$34$	−46.0000	−0.232027
$35$	25.0000	0.120736
$36$	36.0000	0.166667
$37$	43.0000	0.191058	0.0955291	−	0.995427i	$-0.469546\pi$
$37$	43.0000	0.191058	0.0955291	+	0.995427i	$0.469546\pi$
$38$	60.0000	0.256139
$39$	39.0000	0.160128
$40$	−40.0000	−0.158114
$41$	−473.000	−1.80171	−0.900856	−	0.434118i	$-0.857060\pi$
$41$	−473.000	−1.80171	−0.900856	+	0.434118i	$0.857060\pi$
$42$	30.0000	0.110217
$43$	−232.000	−0.822783	−0.411391	−	0.911459i	$-0.634957\pi$
$43$	−232.000	−0.822783	−0.411391	+	0.911459i	$0.634957\pi$
$44$	−140.000	−0.479677
$45$	45.0000	0.149071
$46$	−126.000	−0.403863
$47$	270.000	0.837948	0.418974	−	0.907998i	$-0.362390\pi$
$47$	270.000	0.837948	0.418974	+	0.907998i	$0.362390\pi$
$48$	−48.0000	−0.144338
$49$	−318.000	−0.927114
$50$	−50.0000	−0.141421
$51$	−69.0000	−0.189450
$52$	−52.0000	−0.138675
$53$	−193.000	−0.500200	−0.250100	−	0.968220i	$-0.580463\pi$
$53$	−193.000	−0.500200	−0.250100	+	0.968220i	$0.580463\pi$
$54$	54.0000	0.136083
$55$	−175.000	−0.429036
$56$	−40.0000	−0.0954504
$57$	90.0000	0.209137
$58$	380.000	0.860284
$59$	−200.000	−0.441318	−0.220659	−	0.975351i	$-0.570821\pi$
$59$	−200.000	−0.441318	−0.220659	+	0.975351i	$0.570821\pi$
$60$	−60.0000	−0.129099
$61$	−679.000	−1.42520	−0.712599	−	0.701572i	$-0.752482\pi$
$61$	−679.000	−1.42520	−0.712599	+	0.701572i	$0.752482\pi$
$62$	−660.000	−1.35194
$63$	45.0000	0.0899915
$64$	64.0000	0.125000
$65$	−65.0000	−0.124035
$66$	−210.000	−0.391655
$67$	−12.0000	−0.0218811	−0.0109405	−	0.999940i	$-0.503483\pi$
$67$	−12.0000	−0.0218811	−0.0109405	+	0.999940i	$0.503483\pi$
$68$	92.0000	0.164068
$69$	−189.000	−0.329753
$70$	−50.0000	−0.0853735
$71$	−899.000	−1.50270	−0.751350	−	0.659904i	$-0.770597\pi$
$71$	−899.000	−1.50270	−0.751350	+	0.659904i	$0.770597\pi$
$72$	−72.0000	−0.117851
$73$	154.000	0.246909	0.123454	−	0.992350i	$-0.460603\pi$
$73$	154.000	0.246909	0.123454	+	0.992350i	$0.460603\pi$
$74$	−86.0000	−0.135099
$75$	−75.0000	−0.115470
$76$	−120.000	−0.181118
$77$	−175.000	−0.259001
$78$	−78.0000	−0.113228
$79$	215.000	0.306195	0.153097	−	0.988211i	$-0.451075\pi$
$79$	215.000	0.306195	0.153097	+	0.988211i	$0.451075\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	946.000	1.27400
$83$	−1308.00	−1.72978	−0.864889	−	0.501962i	$-0.832612\pi$
$83$	−1308.00	−1.72978	−0.864889	+	0.501962i	$0.832612\pi$
$84$	−60.0000	−0.0779350
$85$	115.000	0.146747
$86$	464.000	0.581795
$87$	570.000	0.702419
$88$	280.000	0.339183
$89$	−1019.00	−1.21364	−0.606819	−	0.794840i	$-0.707555\pi$
$89$	−1019.00	−1.21364	−0.606819	+	0.794840i	$0.707555\pi$
$90$	−90.0000	−0.105409
$91$	−65.0000	−0.0748775
$92$	252.000	0.285574
$93$	−990.000	−1.10385
$94$	−540.000	−0.592519
$95$	−150.000	−0.161997
$96$	96.0000	0.102062
$97$	−427.000	−0.446962	−0.223481	−	0.974708i	$-0.571742\pi$
$97$	−427.000	−0.446962	−0.223481	+	0.974708i	$0.571742\pi$
$98$	636.000	0.655568
$99$	−315.000	−0.319785
$100$	100.000	0.100000
$101$	−1144.00	−1.12705	−0.563526	−	0.826098i	$-0.690556\pi$
$101$	−1144.00	−1.12705	−0.563526	+	0.826098i	$0.690556\pi$
$102$	138.000	0.133961
$103$	696.000	0.665815	0.332907	−	0.942960i	$-0.391970\pi$
$103$	696.000	0.665815	0.332907	+	0.942960i	$0.391970\pi$
$104$	104.000	0.0980581
$105$	−75.0000	−0.0697071
$106$	386.000	0.353695
$107$	−1353.00	−1.22242	−0.611212	−	0.791467i	$-0.709318\pi$
$107$	−1353.00	−1.22242	−0.611212	+	0.791467i	$0.709318\pi$
$108$	−108.000	−0.0962250
$109$	504.000	0.442885	0.221442	−	0.975173i	$-0.428924\pi$
$109$	504.000	0.442885	0.221442	+	0.975173i	$0.428924\pi$
$110$	350.000	0.303374
$111$	−129.000	−0.110308
$112$	80.0000	0.0674937
$113$	1790.00	1.49017	0.745084	−	0.666970i	$-0.232409\pi$
$113$	1790.00	1.49017	0.745084	+	0.666970i	$0.232409\pi$
$114$	−180.000	−0.147882
$115$	315.000	0.255425
$116$	−760.000	−0.608312
$117$	−117.000	−0.0924500
$118$	400.000	0.312059
$119$	115.000	0.0885885
$120$	120.000	0.0912871
$121$	−106.000	−0.0796394
$122$	1358.00	1.00777
$123$	1419.00	1.04022
$124$	1320.00	0.955964
$125$	125.000	0.0894427
$126$	−90.0000	−0.0636336
$127$	1306.00	0.912510	0.456255	−	0.889849i	$-0.349191\pi$
$127$	1306.00	0.912510	0.456255	+	0.889849i	$0.349191\pi$
$128$	−128.000	−0.0883883
$129$	696.000	0.475034
$130$	130.000	0.0877058
$131$	−1718.00	−1.14582	−0.572910	−	0.819618i	$-0.694185\pi$
$131$	−1718.00	−1.14582	−0.572910	+	0.819618i	$0.694185\pi$
$132$	420.000	0.276942
$133$	−150.000	−0.0977944
$134$	24.0000	0.0154723
$135$	−135.000	−0.0860663
$136$	−184.000	−0.116014
$137$	−438.000	−0.273145	−0.136573	−	0.990630i	$-0.543609\pi$
$137$	−438.000	−0.273145	−0.136573	+	0.990630i	$0.543609\pi$
$138$	378.000	0.233170
$139$	721.000	0.439960	0.219980	−	0.975504i	$-0.429401\pi$
$139$	721.000	0.439960	0.219980	+	0.975504i	$0.429401\pi$
$140$	100.000	0.0603682
$141$	−810.000	−0.483789
$142$	1798.00	1.06257
$143$	455.000	0.266077
$144$	144.000	0.0833333
