LMFDB - Embedded newform 2310.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2310,4,Mod(1,2310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2310, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2310.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2310.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:	$136.294412113$
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	$\chi$	$=$	2310.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} +11.0000 q^{11} +12.0000 q^{12} +54.0000 q^{13} +14.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} -78.0000 q^{17} -18.0000 q^{18} +132.000 q^{19} +20.0000 q^{20} -21.0000 q^{21} -22.0000 q^{22} -80.0000 q^{23} -24.0000 q^{24} +25.0000 q^{25} -108.000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -218.000 q^{29} -30.0000 q^{30} -248.000 q^{31} -32.0000 q^{32} +33.0000 q^{33} +156.000 q^{34} -35.0000 q^{35} +36.0000 q^{36} -378.000 q^{37} -264.000 q^{38} +162.000 q^{39} -40.0000 q^{40} -278.000 q^{41} +42.0000 q^{42} -84.0000 q^{43} +44.0000 q^{44} +45.0000 q^{45} +160.000 q^{46} +360.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} -50.0000 q^{50} -234.000 q^{51} +216.000 q^{52} -114.000 q^{53} -54.0000 q^{54} +55.0000 q^{55} +56.0000 q^{56} +396.000 q^{57} +436.000 q^{58} -92.0000 q^{59} +60.0000 q^{60} +230.000 q^{61} +496.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} +270.000 q^{65} -66.0000 q^{66} +316.000 q^{67} -312.000 q^{68} -240.000 q^{69} +70.0000 q^{70} -920.000 q^{71} -72.0000 q^{72} +1010.00 q^{73} +756.000 q^{74} +75.0000 q^{75} +528.000 q^{76} -77.0000 q^{77} -324.000 q^{78} -1240.00 q^{79} +80.0000 q^{80} +81.0000 q^{81} +556.000 q^{82} -1164.00 q^{83} -84.0000 q^{84} -390.000 q^{85} +168.000 q^{86} -654.000 q^{87} -88.0000 q^{88} +1050.00 q^{89} -90.0000 q^{90} -378.000 q^{91} -320.000 q^{92} -744.000 q^{93} -720.000 q^{94} +660.000 q^{95} -96.0000 q^{96} +626.000 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	12.0000	0.288675
$13$	54.0000	1.15207	0.576035	−	0.817425i	$-0.304599\pi$
$13$	54.0000	1.15207	0.576035	+	0.817425i	$0.304599\pi$
$14$	14.0000	0.267261
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−78.0000	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$18$	−18.0000	−0.235702
$19$	132.000	1.59384	0.796918	−	0.604088i	$-0.206462\pi$
$19$	132.000	1.59384	0.796918	+	0.604088i	$0.206462\pi$
$20$	20.0000	0.223607
$21$	−21.0000	−0.218218
$22$	−22.0000	−0.213201
$23$	−80.0000	−0.725268	−0.362634	−	0.931932i	$-0.618122\pi$
$23$	−80.0000	−0.725268	−0.362634	+	0.931932i	$0.618122\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	−108.000	−0.814636
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−218.000	−1.39592	−0.697958	−	0.716138i	$-0.745908\pi$
$29$	−218.000	−1.39592	−0.697958	+	0.716138i	$0.745908\pi$
$30$	−30.0000	−0.182574
$31$	−248.000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−248.000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−32.0000	−0.176777
$33$	33.0000	0.174078
$34$	156.000	0.786876
$35$	−35.0000	−0.169031
$36$	36.0000	0.166667
$37$	−378.000	−1.67954	−0.839768	−	0.542946i	$-0.817309\pi$
$37$	−378.000	−1.67954	−0.839768	+	0.542946i	$0.817309\pi$
$38$	−264.000	−1.12701
$39$	162.000	0.665148
$40$	−40.0000	−0.158114
$41$	−278.000	−1.05893	−0.529467	−	0.848330i	$-0.677608\pi$
$41$	−278.000	−1.05893	−0.529467	+	0.848330i	$0.677608\pi$
$42$	42.0000	0.154303
$43$	−84.0000	−0.297904	−0.148952	−	0.988844i	$-0.547590\pi$
$43$	−84.0000	−0.297904	−0.148952	+	0.988844i	$0.547590\pi$
$44$	44.0000	0.150756
$45$	45.0000	0.149071
$46$	160.000	0.512842
$47$	360.000	1.11726	0.558632	−	0.829416i	$-0.311326\pi$
$47$	360.000	1.11726	0.558632	+	0.829416i	$0.311326\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−50.0000	−0.141421
$51$	−234.000	−0.642481
$52$	216.000	0.576035
$53$	−114.000	−0.295455	−0.147727	−	0.989028i	$-0.547196\pi$
$53$	−114.000	−0.295455	−0.147727	+	0.989028i	$0.547196\pi$
$54$	−54.0000	−0.136083
$55$	55.0000	0.134840
$56$	56.0000	0.133631
$57$	396.000	0.920201
$58$	436.000	0.987062
$59$	−92.0000	−0.203006	−0.101503	−	0.994835i	$-0.532365\pi$
$59$	−92.0000	−0.203006	−0.101503	+	0.994835i	$0.532365\pi$
$60$	60.0000	0.129099
$61$	230.000	0.482762	0.241381	−	0.970430i	$-0.422400\pi$
$61$	230.000	0.482762	0.241381	+	0.970430i	$0.422400\pi$
$62$	496.000	1.01600
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	270.000	0.515221
$66$	−66.0000	−0.123091
$67$	316.000	0.576202	0.288101	−	0.957600i	$-0.406976\pi$
$67$	316.000	0.576202	0.288101	+	0.957600i	$0.406976\pi$
$68$	−312.000	−0.556405
$69$	−240.000	−0.418733
$70$	70.0000	0.119523
$71$	−920.000	−1.53780	−0.768901	−	0.639368i	$-0.779196\pi$
$71$	−920.000	−1.53780	−0.768901	+	0.639368i	$0.779196\pi$
$72$	−72.0000	−0.117851
$73$	1010.00	1.61934	0.809668	−	0.586888i	$-0.199647\pi$
$73$	1010.00	1.61934	0.809668	+	0.586888i	$0.199647\pi$
$74$	756.000	1.18761
$75$	75.0000	0.115470
$76$	528.000	0.796918
$77$	−77.0000	−0.113961
$78$	−324.000	−0.470330
$79$	−1240.00	−1.76596	−0.882980	−	0.469410i	$-0.844467\pi$
$79$	−1240.00	−1.76596	−0.882980	+	0.469410i	$0.844467\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	556.000	0.748780
$83$	−1164.00	−1.53934	−0.769672	−	0.638439i	$-0.779580\pi$
