LMFDB - Embedded newform 2310.4.a.g.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)

$f(q)$

$=$

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

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} +11.0000 q^{11} -12.0000 q^{12} -22.0000 q^{13} +14.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} -18.0000 q^{17} +18.0000 q^{18} +8.00000 q^{19} -20.0000 q^{20} -21.0000 q^{21} +22.0000 q^{22} +48.0000 q^{23} -24.0000 q^{24} +25.0000 q^{25} -44.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} -186.000 q^{29} +30.0000 q^{30} +128.000 q^{31} +32.0000 q^{32} -33.0000 q^{33} -36.0000 q^{34} -35.0000 q^{35} +36.0000 q^{36} -394.000 q^{37} +16.0000 q^{38} +66.0000 q^{39} -40.0000 q^{40} -462.000 q^{41} -42.0000 q^{42} -52.0000 q^{43} +44.0000 q^{44} -45.0000 q^{45} +96.0000 q^{46} +576.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} +50.0000 q^{50} +54.0000 q^{51} -88.0000 q^{52} +354.000 q^{53} -54.0000 q^{54} -55.0000 q^{55} +56.0000 q^{56} -24.0000 q^{57} -372.000 q^{58} +696.000 q^{59} +60.0000 q^{60} +458.000 q^{61} +256.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} +110.000 q^{65} -66.0000 q^{66} -424.000 q^{67} -72.0000 q^{68} -144.000 q^{69} -70.0000 q^{70} -720.000 q^{71} +72.0000 q^{72} +422.000 q^{73} -788.000 q^{74} -75.0000 q^{75} +32.0000 q^{76} +77.0000 q^{77} +132.000 q^{78} +80.0000 q^{79} -80.0000 q^{80} +81.0000 q^{81} -924.000 q^{82} -348.000 q^{83} -84.0000 q^{84} +90.0000 q^{85} -104.000 q^{86} +558.000 q^{87} +88.0000 q^{88} -234.000 q^{89} -90.0000 q^{90} -154.000 q^{91} +192.000 q^{92} -384.000 q^{93} +1152.00 q^{94} -40.0000 q^{95} -96.0000 q^{96} -1282.00 q^{97} +98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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
$13$	−22.0000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−22.0000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	14.0000	0.267261
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−18.0000	−0.256802	−0.128401	−	0.991722i	$-0.540985\pi$
$17$	−18.0000	−0.256802	−0.128401	+	0.991722i	$0.540985\pi$
$18$	18.0000	0.235702
$19$	8.00000	0.0965961	0.0482980	−	0.998833i	$-0.484620\pi$
$19$	8.00000	0.0965961	0.0482980	+	0.998833i	$0.484620\pi$
$20$	−20.0000	−0.223607
$21$	−21.0000	−0.218218
$22$	22.0000	0.213201
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	−44.0000	−0.331889
$27$	−27.0000	−0.192450
$28$	28.0000	0.188982
$29$	−186.000	−1.19101	−0.595506	−	0.803351i	$-0.703048\pi$
$29$	−186.000	−1.19101	−0.595506	+	0.803351i	$0.703048\pi$
$30$	30.0000	0.182574
$31$	128.000	0.741596	0.370798	−	0.928714i	$-0.379084\pi$
$31$	128.000	0.741596	0.370798	+	0.928714i	$0.379084\pi$
$32$	32.0000	0.176777
$33$	−33.0000	−0.174078
$34$	−36.0000	−0.181587
$35$	−35.0000	−0.169031
$36$	36.0000	0.166667
$37$	−394.000	−1.75063	−0.875314	−	0.483556i	$-0.839345\pi$
$37$	−394.000	−1.75063	−0.875314	+	0.483556i	$0.839345\pi$
$38$	16.0000	0.0683038
$39$	66.0000	0.270986
$40$	−40.0000	−0.158114
$41$	−462.000	−1.75981	−0.879906	−	0.475148i	$-0.842394\pi$
$41$	−462.000	−1.75981	−0.879906	+	0.475148i	$0.842394\pi$
$42$	−42.0000	−0.154303
$43$	−52.0000	−0.184417	−0.0922084	−	0.995740i	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	44.0000	0.150756
$45$	−45.0000	−0.149071
$46$	96.0000	0.307705
$47$	576.000	1.78762	0.893811	−	0.448444i	$-0.148022\pi$
$47$	576.000	1.78762	0.893811	+	0.448444i	$0.148022\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	50.0000	0.141421
$51$	54.0000	0.148265
$52$	−88.0000	−0.234681
$53$	354.000	0.917465	0.458732	−	0.888574i	$-0.348304\pi$
$53$	354.000	0.917465	0.458732	+	0.888574i	$0.348304\pi$
$54$	−54.0000	−0.136083
$55$	−55.0000	−0.134840
$56$	56.0000	0.133631
$57$	−24.0000	−0.0557698
$58$	−372.000	−0.842172
$59$	696.000	1.53579	0.767894	−	0.640577i	$-0.221305\pi$
$59$	696.000	1.53579	0.767894	+	0.640577i	$0.221305\pi$
$60$	60.0000	0.129099
$61$	458.000	0.961326	0.480663	−	0.876905i	$-0.340396\pi$
$61$	458.000	0.961326	0.480663	+	0.876905i	$0.340396\pi$
$62$	256.000	0.524388
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	110.000	0.209905
$66$	−66.0000	−0.123091
$67$	−424.000	−0.773132	−0.386566	−	0.922262i	$-0.626339\pi$
$67$	−424.000	−0.773132	−0.386566	+	0.922262i	$0.626339\pi$
$68$	−72.0000	−0.128401
$69$	−144.000	−0.251240
$70$	−70.0000	−0.119523
$71$	−720.000	−1.20350	−0.601748	−	0.798686i	$-0.705529\pi$
$71$	−720.000	−1.20350	−0.601748	+	0.798686i	$0.705529\pi$
$72$	72.0000	0.117851
$73$	422.000	0.676594	0.338297	−	0.941039i	$-0.390149\pi$
$73$	422.000	0.676594	0.338297	+	0.941039i	$0.390149\pi$
$74$	−788.000	−1.23788
$75$	−75.0000	−0.115470
$76$	32.0000	0.0482980
$77$	77.0000	0.113961
$78$	132.000	0.191616
$79$	80.0000	0.113933	0.0569665	−	0.998376i	$-0.481857\pi$
$79$	80.0000	0.113933	0.0569665	+	0.998376i	$0.481857\pi$
$80$	−80.0000	−0.111803
$81$	81.0000	0.111111
$82$	−924.000	−1.24437
$83$	−348.000	−0.460216	−0.230108	−	0.973165i	$-0.573908\pi$
$83$	−348.000	−0.460216	−0.230108	+	0.973165i	$0.573908\pi$
$84$	−84.0000	−0.109109
$85$	90.0000	0.114846
$86$	−104.000	−0.130402
$87$	558.000	0.687631
$88$	88.0000	0.106600
$89$	−234.000	−0.278696	−0.139348	−	0.990243i	$-0.544501\pi$