$145$	−950.000	−0.544091
$146$	−308.000	−0.174591
$147$	954.000	0.535269
$148$	172.000	0.0955291
$149$	29.0000	0.0159448	0.00797239	−	0.999968i	$-0.497462\pi$
$149$	29.0000	0.0159448	0.00797239	+	0.999968i	$0.497462\pi$
$150$	150.000	0.0816497
$151$	−816.000	−0.439769	−0.219885	−	0.975526i	$-0.570568\pi$
$151$	−816.000	−0.439769	−0.219885	+	0.975526i	$0.570568\pi$
$152$	240.000	0.128070
$153$	207.000	0.109379
$154$	350.000	0.183142
$155$	1650.00	0.855040
$156$	156.000	0.0800641
$157$	−1298.00	−0.659820	−0.329910	−	0.944012i	$-0.607018\pi$
$157$	−1298.00	−0.659820	−0.329910	+	0.944012i	$0.607018\pi$
$158$	−430.000	−0.216512
$159$	579.000	0.288790
$160$	−160.000	−0.0790569
$161$	315.000	0.154196
$162$	−162.000	−0.0785674
$163$	3113.00	1.49588	0.747942	−	0.663764i	$-0.231042\pi$
$163$	3113.00	1.49588	0.747942	+	0.663764i	$0.231042\pi$
$164$	−1892.00	−0.900856
$165$	525.000	0.247704
$166$	2616.00	1.22314
$167$	−172.000	−0.0796992	−0.0398496	−	0.999206i	$-0.512688\pi$
$167$	−172.000	−0.0796992	−0.0398496	+	0.999206i	$0.512688\pi$
$168$	120.000	0.0551083
$169$	169.000	0.0769231
$170$	−230.000	−0.103766
$171$	−270.000	−0.120745
$172$	−928.000	−0.411391
$173$	−1982.00	−0.871033	−0.435516	−	0.900181i	$-0.643434\pi$
$173$	−1982.00	−0.871033	−0.435516	+	0.900181i	$0.643434\pi$
$174$	−1140.00	−0.496685
$175$	125.000	0.0539949
$176$	−560.000	−0.239839
$177$	600.000	0.254795
$178$	2038.00	0.858172
$179$	2990.00	1.24851	0.624254	−	0.781221i	$-0.285403\pi$
$179$	2990.00	1.24851	0.624254	+	0.781221i	$0.285403\pi$
$180$	180.000	0.0745356
$181$	1885.00	0.774094	0.387047	−	0.922060i	$-0.373495\pi$
$181$	1885.00	0.774094	0.387047	+	0.922060i	$0.373495\pi$
$182$	130.000	0.0529464
$183$	2037.00	0.822838
$184$	−504.000	−0.201931
$185$	215.000	0.0854439
$186$	1980.00	0.780541
$187$	−805.000	−0.314799
$188$	1080.00	0.418974
$189$	−135.000	−0.0519566
$190$	300.000	0.114549
$191$	−516.000	−0.195479	−0.0977394	−	0.995212i	$-0.531161\pi$
$191$	−516.000	−0.195479	−0.0977394	+	0.995212i	$0.531161\pi$
$192$	−192.000	−0.0721688
$193$	−2713.00	−1.01184	−0.505922	−	0.862579i	$-0.668848\pi$
$193$	−2713.00	−1.01184	−0.505922	+	0.862579i	$0.668848\pi$
$194$	854.000	0.316050
$195$	195.000	0.0716115
$196$	−1272.00	−0.463557
$197$	1116.00	0.403613	0.201806	−	0.979425i	$-0.435319\pi$
$197$	1116.00	0.403613	0.201806	+	0.979425i	$0.435319\pi$
$198$	630.000	0.226122
$199$	−1164.00	−0.414642	−0.207321	−	0.978273i	$-0.566474\pi$
$199$	−1164.00	−0.414642	−0.207321	+	0.978273i	$0.566474\pi$
$200$	−200.000	−0.0707107
$201$	36.0000	0.0126331
$202$	2288.00	0.796946
$203$	−950.000	−0.328458
$204$	−276.000	−0.0947248
$205$	−2365.00	−0.805750
$206$	−1392.00	−0.470802
$207$	567.000	0.190383
$208$	−208.000	−0.0693375
$209$	1050.00	0.347512
$210$	150.000	0.0492904
$211$	−3572.00	−1.16543	−0.582717	−	0.812675i	$-0.698010\pi$
$211$	−3572.00	−1.16543	−0.582717	+	0.812675i	$0.698010\pi$
$212$	−772.000	−0.250100
$213$	2697.00	0.867584
$214$	2706.00	0.864385
$215$	−1160.00	−0.367960
$216$	216.000	0.0680414
$217$	1650.00	0.516172
$218$	−1008.00	−0.313167
$219$	−462.000	−0.142553
$220$	−700.000	−0.214518
$221$	−299.000	−0.0910087
$222$	258.000	0.0779992
$223$	5248.00	1.57593	0.787964	−	0.615721i	$-0.211135\pi$
$223$	5248.00	1.57593	0.787964	+	0.615721i	$0.211135\pi$
$224$	−160.000	−0.0477252
$225$	225.000	0.0666667
$226$	−3580.00	−1.05371
$227$	5810.00	1.69878	0.849390	−	0.527765i	$-0.176970\pi$
$227$	5810.00	1.69878	0.849390	+	0.527765i	$0.176970\pi$
$228$	360.000	0.104568
$229$	5278.00	1.52306	0.761529	−	0.648131i	$-0.224449\pi$
$229$	5278.00	1.52306	0.761529	+	0.648131i	$0.224449\pi$
$230$	−630.000	−0.180613
$231$	525.000	0.149534
$232$	1520.00	0.430142
$233$	−5091.00	−1.43143	−0.715714	−	0.698394i	$-0.753898\pi$
$233$	−5091.00	−1.43143	−0.715714	+	0.698394i	$0.753898\pi$
$234$	234.000	0.0653720
$235$	1350.00	0.374742
$236$	−800.000	−0.220659
$237$	−645.000	−0.176782
$238$	−230.000	−0.0626415
$239$	4549.00	1.23117	0.615587	−	0.788069i	$-0.288919\pi$
$239$	4549.00	1.23117	0.615587	+	0.788069i	$0.288919\pi$
$240$	−240.000	−0.0645497
$241$	−4534.00	−1.21187	−0.605935	−	0.795514i	$-0.707201\pi$
$241$	−4534.00	−1.21187	−0.605935	+	0.795514i	$0.707201\pi$
$242$	212.000	0.0563135
$243$	−243.000	−0.0641500
$244$	−2716.00	−0.712599
$245$	−1590.00	−0.414618
$246$	−2838.00	−0.735546
$247$	390.000	0.100466
$248$	−2640.00	−0.675968
$249$	3924.00	0.998688
$250$	−250.000	−0.0632456
$251$	−6144.00	−1.54504	−0.772522	−	0.634988i	$-0.781005\pi$
$251$	−6144.00	−1.54504	−0.772522	+	0.634988i	$0.781005\pi$
$252$	180.000	0.0449958
$253$	−2205.00	−0.547933
$254$	−2612.00	−0.645242
$255$	−345.000	−0.0847245
$256$	256.000	0.0625000
$257$	−282.000	−0.0684462	−0.0342231	−	0.999414i	$-0.510896\pi$
$257$	−282.000	−0.0684462	−0.0342231	+	0.999414i	$0.510896\pi$
$258$	−1392.00	−0.335900
$259$	215.000	0.0515809
$260$	−260.000	−0.0620174