$83$	−1164.00	−1.53934	−0.769672	+	0.638439i	$0.779580\pi$
$84$	−84.0000	−0.109109
$85$	−390.000	−0.497664
$86$	168.000	0.210650
$87$	−654.000	−0.805933
$88$	−88.0000	−0.106600
$89$	1050.00	1.25056	0.625280	−	0.780401i	$-0.284985\pi$
$89$	1050.00	1.25056	0.625280	+	0.780401i	$0.284985\pi$
$90$	−90.0000	−0.105409
$91$	−378.000	−0.435441
$92$	−320.000	−0.362634
$93$	−744.000	−0.829561
$94$	−720.000	−0.790025
$95$	660.000	0.712785
$96$	−96.0000	−0.102062
$97$	626.000	0.655265	0.327632	−	0.944805i	$-0.393749\pi$
$97$	626.000	0.655265	0.327632	+	0.944805i	$0.393749\pi$
$98$	−98.0000	−0.101015
$99$	99.0000	0.100504
$100$	100.000	0.100000
$101$	−1170.00	−1.15267	−0.576333	−	0.817215i	$-0.695517\pi$
$101$	−1170.00	−1.15267	−0.576333	+	0.817215i	$0.695517\pi$
$102$	468.000	0.454303
$103$	448.000	0.428570	0.214285	−	0.976771i	$-0.431258\pi$
$103$	448.000	0.428570	0.214285	+	0.976771i	$0.431258\pi$
$104$	−432.000	−0.407318
$105$	−105.000	−0.0975900
$106$	228.000	0.208918
$107$	1068.00	0.964930	0.482465	−	0.875915i	$-0.339742\pi$
$107$	1068.00	0.964930	0.482465	+	0.875915i	$0.339742\pi$
$108$	108.000	0.0962250
$109$	342.000	0.300529	0.150264	−	0.988646i	$-0.451987\pi$
$109$	342.000	0.300529	0.150264	+	0.988646i	$0.451987\pi$
$110$	−110.000	−0.0953463
$111$	−1134.00	−0.969680
$112$	−112.000	−0.0944911
$113$	−86.0000	−0.0715947	−0.0357973	−	0.999359i	$-0.511397\pi$
$113$	−86.0000	−0.0715947	−0.0357973	+	0.999359i	$0.511397\pi$
$114$	−792.000	−0.650681
$115$	−400.000	−0.324349
$116$	−872.000	−0.697958
$117$	486.000	0.384023
$118$	184.000	0.143547
$119$	546.000	0.420603
$120$	−120.000	−0.0912871
$121$	121.000	0.0909091
$122$	−460.000	−0.341364
$123$	−834.000	−0.611376
$124$	−992.000	−0.718421
$125$	125.000	0.0894427
$126$	126.000	0.0890871
$127$	−1888.00	−1.31916	−0.659578	−	0.751636i	$-0.729265\pi$
$127$	−1888.00	−1.31916	−0.659578	+	0.751636i	$0.729265\pi$
$128$	−128.000	−0.0883883
$129$	−252.000	−0.171995
$130$	−540.000	−0.364316
$131$	1004.00	0.669617	0.334809	−	0.942286i	$-0.391328\pi$
$131$	1004.00	0.669617	0.334809	+	0.942286i	$0.391328\pi$
$132$	132.000	0.0870388
$133$	−924.000	−0.602413
$134$	−632.000	−0.407436
$135$	135.000	0.0860663
$136$	624.000	0.393438
$137$	930.000	0.579965	0.289983	−	0.957032i	$-0.406350\pi$
$137$	930.000	0.579965	0.289983	+	0.957032i	$0.406350\pi$
$138$	480.000	0.296089
$139$	−628.000	−0.383211	−0.191605	−	0.981472i	$-0.561369\pi$
$139$	−628.000	−0.383211	−0.191605	+	0.981472i	$0.561369\pi$
$140$	−140.000	−0.0845154
$141$	1080.00	0.645053
$142$	1840.00	1.08739
$143$	594.000	0.347362
$144$	144.000	0.0833333
$145$	−1090.00	−0.624273
$146$	−2020.00	−1.14504
$147$	147.000	0.0824786
$148$	−1512.00	−0.839768
$149$	−1026.00	−0.564115	−0.282058	−	0.959397i	$-0.591017\pi$
$149$	−1026.00	−0.564115	−0.282058	+	0.959397i	$0.591017\pi$
$150$	−150.000	−0.0816497
$151$	−1136.00	−0.612228	−0.306114	−	0.951995i	$-0.599029\pi$
$151$	−1136.00	−0.612228	−0.306114	+	0.951995i	$0.599029\pi$
$152$	−1056.00	−0.563506
$153$	−702.000	−0.370937
$154$	154.000	0.0805823
$155$	−1240.00	−0.642575
$156$	648.000	0.332574
$157$	−978.000	−0.497152	−0.248576	−	0.968612i	$-0.579963\pi$
$157$	−978.000	−0.497152	−0.248576	+	0.968612i	$0.579963\pi$
$158$	2480.00	1.24872
$159$	−342.000	−0.170581
$160$	−160.000	−0.0790569
$161$	560.000	0.274125
$162$	−162.000	−0.0785674
$163$	−3668.00	−1.76258	−0.881288	−	0.472579i	$-0.843323\pi$
$163$	−3668.00	−1.76258	−0.881288	+	0.472579i	$0.843323\pi$
$164$	−1112.00	−0.529467
$165$	165.000	0.0778499
$166$	2328.00	1.08848
$167$	1016.00	0.470781	0.235391	−	0.971901i	$-0.424363\pi$
$167$	1016.00	0.470781	0.235391	+	0.971901i	$0.424363\pi$
$168$	168.000	0.0771517
$169$	719.000	0.327264
$170$	780.000	0.351902
$171$	1188.00	0.531279
$172$	−336.000	−0.148952
$173$	3582.00	1.57419	0.787094	−	0.616833i	$-0.211585\pi$
$173$	3582.00	1.57419	0.787094	+	0.616833i	$0.211585\pi$
$174$	1308.00	0.569881
$175$	−175.000	−0.0755929
$176$	176.000	0.0753778
$177$	−276.000	−0.117206
$178$	−2100.00	−0.884279
$179$	−852.000	−0.355762	−0.177881	−	0.984052i	$-0.556924\pi$
$179$	−852.000	−0.355762	−0.177881	+	0.984052i	$0.556924\pi$
$180$	180.000	0.0745356
$181$	−1458.00	−0.598742	−0.299371	−	0.954137i	$-0.596777\pi$
$181$	−1458.00	−0.598742	−0.299371	+	0.954137i	$0.596777\pi$
$182$	756.000	0.307904
$183$	690.000	0.278723
$184$	640.000	0.256421
$185$	−1890.00	−0.751111
$186$	1488.00	0.586588
$187$	−858.000	−0.335525
$188$	1440.00	0.558632
$189$	−189.000	−0.0727393
$190$	−1320.00	−0.504015
$191$	−3520.00	−1.33350	−0.666749	−	0.745282i	$-0.732315\pi$
$191$	−3520.00	−1.33350	−0.666749	+	0.745282i	$0.732315\pi$
$192$	192.000	0.0721688
$193$	−1318.00	−0.491563	−0.245782	−	0.969325i	$-0.579045\pi$
$193$	−1318.00	−0.491563	−0.245782	+	0.969325i	$0.579045\pi$
$194$	−1252.00	−0.463342
$195$	810.000	0.297463
$196$	196.000	0.0714286
$197$	−3338.00	−1.20722	−0.603611	−	0.797279i	$-0.706272\pi$
$197$	−3338.00	−1.20722	−0.603611	+	0.797279i	$0.706272\pi$
$198$	−198.000	−0.0710669
$199$	2544.00	0.906228	0.453114	−	0.891453i	$-0.350313\pi$
$199$	2544.00	0.906228	0.453114	+	0.891453i	$0.350313\pi$
$200$	−200.000	−0.0707107
$201$	948.000	0.332670
$202$	2340.00	0.815059
$203$	1526.00	0.527607
$204$	−936.000	−0.321241