$89$	−234.000	−0.278696	−0.139348	+	0.990243i	$0.544501\pi$
$90$	−90.0000	−0.105409
$91$	−154.000	−0.177402
$92$	192.000	0.217580
$93$	−384.000	−0.428161
$94$	1152.00	1.26404
$95$	−40.0000	−0.0431991
$96$	−96.0000	−0.102062
$97$	−1282.00	−1.34193	−0.670966	−	0.741488i	$-0.734120\pi$
$97$	−1282.00	−1.34193	−0.670966	+	0.741488i	$0.734120\pi$
$98$	98.0000	0.101015
$99$	99.0000	0.100504
$100$	100.000	0.100000
$101$	−1386.00	−1.36547	−0.682733	−	0.730668i	$-0.739209\pi$
$101$	−1386.00	−1.36547	−0.682733	+	0.730668i	$0.739209\pi$
$102$	108.000	0.104839
$103$	−328.000	−0.313775	−0.156887	−	0.987616i	$-0.550146\pi$
$103$	−328.000	−0.313775	−0.156887	+	0.987616i	$0.550146\pi$
$104$	−176.000	−0.165944
$105$	105.000	0.0975900
$106$	708.000	0.648746
$107$	384.000	0.346941	0.173470	−	0.984839i	$-0.444502\pi$
$107$	384.000	0.346941	0.173470	+	0.984839i	$0.444502\pi$
$108$	−108.000	−0.0962250
$109$	−1234.00	−1.08436	−0.542182	−	0.840261i	$-0.682402\pi$
$109$	−1234.00	−1.08436	−0.542182	+	0.840261i	$0.682402\pi$
$110$	−110.000	−0.0953463
$111$	1182.00	1.01073
$112$	112.000	0.0944911
$113$	−1962.00	−1.63336	−0.816679	−	0.577092i	$-0.804187\pi$
$113$	−1962.00	−1.63336	−0.816679	+	0.577092i	$0.804187\pi$
$114$	−48.0000	−0.0394352
$115$	−240.000	−0.194610
$116$	−744.000	−0.595506
$117$	−198.000	−0.156454
$118$	1392.00	1.08597
$119$	−126.000	−0.0970622
$120$	120.000	0.0912871
$121$	121.000	0.0909091
$122$	916.000	0.679760
$123$	1386.00	1.01603
$124$	512.000	0.370798
$125$	−125.000	−0.0894427
$126$	126.000	0.0890871
$127$	536.000	0.374506	0.187253	−	0.982312i	$-0.440042\pi$
$127$	536.000	0.374506	0.187253	+	0.982312i	$0.440042\pi$
$128$	128.000	0.0883883
$129$	156.000	0.106473
$130$	220.000	0.148425
$131$	2148.00	1.43261	0.716304	−	0.697788i	$-0.245832\pi$
$131$	2148.00	1.43261	0.716304	+	0.697788i	$0.245832\pi$
$132$	−132.000	−0.0870388
$133$	56.0000	0.0365099
$134$	−848.000	−0.546687
$135$	135.000	0.0860663
$136$	−144.000	−0.0907934
$137$	−882.000	−0.550032	−0.275016	−	0.961440i	$-0.588683\pi$
$137$	−882.000	−0.550032	−0.275016	+	0.961440i	$0.588683\pi$
$138$	−288.000	−0.177654
$139$	2288.00	1.39616	0.698078	−	0.716022i	$-0.254039\pi$
$139$	2288.00	1.39616	0.698078	+	0.716022i	$0.254039\pi$
$140$	−140.000	−0.0845154
$141$	−1728.00	−1.03208
$142$	−1440.00	−0.851001
$143$	−242.000	−0.141518
$144$	144.000	0.0833333
$145$	930.000	0.532637
$146$	844.000	0.478424
$147$	−147.000	−0.0824786
$148$	−1576.00	−0.875314
$149$	−2130.00	−1.17112	−0.585558	−	0.810630i	$-0.699125\pi$
$149$	−2130.00	−1.17112	−0.585558	+	0.810630i	$0.699125\pi$
$150$	−150.000	−0.0816497
$151$	−1504.00	−0.810555	−0.405277	−	0.914194i	$-0.632825\pi$
$151$	−1504.00	−0.810555	−0.405277	+	0.914194i	$0.632825\pi$
$152$	64.0000	0.0341519
$153$	−162.000	−0.0856008
$154$	154.000	0.0805823
$155$	−640.000	−0.331652
$156$	264.000	0.135493
$157$	−238.000	−0.120984	−0.0604919	−	0.998169i	$-0.519267\pi$
$157$	−238.000	−0.120984	−0.0604919	+	0.998169i	$0.519267\pi$
$158$	160.000	0.0805628
$159$	−1062.00	−0.529699
$160$	−160.000	−0.0790569
$161$	336.000	0.164475
$162$	162.000	0.0785674
$163$	−3040.00	−1.46080	−0.730402	−	0.683017i	$-0.760667\pi$
$163$	−3040.00	−1.46080	−0.730402	+	0.683017i	$0.760667\pi$
$164$	−1848.00	−0.879906
$165$	165.000	0.0778499
$166$	−696.000	−0.325422
$167$	−1104.00	−0.511557	−0.255779	−	0.966735i	$-0.582332\pi$
$167$	−1104.00	−0.511557	−0.255779	+	0.966735i	$0.582332\pi$
$168$	−168.000	−0.0771517
$169$	−1713.00	−0.779700
$170$	180.000	0.0812081
$171$	72.0000	0.0321987
$172$	−208.000	−0.0922084
$173$	1626.00	0.714581	0.357290	−	0.933993i	$-0.383701\pi$
$173$	1626.00	0.714581	0.357290	+	0.933993i	$0.383701\pi$
$174$	1116.00	0.486228
$175$	175.000	0.0755929
$176$	176.000	0.0753778
$177$	−2088.00	−0.886688
$178$	−468.000	−0.197068
$179$	1068.00	0.445956	0.222978	−	0.974824i	$-0.428422\pi$
$179$	1068.00	0.445956	0.222978	+	0.974824i	$0.428422\pi$
$180$	−180.000	−0.0745356
$181$	3278.00	1.34614	0.673071	−	0.739578i	$-0.264975\pi$
$181$	3278.00	1.34614	0.673071	+	0.739578i	$0.264975\pi$
$182$	−308.000	−0.125442
$183$	−1374.00	−0.555022
$184$	384.000	0.153852
$185$	1970.00	0.782904
$186$	−768.000	−0.302755
$187$	−198.000	−0.0774288
$188$	2304.00	0.893811
$189$	−189.000	−0.0727393
$190$	−80.0000	−0.0305464
$191$	624.000	0.236393	0.118196	−	0.992990i	$-0.462289\pi$
$191$	624.000	0.236393	0.118196	+	0.992990i	$0.462289\pi$
$192$	−192.000	−0.0721688
$193$	−4570.00	−1.70443	−0.852217	−	0.523188i	$-0.824742\pi$
$193$	−4570.00	−1.70443	−0.852217	+	0.523188i	$0.824742\pi$
$194$	−2564.00	−0.948889
$195$	−330.000	−0.121189
$196$	196.000	0.0714286
$197$	2718.00	0.982992	0.491496	−	0.870880i	$-0.336450\pi$
$197$	2718.00	0.982992	0.491496	+	0.870880i	$0.336450\pi$
$198$	198.000	0.0710669
$199$	−4168.00	−1.48473	−0.742366	−	0.669995i	$-0.766296\pi$
$199$	−4168.00	−1.48473	−0.742366	+	0.669995i	$0.766296\pi$
$200$	200.000	0.0707107
$201$	1272.00	0.446368
$202$	−2772.00	−0.965531
$203$	−1302.00	−0.450160
$204$	216.000	0.0741325
$205$	2310.00	0.787012
$206$	−656.000	−0.221872
$207$	432.000	0.145054
$208$	−352.000	−0.117340
$209$	88.0000	0.0291248
$210$	210.000	0.0690066
$211$	−748.000	−0.244049	−0.122025	−	0.992527i	$-0.538939\pi$
