LMFDB - Embedded newform 693.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$693 = 3^{2} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	693.1

$q$ -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} -4.00000 q^{4} -11.0000 q^{5} -7.00000 q^{7} +24.0000 q^{8} +22.0000 q^{10} -11.0000 q^{11} -5.00000 q^{13} +14.0000 q^{14} -16.0000 q^{16} +118.000 q^{17} -105.000 q^{19} +44.0000 q^{20} +22.0000 q^{22} +68.0000 q^{23} -4.00000 q^{25} +10.0000 q^{26} +28.0000 q^{28} +195.000 q^{29} +214.000 q^{31} -160.000 q^{32} -236.000 q^{34} +77.0000 q^{35} +33.0000 q^{37} +210.000 q^{38} -264.000 q^{40} +376.000 q^{41} -168.000 q^{43} +44.0000 q^{44} -136.000 q^{46} -61.0000 q^{47} +49.0000 q^{49} +8.00000 q^{50} +20.0000 q^{52} -24.0000 q^{53} +121.000 q^{55} -168.000 q^{56} -390.000 q^{58} -625.000 q^{59} -558.000 q^{61} -428.000 q^{62} +448.000 q^{64} +55.0000 q^{65} +173.000 q^{67} -472.000 q^{68} -154.000 q^{70} -168.000 q^{71} +973.000 q^{73} -66.0000 q^{74} +420.000 q^{76} +77.0000 q^{77} -1072.00 q^{79} +176.000 q^{80} -752.000 q^{82} -1458.00 q^{83} -1298.00 q^{85} +336.000 q^{86} -264.000 q^{88} +198.000 q^{89} +35.0000 q^{91} -272.000 q^{92} +122.000 q^{94} +1155.00 q^{95} -352.000 q^{97} -98.0000 q^{98} +O(q^{100})

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−2.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	−11.0000	−0.983870	−0.491935	−	0.870632i	$-0.663710\pi$
$5$	−11.0000	−0.983870	−0.491935	+	0.870632i	$0.663710\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	24.0000	1.06066
$9$	0	0
$10$	22.0000	0.695701
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−5.00000	−0.106673	−0.0533366	−	0.998577i	$-0.516986\pi$
$13$	−5.00000	−0.106673	−0.0533366	+	0.998577i	$0.516986\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	−16.0000	−0.250000
$17$	118.000	1.68348	0.841741	−	0.539881i	$-0.181531\pi$
$17$	118.000	1.68348	0.841741	+	0.539881i	$0.181531\pi$
$18$	0	0
$19$	−105.000	−1.26782	−0.633912	−	0.773405i	$-0.718552\pi$
$19$	−105.000	−1.26782	−0.633912	+	0.773405i	$0.718552\pi$
$20$	44.0000	0.491935
$21$	0	0
$22$	22.0000	0.213201
$23$	68.0000	0.616477	0.308239	−	0.951309i	$-0.400260\pi$
$23$	68.0000	0.616477	0.308239	+	0.951309i	$0.400260\pi$
$24$	0	0
$25$	−4.00000	−0.0320000
$26$	10.0000	0.0754293
$27$	0	0
$28$	28.0000	0.188982
$29$	195.000	1.24864	0.624321	−	0.781168i	$-0.285376\pi$
$29$	195.000	1.24864	0.624321	+	0.781168i	$0.285376\pi$
$30$	0	0
$31$	214.000	1.23986	0.619928	−	0.784659i	$-0.287162\pi$
$31$	214.000	1.23986	0.619928	+	0.784659i	$0.287162\pi$
$32$	−160.000	−0.883883
$33$	0	0