$261$	−1710.00	−0.405542
$262$	3436.00	0.810217
$263$	1736.00	0.407020	0.203510	−	0.979073i	$-0.434765\pi$
$263$	1736.00	0.407020	0.203510	+	0.979073i	$0.434765\pi$
$264$	−840.000	−0.195827
$265$	−965.000	−0.223696
$266$	300.000	0.0691511
$267$	3057.00	0.700694
$268$	−48.0000	−0.0109405
$269$	6636.00	1.50410	0.752052	−	0.659104i	$-0.229064\pi$
$269$	6636.00	1.50410	0.752052	+	0.659104i	$0.229064\pi$
$270$	270.000	0.0608581
$271$	106.000	0.0237603	0.0118802	−	0.999929i	$-0.496218\pi$
$271$	106.000	0.0237603	0.0118802	+	0.999929i	$0.496218\pi$
$272$	368.000	0.0820341
$273$	195.000	0.0432305
$274$	876.000	0.193143
$275$	−875.000	−0.191871
$276$	−756.000	−0.164876
$277$	−4366.00	−0.947031	−0.473515	−	0.880786i	$-0.657015\pi$
$277$	−4366.00	−0.947031	−0.473515	+	0.880786i	$0.657015\pi$
$278$	−1442.00	−0.311099
$279$	2970.00	0.637309
$280$	−200.000	−0.0426867
$281$	−4266.00	−0.905652	−0.452826	−	0.891599i	$-0.649584\pi$
$281$	−4266.00	−0.905652	−0.452826	+	0.891599i	$0.649584\pi$
$282$	1620.00	0.342091
$283$	−3076.00	−0.646110	−0.323055	−	0.946380i	$-0.604710\pi$
$283$	−3076.00	−0.646110	−0.323055	+	0.946380i	$0.604710\pi$
$284$	−3596.00	−0.751350
$285$	450.000	0.0935288
$286$	−910.000	−0.188145
$287$	−2365.00	−0.486417
$288$	−288.000	−0.0589256
$289$	−4384.00	−0.892326
$290$	1900.00	0.384730
$291$	1281.00	0.258053
$292$	616.000	0.123454
$293$	7048.00	1.40529	0.702643	−	0.711543i	$-0.252003\pi$
$293$	7048.00	1.40529	0.702643	+	0.711543i	$0.252003\pi$
$294$	−1908.00	−0.378493
$295$	−1000.00	−0.197364
$296$	−344.000	−0.0675493
$297$	945.000	0.184628
$298$	−58.0000	−0.0112747
$299$	−819.000	−0.158408
$300$	−300.000	−0.0577350
$301$	−1160.00	−0.222131
$302$	1632.00	0.310964
$303$	3432.00	0.650704
$304$	−480.000	−0.0905588
$305$	−3395.00	−0.637368
$306$	−414.000	−0.0773425
$307$	−4263.00	−0.792516	−0.396258	−	0.918139i	$-0.629691\pi$
$307$	−4263.00	−0.792516	−0.396258	+	0.918139i	$0.629691\pi$
$308$	−700.000	−0.129501
$309$	−2088.00	−0.384408
$310$	−3300.00	−0.604605
$311$	444.000	0.0809548	0.0404774	−	0.999180i	$-0.487112\pi$
$311$	444.000	0.0809548	0.0404774	+	0.999180i	$0.487112\pi$
$312$	−312.000	−0.0566139
$313$	−4242.00	−0.766045	−0.383022	−	0.923739i	$-0.625117\pi$
$313$	−4242.00	−0.766045	−0.383022	+	0.923739i	$0.625117\pi$
$314$	2596.00	0.466563
$315$	225.000	0.0402454
$316$	860.000	0.153097
$317$	−2736.00	−0.484760	−0.242380	−	0.970181i	$-0.577928\pi$
$317$	−2736.00	−0.484760	−0.242380	+	0.970181i	$0.577928\pi$
$318$	−1158.00	−0.204206
$319$	6650.00	1.16717
$320$	320.000	0.0559017
$321$	4059.00	0.705767
$322$	−630.000	−0.109033
$323$	−690.000	−0.118863
$324$	324.000	0.0555556
$325$	−325.000	−0.0554700
$326$	−6226.00	−1.05775
$327$	−1512.00	−0.255700
$328$	3784.00	0.637001
$329$	1350.00	0.226225
$330$	−1050.00	−0.175153
$331$	6148.00	1.02092	0.510460	−	0.859901i	$-0.329475\pi$
$331$	6148.00	1.02092	0.510460	+	0.859901i	$0.329475\pi$
$332$	−5232.00	−0.864889
$333$	387.000	0.0636861
$334$	344.000	0.0563558
$335$	−60.0000	−0.00978552
$336$	−240.000	−0.0389675
$337$	4016.00	0.649156	0.324578	−	0.945859i	$-0.394778\pi$
$337$	4016.00	0.649156	0.324578	+	0.945859i	$0.394778\pi$
$338$	−338.000	−0.0543928
$339$	−5370.00	−0.860349
$340$	460.000	0.0733735
$341$	−11550.0	−1.83422
$342$	540.000	0.0853797
$343$	−3305.00	−0.520272
$344$	1856.00	0.290898
$345$	−945.000	−0.147470
$346$	3964.00	0.615913
$347$	7299.00	1.12920	0.564598	−	0.825366i	$-0.309031\pi$
$347$	7299.00	1.12920	0.564598	+	0.825366i	$0.309031\pi$
$348$	2280.00	0.351209
$349$	116.000	0.0177918	0.00889590	−	0.999960i	$-0.497168\pi$
$349$	116.000	0.0177918	0.00889590	+	0.999960i	$0.497168\pi$
$350$	−250.000	−0.0381802
$351$	351.000	0.0533761
$352$	1120.00	0.169591
$353$	−6878.00	−1.03705	−0.518525	−	0.855062i	$-0.673519\pi$
$353$	−6878.00	−1.03705	−0.518525	+	0.855062i	$0.673519\pi$
$354$	−1200.00	−0.180167
$355$	−4495.00	−0.672028
$356$	−4076.00	−0.606819
$357$	−345.000	−0.0511466
$358$	−5980.00	−0.882829
$359$	4656.00	0.684497	0.342248	−	0.939610i	$-0.388812\pi$
$359$	4656.00	0.684497	0.342248	+	0.939610i	$0.388812\pi$
$360$	−360.000	−0.0527046
$361$	−5959.00	−0.868786
$362$	−3770.00	−0.547367
$363$	318.000	0.0459798
$364$	−260.000	−0.0374387
$365$	770.000	0.110421
$366$	−4074.00	−0.581834
$367$	−5640.00	−0.802195	−0.401098	−	0.916035i	$-0.631371\pi$
$367$	−5640.00	−0.802195	−0.401098	+	0.916035i	$0.631371\pi$
$368$	1008.00	0.142787
$369$	−4257.00	−0.600571
$370$	−430.000	−0.0604179
$371$	−965.000	−0.135041
$372$	−3960.00	−0.551926
$373$	−4160.00	−0.577471	−0.288735	−	0.957409i	$-0.593235\pi$
$373$	−4160.00	−0.577471	−0.288735	+	0.957409i	$0.593235\pi$
$374$	1610.00	0.222597
$375$	−375.000	−0.0516398
$376$	−2160.00	−0.296259
$377$	2470.00	0.337431
$378$	270.000	0.0367389
$379$	−7854.00	−1.06447	−0.532233	−	0.846598i	$-0.678647\pi$
$379$	−7854.00	−1.06447	−0.532233	+	0.846598i	$0.678647\pi$