$211$	−748.000	−0.244049	−0.122025	+	0.992527i	$0.538939\pi$
$212$	1416.00	0.458732
$213$	2160.00	0.694839
$214$	768.000	0.245324
$215$	260.000	0.0824737
$216$	−216.000	−0.0680414
$217$	896.000	0.280297
$218$	−2468.00	−0.766762
$219$	−1266.00	−0.390632
$220$	−220.000	−0.0674200
$221$	396.000	0.120533
$222$	2364.00	0.714691
$223$	−3448.00	−1.03540	−0.517702	−	0.855561i	$-0.673212\pi$
$223$	−3448.00	−1.03540	−0.517702	+	0.855561i	$0.673212\pi$
$224$	224.000	0.0668153
$225$	225.000	0.0666667
$226$	−3924.00	−1.15496
$227$	−4236.00	−1.23856	−0.619280	−	0.785170i	$-0.712575\pi$
$227$	−4236.00	−1.23856	−0.619280	+	0.785170i	$0.712575\pi$
$228$	−96.0000	−0.0278849
$229$	−4378.00	−1.26335	−0.631673	−	0.775235i	$-0.717632\pi$
$229$	−4378.00	−1.26335	−0.631673	+	0.775235i	$0.717632\pi$
$230$	−480.000	−0.137610
$231$	−231.000	−0.0657952
$232$	−1488.00	−0.421086
$233$	−462.000	−0.129900	−0.0649498	−	0.997889i	$-0.520689\pi$
$233$	−462.000	−0.129900	−0.0649498	+	0.997889i	$0.520689\pi$
$234$	−396.000	−0.110630
$235$	−2880.00	−0.799449
$236$	2784.00	0.767894
$237$	−240.000	−0.0657792
$238$	−252.000	−0.0686333
$239$	−720.000	−0.194866	−0.0974329	−	0.995242i	$-0.531063\pi$
$239$	−720.000	−0.194866	−0.0974329	+	0.995242i	$0.531063\pi$
$240$	240.000	0.0645497
$241$	3278.00	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$241$	3278.00	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$242$	242.000	0.0642824
$243$	−243.000	−0.0641500
$244$	1832.00	0.480663
$245$	−245.000	−0.0638877
$246$	2772.00	0.718440
$247$	−176.000	−0.0453385
$248$	1024.00	0.262194
$249$	1044.00	0.265706
$250$	−250.000	−0.0632456
$251$	−1176.00	−0.295731	−0.147865	−	0.989007i	$-0.547240\pi$
$251$	−1176.00	−0.295731	−0.147865	+	0.989007i	$0.547240\pi$
$252$	252.000	0.0629941
$253$	528.000	0.131206
$254$	1072.00	0.264816
$255$	−270.000	−0.0663061
$256$	256.000	0.0625000
$257$	−5826.00	−1.41407	−0.707035	−	0.707179i	$-0.749968\pi$
$257$	−5826.00	−1.41407	−0.707035	+	0.707179i	$0.749968\pi$
$258$	312.000	0.0752879
$259$	−2758.00	−0.661675
$260$	440.000	0.104952
$261$	−1674.00	−0.397004
$262$	4296.00	1.01301
$263$	−6924.00	−1.62339	−0.811696	−	0.584080i	$-0.801455\pi$
$263$	−6924.00	−1.62339	−0.811696	+	0.584080i	$0.801455\pi$
$264$	−264.000	−0.0615457
$265$	−1770.00	−0.410303
$266$	112.000	0.0258164
$267$	702.000	0.160905
$268$	−1696.00	−0.386566
$269$	−1182.00	−0.267910	−0.133955	−	0.990987i	$-0.542768\pi$
$269$	−1182.00	−0.267910	−0.133955	+	0.990987i	$0.542768\pi$
$270$	270.000	0.0608581
$271$	−7684.00	−1.72240	−0.861199	−	0.508268i	$-0.830286\pi$
$271$	−7684.00	−1.72240	−0.861199	+	0.508268i	$0.830286\pi$
$272$	−288.000	−0.0642006
$273$	462.000	0.102423
$274$	−1764.00	−0.388931
$275$	275.000	0.0603023
$276$	−576.000	−0.125620
$277$	−1534.00	−0.332741	−0.166370	−	0.986063i	$-0.553205\pi$
$277$	−1534.00	−0.332741	−0.166370	+	0.986063i	$0.553205\pi$
$278$	4576.00	0.987231
$279$	1152.00	0.247199
$280$	−280.000	−0.0597614
$281$	3498.00	0.742609	0.371305	−	0.928511i	$-0.378911\pi$
$281$	3498.00	0.742609	0.371305	+	0.928511i	$0.378911\pi$
$282$	−3456.00	−0.729794
$283$	2372.00	0.498236	0.249118	−	0.968473i	$-0.419859\pi$
$283$	2372.00	0.498236	0.249118	+	0.968473i	$0.419859\pi$
$284$	−2880.00	−0.601748
$285$	120.000	0.0249410
$286$	−484.000	−0.100068
$287$	−3234.00	−0.665146
$288$	288.000	0.0589256
$289$	−4589.00	−0.934053
$290$	1860.00	0.376631
$291$	3846.00	0.774765
$292$	1688.00	0.338297
$293$	−4398.00	−0.876908	−0.438454	−	0.898754i	$-0.644474\pi$
$293$	−4398.00	−0.876908	−0.438454	+	0.898754i	$0.644474\pi$
$294$	−294.000	−0.0583212
$295$	−3480.00	−0.686825
$296$	−3152.00	−0.618940
$297$	−297.000	−0.0580259
$298$	−4260.00	−0.828105
$299$	−1056.00	−0.204248
$300$	−300.000	−0.0577350
$301$	−364.000	−0.0697030
$302$	−3008.00	−0.573149
$303$	4158.00	0.788353
$304$	128.000	0.0241490
$305$	−2290.00	−0.429918
$306$	−324.000	−0.0605289
$307$	4412.00	0.820215	0.410108	−	0.912037i	$-0.365491\pi$
$307$	4412.00	0.820215	0.410108	+	0.912037i	$0.365491\pi$
$308$	308.000	0.0569803
$309$	984.000	0.181158
$310$	−1280.00	−0.234513
$311$	564.000	0.102834	0.0514172	−	0.998677i	$-0.483626\pi$
$311$	564.000	0.102834	0.0514172	+	0.998677i	$0.483626\pi$
$312$	528.000	0.0958081
$313$	3998.00	0.721982	0.360991	−	0.932569i	$-0.382439\pi$
$313$	3998.00	0.721982	0.360991	+	0.932569i	$0.382439\pi$
$314$	−476.000	−0.0855485
$315$	−315.000	−0.0563436
$316$	320.000	0.0569665
$317$	−3342.00	−0.592131	−0.296065	−	0.955168i	$-0.595675\pi$
$317$	−3342.00	−0.592131	−0.296065	+	0.955168i	$0.595675\pi$
$318$	−2124.00	−0.374553
$319$	−2046.00	−0.359103
$320$	−320.000	−0.0559017
$321$	−1152.00	−0.200306
$322$	672.000	0.116302
$323$	−144.000	−0.0248061
$324$	324.000	0.0555556
$325$	−550.000	−0.0938723
$326$	−6080.00	−1.03294
$327$	3702.00	0.626058
$328$	−3696.00	−0.622187
$329$	4032.00	0.675658
$330$	330.000	0.0550482
$331$	−4108.00	−0.682163	−0.341082	−	0.940034i	$-0.610793\pi$
$331$	−4108.00	−0.682163	−0.341082	+	0.940034i	$0.610793\pi$
$332$	−1392.00	−0.230108
$333$	−3546.00	−0.583542
$334$	−2208.00	−0.361726
$335$	2120.00	0.345755
$336$	−336.000	−0.0545545
$337$	−6226.00	−1.00639	−0.503193	−	0.864174i	$-0.667841\pi$
$337$	−6226.00	−1.00639	−0.503193	+	0.864174i	$0.667841\pi$