$34$	−236.000	−1.19040
$35$	77.0000	0.371868
$36$	0	0
$37$	33.0000	0.146626	0.0733131	−	0.997309i	$-0.476643\pi$
$37$	33.0000	0.146626	0.0733131	+	0.997309i	$0.476643\pi$
$38$	210.000	0.896487
$39$	0	0
$40$	−264.000	−1.04355
$41$	376.000	1.43223	0.716114	−	0.697984i	$-0.245919\pi$
$41$	376.000	1.43223	0.716114	+	0.697984i	$0.245919\pi$
$42$	0	0
$43$	−168.000	−0.595808	−0.297904	−	0.954596i	$-0.596288\pi$
$43$	−168.000	−0.595808	−0.297904	+	0.954596i	$0.596288\pi$
$44$	44.0000	0.150756
$45$	0	0
$46$	−136.000	−0.435915
$47$	−61.0000	−0.189314	−0.0946571	−	0.995510i	$-0.530175\pi$
$47$	−61.0000	−0.189314	−0.0946571	+	0.995510i	$0.530175\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	8.00000	0.0226274
$51$	0	0
$52$	20.0000	0.0533366
$53$	−24.0000	−0.0622010	−0.0311005	−	0.999516i	$-0.509901\pi$
$53$	−24.0000	−0.0622010	−0.0311005	+	0.999516i	$0.509901\pi$
$54$	0	0
$55$	121.000	0.296648
$56$	−168.000	−0.400892
$57$	0	0
$58$	−390.000	−0.882923
$59$	−625.000	−1.37912	−0.689560	−	0.724229i	$-0.742196\pi$
$59$	−625.000	−1.37912	−0.689560	+	0.724229i	$0.742196\pi$
$60$	0	0
$61$	−558.000	−1.17122	−0.585611	−	0.810592i	$-0.699145\pi$
$61$	−558.000	−1.17122	−0.585611	+	0.810592i	$0.699145\pi$
$62$	−428.000	−0.876711
$63$	0	0
$64$	448.000	0.875000
$65$	55.0000	0.104952
$66$	0	0
$67$	173.000	0.315452	0.157726	−	0.987483i	$-0.449584\pi$
$67$	173.000	0.315452	0.157726	+	0.987483i	$0.449584\pi$
$68$	−472.000	−0.841741
$69$	0	0
$70$	−154.000	−0.262950
$71$	−168.000	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$71$	−168.000	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$72$	0	0
$73$	973.000	1.56001	0.780007	−	0.625771i	$-0.215215\pi$
$73$	973.000	1.56001	0.780007	+	0.625771i	$0.215215\pi$
$74$	−66.0000	−0.103680
$75$	0	0
$76$	420.000	0.633912
$77$	77.0000	0.113961
$78$	0	0
$79$	−1072.00	−1.52670	−0.763351	−	0.645984i	$-0.776447\pi$
$79$	−1072.00	−1.52670	−0.763351	+	0.645984i	$0.776447\pi$
$80$	176.000	0.245967
$81$	0	0
$82$	−752.000	−1.01274
$83$	−1458.00	−1.92815	−0.964074	−	0.265634i	$-0.914419\pi$
$83$	−1458.00	−1.92815	−0.964074	+	0.265634i	$0.914419\pi$
$84$	0	0
$85$	−1298.00	−1.65633
$86$	336.000	0.421300
$87$	0	0
$88$	−264.000	−0.319801
$89$	198.000	0.235820	0.117910	−	0.993024i	$-0.462381\pi$
$89$	198.000	0.235820	0.117910	+	0.993024i	$0.462381\pi$
$90$	0	0
$91$	35.0000	0.0403186
$92$	−272.000	−0.308239
$93$	0	0
$94$	122.000	0.133865
$95$	1155.00	1.24737
$96$	0	0
$97$	−352.000	−0.368456	−0.184228	−	0.982884i	$-0.558978\pi$
$97$	−352.000	−0.368456	−0.184228	+	0.982884i	$0.558978\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	16.0000	0.0160000