$380$	−600.000	−0.0809983
$381$	−3918.00	−0.526838
$382$	1032.00	0.138224
$383$	1206.00	0.160897	0.0804487	−	0.996759i	$-0.474365\pi$
$383$	1206.00	0.160897	0.0804487	+	0.996759i	$0.474365\pi$
$384$	384.000	0.0510310
$385$	−875.000	−0.115829
$386$	5426.00	0.715482
$387$	−2088.00	−0.274261
$388$	−1708.00	−0.223481
$389$	−952.000	−0.124083	−0.0620415	−	0.998074i	$-0.519761\pi$
$389$	−952.000	−0.124083	−0.0620415	+	0.998074i	$0.519761\pi$
$390$	−390.000	−0.0506370
$391$	1449.00	0.187415
$392$	2544.00	0.327784
$393$	5154.00	0.661539
$394$	−2232.00	−0.285397
$395$	1075.00	0.136934
$396$	−1260.00	−0.159892
$397$	5747.00	0.726533	0.363267	−	0.931685i	$-0.381661\pi$
$397$	5747.00	0.726533	0.363267	+	0.931685i	$0.381661\pi$
$398$	2328.00	0.293196
$399$	450.000	0.0564616
$400$	400.000	0.0500000
$401$	11958.0	1.48916	0.744581	−	0.667532i	$-0.232649\pi$
$401$	11958.0	1.48916	0.744581	+	0.667532i	$0.232649\pi$
$402$	−72.0000	−0.00893292
$403$	−4290.00	−0.530273
$404$	−4576.00	−0.563526
$405$	405.000	0.0496904
$406$	1900.00	0.232255
$407$	−1505.00	−0.183293
$408$	552.000	0.0669806
$409$	14878.0	1.79870	0.899352	−	0.437226i	$-0.144039\pi$
$409$	14878.0	1.79870	0.899352	+	0.437226i	$0.144039\pi$
$410$	4730.00	0.569751
$411$	1314.00	0.157700
$412$	2784.00	0.332907
$413$	−1000.00	−0.119145
$414$	−1134.00	−0.134621
$415$	−6540.00	−0.773581
$416$	416.000	0.0490290
$417$	−2163.00	−0.254011
$418$	−2100.00	−0.245728
$419$	3178.00	0.370538	0.185269	−	0.982688i	$-0.440684\pi$
$419$	3178.00	0.370538	0.185269	+	0.982688i	$0.440684\pi$
$420$	−300.000	−0.0348536
$421$	−4412.00	−0.510755	−0.255377	−	0.966841i	$-0.582200\pi$
$421$	−4412.00	−0.510755	−0.255377	+	0.966841i	$0.582200\pi$
$422$	7144.00	0.824086
$423$	2430.00	0.279316
$424$	1544.00	0.176847
$425$	575.000	0.0656273
$426$	−5394.00	−0.613475
$427$	−3395.00	−0.384767
$428$	−5412.00	−0.611212
$429$	−1365.00	−0.153620
$430$	2320.00	0.260187
$431$	9216.00	1.02997	0.514987	−	0.857198i	$-0.327797\pi$
$431$	9216.00	1.02997	0.514987	+	0.857198i	$0.327797\pi$
$432$	−432.000	−0.0481125
$433$	2536.00	0.281460	0.140730	−	0.990048i	$-0.455055\pi$
$433$	2536.00	0.281460	0.140730	+	0.990048i	$0.455055\pi$
$434$	−3300.00	−0.364989
$435$	2850.00	0.314131
$436$	2016.00	0.221442
$437$	−1890.00	−0.206890
$438$	924.000	0.100800
$439$	−1811.00	−0.196889	−0.0984445	−	0.995143i	$-0.531387\pi$
$439$	−1811.00	−0.196889	−0.0984445	+	0.995143i	$0.531387\pi$
$440$	1400.00	0.151687
$441$	−2862.00	−0.309038
$442$	598.000	0.0643528
$443$	14743.0	1.58118	0.790588	−	0.612348i	$-0.209775\pi$
$443$	14743.0	1.58118	0.790588	+	0.612348i	$0.209775\pi$
$444$	−516.000	−0.0551538
$445$	−5095.00	−0.542755
$446$	−10496.0	−1.11435
$447$	−87.0000	−0.00920572
$448$	320.000	0.0337468
$449$	999.000	0.105002	0.0525008	−	0.998621i	$-0.483281\pi$
$449$	999.000	0.105002	0.0525008	+	0.998621i	$0.483281\pi$
$450$	−450.000	−0.0471405
$451$	16555.0	1.72848
$452$	7160.00	0.745084
$453$	2448.00	0.253901
$454$	−11620.0	−1.20122
$455$	−325.000	−0.0334862
$456$	−720.000	−0.0739410
$457$	18497.0	1.89333	0.946666	−	0.322215i	$-0.104427\pi$
$457$	18497.0	1.89333	0.946666	+	0.322215i	$0.104427\pi$
$458$	−10556.0	−1.07696
$459$	−621.000	−0.0631499
$460$	1260.00	0.127713
$461$	3951.00	0.399168	0.199584	−	0.979881i	$-0.436041\pi$
$461$	3951.00	0.399168	0.199584	+	0.979881i	$0.436041\pi$
$462$	−1050.00	−0.105737
$463$	−947.000	−0.0950558	−0.0475279	−	0.998870i	$-0.515134\pi$
$463$	−947.000	−0.0950558	−0.0475279	+	0.998870i	$0.515134\pi$
$464$	−3040.00	−0.304156
$465$	−4950.00	−0.493658
$466$	10182.0	1.01217
$467$	4159.00	0.412110	0.206055	−	0.978540i	$-0.433937\pi$
$467$	4159.00	0.412110	0.206055	+	0.978540i	$0.433937\pi$
$468$	−468.000	−0.0462250
$469$	−60.0000	−0.00590734
$470$	−2700.00	−0.264982
$471$	3894.00	0.380947
$472$	1600.00	0.156030
$473$	8120.00	0.789340
$474$	1290.00	0.125004
$475$	−750.000	−0.0724471
$476$	460.000	0.0442943
$477$	−1737.00	−0.166733
$478$	−9098.00	−0.870571
$479$	14135.0	1.34832	0.674159	−	0.738586i	$-0.264506\pi$
$479$	14135.0	1.34832	0.674159	+	0.738586i	$0.264506\pi$
$480$	480.000	0.0456435
$481$	−559.000	−0.0529900
$482$	9068.00	0.856921
$483$	−945.000	−0.0890248
$484$	−424.000	−0.0398197
$485$	−2135.00	−0.199887
$486$	486.000	0.0453609
$487$	−7609.00	−0.708001	−0.354001	−	0.935245i	$-0.615179\pi$
$487$	−7609.00	−0.708001	−0.354001	+	0.935245i	$0.615179\pi$
$488$	5432.00	0.503883
$489$	−9339.00	−0.863649
$490$	3180.00	0.293179
$491$	−13320.0	−1.22428	−0.612142	−	0.790748i	$-0.709692\pi$
$491$	−13320.0	−1.22428	−0.612142	+	0.790748i	$0.709692\pi$
$492$	5676.00	0.520109
$493$	−4370.00	−0.399219
$494$	−780.000	−0.0710402
$495$	−1575.00	−0.143012
$496$	5280.00	0.477982
$497$	−4495.00	−0.405691
$498$	−7848.00	−0.706179
$499$	−3010.00	−0.270032	−0.135016	−	0.990843i	$-0.543109\pi$
$499$	−3010.00	−0.270032	−0.135016	+	0.990843i	$0.543109\pi$