$338$	−3426.00	−0.551331
$339$	5886.00	0.943020
$340$	360.000	0.0574228
$341$	1408.00	0.223600
$342$	144.000	0.0227679
$343$	343.000	0.0539949
$344$	−416.000	−0.0652012
$345$	720.000	0.112358
$346$	3252.00	0.505285
$347$	12312.0	1.90473	0.952367	−	0.304954i	$-0.0986410\pi$
$347$	12312.0	1.90473	0.952367	+	0.304954i	$0.0986410\pi$
$348$	2232.00	0.343815
$349$	−646.000	−0.0990819	−0.0495410	−	0.998772i	$-0.515776\pi$
$349$	−646.000	−0.0990819	−0.0495410	+	0.998772i	$0.515776\pi$
$350$	350.000	0.0534522
$351$	594.000	0.0903287
$352$	352.000	0.0533002
$353$	8430.00	1.27106	0.635529	−	0.772077i	$-0.280782\pi$
$353$	8430.00	1.27106	0.635529	+	0.772077i	$0.280782\pi$
$354$	−4176.00	−0.626983
$355$	3600.00	0.538220
$356$	−936.000	−0.139348
$357$	378.000	0.0560389
$358$	2136.00	0.315338
$359$	2640.00	0.388117	0.194058	−	0.980990i	$-0.437835\pi$
$359$	2640.00	0.388117	0.194058	+	0.980990i	$0.437835\pi$
$360$	−360.000	−0.0527046
$361$	−6795.00	−0.990669
$362$	6556.00	0.951867
$363$	−363.000	−0.0524864
$364$	−616.000	−0.0887010
$365$	−2110.00	−0.302582
$366$	−2748.00	−0.392460
$367$	−8800.00	−1.25165	−0.625826	−	0.779963i	$-0.715238\pi$
$367$	−8800.00	−1.25165	−0.625826	+	0.779963i	$0.715238\pi$
$368$	768.000	0.108790
$369$	−4158.00	−0.586604
$370$	3940.00	0.553597
$371$	2478.00	0.346769
$372$	−1536.00	−0.214080
$373$	−1774.00	−0.246258	−0.123129	−	0.992391i	$-0.539293\pi$
$373$	−1774.00	−0.246258	−0.123129	+	0.992391i	$0.539293\pi$
$374$	−396.000	−0.0547505
$375$	375.000	0.0516398
$376$	4608.00	0.632020
$377$	4092.00	0.559015
$378$	−378.000	−0.0514344
$379$	9812.00	1.32984	0.664919	−	0.746915i	$-0.268466\pi$
$379$	9812.00	1.32984	0.664919	+	0.746915i	$0.268466\pi$
$380$	−160.000	−0.0215995
$381$	−1608.00	−0.216221
$382$	1248.00	0.167155
$383$	2856.00	0.381031	0.190515	−	0.981684i	$-0.438984\pi$
$383$	2856.00	0.381031	0.190515	+	0.981684i	$0.438984\pi$
$384$	−384.000	−0.0510310
$385$	−385.000	−0.0509647
$386$	−9140.00	−1.20522
$387$	−468.000	−0.0614723
$388$	−5128.00	−0.670966
$389$	−6090.00	−0.793767	−0.396883	−	0.917869i	$-0.629908\pi$
$389$	−6090.00	−0.793767	−0.396883	+	0.917869i	$0.629908\pi$
$390$	−660.000	−0.0856933
$391$	−864.000	−0.111750
$392$	392.000	0.0505076
$393$	−6444.00	−0.827117
$394$	5436.00	0.695081
$395$	−400.000	−0.0509524
$396$	396.000	0.0502519
$397$	13034.0	1.64775	0.823876	−	0.566770i	$-0.191807\pi$
$397$	13034.0	1.64775	0.823876	+	0.566770i	$0.191807\pi$
$398$	−8336.00	−1.04986
$399$	−168.000	−0.0210790
$400$	400.000	0.0500000
$401$	5610.00	0.698629	0.349314	−	0.937006i	$-0.386415\pi$
$401$	5610.00	0.698629	0.349314	+	0.937006i	$0.386415\pi$
$402$	2544.00	0.315630
$403$	−2816.00	−0.348077
$404$	−5544.00	−0.682733
$405$	−405.000	−0.0496904
$406$	−2604.00	−0.318311
$407$	−4334.00	−0.527834
$408$	432.000	0.0524196
$409$	254.000	0.0307078	0.0153539	−	0.999882i	$-0.495113\pi$
$409$	254.000	0.0307078	0.0153539	+	0.999882i	$0.495113\pi$
$410$	4620.00	0.556501
$411$	2646.00	0.317561
$412$	−1312.00	−0.156887
$413$	4872.00	0.580473
$414$	864.000	0.102568
$415$	1740.00	0.205815
$416$	−704.000	−0.0829722
$417$	−6864.00	−0.806071
$418$	176.000	0.0205944
$419$	−1056.00	−0.123124	−0.0615620	−	0.998103i	$-0.519608\pi$
$419$	−1056.00	−0.123124	−0.0615620	+	0.998103i	$0.519608\pi$
$420$	420.000	0.0487950
$421$	−2914.00	−0.337339	−0.168669	−	0.985673i	$-0.553947\pi$
$421$	−2914.00	−0.337339	−0.168669	+	0.985673i	$0.553947\pi$
$422$	−1496.00	−0.172569
$423$	5184.00	0.595874
$424$	2832.00	0.324373
$425$	−450.000	−0.0513605
$426$	4320.00	0.491326
$427$	3206.00	0.363347
$428$	1536.00	0.173470
$429$	726.000	0.0817054
$430$	520.000	0.0583177
$431$	14016.0	1.56642	0.783210	−	0.621757i	$-0.213581\pi$
$431$	14016.0	1.56642	0.783210	+	0.621757i	$0.213581\pi$
$432$	−432.000	−0.0481125
$433$	−13642.0	−1.51407	−0.757035	−	0.653374i	$-0.773353\pi$
$433$	−13642.0	−1.51407	−0.757035	+	0.653374i	$0.773353\pi$
$434$	1792.00	0.198200
$435$	−2790.00	−0.307518
$436$	−4936.00	−0.542182
$437$	384.000	0.0420348
$438$	−2532.00	−0.276218
$439$	−8476.00	−0.921498	−0.460749	−	0.887531i	$-0.652419\pi$
$439$	−8476.00	−0.921498	−0.460749	+	0.887531i	$0.652419\pi$
$440$	−440.000	−0.0476731
$441$	441.000	0.0476190
$442$	792.000	0.0852299
$443$	−11748.0	−1.25996	−0.629982	−	0.776609i	$-0.716938\pi$
$443$	−11748.0	−1.25996	−0.629982	+	0.776609i	$0.716938\pi$
$444$	4728.00	0.505363
$445$	1170.00	0.124637
$446$	−6896.00	−0.732141
$447$	6390.00	0.676145
$448$	448.000	0.0472456
$449$	−13326.0	−1.40065	−0.700326	−	0.713823i	$-0.746962\pi$
$449$	−13326.0	−1.40065	−0.700326	+	0.713823i	$0.746962\pi$
$450$	450.000	0.0471405
$451$	−5082.00	−0.530603
$452$	−7848.00	−0.816679
$453$	4512.00	0.467974
$454$	−8472.00	−0.875794
$455$	770.000	0.0793366
$456$	−192.000	−0.0197176
$457$	6686.00	0.684372	0.342186	−	0.939632i	$-0.388833\pi$
$457$	6686.00	0.684372	0.342186	+	0.939632i	$0.388833\pi$
$458$	−8756.00	−0.893321
$459$	486.000	0.0494217
$460$	−960.000	−0.0973048
$461$	8070.00	0.815309	0.407654	−	0.913136i	$-0.366347\pi$
$461$	8070.00	0.815309	0.407654	+	0.913136i	$0.366347\pi$
$462$	−462.000	−0.0465242
$463$	1868.00	0.187502	0.0937509	−	0.995596i	$-0.470114\pi$
$463$	1868.00	0.187502	0.0937509	+	0.995596i	$0.470114\pi$