$101$	−706.000	−0.695541	−0.347770	−	0.937580i	$-0.613061\pi$
$101$	−706.000	−0.695541	−0.347770	+	0.937580i	$0.613061\pi$
$102$	0	0
$103$	1034.00	0.989156	0.494578	−	0.869133i	$-0.335323\pi$
$103$	1034.00	0.989156	0.494578	+	0.869133i	$0.335323\pi$
$104$	−120.000	−0.113144
$105$	0	0
$106$	48.0000	0.0439828
$107$	675.000	0.609857	0.304929	−	0.952375i	$-0.401367\pi$
$107$	675.000	0.609857	0.304929	+	0.952375i	$0.401367\pi$
$108$	0	0
$109$	−1142.00	−1.00352	−0.501760	−	0.865007i	$-0.667314\pi$
$109$	−1142.00	−1.00352	−0.501760	+	0.865007i	$0.667314\pi$
$110$	−242.000	−0.209762
$111$	0	0
$112$	112.000	0.0944911
$113$	−162.000	−0.134864	−0.0674322	−	0.997724i	$-0.521481\pi$
$113$	−162.000	−0.134864	−0.0674322	+	0.997724i	$0.521481\pi$
$114$	0	0
$115$	−748.000	−0.606534
$116$	−780.000	−0.624321
$117$	0	0
$118$	1250.00	0.975185
$119$	−826.000	−0.636297
$120$	0	0
$121$	121.000	0.0909091
$122$	1116.00	0.828179
$123$	0	0
$124$	−856.000	−0.619928
$125$	1419.00	1.01535
$126$	0	0
$127$	1130.00	0.789538	0.394769	−	0.918780i	$-0.370825\pi$
$127$	1130.00	0.789538	0.394769	+	0.918780i	$0.370825\pi$
$128$	384.000	0.265165
$129$	0	0
$130$	−110.000	−0.0742126
$131$	−2208.00	−1.47262	−0.736312	−	0.676642i	$-0.763435\pi$
$131$	−2208.00	−1.47262	−0.736312	+	0.676642i	$0.763435\pi$
$132$	0	0
$133$	735.000	0.479192
$134$	−346.000	−0.223059
$135$	0	0
$136$	2832.00	1.78560
$137$	1658.00	1.03396	0.516980	−	0.855998i	$-0.327056\pi$
$137$	1658.00	1.03396	0.516980	+	0.855998i	$0.327056\pi$
$138$	0	0
$139$	−2120.00	−1.29364	−0.646820	−	0.762642i	$-0.723902\pi$
$139$	−2120.00	−1.29364	−0.646820	+	0.762642i	$0.723902\pi$
$140$	−308.000	−0.185934
$141$	0	0
$142$	336.000	0.198567
$143$	55.0000	0.0321632
$144$	0	0
$145$	−2145.00	−1.22850
$146$	−1946.00	−1.10310
$147$	0	0
$148$	−132.000	−0.0733131
$149$	23.0000	0.0126459	0.00632293	−	0.999980i	$-0.497987\pi$
$149$	23.0000	0.0126459	0.00632293	+	0.999980i	$0.497987\pi$
$150$	0	0
$151$	−1810.00	−0.975468	−0.487734	−	0.872992i	$-0.662176\pi$
$151$	−1810.00	−0.975468	−0.487734	+	0.872992i	$0.662176\pi$
$152$	−2520.00	−1.34473
$153$	0	0
$154$	−154.000	−0.0805823
$155$	−2354.00	−1.21986
$156$	0	0
$157$	−868.000	−0.441235	−0.220618	−	0.975360i	$-0.570807\pi$
$157$	−868.000	−0.441235	−0.220618	+	0.975360i	$0.570807\pi$
$158$	2144.00	1.07954
$159$	0	0
$160$	1760.00	0.869626
$161$	−476.000	−0.233007
$162$	0	0
$163$	−2267.00	−1.08936	−0.544678	−	0.838645i	$-0.683348\pi$
$163$	−2267.00	−1.08936	−0.544678	+	0.838645i	$0.683348\pi$
$164$	−1504.00	−0.716114
$165$	0	0
$166$	2916.00	1.36341