$500$	500.000	0.0447214
$501$	516.000	0.0460143
$502$	12288.0	1.09251
$503$	16252.0	1.44064	0.720319	−	0.693643i	$-0.243995\pi$
$503$	16252.0	1.44064	0.720319	+	0.693643i	$0.243995\pi$
$504$	−360.000	−0.0318168
$505$	−5720.00	−0.504033
$506$	4410.00	0.387447
$507$	−507.000	−0.0444116
$508$	5224.00	0.456255
$509$	15873.0	1.38224	0.691118	−	0.722742i	$-0.257118\pi$
$509$	15873.0	1.38224	0.691118	+	0.722742i	$0.257118\pi$
$510$	690.000	0.0599092
$511$	770.000	0.0666591
$512$	−512.000	−0.0441942
$513$	810.000	0.0697122
$514$	564.000	0.0483988
$515$	3480.00	0.297761
$516$	2784.00	0.237517
$517$	−9450.00	−0.803889
$518$	−430.000	−0.0364732
$519$	5946.00	0.502891
$520$	520.000	0.0438529
$521$	−7910.00	−0.665150	−0.332575	−	0.943077i	$-0.607918\pi$
$521$	−7910.00	−0.665150	−0.332575	+	0.943077i	$0.607918\pi$
$522$	3420.00	0.286761
$523$	−7020.00	−0.586928	−0.293464	−	0.955970i	$-0.594808\pi$
$523$	−7020.00	−0.586928	−0.293464	+	0.955970i	$0.594808\pi$
$524$	−6872.00	−0.572910
$525$	−375.000	−0.0311740
$526$	−3472.00	−0.287807
$527$	7590.00	0.627373
$528$	1680.00	0.138471
$529$	−8198.00	−0.673790
$530$	1930.00	0.158177
$531$	−1800.00	−0.147106
$532$	−600.000	−0.0488972
$533$	6149.00	0.499705
$534$	−6114.00	−0.495466
$535$	−6765.00	−0.546685
$536$	96.0000	0.00773614
$537$	−8970.00	−0.720827
$538$	−13272.0	−1.06356
$539$	11130.0	0.889430
$540$	−540.000	−0.0430331
$541$	7018.00	0.557722	0.278861	−	0.960332i	$-0.410043\pi$
$541$	7018.00	0.557722	0.278861	+	0.960332i	$0.410043\pi$
$542$	−212.000	−0.0168011
$543$	−5655.00	−0.446923
$544$	−736.000	−0.0580069
$545$	2520.00	0.198064
$546$	−390.000	−0.0305686
$547$	21136.0	1.65212	0.826060	−	0.563582i	$-0.190577\pi$
$547$	21136.0	1.65212	0.826060	+	0.563582i	$0.190577\pi$
$548$	−1752.00	−0.136573
$549$	−6111.00	−0.475066
$550$	1750.00	0.135673
$551$	5700.00	0.440704
$552$	1512.00	0.116585
$553$	1075.00	0.0826648
$554$	8732.00	0.669652
$555$	−645.000	−0.0493310
$556$	2884.00	0.219980
$557$	13070.0	0.994244	0.497122	−	0.867681i	$-0.334390\pi$
$557$	13070.0	0.994244	0.497122	+	0.867681i	$0.334390\pi$
$558$	−5940.00	−0.450646
$559$	3016.00	0.228199
$560$	400.000	0.0301841
$561$	2415.00	0.181749
$562$	8532.00	0.640393
$563$	−15669.0	−1.17295	−0.586474	−	0.809968i	$-0.699484\pi$
$563$	−15669.0	−1.17295	−0.586474	+	0.809968i	$0.699484\pi$
$564$	−3240.00	−0.241895
$565$	8950.00	0.666424
$566$	6152.00	0.456869
$567$	405.000	0.0299972
$568$	7192.00	0.531285
$569$	−1404.00	−0.103442	−0.0517212	−	0.998662i	$-0.516471\pi$
$569$	−1404.00	−0.103442	−0.0517212	+	0.998662i	$0.516471\pi$
$570$	−900.000	−0.0661348
$571$	8791.00	0.644294	0.322147	−	0.946690i	$-0.395595\pi$
$571$	8791.00	0.644294	0.322147	+	0.946690i	$0.395595\pi$
$572$	1820.00	0.133039
$573$	1548.00	0.112860
$574$	4730.00	0.343948
$575$	1575.00	0.114230
$576$	576.000	0.0416667
$577$	−8511.00	−0.614069	−0.307034	−	0.951698i	$-0.599337\pi$
$577$	−8511.00	−0.614069	−0.307034	+	0.951698i	$0.599337\pi$
$578$	8768.00	0.630970
$579$	8139.00	0.584189
$580$	−3800.00	−0.272046
$581$	−6540.00	−0.466996
$582$	−2562.00	−0.182471
$583$	6755.00	0.479869
$584$	−1232.00	−0.0872954
$585$	−585.000	−0.0413449
$586$	−14096.0	−0.993687
$587$	−10494.0	−0.737877	−0.368938	−	0.929454i	$-0.620279\pi$
$587$	−10494.0	−0.737877	−0.368938	+	0.929454i	$0.620279\pi$
$588$	3816.00	0.267635
$589$	−9900.00	−0.692568
$590$	2000.00	0.139557
$591$	−3348.00	−0.233026
$592$	688.000	0.0477646
$593$	−22052.0	−1.52709	−0.763547	−	0.645752i	$-0.776544\pi$
$593$	−22052.0	−1.52709	−0.763547	+	0.645752i	$0.776544\pi$
$594$	−1890.00	−0.130552
$595$	575.000	0.0396180
$596$	116.000	0.00797239
$597$	3492.00	0.239394
$598$	1638.00	0.112011
$599$	−6384.00	−0.435464	−0.217732	−	0.976009i	$-0.569866\pi$
$599$	−6384.00	−0.435464	−0.217732	+	0.976009i	$0.569866\pi$
$600$	600.000	0.0408248
$601$	10627.0	0.721272	0.360636	−	0.932707i	$-0.382560\pi$
$601$	10627.0	0.721272	0.360636	+	0.932707i	$0.382560\pi$
$602$	2320.00	0.157070
$603$	−108.000	−0.00729370
$604$	−3264.00	−0.219885
$605$	−530.000	−0.0356158
$606$	−6864.00	−0.460117
$607$	12668.0	0.847081	0.423541	−	0.905877i	$-0.360787\pi$
$607$	12668.0	0.847081	0.423541	+	0.905877i	$0.360787\pi$
$608$	960.000	0.0640348
$609$	2850.00	0.189635
$610$	6790.00	0.450687
$611$	−3510.00	−0.232405
$612$	828.000	0.0546894
$613$	−11491.0	−0.757124	−0.378562	−	0.925576i	$-0.623581\pi$
$613$	−11491.0	−0.757124	−0.378562	+	0.925576i	$0.623581\pi$
$614$	8526.00	0.560393
$615$	7095.00	0.465200
$616$	1400.00	0.0915708
$617$	−23250.0	−1.51703	−0.758517	−	0.651653i	$-0.774076\pi$
$617$	−23250.0	−1.51703	−0.758517	+	0.651653i	$0.774076\pi$
$618$	4176.00	0.271818
$619$	−20050.0	−1.30190	−0.650951	−	0.759120i	$-0.725630\pi$
$619$	−20050.0	−1.30190	−0.650951	+	0.759120i	$0.725630\pi$
$620$	6600.00	0.427520
$621$	−1701.00	−0.109918
$622$	−888.000	−0.0572437
$623$	−5095.00	−0.327651