$464$	−2976.00	−0.297753
$465$	1920.00	0.191479
$466$	−924.000	−0.0918529
$467$	−16356.0	−1.62070	−0.810348	−	0.585948i	$-0.800722\pi$
$467$	−16356.0	−1.62070	−0.810348	+	0.585948i	$0.800722\pi$
$468$	−792.000	−0.0782270
$469$	−2968.00	−0.292216
$470$	−5760.00	−0.565296
$471$	714.000	0.0698501
$472$	5568.00	0.542983
$473$	−572.000	−0.0556038
$474$	−480.000	−0.0465129
$475$	200.000	0.0193192
$476$	−504.000	−0.0485311
$477$	3186.00	0.305822
$478$	−1440.00	−0.137791
$479$	−10728.0	−1.02333	−0.511665	−	0.859185i	$-0.670971\pi$
$479$	−10728.0	−1.02333	−0.511665	+	0.859185i	$0.670971\pi$
$480$	480.000	0.0456435
$481$	8668.00	0.821677
$482$	6556.00	0.619539
$483$	−1008.00	−0.0949598
$484$	484.000	0.0454545
$485$	6410.00	0.600130
$486$	−486.000	−0.0453609
$487$	17156.0	1.59633	0.798165	−	0.602439i	$-0.205804\pi$
$487$	17156.0	1.59633	0.798165	+	0.602439i	$0.205804\pi$
$488$	3664.00	0.339880
$489$	9120.00	0.843396
$490$	−490.000	−0.0451754
$491$	8484.00	0.779791	0.389896	−	0.920859i	$-0.372511\pi$
$491$	8484.00	0.779791	0.389896	+	0.920859i	$0.372511\pi$
$492$	5544.00	0.508014
$493$	3348.00	0.305855
$494$	−352.000	−0.0320592
$495$	−495.000	−0.0449467
$496$	2048.00	0.185399
$497$	−5040.00	−0.454879
$498$	2088.00	0.187883
$499$	10460.0	0.938385	0.469192	−	0.883096i	$-0.344545\pi$
$499$	10460.0	0.938385	0.469192	+	0.883096i	$0.344545\pi$
$500$	−500.000	−0.0447214
$501$	3312.00	0.295348
$502$	−2352.00	−0.209113
$503$	10776.0	0.955225	0.477612	−	0.878571i	$-0.341502\pi$
$503$	10776.0	0.955225	0.477612	+	0.878571i	$0.341502\pi$
$504$	504.000	0.0445435
$505$	6930.00	0.610655
$506$	1056.00	0.0927765
$507$	5139.00	0.450160
$508$	2144.00	0.187253
$509$	−3390.00	−0.295205	−0.147602	−	0.989047i	$-0.547156\pi$
$509$	−3390.00	−0.295205	−0.147602	+	0.989047i	$0.547156\pi$
$510$	−540.000	−0.0468855
$511$	2954.00	0.255729
$512$	512.000	0.0441942
$513$	−216.000	−0.0185899
$514$	−11652.0	−0.999898
$515$	1640.00	0.140324
$516$	624.000	0.0532366
$517$	6336.00	0.538988
$518$	−5516.00	−0.467875
$519$	−4878.00	−0.412563
$520$	880.000	0.0742126
$521$	−42.0000	−0.00353177	−0.00176589	−	0.999998i	$-0.500562\pi$
$521$	−42.0000	−0.00353177	−0.00176589	+	0.999998i	$0.500562\pi$
$522$	−3348.00	−0.280724
$523$	−8548.00	−0.714681	−0.357340	−	0.933974i	$-0.616316\pi$
$523$	−8548.00	−0.714681	−0.357340	+	0.933974i	$0.616316\pi$
$524$	8592.00	0.716304
$525$	−525.000	−0.0436436
$526$	−13848.0	−1.14791
$527$	−2304.00	−0.190444
$528$	−528.000	−0.0435194
$529$	−9863.00	−0.810635
$530$	−3540.00	−0.290128
$531$	6264.00	0.511929
$532$	224.000	0.0182549
$533$	10164.0	0.825988
$534$	1404.00	0.113777
$535$	−1920.00	−0.155157
$536$	−3392.00	−0.273343
$537$	−3204.00	−0.257473
$538$	−2364.00	−0.189441
$539$	539.000	0.0430730
$540$	540.000	0.0430331
$541$	758.000	0.0602384	0.0301192	−	0.999546i	$-0.490411\pi$
$541$	758.000	0.0602384	0.0301192	+	0.999546i	$0.490411\pi$
$542$	−15368.0	−1.21792
$543$	−9834.00	−0.777196
$544$	−576.000	−0.0453967
$545$	6170.00	0.484943
$546$	924.000	0.0724241
$547$	−14596.0	−1.14091	−0.570457	−	0.821328i	$-0.693234\pi$
$547$	−14596.0	−1.14091	−0.570457	+	0.821328i	$0.693234\pi$
$548$	−3528.00	−0.275016
$549$	4122.00	0.320442
$550$	550.000	0.0426401
$551$	−1488.00	−0.115047
$552$	−1152.00	−0.0888268
$553$	560.000	0.0430626
$554$	−3068.00	−0.235283
$555$	−5910.00	−0.452010
$556$	9152.00	0.698078
$557$	22734.0	1.72939	0.864695	−	0.502297i	$-0.167512\pi$
$557$	22734.0	1.72939	0.864695	+	0.502297i	$0.167512\pi$
$558$	2304.00	0.174796
$559$	1144.00	0.0865582
$560$	−560.000	−0.0422577
$561$	594.000	0.0447036
$562$	6996.00	0.525104
$563$	8700.00	0.651263	0.325632	−	0.945497i	$-0.394423\pi$
$563$	8700.00	0.651263	0.325632	+	0.945497i	$0.394423\pi$
$564$	−6912.00	−0.516042
$565$	9810.00	0.730460
$566$	4744.00	0.352306
$567$	567.000	0.0419961
$568$	−5760.00	−0.425500
$569$	−246.000	−0.0181245	−0.00906226	−	0.999959i	$-0.502885\pi$
$569$	−246.000	−0.0181245	−0.00906226	+	0.999959i	$0.502885\pi$
$570$	240.000	0.0176360
$571$	3764.00	0.275864	0.137932	−	0.990442i	$-0.455954\pi$
$571$	3764.00	0.275864	0.137932	+	0.990442i	$0.455954\pi$
$572$	−968.000	−0.0707589
$573$	−1872.00	−0.136482
$574$	−6468.00	−0.470329
$575$	1200.00	0.0870321
$576$	576.000	0.0416667
$577$	−17242.0	−1.24401	−0.622005	−	0.783013i	$-0.713682\pi$
$577$	−17242.0	−1.24401	−0.622005	+	0.783013i	$0.713682\pi$
$578$	−9178.00	−0.660475
$579$	13710.0	0.984056
$580$	3720.00	0.266318
$581$	−2436.00	−0.173945
$582$	7692.00	0.547841
$583$	3894.00	0.276626
$584$	3376.00	0.239212
$585$	990.000	0.0699683
$586$	−8796.00	−0.620067
$587$	−11460.0	−0.805800	−0.402900	−	0.915244i	$-0.631998\pi$
$587$	−11460.0	−0.805800	−0.402900	+	0.915244i	$0.631998\pi$
$588$	−588.000	−0.0412393
$589$	1024.00	0.0716353
$590$	−6960.00	−0.485659
$591$	−8154.00	−0.567531
$592$	−6304.00	−0.437657
$593$	−21114.0	−1.46214	−0.731069	−	0.682303i	$-0.760978\pi$
$593$	−21114.0	−1.46214	−0.731069	+	0.682303i	$0.760978\pi$
$594$	−594.000	−0.0410305
$595$	630.000	0.0434075
$596$	−8520.00	−0.585558
$597$	12504.0	0.857210
$598$	−2112.00	−0.144425
$599$	−28584.0	−1.94977	−0.974884	−	0.222715i	$-0.928508\pi$
$599$	−28584.0	−1.94977	−0.974884	+	0.222715i	$0.928508\pi$
$600$	−600.000	−0.0408248