$205$	−1390.00	−0.473570
$206$	−896.000	−0.303045
$207$	−720.000	−0.241756
$208$	864.000	0.288017
$209$	1452.00	0.480560
$210$	210.000	0.0690066
$211$	3252.00	1.06103	0.530514	−	0.847676i	$-0.321999\pi$
$211$	3252.00	1.06103	0.530514	+	0.847676i	$0.321999\pi$
$212$	−456.000	−0.147727
$213$	−2760.00	−0.887850
$214$	−2136.00	−0.682308
$215$	−420.000	−0.133227
$216$	−216.000	−0.0680414
$217$	1736.00	0.543075
$218$	−684.000	−0.212506
$219$	3030.00	0.934924
$220$	220.000	0.0674200
$221$	−4212.00	−1.28204
$222$	2268.00	0.685668
$223$	−1368.00	−0.410798	−0.205399	−	0.978678i	$-0.565849\pi$
$223$	−1368.00	−0.410798	−0.205399	+	0.978678i	$0.565849\pi$
$224$	224.000	0.0668153
$225$	225.000	0.0666667
$226$	172.000	0.0506251
$227$	2900.00	0.847928	0.423964	−	0.905679i	$-0.360638\pi$
$227$	2900.00	0.847928	0.423964	+	0.905679i	$0.360638\pi$
$228$	1584.00	0.460101
$229$	−3346.00	−0.965545	−0.482773	−	0.875746i	$-0.660370\pi$
$229$	−3346.00	−0.965545	−0.482773	+	0.875746i	$0.660370\pi$
$230$	800.000	0.229350
$231$	−231.000	−0.0657952
$232$	1744.00	0.493531
$233$	−294.000	−0.0826634	−0.0413317	−	0.999145i	$-0.513160\pi$
$233$	−294.000	−0.0826634	−0.0413317	+	0.999145i	$0.513160\pi$
$234$	−972.000	−0.271545
$235$	1800.00	0.499656
$236$	−368.000	−0.101503
$237$	−3720.00	−1.01958
$238$	−1092.00	−0.297411
$239$	5424.00	1.46799	0.733995	−	0.679155i	$-0.237654\pi$
$239$	5424.00	1.46799	0.733995	+	0.679155i	$0.237654\pi$
$240$	240.000	0.0645497
$241$	4994.00	1.33482	0.667410	−	0.744690i	$-0.267403\pi$
$241$	4994.00	1.33482	0.667410	+	0.744690i	$0.267403\pi$
$242$	−242.000	−0.0642824
$243$	243.000	0.0641500
$244$	920.000	0.241381
$245$	245.000	0.0638877
$246$	1668.00	0.432308
$247$	7128.00	1.83621
$248$	1984.00	0.508001
$249$	−3492.00	−0.888741
$250$	−250.000	−0.0632456
$251$	−3148.00	−0.791633	−0.395817	−	0.918330i	$-0.629538\pi$
$251$	−3148.00	−0.791633	−0.395817	+	0.918330i	$0.629538\pi$
$252$	−252.000	−0.0629941
$253$	−880.000	−0.218676
$254$	3776.00	0.932785
$255$	−1170.00	−0.287326
$256$	256.000	0.0625000
$257$	−5398.00	−1.31019	−0.655093	−	0.755548i	$-0.727371\pi$
$257$	−5398.00	−1.31019	−0.655093	+	0.755548i	$0.727371\pi$
$258$	504.000	0.121619
$259$	2646.00	0.634805
$260$	1080.00	0.257611
$261$	−1962.00	−0.465306
$262$	−2008.00	−0.473491
$263$	−4488.00	−1.05225	−0.526125	−	0.850407i	$-0.676356\pi$
$263$	−4488.00	−1.05225	−0.526125	+	0.850407i	$0.676356\pi$
$264$	−264.000	−0.0615457
$265$	−570.000	−0.132131
$266$	1848.00	0.425970
$267$	3150.00	0.722011
$268$	1264.00	0.288101
$269$	−2026.00	−0.459210	−0.229605	−	0.973284i	$-0.573743\pi$
$269$	−2026.00	−0.459210	−0.229605	+	0.973284i	$0.573743\pi$
$270$	−270.000	−0.0608581
$271$	−5992.00	−1.34313	−0.671565	−	0.740946i	$-0.734377\pi$
$271$	−5992.00	−1.34313	−0.671565	+	0.740946i	$0.734377\pi$
$272$	−1248.00	−0.278203
$273$	−1134.00	−0.251402
$274$	−1860.00	−0.410097
$275$	275.000	0.0603023
$276$	−960.000	−0.209367
$277$	4894.00	1.06156	0.530780	−	0.847510i	$-0.321899\pi$
$277$	4894.00	1.06156	0.530780	+	0.847510i	$0.321899\pi$
$278$	1256.00	0.270971
$279$	−2232.00	−0.478947
$280$	280.000	0.0597614
$281$	−1606.00	−0.340946	−0.170473	−	0.985362i	$-0.554530\pi$
$281$	−1606.00	−0.340946	−0.170473	+	0.985362i	$0.554530\pi$
$282$	−2160.00	−0.456121
$283$	4348.00	0.913292	0.456646	−	0.889648i	$-0.349051\pi$
$283$	4348.00	0.913292	0.456646	+	0.889648i	$0.349051\pi$
$284$	−3680.00	−0.768901
$285$	1980.00	0.411527
$286$	−1188.00	−0.245622
$287$	1946.00	0.400240
$288$	−288.000	−0.0589256
$289$	1171.00	0.238347
$290$	2180.00	0.441428
$291$	1878.00	0.378317
$292$	4040.00	0.809668
$293$	1606.00	0.320217	0.160108	−	0.987099i	$-0.448816\pi$
$293$	1606.00	0.320217	0.160108	+	0.987099i	$0.448816\pi$
$294$	−294.000	−0.0583212
$295$	−460.000	−0.0907872
$296$	3024.00	0.593806
$297$	297.000	0.0580259
$298$	2052.00	0.398890
$299$	−4320.00	−0.835559
$300$	300.000	0.0577350
$301$	588.000	0.112597
$302$	2272.00	0.432910
$303$	−3510.00	−0.665493
$304$	2112.00	0.398459
$305$	1150.00	0.215898
$306$	1404.00	0.262292
$307$	1396.00	0.259524	0.129762	−	0.991545i	$-0.458579\pi$
$307$	1396.00	0.259524	0.129762	+	0.991545i	$0.458579\pi$
$308$	−308.000	−0.0569803
$309$	1344.00	0.247435
$310$	2480.00	0.454369
$311$	−3768.00	−0.687021	−0.343511	−	0.939149i	$-0.611616\pi$
$311$	−3768.00	−0.687021	−0.343511	+	0.939149i	$0.611616\pi$
$312$	−1296.00	−0.235165
$313$	9050.00	1.63430	0.817151	−	0.576424i	$-0.195552\pi$
$313$	9050.00	1.63430	0.817151	+	0.576424i	$0.195552\pi$
$314$	1956.00	0.351540
$315$	−315.000	−0.0563436
$316$	−4960.00	−0.882980
$317$	−1066.00	−0.188872	−0.0944362	−	0.995531i	$-0.530105\pi$
$317$	−1066.00	−0.188872	−0.0944362	+	0.995531i	$0.530105\pi$
$318$	684.000	0.120619
$319$	−2398.00	−0.420885
$320$	320.000	0.0559017
$321$	3204.00	0.557102
$322$	−1120.00	−0.193836
$323$	−10296.0	−1.77364
$324$	324.000	0.0555556
$325$	1350.00	0.230414
$326$	7336.00	1.24633
$327$	1026.00	0.173510
$328$	2224.00	0.374390
$329$	−2520.00	−0.422286
$330$	−330.000	−0.0550482
$331$	−1412.00	−0.234473	−0.117236	−	0.993104i	$-0.537404\pi$
$331$	−1412.00	−0.234473	−0.117236	+	0.993104i	$0.537404\pi$
$332$	−4656.00	−0.769672
$333$	−3402.00	−0.559845
$334$	−2032.00	−0.332892
$335$	1580.00	0.257685
$336$	−336.000	−0.0545545