$624$	624.000	0.0400320
$625$	625.000	0.0400000
$626$	8484.00	0.541675
$627$	−3150.00	−0.200636
$628$	−5192.00	−0.329910
$629$	989.000	0.0626932
$630$	−450.000	−0.0284578
$631$	−48.0000	−0.00302829	−0.00151414	−	0.999999i	$-0.500482\pi$
$631$	−48.0000	−0.00302829	−0.00151414	+	0.999999i	$0.500482\pi$
$632$	−1720.00	−0.108256
$633$	10716.0	0.672864
$634$	5472.00	0.342777
$635$	6530.00	0.408087
$636$	2316.00	0.144395
$637$	4134.00	0.257135
$638$	−13300.0	−0.825317
$639$	−8091.00	−0.500900
$640$	−640.000	−0.0395285
$641$	912.000	0.0561963	0.0280982	−	0.999605i	$-0.491055\pi$
$641$	912.000	0.0561963	0.0280982	+	0.999605i	$0.491055\pi$
$642$	−8118.00	−0.499053
$643$	13877.0	0.851097	0.425549	−	0.904936i	$-0.360081\pi$
$643$	13877.0	0.851097	0.425549	+	0.904936i	$0.360081\pi$
$644$	1260.00	0.0770978
$645$	3480.00	0.212442
$646$	1380.00	0.0840486
$647$	−15987.0	−0.971428	−0.485714	−	0.874118i	$-0.661440\pi$
$647$	−15987.0	−0.971428	−0.485714	+	0.874118i	$0.661440\pi$
$648$	−648.000	−0.0392837
$649$	7000.00	0.423381
$650$	650.000	0.0392232
$651$	−4950.00	−0.298012
$652$	12452.0	0.747942
$653$	10542.0	0.631762	0.315881	−	0.948799i	$-0.397700\pi$
$653$	10542.0	0.631762	0.315881	+	0.948799i	$0.397700\pi$
$654$	3024.00	0.180807
$655$	−8590.00	−0.512426
$656$	−7568.00	−0.450428
$657$	1386.00	0.0823029
$658$	−2700.00	−0.159965
$659$	22820.0	1.34892	0.674462	−	0.738310i	$-0.264376\pi$
$659$	22820.0	1.34892	0.674462	+	0.738310i	$0.264376\pi$
$660$	2100.00	0.123852
$661$	6360.00	0.374244	0.187122	−	0.982337i	$-0.440084\pi$
$661$	6360.00	0.374244	0.187122	+	0.982337i	$0.440084\pi$
$662$	−12296.0	−0.721900
$663$	897.000	0.0525439
$664$	10464.0	0.611569
$665$	−750.000	−0.0437350
$666$	−774.000	−0.0450329
$667$	−11970.0	−0.694873
$668$	−688.000	−0.0398496
$669$	−15744.0	−0.909863
$670$	120.000	0.00691941
$671$	23765.0	1.36727
$672$	480.000	0.0275542
$673$	−32554.0	−1.86458	−0.932292	−	0.361708i	$-0.882194\pi$
$673$	−32554.0	−1.86458	−0.932292	+	0.361708i	$0.882194\pi$
$674$	−8032.00	−0.459022
$675$	−675.000	−0.0384900
$676$	676.000	0.0384615
$677$	−7065.00	−0.401078	−0.200539	−	0.979686i	$-0.564269\pi$
$677$	−7065.00	−0.401078	−0.200539	+	0.979686i	$0.564269\pi$
$678$	10740.0	0.608359
$679$	−2135.00	−0.120668
$680$	−920.000	−0.0518829
$681$	−17430.0	−0.980792
$682$	23100.0	1.29699
$683$	8236.00	0.461408	0.230704	−	0.973024i	$-0.425897\pi$
$683$	8236.00	0.461408	0.230704	+	0.973024i	$0.425897\pi$
$684$	−1080.00	−0.0603726
$685$	−2190.00	−0.122154
$686$	6610.00	0.367888
$687$	−15834.0	−0.879337
$688$	−3712.00	−0.205696
$689$	2509.00	0.138730
$690$	1890.00	0.104277
$691$	23242.0	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$691$	23242.0	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$692$	−7928.00	−0.435516
$693$	−1575.00	−0.0863338
$694$	−14598.0	−0.798462
$695$	3605.00	0.196756
$696$	−4560.00	−0.248342
$697$	−10879.0	−0.591207
$698$	−232.000	−0.0125807
$699$	15273.0	0.826435
$700$	500.000	0.0269975
$701$	−17308.0	−0.932545	−0.466273	−	0.884641i	$-0.654403\pi$
$701$	−17308.0	−0.932545	−0.466273	+	0.884641i	$0.654403\pi$
$702$	−702.000	−0.0377426
$703$	−1290.00	−0.0692081
$704$	−2240.00	−0.119919
$705$	−4050.00	−0.216357
$706$	13756.0	0.733306
$707$	−5720.00	−0.304275
$708$	2400.00	0.127398
$709$	−4252.00	−0.225229	−0.112614	−	0.993639i	$-0.535922\pi$
$709$	−4252.00	−0.225229	−0.112614	+	0.993639i	$0.535922\pi$
$710$	8990.00	0.475195
$711$	1935.00	0.102065
$712$	8152.00	0.429086
$713$	20790.0	1.09199
$714$	690.000	0.0361661
$715$	2275.00	0.118993
$716$	11960.0	0.624254
$717$	−13647.0	−0.710818
$718$	−9312.00	−0.484012
$719$	−35424.0	−1.83740	−0.918701	−	0.394953i	$-0.870761\pi$
$719$	−35424.0	−1.83740	−0.918701	+	0.394953i	$0.870761\pi$
$720$	720.000	0.0372678
$721$	3480.00	0.179753
$722$	11918.0	0.614324
$723$	13602.0	0.699673
$724$	7540.00	0.387047
$725$	−4750.00	−0.243325
$726$	−636.000	−0.0325126
$727$	3754.00	0.191511	0.0957553	−	0.995405i	$-0.469473\pi$
$727$	3754.00	0.191511	0.0957553	+	0.995405i	$0.469473\pi$
$728$	520.000	0.0264732
$729$	729.000	0.0370370
$730$	−1540.00	−0.0780794
$731$	−5336.00	−0.269985
$732$	8148.00	0.411419
$733$	−35425.0	−1.78506	−0.892532	−	0.450984i	$-0.851073\pi$
$733$	−35425.0	−1.78506	−0.892532	+	0.450984i	$0.851073\pi$
$734$	11280.0	0.567238
$735$	4770.00	0.239380
$736$	−2016.00	−0.100966
$737$	420.000	0.0209917
$738$	8514.00	0.424668
$739$	−7170.00	−0.356905	−0.178452	−	0.983949i	$-0.557109\pi$
$739$	−7170.00	−0.356905	−0.178452	+	0.983949i	$0.557109\pi$
$740$	860.000	0.0427219
$741$	−1170.00	−0.0580041
$742$	1930.00	0.0954886
$743$	35946.0	1.77487	0.887437	−	0.460930i	$-0.152484\pi$
$743$	35946.0	1.77487	0.887437	+	0.460930i	$0.152484\pi$
$744$	7920.00	0.390271
$745$	145.000	0.00713072
$746$	8320.00	0.408334
$747$	−11772.0	−0.576593
$748$	−3220.00	−0.157400
$749$	−6765.00	−0.330024