$601$	−15202.0	−1.03178	−0.515892	−	0.856653i	$-0.672540\pi$
$601$	−15202.0	−1.03178	−0.515892	+	0.856653i	$0.672540\pi$
$602$	−728.000	−0.0492875
$603$	−3816.00	−0.257711
$604$	−6016.00	−0.405277
$605$	−605.000	−0.0406558
$606$	8316.00	0.557450
$607$	−6496.00	−0.434373	−0.217187	−	0.976130i	$-0.569688\pi$
$607$	−6496.00	−0.434373	−0.217187	+	0.976130i	$0.569688\pi$
$608$	256.000	0.0170759
$609$	3906.00	0.259900
$610$	−4580.00	−0.303998
$611$	−12672.0	−0.839041
$612$	−648.000	−0.0428004
$613$	−4174.00	−0.275018	−0.137509	−	0.990500i	$-0.543910\pi$
$613$	−4174.00	−0.275018	−0.137509	+	0.990500i	$0.543910\pi$
$614$	8824.00	0.579980
$615$	−6930.00	−0.454381
$616$	616.000	0.0402911
$617$	7926.00	0.517162	0.258581	−	0.965990i	$-0.416745\pi$
$617$	7926.00	0.517162	0.258581	+	0.965990i	$0.416745\pi$
$618$	1968.00	0.128098
$619$	12404.0	0.805426	0.402713	−	0.915326i	$-0.368067\pi$
$619$	12404.0	0.805426	0.402713	+	0.915326i	$0.368067\pi$
$620$	−2560.00	−0.165826
$621$	−1296.00	−0.0837467
$622$	1128.00	0.0727149
$623$	−1638.00	−0.105337
$624$	1056.00	0.0677465
$625$	625.000	0.0400000
$626$	7996.00	0.510518
$627$	−264.000	−0.0168152
$628$	−952.000	−0.0604919
$629$	7092.00	0.449565
$630$	−630.000	−0.0398410
$631$	−1240.00	−0.0782308	−0.0391154	−	0.999235i	$-0.512454\pi$
$631$	−1240.00	−0.0782308	−0.0391154	+	0.999235i	$0.512454\pi$
$632$	640.000	0.0402814
$633$	2244.00	0.140902
$634$	−6684.00	−0.418700
$635$	−2680.00	−0.167484
$636$	−4248.00	−0.264849
$637$	−1078.00	−0.0670517
$638$	−4092.00	−0.253925
$639$	−6480.00	−0.401166
$640$	−640.000	−0.0395285
$641$	15954.0	0.983066	0.491533	−	0.870859i	$-0.336437\pi$
$641$	15954.0	0.983066	0.491533	+	0.870859i	$0.336437\pi$
$642$	−2304.00	−0.141638
$643$	17948.0	1.10078	0.550389	−	0.834908i	$-0.314479\pi$
$643$	17948.0	1.10078	0.550389	+	0.834908i	$0.314479\pi$
$644$	1344.00	0.0822376
$645$	−780.000	−0.0476162
$646$	−288.000	−0.0175406
$647$	29280.0	1.77916	0.889579	−	0.456781i	$-0.150998\pi$
$647$	29280.0	1.77916	0.889579	+	0.456781i	$0.150998\pi$
$648$	648.000	0.0392837
$649$	7656.00	0.463057
$650$	−1100.00	−0.0663778
$651$	−2688.00	−0.161830
$652$	−12160.0	−0.730402
$653$	−19110.0	−1.14523	−0.572613	−	0.819826i	$-0.694070\pi$
$653$	−19110.0	−1.14523	−0.572613	+	0.819826i	$0.694070\pi$
$654$	7404.00	0.442690
$655$	−10740.0	−0.640682
$656$	−7392.00	−0.439953
$657$	3798.00	0.225531
$658$	8064.00	0.477762
$659$	23220.0	1.37257	0.686284	−	0.727333i	$-0.259241\pi$
$659$	23220.0	1.37257	0.686284	+	0.727333i	$0.259241\pi$
$660$	660.000	0.0389249
$661$	31814.0	1.87204	0.936022	−	0.351941i	$-0.114478\pi$
$661$	31814.0	1.87204	0.936022	+	0.351941i	$0.114478\pi$
$662$	−8216.00	−0.482362
$663$	−1188.00	−0.0695899
$664$	−2784.00	−0.162711
$665$	−280.000	−0.0163277
$666$	−7092.00	−0.412627
$667$	−8928.00	−0.518281
$668$	−4416.00	−0.255779
$669$	10344.0	0.597791
$670$	4240.00	0.244486
$671$	5038.00	0.289851
$672$	−672.000	−0.0385758
$673$	−3082.00	−0.176527	−0.0882633	−	0.996097i	$-0.528132\pi$
$673$	−3082.00	−0.176527	−0.0882633	+	0.996097i	$0.528132\pi$
$674$	−12452.0	−0.711622
$675$	−675.000	−0.0384900
$676$	−6852.00	−0.389850
$677$	11466.0	0.650922	0.325461	−	0.945555i	$-0.394481\pi$
$677$	11466.0	0.650922	0.325461	+	0.945555i	$0.394481\pi$
$678$	11772.0	0.666816
$679$	−8974.00	−0.507203
$680$	720.000	0.0406040
$681$	12708.0	0.715083
$682$	2816.00	0.158109
$683$	−13548.0	−0.759004	−0.379502	−	0.925191i	$-0.623905\pi$
$683$	−13548.0	−0.759004	−0.379502	+	0.925191i	$0.623905\pi$
$684$	288.000	0.0160993
$685$	4410.00	0.245982
$686$	686.000	0.0381802
$687$	13134.0	0.729394
$688$	−832.000	−0.0461042
$689$	−7788.00	−0.430623
$690$	1440.00	0.0794491
$691$	26684.0	1.46904	0.734520	−	0.678587i	$-0.237407\pi$
$691$	26684.0	1.46904	0.734520	+	0.678587i	$0.237407\pi$
$692$	6504.00	0.357290
$693$	693.000	0.0379869
$694$	24624.0	1.34685
$695$	−11440.0	−0.624380
$696$	4464.00	0.243114
$697$	8316.00	0.451924
$698$	−1292.00	−0.0700615
$699$	1386.00	0.0749976
$700$	700.000	0.0377964
$701$	16230.0	0.874463	0.437232	−	0.899349i	$-0.355959\pi$
$701$	16230.0	0.874463	0.437232	+	0.899349i	$0.355959\pi$
$702$	1188.00	0.0638720
$703$	−3152.00	−0.169104
$704$	704.000	0.0376889
$705$	8640.00	0.461562
$706$	16860.0	0.898774
$707$	−9702.00	−0.516098
$708$	−8352.00	−0.443344
$709$	18878.0	0.999969	0.499985	−	0.866034i	$-0.333339\pi$
$709$	18878.0	0.999969	0.499985	+	0.866034i	$0.333339\pi$
$710$	7200.00	0.380579
$711$	720.000	0.0379777
$712$	−1872.00	−0.0985339
$713$	6144.00	0.322713
$714$	756.000	0.0396255
$715$	1210.00	0.0632887
$716$	4272.00	0.222978
$717$	2160.00	0.112506
$718$	5280.00	0.274440
$719$	36228.0	1.87910	0.939552	−	0.342405i	$-0.111241\pi$
$719$	36228.0	1.87910	0.939552	+	0.342405i	$0.111241\pi$
$720$	−720.000	−0.0372678
$721$	−2296.00	−0.118596
$722$	−13590.0	−0.700509
$723$	−9834.00	−0.505851
$724$	13112.0	0.673071
$725$	−4650.00	−0.238202
$726$	−726.000	−0.0371135
$727$	−6496.00	−0.331394	−0.165697	−	0.986177i	$-0.552987\pi$
$727$	−6496.00	−0.331394	−0.165697	+	0.986177i	$0.552987\pi$
$728$	−1232.00	−0.0627211
$729$	729.000	0.0370370
$730$	−4220.00	−0.213958
$731$	936.000	0.0473587
$732$	−5496.00	−0.277511
$733$	−36022.0	−1.81515	−0.907574	−	0.419893i	$-0.862068\pi$