$337$	−5206.00	−0.841510	−0.420755	−	0.907174i	$-0.638235\pi$
$337$	−5206.00	−0.841510	−0.420755	+	0.907174i	$0.638235\pi$
$338$	−1438.00	−0.231411
$339$	−258.000	−0.0413352
$340$	−1560.00	−0.248832
$341$	−2728.00	−0.433224
$342$	−2376.00	−0.375671
$343$	−343.000	−0.0539949
$344$	672.000	0.105325
$345$	−1200.00	−0.187263
$346$	−7164.00	−1.11312
$347$	2588.00	0.400378	0.200189	−	0.979757i	$-0.435844\pi$
$347$	2588.00	0.400378	0.200189	+	0.979757i	$0.435844\pi$
$348$	−2616.00	−0.402966
$349$	4870.00	0.746949	0.373474	−	0.927640i	$-0.378166\pi$
$349$	4870.00	0.746949	0.373474	+	0.927640i	$0.378166\pi$
$350$	350.000	0.0534522
$351$	1458.00	0.221716
$352$	−352.000	−0.0533002
$353$	−11718.0	−1.76682	−0.883408	−	0.468604i	$-0.844757\pi$
$353$	−11718.0	−1.76682	−0.883408	+	0.468604i	$0.844757\pi$
$354$	552.000	0.0828770
$355$	−4600.00	−0.687726
$356$	4200.00	0.625280
$357$	1638.00	0.242835
$358$	1704.00	0.251562
$359$	104.000	0.0152894	0.00764472	−	0.999971i	$-0.497567\pi$
$359$	104.000	0.0152894	0.00764472	+	0.999971i	$0.497567\pi$
$360$	−360.000	−0.0527046
$361$	10565.0	1.54031
$362$	2916.00	0.423374
$363$	363.000	0.0524864
$364$	−1512.00	−0.217721
$365$	5050.00	0.724189
$366$	−1380.00	−0.197087
$367$	4856.00	0.690684	0.345342	−	0.938477i	$-0.387763\pi$
$367$	4856.00	0.690684	0.345342	+	0.938477i	$0.387763\pi$
$368$	−1280.00	−0.181317
$369$	−2502.00	−0.352978
$370$	3780.00	0.531116
$371$	798.000	0.111671
$372$	−2976.00	−0.414781
$373$	3278.00	0.455036	0.227518	−	0.973774i	$-0.426939\pi$
$373$	3278.00	0.455036	0.227518	+	0.973774i	$0.426939\pi$
$374$	1716.00	0.237252
$375$	375.000	0.0516398
$376$	−2880.00	−0.395012
$377$	−11772.0	−1.60819
$378$	378.000	0.0514344
$379$	812.000	0.110052	0.0550259	−	0.998485i	$-0.482476\pi$
$379$	812.000	0.110052	0.0550259	+	0.998485i	$0.482476\pi$
$380$	2640.00	0.356392
$381$	−5664.00	−0.761616
$382$	7040.00	0.942926
$383$	−11608.0	−1.54867	−0.774336	−	0.632775i	$-0.781916\pi$
$383$	−11608.0	−1.54867	−0.774336	+	0.632775i	$0.781916\pi$
$384$	−384.000	−0.0510310
$385$	−385.000	−0.0509647
$386$	2636.00	0.347588
$387$	−756.000	−0.0993014
$388$	2504.00	0.327632
$389$	−6466.00	−0.842774	−0.421387	−	0.906881i	$-0.638457\pi$
$389$	−6466.00	−0.842774	−0.421387	+	0.906881i	$0.638457\pi$
$390$	−1620.00	−0.210338
$391$	6240.00	0.807085
$392$	−392.000	−0.0505076
$393$	3012.00	0.386604
$394$	6676.00	0.853635
$395$	−6200.00	−0.789762
$396$	396.000	0.0502519
$397$	−5266.00	−0.665725	−0.332863	−	0.942975i	$-0.608015\pi$
$397$	−5266.00	−0.665725	−0.332863	+	0.942975i	$0.608015\pi$
$398$	−5088.00	−0.640800
$399$	−2772.00	−0.347803
$400$	400.000	0.0500000
$401$	−13902.0	−1.73125	−0.865627	−	0.500690i	$-0.833080\pi$
$401$	−13902.0	−1.73125	−0.865627	+	0.500690i	$0.833080\pi$
$402$	−1896.00	−0.235234
$403$	−13392.0	−1.65534
$404$	−4680.00	−0.576333
$405$	405.000	0.0496904
$406$	−3052.00	−0.373074
$407$	−4158.00	−0.506399
$408$	1872.00	0.227151
$409$	−10934.0	−1.32189	−0.660943	−	0.750436i	$-0.729844\pi$
$409$	−10934.0	−1.32189	−0.660943	+	0.750436i	$0.729844\pi$
$410$	2780.00	0.334864
$411$	2790.00	0.334843
$412$	1792.00	0.214285
$413$	644.000	0.0767292
$414$	1440.00	0.170947
$415$	−5820.00	−0.688416
$416$	−1728.00	−0.203659
$417$	−1884.00	−0.221247
$418$	−2904.00	−0.339807
$419$	11708.0	1.36509	0.682546	−	0.730843i	$-0.260873\pi$
$419$	11708.0	1.36509	0.682546	+	0.730843i	$0.260873\pi$
$420$	−420.000	−0.0487950
$421$	−2802.00	−0.324373	−0.162187	−	0.986760i	$-0.551855\pi$
$421$	−2802.00	−0.324373	−0.162187	+	0.986760i	$0.551855\pi$
$422$	−6504.00	−0.750260
$423$	3240.00	0.372421
$424$	912.000	0.104459
$425$	−1950.00	−0.222562
$426$	5520.00	0.627805
$427$	−1610.00	−0.182467
$428$	4272.00	0.482465
$429$	1782.00	0.200550
$430$	840.000	0.0942056
$431$	−16976.0	−1.89723	−0.948614	−	0.316436i	$-0.897514\pi$
$431$	−16976.0	−1.89723	−0.948614	+	0.316436i	$0.897514\pi$
$432$	432.000	0.0481125
$433$	7250.00	0.804648	0.402324	−	0.915497i	$-0.368202\pi$
$433$	7250.00	0.804648	0.402324	+	0.915497i	$0.368202\pi$
$434$	−3472.00	−0.384012
$435$	−3270.00	−0.360424
$436$	1368.00	0.150264
$437$	−10560.0	−1.15596
$438$	−6060.00	−0.661091
$439$	4960.00	0.539243	0.269622	−	0.962966i	$-0.413101\pi$
$439$	4960.00	0.539243	0.269622	+	0.962966i	$0.413101\pi$
$440$	−440.000	−0.0476731
$441$	441.000	0.0476190
$442$	8424.00	0.906536
$443$	−12076.0	−1.29514	−0.647571	−	0.762005i	$-0.724215\pi$
$443$	−12076.0	−1.29514	−0.647571	+	0.762005i	$0.724215\pi$
$444$	−4536.00	−0.484840
$445$	5250.00	0.559267
$446$	2736.00	0.290478
$447$	−3078.00	−0.325692
$448$	−448.000	−0.0472456
$449$	−158.000	−0.0166069	−0.00830343	−	0.999966i	$-0.502643\pi$
$449$	−158.000	−0.0166069	−0.00830343	+	0.999966i	$0.502643\pi$
$450$	−450.000	−0.0471405
$451$	−3058.00	−0.319281
$452$	−344.000	−0.0357973
$453$	−3408.00	−0.353470
$454$	−5800.00	−0.599576
$455$	−1890.00	−0.194735
$456$	−3168.00	−0.325340
$457$	−8878.00	−0.908743	−0.454371	−	0.890812i	$-0.650136\pi$
$457$	−8878.00	−0.908743	−0.454371	+	0.890812i	$0.650136\pi$
$458$	6692.00	0.682744
$459$	−2106.00	−0.214160
$460$	−1600.00	−0.162175
$461$	5222.00	0.527577	0.263788	−	0.964581i	$-0.415028\pi$
$461$	5222.00	0.527577	0.263788	+	0.964581i	$0.415028\pi$
$462$	462.000	0.0465242