$750$	750.000	0.0365148
$751$	−20791.0	−1.01022	−0.505109	−	0.863055i	$-0.668548\pi$
$751$	−20791.0	−1.01022	−0.505109	+	0.863055i	$0.668548\pi$
$752$	4320.00	0.209487
$753$	18432.0	0.892031
$754$	−4940.00	−0.238600
$755$	−4080.00	−0.196671
$756$	−540.000	−0.0259783
$757$	−10536.0	−0.505862	−0.252931	−	0.967484i	$-0.581395\pi$
$757$	−10536.0	−0.505862	−0.252931	+	0.967484i	$0.581395\pi$
$758$	15708.0	0.752692
$759$	6615.00	0.316350
$760$	1200.00	0.0572744
$761$	6778.00	0.322868	0.161434	−	0.986884i	$-0.448388\pi$
$761$	6778.00	0.322868	0.161434	+	0.986884i	$0.448388\pi$
$762$	7836.00	0.372531
$763$	2520.00	0.119568
$764$	−2064.00	−0.0977394
$765$	1035.00	0.0489157
$766$	−2412.00	−0.113772
$767$	2600.00	0.122400
$768$	−768.000	−0.0360844
$769$	33436.0	1.56792	0.783962	−	0.620809i	$-0.213196\pi$
$769$	33436.0	1.56792	0.783962	+	0.620809i	$0.213196\pi$
$770$	1750.00	0.0819034
$771$	846.000	0.0395174
$772$	−10852.0	−0.505922
$773$	−13972.0	−0.650114	−0.325057	−	0.945694i	$-0.605383\pi$
$773$	−13972.0	−0.650114	−0.325057	+	0.945694i	$0.605383\pi$
$774$	4176.00	0.193932
$775$	8250.00	0.382385
$776$	3416.00	0.158025
$777$	−645.000	−0.0297802
$778$	1904.00	0.0877400
$779$	14190.0	0.652644
$780$	780.000	0.0358057
$781$	31465.0	1.44162
$782$	−2898.00	−0.132522
$783$	5130.00	0.234140
$784$	−5088.00	−0.231778
$785$	−6490.00	−0.295080
$786$	−10308.0	−0.467779
$787$	−38132.0	−1.72714	−0.863570	−	0.504229i	$-0.831777\pi$
$787$	−38132.0	−1.72714	−0.863570	+	0.504229i	$0.831777\pi$
$788$	4464.00	0.201806
$789$	−5208.00	−0.234993
$790$	−2150.00	−0.0968273
$791$	8950.00	0.402308
$792$	2520.00	0.113061
$793$	8827.00	0.395279
$794$	−11494.0	−0.513737
$795$	2895.00	0.129151
$796$	−4656.00	−0.207321
$797$	−6259.00	−0.278175	−0.139087	−	0.990280i	$-0.544417\pi$
$797$	−6259.00	−0.278175	−0.139087	+	0.990280i	$0.544417\pi$
$798$	−900.000	−0.0399244
$799$	6210.00	0.274961
$800$	−800.000	−0.0353553
$801$	−9171.00	−0.404546
$802$	−23916.0	−1.05300
$803$	−5390.00	−0.236873
$804$	144.000	0.00631653
$805$	1575.00	0.0689583
$806$	8580.00	0.374960
$807$	−19908.0	−0.868395
$808$	9152.00	0.398473
$809$	−2026.00	−0.0880474	−0.0440237	−	0.999030i	$-0.514018\pi$
$809$	−2026.00	−0.0880474	−0.0440237	+	0.999030i	$0.514018\pi$
$810$	−810.000	−0.0351364
$811$	−5888.00	−0.254939	−0.127470	−	0.991842i	$-0.540686\pi$
$811$	−5888.00	−0.254939	−0.127470	+	0.991842i	$0.540686\pi$
$812$	−3800.00	−0.164229
$813$	−318.000	−0.0137180
$814$	3010.00	0.129607
$815$	15565.0	0.668979
$816$	−1104.00	−0.0473624
$817$	6960.00	0.298041
$818$	−29756.0	−1.27188
$819$	−585.000	−0.0249592
$820$	−9460.00	−0.402875
$821$	−31957.0	−1.35847	−0.679237	−	0.733919i	$-0.737689\pi$
$821$	−31957.0	−1.35847	−0.679237	+	0.733919i	$0.737689\pi$
$822$	−2628.00	−0.111511
$823$	−40972.0	−1.73535	−0.867676	−	0.497131i	$-0.834387\pi$
$823$	−40972.0	−1.73535	−0.867676	+	0.497131i	$0.834387\pi$
$824$	−5568.00	−0.235401
$825$	2625.00	0.110777
$826$	2000.00	0.0842481
$827$	−20934.0	−0.880226	−0.440113	−	0.897943i	$-0.645062\pi$
$827$	−20934.0	−0.880226	−0.440113	+	0.897943i	$0.645062\pi$
$828$	2268.00	0.0951914
$829$	−21994.0	−0.921451	−0.460726	−	0.887543i	$-0.652411\pi$
$829$	−21994.0	−0.921451	−0.460726	+	0.887543i	$0.652411\pi$
$830$	13080.0	0.547004
$831$	13098.0	0.546768
$832$	−832.000	−0.0346688
$833$	−7314.00	−0.304220
$834$	4326.00	0.179613
$835$	−860.000	−0.0356425
$836$	4200.00	0.173756
$837$	−8910.00	−0.367951
$838$	−6356.00	−0.262010
$839$	19925.0	0.819890	0.409945	−	0.912110i	$-0.365548\pi$
$839$	19925.0	0.819890	0.409945	+	0.912110i	$0.365548\pi$
$840$	600.000	0.0246452
$841$	11711.0	0.480175
$842$	8824.00	0.361158
$843$	12798.0	0.522878
$844$	−14288.0	−0.582717
$845$	845.000	0.0344010
$846$	−4860.00	−0.197506
$847$	−530.000	−0.0215006
$848$	−3088.00	−0.125050
$849$	9228.00	0.373032
$850$	−1150.00	−0.0464055
$851$	2709.00	0.109123
$852$	10788.0	0.433792
$853$	29821.0	1.19701	0.598506	−	0.801118i	$-0.295761\pi$
$853$	29821.0	1.19701	0.598506	+	0.801118i	$0.295761\pi$
$854$	6790.00	0.272071
$855$	−1350.00	−0.0539989
$856$	10824.0	0.432192
$857$	469.000	0.0186940	0.00934699	−	0.999956i	$-0.497025\pi$
$857$	469.000	0.0186940	0.00934699	+	0.999956i	$0.497025\pi$
$858$	2730.00	0.108625
$859$	−26595.0	−1.05636	−0.528178	−	0.849134i	$-0.677125\pi$
$859$	−26595.0	−1.05636	−0.528178	+	0.849134i	$0.677125\pi$
$860$	−4640.00	−0.183980
$861$	7095.00	0.280833
$862$	−18432.0	−0.728302
$863$	174.000	0.00686330	0.00343165	−	0.999994i	$-0.498908\pi$
$863$	174.000	0.00686330	0.00343165	+	0.999994i	$0.498908\pi$
$864$	864.000	0.0340207
$865$	−9910.00	−0.389538
$866$	−5072.00	−0.199023
$867$	13152.0	0.515185
$868$	6600.00	0.258086
$869$	−7525.00	−0.293749
$870$	−5700.00	−0.222124
$871$	156.000	0.00606872
$872$	−4032.00	−0.156583
$873$	−3843.00	−0.148987
$874$	3780.00	0.146293
$875$	625.000	0.0241473
$876$	−1848.00	−0.0712764