$733$	−36022.0	−1.81515	−0.907574	+	0.419893i	$0.862068\pi$
$734$	−17600.0	−0.885052
$735$	735.000	0.0368856
$736$	1536.00	0.0769262
$737$	−4664.00	−0.233108
$738$	−8316.00	−0.414792
$739$	13604.0	0.677174	0.338587	−	0.940935i	$-0.390051\pi$
$739$	13604.0	0.677174	0.338587	+	0.940935i	$0.390051\pi$
$740$	7880.00	0.391452
$741$	528.000	0.0261762
$742$	4956.00	0.245203
$743$	−11652.0	−0.575330	−0.287665	−	0.957731i	$-0.592879\pi$
$743$	−11652.0	−0.575330	−0.287665	+	0.957731i	$0.592879\pi$
$744$	−3072.00	−0.151378
$745$	10650.0	0.523739
$746$	−3548.00	−0.174131
$747$	−3132.00	−0.153405
$748$	−792.000	−0.0387144
$749$	2688.00	0.131131
$750$	750.000	0.0365148
$751$	11072.0	0.537980	0.268990	−	0.963143i	$-0.413310\pi$
$751$	11072.0	0.537980	0.268990	+	0.963143i	$0.413310\pi$
$752$	9216.00	0.446906
$753$	3528.00	0.170740
$754$	8184.00	0.395283
$755$	7520.00	0.362491
$756$	−756.000	−0.0363696
$757$	−13978.0	−0.671122	−0.335561	−	0.942019i	$-0.608926\pi$
$757$	−13978.0	−0.671122	−0.335561	+	0.942019i	$0.608926\pi$
$758$	19624.0	0.940337
$759$	−1584.00	−0.0757517
$760$	−320.000	−0.0152732
$761$	8610.00	0.410134	0.205067	−	0.978748i	$-0.434259\pi$
$761$	8610.00	0.410134	0.205067	+	0.978748i	$0.434259\pi$
$762$	−3216.00	−0.152892
$763$	−8638.00	−0.409851
$764$	2496.00	0.118196
$765$	810.000	0.0382818
$766$	5712.00	0.269429
$767$	−15312.0	−0.720840
$768$	−768.000	−0.0360844
$769$	36830.0	1.72708	0.863540	−	0.504281i	$-0.168242\pi$
$769$	36830.0	1.72708	0.863540	+	0.504281i	$0.168242\pi$
$770$	−770.000	−0.0360375
$771$	17478.0	0.816413
$772$	−18280.0	−0.852217
$773$	−32958.0	−1.53353	−0.766764	−	0.641929i	$-0.778134\pi$
$773$	−32958.0	−1.53353	−0.766764	+	0.641929i	$0.778134\pi$
$774$	−936.000	−0.0434675
$775$	3200.00	0.148319
$776$	−10256.0	−0.474445
$777$	8274.00	0.382018
$778$	−12180.0	−0.561278
$779$	−3696.00	−0.169991
$780$	−1320.00	−0.0605943
$781$	−7920.00	−0.362868
$782$	−1728.00	−0.0790194
$783$	5022.00	0.229210
$784$	784.000	0.0357143
$785$	1190.00	0.0541056
$786$	−12888.0	−0.584860
$787$	−12052.0	−0.545880	−0.272940	−	0.962031i	$-0.587996\pi$
$787$	−12052.0	−0.545880	−0.272940	+	0.962031i	$0.587996\pi$
$788$	10872.0	0.491496
$789$	20772.0	0.937266
$790$	−800.000	−0.0360288
$791$	−13734.0	−0.617351
$792$	792.000	0.0355335
$793$	−10076.0	−0.451210
$794$	26068.0	1.16514
$795$	5310.00	0.236888
$796$	−16672.0	−0.742366
$797$	28218.0	1.25412	0.627059	−	0.778971i	$-0.284258\pi$
$797$	28218.0	1.25412	0.627059	+	0.778971i	$0.284258\pi$
$798$	−336.000	−0.0149051
$799$	−10368.0	−0.459066
$800$	800.000	0.0353553
$801$	−2106.00	−0.0928987
$802$	11220.0	0.494005
$803$	4642.00	0.204001
$804$	5088.00	0.223184
$805$	−1680.00	−0.0735556
$806$	−5632.00	−0.246127
$807$	3546.00	0.154678
$808$	−11088.0	−0.482765
$809$	4746.00	0.206255	0.103128	−	0.994668i	$-0.467115\pi$
$809$	4746.00	0.206255	0.103128	+	0.994668i	$0.467115\pi$
$810$	−810.000	−0.0351364
$811$	1664.00	0.0720480	0.0360240	−	0.999351i	$-0.488531\pi$
$811$	1664.00	0.0720480	0.0360240	+	0.999351i	$0.488531\pi$
$812$	−5208.00	−0.225080
$813$	23052.0	0.994427
$814$	−8668.00	−0.373235
$815$	15200.0	0.653292
$816$	864.000	0.0370662
$817$	−416.000	−0.0178140
$818$	508.000	0.0217137
$819$	−1386.00	−0.0591340
$820$	9240.00	0.393506
$821$	−3762.00	−0.159920	−0.0799602	−	0.996798i	$-0.525479\pi$
$821$	−3762.00	−0.159920	−0.0799602	+	0.996798i	$0.525479\pi$
$822$	5292.00	0.224550
$823$	−13852.0	−0.586695	−0.293348	−	0.956006i	$-0.594769\pi$
$823$	−13852.0	−0.586695	−0.293348	+	0.956006i	$0.594769\pi$
$824$	−2624.00	−0.110936
$825$	−825.000	−0.0348155
$826$	9744.00	0.410457
$827$	43944.0	1.84774	0.923871	−	0.382704i	$-0.125007\pi$
$827$	43944.0	1.84774	0.923871	+	0.382704i	$0.125007\pi$
$828$	1728.00	0.0725268
$829$	29390.0	1.23131	0.615656	−	0.788015i	$-0.288891\pi$
$829$	29390.0	1.23131	0.615656	+	0.788015i	$0.288891\pi$
$830$	3480.00	0.145533
$831$	4602.00	0.192108
$832$	−1408.00	−0.0586702
$833$	−882.000	−0.0366861
$834$	−13728.0	−0.569978
$835$	5520.00	0.228775
$836$	352.000	0.0145624
$837$	−3456.00	−0.142720
$838$	−2112.00	−0.0870618
$839$	38556.0	1.58653	0.793266	−	0.608875i	$-0.208379\pi$
$839$	38556.0	1.58653	0.793266	+	0.608875i	$0.208379\pi$
$840$	840.000	0.0345033
$841$	10207.0	0.418508
$842$	−5828.00	−0.238535
$843$	−10494.0	−0.428746
$844$	−2992.00	−0.122025
$845$	8565.00	0.348692
$846$	10368.0	0.421347
$847$	847.000	0.0343604
$848$	5664.00	0.229366
$849$	−7116.00	−0.287657
$850$	−900.000	−0.0363173
$851$	−18912.0	−0.761804
$852$	8640.00	0.347420
$853$	−30958.0	−1.24265	−0.621326	−	0.783552i	$-0.713406\pi$
$853$	−30958.0	−1.24265	−0.621326	+	0.783552i	$0.713406\pi$
$854$	6412.00	0.256925
$855$	−360.000	−0.0143997
$856$	3072.00	0.122662
$857$	36654.0	1.46100	0.730500	−	0.682913i	$-0.239287\pi$
$857$	36654.0	1.46100	0.730500	+	0.682913i	$0.239287\pi$
$858$	1452.00	0.0577744
$859$	−25468.0	−1.01159	−0.505796	−	0.862653i	$-0.668801\pi$
$859$	−25468.0	−1.01159	−0.505796	+	0.862653i	$0.668801\pi$
$860$	1040.00	0.0412369
$861$	9702.00	0.384022
$862$	28032.0	1.10763
$863$	25704.0	1.01388	0.506938	−	0.861983i	$-0.330777\pi$
$863$	25704.0	1.01388	0.506938	+	0.861983i	$0.330777\pi$
$864$	−864.000	−0.0340207