$463$	−5528.00	−0.554877	−0.277438	−	0.960743i	$-0.589485\pi$
$463$	−5528.00	−0.554877	−0.277438	+	0.960743i	$0.589485\pi$
$464$	−3488.00	−0.348979
$465$	−3720.00	−0.370991
$466$	588.000	0.0584519
$467$	3420.00	0.338884	0.169442	−	0.985540i	$-0.445803\pi$
$467$	3420.00	0.338884	0.169442	+	0.985540i	$0.445803\pi$
$468$	1944.00	0.192012
$469$	−2212.00	−0.217784
$470$	−3600.00	−0.353310
$471$	−2934.00	−0.287031
$472$	736.000	0.0717736
$473$	−924.000	−0.0898215
$474$	7440.00	0.720950
$475$	3300.00	0.318767
$476$	2184.00	0.210301
$477$	−1026.00	−0.0984849
$478$	−10848.0	−1.03803
$479$	4720.00	0.450234	0.225117	−	0.974332i	$-0.427723\pi$
$479$	4720.00	0.450234	0.225117	+	0.974332i	$0.427723\pi$
$480$	−480.000	−0.0456435
$481$	−20412.0	−1.93494
$482$	−9988.00	−0.943861
$483$	1680.00	0.158266
$484$	484.000	0.0454545
$485$	3130.00	0.293043
$486$	−486.000	−0.0453609
$487$	−4480.00	−0.416855	−0.208427	−	0.978038i	$-0.566834\pi$
$487$	−4480.00	−0.416855	−0.208427	+	0.978038i	$0.566834\pi$
$488$	−1840.00	−0.170682
$489$	−11004.0	−1.01762
$490$	−490.000	−0.0451754
$491$	16548.0	1.52098	0.760490	−	0.649350i	$-0.224959\pi$
$491$	16548.0	1.52098	0.760490	+	0.649350i	$0.224959\pi$
$492$	−3336.00	−0.305688
$493$	17004.0	1.55339
$494$	−14256.0	−1.29840
$495$	495.000	0.0449467
$496$	−3968.00	−0.359211
$497$	6440.00	0.581234
$498$	6984.00	0.628435
$499$	−9052.00	−0.812070	−0.406035	−	0.913857i	$-0.633089\pi$
$499$	−9052.00	−0.812070	−0.406035	+	0.913857i	$0.633089\pi$
$500$	500.000	0.0447214
$501$	3048.00	0.271806
$502$	6296.00	0.559769
$503$	11352.0	1.00628	0.503142	−	0.864204i	$-0.332177\pi$
$503$	11352.0	1.00628	0.503142	+	0.864204i	$0.332177\pi$
$504$	504.000	0.0445435
$505$	−5850.00	−0.515488
$506$	1760.00	0.154628
$507$	2157.00	0.188946
$508$	−7552.00	−0.659578
$509$	−14490.0	−1.26180	−0.630902	−	0.775863i	$-0.717315\pi$
$509$	−14490.0	−1.26180	−0.630902	+	0.775863i	$0.717315\pi$
$510$	2340.00	0.203170
$511$	−7070.00	−0.612052
$512$	−512.000	−0.0441942
$513$	3564.00	0.306734
$514$	10796.0	0.926442
$515$	2240.00	0.191663
$516$	−1008.00	−0.0859975
$517$	3960.00	0.336868
$518$	−5292.00	−0.448875
$519$	10746.0	0.908858
$520$	−2160.00	−0.182158
$521$	−17606.0	−1.48049	−0.740243	−	0.672340i	$-0.765289\pi$
$521$	−17606.0	−1.48049	−0.740243	+	0.672340i	$0.765289\pi$
$522$	3924.00	0.329021
$523$	17068.0	1.42702	0.713510	−	0.700645i	$-0.247104\pi$
$523$	17068.0	1.42702	0.713510	+	0.700645i	$0.247104\pi$
$524$	4016.00	0.334809
$525$	−525.000	−0.0436436
$526$	8976.00	0.744054
$527$	19344.0	1.59893
$528$	528.000	0.0435194
$529$	−5767.00	−0.473987
$530$	1140.00	0.0934310
$531$	−828.000	−0.0676688
$532$	−3696.00	−0.301207
$533$	−15012.0	−1.21997
$534$	−6300.00	−0.510539
$535$	5340.00	0.431530
$536$	−2528.00	−0.203718
$537$	−2556.00	−0.205400
$538$	4052.00	0.324710
$539$	539.000	0.0430730
$540$	540.000	0.0430331
$541$	6086.00	0.483655	0.241828	−	0.970319i	$-0.422253\pi$
$541$	6086.00	0.483655	0.241828	+	0.970319i	$0.422253\pi$
$542$	11984.0	0.949736
$543$	−4374.00	−0.345684
$544$	2496.00	0.196719
$545$	1710.00	0.134401
$546$	2268.00	0.177768
$547$	23812.0	1.86129	0.930647	−	0.365919i	$-0.119245\pi$
$547$	23812.0	1.86129	0.930647	+	0.365919i	$0.119245\pi$
$548$	3720.00	0.289983
$549$	2070.00	0.160921
$550$	−550.000	−0.0426401
$551$	−28776.0	−2.22486
$552$	1920.00	0.148045
$553$	8680.00	0.667470
$554$	−9788.00	−0.750636
$555$	−5670.00	−0.433654
$556$	−2512.00	−0.191605
$557$	18558.0	1.41172	0.705860	−	0.708352i	$-0.250561\pi$
$557$	18558.0	1.41172	0.705860	+	0.708352i	$0.250561\pi$
$558$	4464.00	0.338667
$559$	−4536.00	−0.343206
$560$	−560.000	−0.0422577
$561$	−2574.00	−0.193715
$562$	3212.00	0.241085
$563$	−17516.0	−1.31121	−0.655605	−	0.755104i	$-0.727586\pi$
$563$	−17516.0	−1.31121	−0.655605	+	0.755104i	$0.727586\pi$
$564$	4320.00	0.322526
$565$	−430.000	−0.0320181
$566$	−8696.00	−0.645795
$567$	−567.000	−0.0419961
$568$	7360.00	0.543695
$569$	−19398.0	−1.42919	−0.714593	−	0.699541i	$-0.753388\pi$
$569$	−19398.0	−1.42919	−0.714593	+	0.699541i	$0.753388\pi$
$570$	−3960.00	−0.290993
$571$	18524.0	1.35763	0.678814	−	0.734311i	$-0.262494\pi$
$571$	18524.0	1.35763	0.678814	+	0.734311i	$0.262494\pi$
$572$	2376.00	0.173681
$573$	−10560.0	−0.769896
$574$	−3892.00	−0.283012
$575$	−2000.00	−0.145054
$576$	576.000	0.0416667
$577$	11410.0	0.823231	0.411616	−	0.911358i	$-0.364965\pi$
$577$	11410.0	0.823231	0.411616	+	0.911358i	$0.364965\pi$
$578$	−2342.00	−0.168537
$579$	−3954.00	−0.283804
$580$	−4360.00	−0.312136
$581$	8148.00	0.581818
$582$	−3756.00	−0.267511
$583$	−1254.00	−0.0890830
$584$	−8080.00	−0.572522
$585$	2430.00	0.171740
$586$	−3212.00	−0.226427
$587$	−2012.00	−0.141472	−0.0707361	−	0.997495i	$-0.522535\pi$
$587$	−2012.00	−0.141472	−0.0707361	+	0.997495i	$0.522535\pi$
$588$	588.000	0.0412393
$589$	−32736.0	−2.29009
$590$	920.000	0.0641963
$591$	−10014.0	−0.696990
$592$	−6048.00	−0.419884
$593$	−366.000	−0.0253454	−0.0126727	−	0.999920i	$-0.504034\pi$
$593$	−366.000	−0.0253454	−0.0126727	+	0.999920i	$0.504034\pi$
$594$	−594.000	−0.0410305
$595$	2730.00	0.188099
$596$	−4104.00	−0.282058
$597$	7632.00	0.523211
$598$	8640.00	0.590829
$599$	−12440.0	−0.848555	−0.424278	−	0.905532i	$-0.639472\pi$
$599$	−12440.0	−0.848555	−0.424278	+	0.905532i	$0.639472\pi$