$877$	−18398.0	−0.708388	−0.354194	−	0.935172i	$-0.615245\pi$
$877$	−18398.0	−0.708388	−0.354194	+	0.935172i	$0.615245\pi$
$878$	3622.00	0.139222
$879$	−21144.0	−0.811342
$880$	−2800.00	−0.107259
$881$	−17946.0	−0.686284	−0.343142	−	0.939284i	$-0.611491\pi$
$881$	−17946.0	−0.686284	−0.343142	+	0.939284i	$0.611491\pi$
$882$	5724.00	0.218523
$883$	−27544.0	−1.04975	−0.524875	−	0.851179i	$-0.675888\pi$
$883$	−27544.0	−1.04975	−0.524875	+	0.851179i	$0.675888\pi$
$884$	−1196.00	−0.0455043
$885$	3000.00	0.113948
$886$	−29486.0	−1.11806
$887$	−10967.0	−0.415147	−0.207574	−	0.978219i	$-0.566557\pi$
$887$	−10967.0	−0.415147	−0.207574	+	0.978219i	$0.566557\pi$
$888$	1032.00	0.0389996
$889$	6530.00	0.246355
$890$	10190.0	0.383786
$891$	−2835.00	−0.106595
$892$	20992.0	0.787964
$893$	−8100.00	−0.303534
$894$	174.000	0.00650943
$895$	14950.0	0.558350
$896$	−640.000	−0.0238626
$897$	2457.00	0.0914569
$898$	−1998.00	−0.0742474
$899$	−62700.0	−2.32610
$900$	900.000	0.0333333
$901$	−4439.00	−0.164134
$902$	−33110.0	−1.22222
$903$	3480.00	0.128247
$904$	−14320.0	−0.526854
$905$	9425.00	0.346185
$906$	−4896.00	−0.179535
$907$	−23414.0	−0.857166	−0.428583	−	0.903503i	$-0.640987\pi$
$907$	−23414.0	−0.857166	−0.428583	+	0.903503i	$0.640987\pi$
$908$	23240.0	0.849390
$909$	−10296.0	−0.375684
$910$	650.000	0.0236783
$911$	−14000.0	−0.509156	−0.254578	−	0.967052i	$-0.581936\pi$
$911$	−14000.0	−0.509156	−0.254578	+	0.967052i	$0.581936\pi$
$912$	1440.00	0.0522842
$913$	45780.0	1.65947
$914$	−36994.0	−1.33879
$915$	10185.0	0.367984
$916$	21112.0	0.761529
$917$	−8590.00	−0.309342
$918$	1242.00	0.0446537
$919$	7471.00	0.268167	0.134084	−	0.990970i	$-0.457191\pi$
$919$	7471.00	0.268167	0.134084	+	0.990970i	$0.457191\pi$
$920$	−2520.00	−0.0903065
$921$	12789.0	0.457559
$922$	−7902.00	−0.282254
$923$	11687.0	0.416774
$924$	2100.00	0.0747672
$925$	1075.00	0.0382117
$926$	1894.00	0.0672146
$927$	6264.00	0.221938
$928$	6080.00	0.215071
$929$	15861.0	0.560153	0.280077	−	0.959978i	$-0.409640\pi$
$929$	15861.0	0.560153	0.280077	+	0.959978i	$0.409640\pi$
$930$	9900.00	0.349069
$931$	9540.00	0.335833
$932$	−20364.0	−0.715714
$933$	−1332.00	−0.0467393
$934$	−8318.00	−0.291406
$935$	−4025.00	−0.140782
$936$	936.000	0.0326860
$937$	−7494.00	−0.261279	−0.130639	−	0.991430i	$-0.541703\pi$
$937$	−7494.00	−0.261279	−0.130639	+	0.991430i	$0.541703\pi$
$938$	120.000	0.00417712
$939$	12726.0	0.442276
$940$	5400.00	0.187371
$941$	−29995.0	−1.03912	−0.519558	−	0.854435i	$-0.673904\pi$
$941$	−29995.0	−1.03912	−0.519558	+	0.854435i	$0.673904\pi$
$942$	−7788.00	−0.269370
$943$	−29799.0	−1.02904
$944$	−3200.00	−0.110330
$945$	−675.000	−0.0232357
$946$	−16240.0	−0.558148
$947$	−20792.0	−0.713463	−0.356731	−	0.934207i	$-0.616109\pi$
$947$	−20792.0	−0.713463	−0.356731	+	0.934207i	$0.616109\pi$
$948$	−2580.00	−0.0883908
$949$	−2002.00	−0.0684802
$950$	1500.00	0.0512278
$951$	8208.00	0.279877
$952$	−920.000	−0.0313208
$953$	32895.0	1.11813	0.559063	−	0.829125i	$-0.311161\pi$
$953$	32895.0	1.11813	0.559063	+	0.829125i	$0.311161\pi$
$954$	3474.00	0.117898
$955$	−2580.00	−0.0874208
$956$	18196.0	0.615587
$957$	−19950.0	−0.673868
$958$	−28270.0	−0.953405
$959$	−2190.00	−0.0737422
$960$	−960.000	−0.0322749
$961$	79109.0	2.65547
$962$	1118.00	0.0374696
$963$	−12177.0	−0.407475
$964$	−18136.0	−0.605935
$965$	−13565.0	−0.452511
$966$	1890.00	0.0629501
$967$	−37304.0	−1.24055	−0.620277	−	0.784383i	$-0.712980\pi$
$967$	−37304.0	−1.24055	−0.620277	+	0.784383i	$0.712980\pi$
$968$	848.000	0.0281568
$969$	2070.00	0.0686254
$970$	4270.00	0.141342
$971$	−26248.0	−0.867496	−0.433748	−	0.901034i	$-0.642809\pi$
$971$	−26248.0	−0.867496	−0.433748	+	0.901034i	$0.642809\pi$
$972$	−972.000	−0.0320750
$973$	3605.00	0.118778
$974$	15218.0	0.500633
$975$	975.000	0.0320256
$976$	−10864.0	−0.356299
$977$	−18804.0	−0.615756	−0.307878	−	0.951426i	$-0.599619\pi$
$977$	−18804.0	−0.615756	−0.307878	+	0.951426i	$0.599619\pi$
$978$	18678.0	0.610692
$979$	35665.0	1.16431
$980$	−6360.00	−0.207309
$981$	4536.00	0.147628
$982$	26640.0	0.865699
$983$	26228.0	0.851010	0.425505	−	0.904956i	$-0.360096\pi$
$983$	26228.0	0.851010	0.425505	+	0.904956i	$0.360096\pi$
$984$	−11352.0	−0.367773
$985$	5580.00	0.180501
$986$	8740.00	0.282290
$987$	−4050.00	−0.130611
$988$	1560.00	0.0502330
$989$	−14616.0	−0.469931
$990$	3150.00	0.101125
$991$	23757.0	0.761520	0.380760	−	0.924674i	$-0.375662\pi$
$991$	23757.0	0.761520	0.380760	+	0.924674i	$0.375662\pi$
$992$	−10560.0	−0.337984
$993$	−18444.0	−0.589429
$994$	8990.00	0.286867
$995$	−5820.00	−0.185434
$996$	15696.0	0.499344
$997$	38356.0	1.21840	0.609201	−	0.793016i	$-0.291490\pi$
$997$	38356.0	1.21840	0.609201	+	0.793016i	$0.291490\pi$
$998$	6020.00	0.190942
$999$	−1161.00	−0.0367692

Twists

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

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