$865$	−8130.00	−0.319570
$866$	−27284.0	−1.07061
$867$	13767.0	0.539275
$868$	3584.00	0.140148
$869$	880.000	0.0343521
$870$	−5580.00	−0.217448
$871$	9328.00	0.362879
$872$	−9872.00	−0.383381
$873$	−11538.0	−0.447311
$874$	768.000	0.0297231
$875$	−875.000	−0.0338062
$876$	−5064.00	−0.195316
$877$	26714.0	1.02858	0.514292	−	0.857615i	$-0.328055\pi$
$877$	26714.0	1.02858	0.514292	+	0.857615i	$0.328055\pi$
$878$	−16952.0	−0.651597
$879$	13194.0	0.506283
$880$	−880.000	−0.0337100
$881$	−25314.0	−0.968048	−0.484024	−	0.875055i	$-0.660825\pi$
$881$	−25314.0	−0.968048	−0.484024	+	0.875055i	$0.660825\pi$
$882$	882.000	0.0336718
$883$	−19000.0	−0.724123	−0.362062	−	0.932154i	$-0.617927\pi$
$883$	−19000.0	−0.724123	−0.362062	+	0.932154i	$0.617927\pi$
$884$	1584.00	0.0602666
$885$	10440.0	0.396539
$886$	−23496.0	−0.890930
$887$	28368.0	1.07385	0.536925	−	0.843630i	$-0.319586\pi$
$887$	28368.0	1.07385	0.536925	+	0.843630i	$0.319586\pi$
$888$	9456.00	0.357345
$889$	3752.00	0.141550
$890$	2340.00	0.0881314
$891$	891.000	0.0335013
$892$	−13792.0	−0.517702
$893$	4608.00	0.172677
$894$	12780.0	0.478106
$895$	−5340.00	−0.199437
$896$	896.000	0.0334077
$897$	3168.00	0.117922
$898$	−26652.0	−0.990411
$899$	−23808.0	−0.883249
$900$	900.000	0.0333333
$901$	−6372.00	−0.235607
$902$	−10164.0	−0.375193
$903$	1092.00	0.0402431
$904$	−15696.0	−0.577479
$905$	−16390.0	−0.602013
$906$	9024.00	0.330908
$907$	21896.0	0.801593	0.400796	−	0.916167i	$-0.368733\pi$
$907$	21896.0	0.801593	0.400796	+	0.916167i	$0.368733\pi$
$908$	−16944.0	−0.619280
$909$	−12474.0	−0.455156
$910$	1540.00	0.0560995
$911$	−2232.00	−0.0811739	−0.0405870	−	0.999176i	$-0.512923\pi$
$911$	−2232.00	−0.0811739	−0.0405870	+	0.999176i	$0.512923\pi$
$912$	−384.000	−0.0139424
$913$	−3828.00	−0.138760
$914$	13372.0	0.483924
$915$	6870.00	0.248213
$916$	−17512.0	−0.631673
$917$	15036.0	0.541475
$918$	972.000	0.0349464
$919$	−15280.0	−0.548466	−0.274233	−	0.961663i	$-0.588424\pi$
$919$	−15280.0	−0.548466	−0.274233	+	0.961663i	$0.588424\pi$
$920$	−1920.00	−0.0688049
$921$	−13236.0	−0.473552
$922$	16140.0	0.576510
$923$	15840.0	0.564875
$924$	−924.000	−0.0328976
$925$	−9850.00	−0.350125
$926$	3736.00	0.132584
$927$	−2952.00	−0.104592
$928$	−5952.00	−0.210543
$929$	48054.0	1.69709	0.848547	−	0.529120i	$-0.177478\pi$
$929$	48054.0	1.69709	0.848547	+	0.529120i	$0.177478\pi$
$930$	3840.00	0.135396
$931$	392.000	0.0137994
$932$	−1848.00	−0.0649498
$933$	−1692.00	−0.0593715
$934$	−32712.0	−1.14601
$935$	990.000	0.0346272
$936$	−1584.00	−0.0553148
$937$	20630.0	0.719267	0.359633	−	0.933094i	$-0.382902\pi$
$937$	20630.0	0.719267	0.359633	+	0.933094i	$0.382902\pi$
$938$	−5936.00	−0.206628
$939$	−11994.0	−0.416836
$940$	−11520.0	−0.399724
$941$	−11586.0	−0.401374	−0.200687	−	0.979655i	$-0.564317\pi$
$941$	−11586.0	−0.401374	−0.200687	+	0.979655i	$0.564317\pi$
$942$	1428.00	0.0493915
$943$	−22176.0	−0.765801
$944$	11136.0	0.383947
$945$	945.000	0.0325300
$946$	−1144.00	−0.0393178
$947$	−3300.00	−0.113237	−0.0566186	−	0.998396i	$-0.518032\pi$
$947$	−3300.00	−0.113237	−0.0566186	+	0.998396i	$0.518032\pi$
$948$	−960.000	−0.0328896
$949$	−9284.00	−0.317567
$950$	400.000	0.0136608
$951$	10026.0	0.341867
$952$	−1008.00	−0.0343167
$953$	22386.0	0.760917	0.380458	−	0.924798i	$-0.375766\pi$
$953$	22386.0	0.760917	0.380458	+	0.924798i	$0.375766\pi$
$954$	6372.00	0.216249
$955$	−3120.00	−0.105718
$956$	−2880.00	−0.0974329
$957$	6138.00	0.207328
$958$	−21456.0	−0.723603
$959$	−6174.00	−0.207892
$960$	960.000	0.0322749
$961$	−13407.0	−0.450035
$962$	17336.0	0.581014
$963$	3456.00	0.115647
$964$	13112.0	0.438080
$965$	22850.0	0.762246
$966$	−2016.00	−0.0671467
$967$	−26464.0	−0.880067	−0.440034	−	0.897981i	$-0.645033\pi$
$967$	−26464.0	−0.880067	−0.440034	+	0.897981i	$0.645033\pi$
$968$	968.000	0.0321412
$969$	432.000	0.0143218
$970$	12820.0	0.424356
$971$	−12648.0	−0.418016	−0.209008	−	0.977914i	$-0.567023\pi$
$971$	−12648.0	−0.418016	−0.209008	+	0.977914i	$0.567023\pi$
$972$	−972.000	−0.0320750
$973$	16016.0	0.527697
$974$	34312.0	1.12878
$975$	1650.00	0.0541972
$976$	7328.00	0.240332
$977$	12174.0	0.398650	0.199325	−	0.979933i	$-0.436125\pi$
$977$	12174.0	0.398650	0.199325	+	0.979933i	$0.436125\pi$
$978$	18240.0	0.596371
$979$	−2574.00	−0.0840300
$980$	−980.000	−0.0319438
$981$	−11106.0	−0.361455
$982$	16968.0	0.551396
$983$	20688.0	0.671256	0.335628	−	0.941995i	$-0.391051\pi$
$983$	20688.0	0.671256	0.335628	+	0.941995i	$0.391051\pi$
$984$	11088.0	0.359220
$985$	−13590.0	−0.439608
$986$	6696.00	0.216272
$987$	−12096.0	−0.390091
$988$	−704.000	−0.0226693
$989$	−2496.00	−0.0802509
$990$	−990.000	−0.0317821
$991$	−47896.0	−1.53528	−0.767642	−	0.640879i	$-0.778570\pi$
$991$	−47896.0	−1.53528	−0.767642	+	0.640879i	$0.778570\pi$
$992$	4096.00	0.131097
$993$	12324.0	0.393847
$994$	−10080.0	−0.321648
$995$	20840.0	0.663992
$996$	4176.00	0.132853
$997$	−8062.00	−0.256094	−0.128047	−	0.991768i	$-0.540871\pi$
$997$	−8062.00	−0.256094	−0.128047	+	0.991768i	$0.540871\pi$
$998$	20920.0	0.663538
$999$	10638.0	0.336908

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.4.a.g.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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