$600$	−600.000	−0.0408248
$601$	−9350.00	−0.634600	−0.317300	−	0.948325i	$-0.602776\pi$
$601$	−9350.00	−0.634600	−0.317300	+	0.948325i	$0.602776\pi$
$602$	−1176.00	−0.0796182
$603$	2844.00	0.192067
$604$	−4544.00	−0.306114
$605$	605.000	0.0406558
$606$	7020.00	0.470574
$607$	−4848.00	−0.324175	−0.162088	−	0.986776i	$-0.551823\pi$
$607$	−4848.00	−0.324175	−0.162088	+	0.986776i	$0.551823\pi$
$608$	−4224.00	−0.281753
$609$	4578.00	0.304614
$610$	−2300.00	−0.152663
$611$	19440.0	1.28717
$612$	−2808.00	−0.185468
$613$	−28530.0	−1.87980	−0.939899	−	0.341453i	$-0.889081\pi$
$613$	−28530.0	−1.87980	−0.939899	+	0.341453i	$0.889081\pi$
$614$	−2792.00	−0.183511
$615$	−4170.00	−0.273416
$616$	616.000	0.0402911
$617$	−12702.0	−0.828790	−0.414395	−	0.910097i	$-0.636007\pi$
$617$	−12702.0	−0.828790	−0.414395	+	0.910097i	$0.636007\pi$
$618$	−2688.00	−0.174963
$619$	−484.000	−0.0314275	−0.0157137	−	0.999877i	$-0.505002\pi$
$619$	−484.000	−0.0314275	−0.0157137	+	0.999877i	$0.505002\pi$
$620$	−4960.00	−0.321288
$621$	−2160.00	−0.139578
$622$	7536.00	0.485798
$623$	−7350.00	−0.472667
$624$	2592.00	0.166287
$625$	625.000	0.0400000
$626$	−18100.0	−1.15563
$627$	4356.00	0.277451
$628$	−3912.00	−0.248576
$629$	29484.0	1.86900
$630$	630.000	0.0398410
$631$	−6896.00	−0.435064	−0.217532	−	0.976053i	$-0.569801\pi$
$631$	−6896.00	−0.435064	−0.217532	+	0.976053i	$0.569801\pi$
$632$	9920.00	0.624361
$633$	9756.00	0.612585
$634$	2132.00	0.133553
$635$	−9440.00	−0.589945
$636$	−1368.00	−0.0852905
$637$	2646.00	0.164581
$638$	4796.00	0.297610
$639$	−8280.00	−0.512601
$640$	−640.000	−0.0395285
$641$	−19630.0	−1.20958	−0.604788	−	0.796386i	$-0.706742\pi$
$641$	−19630.0	−1.20958	−0.604788	+	0.796386i	$0.706742\pi$
$642$	−6408.00	−0.393931
$643$	−5780.00	−0.354496	−0.177248	−	0.984166i	$-0.556719\pi$
$643$	−5780.00	−0.354496	−0.177248	+	0.984166i	$0.556719\pi$
$644$	2240.00	0.137063
$645$	−1260.00	−0.0769185
$646$	20592.0	1.25415
$647$	25872.0	1.57208	0.786038	−	0.618178i	$-0.212129\pi$
$647$	25872.0	1.57208	0.786038	+	0.618178i	$0.212129\pi$
$648$	−648.000	−0.0392837
$649$	−1012.00	−0.0612087
$650$	−2700.00	−0.162927
$651$	5208.00	0.313545
$652$	−14672.0	−0.881288
$653$	−21562.0	−1.29217	−0.646084	−	0.763266i	$-0.723595\pi$
$653$	−21562.0	−1.29217	−0.646084	+	0.763266i	$0.723595\pi$
$654$	−2052.00	−0.122690
$655$	5020.00	0.299462
$656$	−4448.00	−0.264734
$657$	9090.00	0.539779
$658$	5040.00	0.298601
$659$	−3940.00	−0.232899	−0.116450	−	0.993197i	$-0.537151\pi$
$659$	−3940.00	−0.232899	−0.116450	+	0.993197i	$0.537151\pi$
$660$	660.000	0.0389249
$661$	−11330.0	−0.666696	−0.333348	−	0.942804i	$-0.608178\pi$
$661$	−11330.0	−0.666696	−0.333348	+	0.942804i	$0.608178\pi$
$662$	2824.00	0.165797
$663$	−12636.0	−0.740183
$664$	9312.00	0.544240
$665$	−4620.00	−0.269407
$666$	6804.00	0.395870
$667$	17440.0	1.01241
$668$	4064.00	0.235391
$669$	−4104.00	−0.237175
$670$	−3160.00	−0.182211
$671$	2530.00	0.145558
$672$	672.000	0.0385758
$673$	16730.0	0.958238	0.479119	−	0.877750i	$-0.340956\pi$
$673$	16730.0	0.958238	0.479119	+	0.877750i	$0.340956\pi$
$674$	10412.0	0.595037
$675$	675.000	0.0384900
$676$	2876.00	0.163632
$677$	822.000	0.0466647	0.0233324	−	0.999728i	$-0.492572\pi$
$677$	822.000	0.0466647	0.0233324	+	0.999728i	$0.492572\pi$
$678$	516.000	0.0292284
$679$	−4382.00	−0.247667
$680$	3120.00	0.175951
$681$	8700.00	0.489552
$682$	5456.00	0.306336
$683$	5476.00	0.306784	0.153392	−	0.988165i	$-0.450980\pi$
$683$	5476.00	0.306784	0.153392	+	0.988165i	$0.450980\pi$
$684$	4752.00	0.265639
$685$	4650.00	0.259368
$686$	686.000	0.0381802
$687$	−10038.0	−0.557458
$688$	−1344.00	−0.0744760
$689$	−6156.00	−0.340385
$690$	2400.00	0.132415
$691$	13092.0	0.720757	0.360379	−	0.932806i	$-0.382648\pi$
$691$	13092.0	0.720757	0.360379	+	0.932806i	$0.382648\pi$
$692$	14328.0	0.787094
$693$	−693.000	−0.0379869
$694$	−5176.00	−0.283110
$695$	−3140.00	−0.171377
$696$	5232.00	0.284940
$697$	21684.0	1.17839
$698$	−9740.00	−0.528173
$699$	−882.000	−0.0477258
$700$	−700.000	−0.0377964
$701$	10982.0	0.591704	0.295852	−	0.955234i	$-0.404396\pi$
$701$	10982.0	0.591704	0.295852	+	0.955234i	$0.404396\pi$
$702$	−2916.00	−0.156777
$703$	−49896.0	−2.67690
$704$	704.000	0.0376889
$705$	5400.00	0.288476
$706$	23436.0	1.24933
$707$	8190.00	0.435667
$708$	−1104.00	−0.0586029
$709$	−7122.00	−0.377253	−0.188626	−	0.982049i	$-0.560404\pi$
$709$	−7122.00	−0.377253	−0.188626	+	0.982049i	$0.560404\pi$
$710$	9200.00	0.486296
$711$	−11160.0	−0.588654
$712$	−8400.00	−0.442139
$713$	19840.0	1.04210
$714$	−3276.00	−0.171710
$715$	2970.00	0.155345
$716$	−3408.00	−0.177881
$717$	16272.0	0.847544
$718$	−208.000	−0.0108113
$719$	−8288.00	−0.429889	−0.214945	−	0.976626i	$-0.568957\pi$
$719$	−8288.00	−0.429889	−0.214945	+	0.976626i	$0.568957\pi$
$720$	720.000	0.0372678
$721$	−3136.00	−0.161984
$722$	−21130.0	−1.08917
$723$	14982.0	0.770659
$724$	−5832.00	−0.299371
$725$	−5450.00	−0.279183
$726$	−726.000	−0.0371135
$727$	21632.0	1.10356	0.551779	−	0.833990i	$-0.313949\pi$
$727$	21632.0	1.10356	0.551779	+	0.833990i	$0.313949\pi$
$728$	3024.00	0.153952
$729$	729.000	0.0370370
$730$	−10100.0	−0.512079
$731$	6552.00	0.331511
$732$	2760.00	0.139361
$733$	8326.00	0.419547	0.209773	−	0.977750i	$-0.432727\pi$
$733$	8326.00	0.419547	0.209773	+	0.977750i	$0.432727\pi$
$734$	−9712.00	−0.488388
$735$	735.000	0.0368856
$736$	2560.00	0.128210
$737$	3476.00	0.173731
$738$	5004.00	0.249593
$739$	13812.0	0.687527	0.343764	−	0.939056i	$-0.388298\pi$
$739$	13812.0	0.687527	0.343764	+	0.939056i	$0.388298\pi$
$740$	−7560.00	−0.375556
$741$	21384.0	1.06014
$742$	−1596.00	−0.0789636
$743$	−11608.0	−0.573158	−0.286579	−	0.958057i	$-0.592518\pi$
$743$	−11608.0	−0.573158	−0.286579	+	0.958057i	$0.592518\pi$
$744$	5952.00	0.293294
$745$	−5130.00	−0.252280
$746$	−6556.00	−0.321759
$747$	−10476.0	−0.513115
$748$	−3432.00	−0.167762
$749$	−7476.00	−0.364709
$750$	−750.000	−0.0365148
$751$	11944.0	0.580350	0.290175	−	0.956974i	$-0.406287\pi$
$751$	11944.0	0.580350	0.290175	+	0.956974i	$0.406287\pi$
$752$	5760.00	0.279316
$753$	−9444.00	−0.457050
$754$	23544.0	1.13716
$755$	−5680.00	−0.273797
$756$	−756.000	−0.0363696
$757$	−15530.0	−0.745637	−0.372819	−	0.927904i	$-0.621609\pi$
$757$	−15530.0	−0.745637	−0.372819	+	0.927904i	$0.621609\pi$
$758$	−1624.00	−0.0778184
$759$	−2640.00	−0.126253
$760$	−5280.00	−0.252008
$761$	3834.00	0.182631	0.0913156	−	0.995822i	$-0.470893\pi$
$761$	3834.00	0.182631	0.0913156	+	0.995822i	$0.470893\pi$
$762$	11328.0	0.538543
$763$	−2394.00	−0.113589
$764$	−14080.0	−0.666749
$765$	−3510.00	−0.165888
$766$	23216.0	1.09508
$767$	−4968.00	−0.233878
$768$	768.000	0.0360844
$769$	−6510.00	−0.305275	−0.152638	−	0.988282i	$-0.548777\pi$
$769$	−6510.00	−0.305275	−0.152638	+	0.988282i	$0.548777\pi$
$770$	770.000	0.0360375
$771$	−16194.0	−0.756437
$772$	−5272.00	−0.245782
$773$	11646.0	0.541886	0.270943	−	0.962595i	$-0.412665\pi$
$773$	11646.0	0.541886	0.270943	+	0.962595i	$0.412665\pi$
$774$	1512.00	0.0702167
$775$	−6200.00	−0.287368
$776$	−5008.00	−0.231671
$777$	7938.00	0.366505
$778$	12932.0	0.595931
$779$	−36696.0	−1.68777
$780$	3240.00	0.148732
$781$	−10120.0	−0.463665
$782$	−12480.0	−0.570696
$783$	−5886.00	−0.268644
$784$	784.000	0.0357143
$785$	−4890.00	−0.222333
$786$	−6024.00	−0.273370
$787$	−17004.0	−0.770174	−0.385087	−	0.922880i	$-0.625829\pi$
$787$	−17004.0	−0.770174	−0.385087	+	0.922880i	$0.625829\pi$
$788$	−13352.0	−0.603611
$789$	−13464.0	−0.607517
$790$	12400.0	0.558446
$791$	602.000	0.0270602
$792$	−792.000	−0.0355335
$793$	12420.0	0.556175
$794$	10532.0	0.470739
$795$	−1710.00	−0.0762861
$796$	10176.0	0.453114
$797$	−5082.00	−0.225864	−0.112932	−	0.993603i	$-0.536024\pi$
$797$	−5082.00	−0.225864	−0.112932	+	0.993603i	$0.536024\pi$
$798$	5544.00	0.245934
$799$	−28080.0	−1.24330
$800$	−800.000	−0.0353553
$801$	9450.00	0.416853
$802$	27804.0	1.22418
$803$	11110.0	0.488248
$804$	3792.00	0.166335
$805$	2800.00	0.122593
$806$	26784.0	1.17050
$807$	−6078.00	−0.265125
$808$	9360.00	0.407529
$809$	−24918.0	−1.08291	−0.541453	−	0.840731i	$-0.682125\pi$
$809$	−24918.0	−1.08291	−0.541453	+	0.840731i	$0.682125\pi$
$810$	−810.000	−0.0351364
$811$	35212.0	1.52461	0.762306	−	0.647217i	$-0.224067\pi$
$811$	35212.0	1.52461	0.762306	+	0.647217i	$0.224067\pi$
$812$	6104.00	0.263803
$813$	−17976.0	−0.775456
$814$	8316.00	0.358078
$815$	−18340.0	−0.788248
$816$	−3744.00	−0.160620
$817$	−11088.0	−0.474810
$818$	21868.0	0.934715
$819$	−3402.00	−0.145147
$820$	−5560.00	−0.236785
$821$	38350.0	1.63024	0.815118	−	0.579295i	$-0.196672\pi$
$821$	38350.0	1.63024	0.815118	+	0.579295i	$0.196672\pi$
$822$	−5580.00	−0.236770
$823$	10352.0	0.438454	0.219227	−	0.975674i	$-0.429646\pi$
$823$	10352.0	0.438454	0.219227	+	0.975674i	$0.429646\pi$
$824$	−3584.00	−0.151523
$825$	825.000	0.0348155
$826$	−1288.00	−0.0542558
$827$	−16788.0	−0.705896	−0.352948	−	0.935643i	$-0.614821\pi$
$827$	−16788.0	−0.705896	−0.352948	+	0.935643i	$0.614821\pi$
$828$	−2880.00	−0.120878
$829$	−14746.0	−0.617792	−0.308896	−	0.951096i	$-0.599960\pi$
$829$	−14746.0	−0.617792	−0.308896	+	0.951096i	$0.599960\pi$
$830$	11640.0	0.486783
$831$	14682.0	0.612892
$832$	3456.00	0.144009
$833$	−3822.00	−0.158973
$834$	3768.00	0.156445
$835$	5080.00	0.210540
$836$	5808.00	0.240280
$837$	−6696.00	−0.276520
$838$	−23416.0	−0.965265
$839$	31240.0	1.28549	0.642744	−	0.766081i	$-0.277796\pi$
$839$	31240.0	1.28549	0.642744	+	0.766081i	$0.277796\pi$
$840$	840.000	0.0345033
$841$	23135.0	0.948583
$842$	5604.00	0.229367
$843$	−4818.00	−0.196845
$844$	13008.0	0.530514
$845$	3595.00	0.146357
$846$	−6480.00	−0.263342
$847$	−847.000	−0.0343604
$848$	−1824.00	−0.0738637
$849$	13044.0	0.527290
$850$	3900.00	0.157375
$851$	30240.0	1.21811
$852$	−11040.0	−0.443925
$853$	49246.0	1.97673	0.988365	−	0.152100i	$-0.0486035\pi$
$853$	49246.0	1.97673	0.988365	+	0.152100i	$0.0486035\pi$
$854$	3220.00	0.129024
$855$	5940.00	0.237595
$856$	−8544.00	−0.341154
$857$	16938.0	0.675135	0.337568	−	0.941301i	$-0.390396\pi$
$857$	16938.0	0.675135	0.337568	+	0.941301i	$0.390396\pi$
$858$	−3564.00	−0.141810
$859$	−7972.00	−0.316649	−0.158324	−	0.987387i	$-0.550609\pi$
$859$	−7972.00	−0.316649	−0.158324	+	0.987387i	$0.550609\pi$
$860$	−1680.00	−0.0666134
$861$	5838.00	0.231078
$862$	33952.0	1.34154
$863$	5048.00	0.199115	0.0995573	−	0.995032i	$-0.468257\pi$
$863$	5048.00	0.199115	0.0995573	+	0.995032i	$0.468257\pi$
$864$	−864.000	−0.0340207
$865$	17910.0	0.703998
$866$	−14500.0	−0.568972
$867$	3513.00	0.137610
$868$	6944.00	0.271538
$869$	−13640.0	−0.532457
$870$	6540.00	0.254858
$871$	17064.0	0.663825
$872$	−2736.00	−0.106253
$873$	5634.00	0.218422
$874$	21120.0	0.817385
$875$	−875.000	−0.0338062
$876$	12120.0	0.467462
$877$	35542.0	1.36849	0.684246	−	0.729251i	$-0.260131\pi$
$877$	35542.0	1.36849	0.684246	+	0.729251i	$0.260131\pi$
$878$	−9920.00	−0.381303
$879$	4818.00	0.184877
$880$	880.000	0.0337100
$881$	13826.0	0.528728	0.264364	−	0.964423i	$-0.414838\pi$
$881$	13826.0	0.528728	0.264364	+	0.964423i	$0.414838\pi$
$882$	−882.000	−0.0336718
$883$	20764.0	0.791352	0.395676	−	0.918390i	$-0.370510\pi$
$883$	20764.0	0.791352	0.395676	+	0.918390i	$0.370510\pi$
$884$	−16848.0	−0.641018
$885$	−1380.00	−0.0524160
$886$	24152.0	0.915804
$887$	−2456.00	−0.0929700	−0.0464850	−	0.998919i	$-0.514802\pi$
$887$	−2456.00	−0.0929700	−0.0464850	+	0.998919i	$0.514802\pi$
$888$	9072.00	0.342834
$889$	13216.0	0.498594
$890$	−10500.0	−0.395462
$891$	891.000	0.0335013
$892$	−5472.00	−0.205399
$893$	47520.0	1.78073
$894$	6156.00	0.230299
$895$	−4260.00	−0.159102
$896$	896.000	0.0334077
$897$	−12960.0	−0.482410
$898$	316.000	0.0117428
$899$	54064.0	2.00571
$900$	900.000	0.0333333
$901$	8892.00	0.328785
$902$	6116.00	0.225766
$903$	1764.00	0.0650080
$904$	688.000	0.0253125
$905$	−7290.00	−0.267766
$906$	6816.00	0.249941
$907$	−10716.0	−0.392303	−0.196152	−	0.980574i	$-0.562844\pi$
$907$	−10716.0	−0.392303	−0.196152	+	0.980574i	$0.562844\pi$
$908$	11600.0	0.423964
$909$	−10530.0	−0.384222
$910$	3780.00	0.137699
$911$	1280.00	0.0465514	0.0232757	−	0.999729i	$-0.492590\pi$
$911$	1280.00	0.0465514	0.0232757	+	0.999729i	$0.492590\pi$
$912$	6336.00	0.230050
$913$	−12804.0	−0.464130
$914$	17756.0	0.642578
$915$	3450.00	0.124649
$916$	−13384.0	−0.482773
$917$	−7028.00	−0.253092
$918$	4212.00	0.151434
$919$	−50720.0	−1.82056	−0.910282	−	0.413989i	$-0.864135\pi$
$919$	−50720.0	−1.82056	−0.910282	+	0.413989i	$0.864135\pi$
$920$	3200.00	0.114675
$921$	4188.00	0.149836
$922$	−10444.0	−0.373053
$923$	−49680.0	−1.77165
$924$	−924.000	−0.0328976
$925$	−9450.00	−0.335907
$926$	11056.0	0.392357
$927$	4032.00	0.142857
$928$	6976.00	0.246766
$929$	−16350.0	−0.577423	−0.288712	−	0.957416i	$-0.593227\pi$
$929$	−16350.0	−0.577423	−0.288712	+	0.957416i	$0.593227\pi$
$930$	7440.00	0.262330
$931$	6468.00	0.227691
$932$	−1176.00	−0.0413317
$933$	−11304.0	−0.396652
$934$	−6840.00	−0.239627
$935$	−4290.00	−0.150051
$936$	−3888.00	−0.135773
$937$	−21006.0	−0.732376	−0.366188	−	0.930541i	$-0.619337\pi$
$937$	−21006.0	−0.732376	−0.366188	+	0.930541i	$0.619337\pi$
$938$	4424.00	0.153997
$939$	27150.0	0.943564
$940$	7200.00	0.249828
$941$	39942.0	1.38371	0.691855	−	0.722036i	$-0.256794\pi$
$941$	39942.0	1.38371	0.691855	+	0.722036i	$0.256794\pi$
$942$	5868.00	0.202962
$943$	22240.0	0.768011
$944$	−1472.00	−0.0507516
$945$	−945.000	−0.0325300
$946$	1848.00	0.0635134
$947$	−6180.00	−0.212062	−0.106031	−	0.994363i	$-0.533814\pi$
$947$	−6180.00	−0.212062	−0.106031	+	0.994363i	$0.533814\pi$
$948$	−14880.0	−0.509789
$949$	54540.0	1.86559
$950$	−6600.00	−0.225402
$951$	−3198.00	−0.109045
$952$	−4368.00	−0.148706
$953$	−44310.0	−1.50613	−0.753065	−	0.657946i	$-0.771425\pi$
$953$	−44310.0	−1.50613	−0.753065	+	0.657946i	$0.771425\pi$
$954$	2052.00	0.0696394
$955$	−17600.0	−0.596359
$956$	21696.0	0.733995
$957$	−7194.00	−0.242998
$958$	−9440.00	−0.318364
$959$	−6510.00	−0.219206
$960$	960.000	0.0322749
$961$	31713.0	1.06452
$962$	40824.0	1.36821
$963$	9612.00	0.321643
$964$	19976.0	0.667410
$965$	−6590.00	−0.219834
$966$	−3360.00	−0.111911
$967$	−13944.0	−0.463711	−0.231856	−	0.972750i	$-0.574480\pi$
$967$	−13944.0	−0.463711	−0.231856	+	0.972750i	$0.574480\pi$
$968$	−968.000	−0.0321412
$969$	−30888.0	−1.02401
$970$	−6260.00	−0.207213
$971$	27604.0	0.912312	0.456156	−	0.889900i	$-0.349226\pi$
$971$	27604.0	0.912312	0.456156	+	0.889900i	$0.349226\pi$
$972$	972.000	0.0320750
$973$	4396.00	0.144840
$974$	8960.00	0.294761
$975$	4050.00	0.133030
$976$	3680.00	0.120691
$977$	−39318.0	−1.28751	−0.643753	−	0.765233i	$-0.722624\pi$
$977$	−39318.0	−1.28751	−0.643753	+	0.765233i	$0.722624\pi$
$978$	22008.0	0.719569
$979$	11550.0	0.377058
$980$	980.000	0.0319438
$981$	3078.00	0.100176
$982$	−33096.0	−1.07549
$983$	12352.0	0.400781	0.200390	−	0.979716i	$-0.435779\pi$
$983$	12352.0	0.400781	0.200390	+	0.979716i	$0.435779\pi$
$984$	6672.00	0.216154
$985$	−16690.0	−0.539886
$986$	−34008.0	−1.09841
$987$	−7560.00	−0.243807
$988$	28512.0	0.918105
$989$	6720.00	0.216060
$990$	−990.000	−0.0317821
$991$	34712.0	1.11268	0.556339	−	0.830956i	$-0.312206\pi$
$991$	34712.0	1.11268	0.556339	+	0.830956i	$0.312206\pi$
$992$	7936.00	0.254000
$993$	−4236.00	−0.135373
$994$	−12880.0	−0.410995
$995$	12720.0	0.405277
$996$	−13968.0	−0.444370
$997$	53598.0	1.70257	0.851287	−	0.524701i	$-0.175823\pi$
$997$	53598.0	1.70257	0.851287	+	0.524701i	$0.175823\pi$
$998$	18104.0	0.574221
$999$	−10206.0	−0.323227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2310.4.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.4.a.d.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2310.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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