$167$	−3024.00	−1.40122	−0.700611	−	0.713543i	$-0.747089\pi$
$167$	−3024.00	−1.40122	−0.700611	+	0.713543i	$0.747089\pi$
$168$	0	0
$169$	−2172.00	−0.988621
$170$	2596.00	1.17120
$171$	0	0
$172$	672.000	0.297904
$173$	−1272.00	−0.559008	−0.279504	−	0.960145i	$-0.590170\pi$
$173$	−1272.00	−0.559008	−0.279504	+	0.960145i	$0.590170\pi$
$174$	0	0
$175$	28.0000	0.0120949
$176$	176.000	0.0753778
$177$	0	0
$178$	−396.000	−0.166750
$179$	3228.00	1.34789	0.673944	−	0.738782i	$-0.264599\pi$
$179$	3228.00	1.34789	0.673944	+	0.738782i	$0.264599\pi$
$180$	0	0
$181$	−3452.00	−1.41760	−0.708799	−	0.705411i	$-0.750763\pi$
$181$	−3452.00	−1.41760	−0.708799	+	0.705411i	$0.750763\pi$
$182$	−70.0000	−0.0285096
$183$	0	0
$184$	1632.00	0.653873
$185$	−363.000	−0.144261
$186$	0	0
$187$	−1298.00	−0.507589
$188$	244.000	0.0946571
$189$	0	0
$190$	−2310.00	−0.882026
$191$	−2074.00	−0.785704	−0.392852	−	0.919602i	$-0.628511\pi$
$191$	−2074.00	−0.785704	−0.392852	+	0.919602i	$0.628511\pi$
$192$	0	0
$193$	1204.00	0.449046	0.224523	−	0.974469i	$-0.427918\pi$
$193$	1204.00	0.449046	0.224523	+	0.974469i	$0.427918\pi$
$194$	704.000	0.260537
$195$	0	0
$196$	−196.000	−0.0714286
$197$	2086.00	0.754423	0.377212	−	0.926127i	$-0.376883\pi$
$197$	2086.00	0.754423	0.377212	+	0.926127i	$0.376883\pi$
$198$	0	0
$199$	1344.00	0.478762	0.239381	−	0.970926i	$-0.423056\pi$
$199$	1344.00	0.478762	0.239381	+	0.970926i	$0.423056\pi$
$200$	−96.0000	−0.0339411
$201$	0	0
$202$	1412.00	0.491822
$203$	−1365.00	−0.471942
$204$	0	0
$205$	−4136.00	−1.40913
$206$	−2068.00	−0.699439
$207$	0	0
$208$	80.0000	0.0266683
$209$	1155.00	0.382263
$210$	0	0
$211$	1302.00	0.424803	0.212401	−	0.977183i	$-0.431872\pi$
$211$	1302.00	0.424803	0.212401	+	0.977183i	$0.431872\pi$
$212$	96.0000	0.0311005
$213$	0	0
$214$	−1350.00	−0.431234
$215$	1848.00	0.586198
$216$	0	0
$217$	−1498.00	−0.468622
$218$	2284.00	0.709596
$219$	0	0
$220$	−484.000	−0.148324
$221$	−590.000	−0.179582
$222$	0	0
$223$	−926.000	−0.278070	−0.139035	−	0.990287i	$-0.544400\pi$
$223$	−926.000	−0.278070	−0.139035	+	0.990287i	$0.544400\pi$
$224$	1120.00	0.334077
$225$	0	0
$226$	324.000	0.0953635
$227$	4206.00	1.22979	0.614894	−	0.788610i	$-0.289199\pi$
$227$	4206.00	1.22979	0.614894	+	0.788610i	$0.289199\pi$
$228$	0	0
$229$	−1608.00	−0.464016	−0.232008	−	0.972714i	$-0.574530\pi$
$229$	−1608.00	−0.464016	−0.232008	+	0.972714i	$0.574530\pi$
$230$	1496.00	0.428884
$231$	0	0
$232$	4680.00	1.32438
$233$	−6402.00	−1.80004	−0.900019	−	0.435850i	$-0.856448\pi$
$233$	−6402.00	−1.80004	−0.900019	+	0.435850i	$0.856448\pi$
$234$	0	0
$235$	671.000	0.186260
$236$	2500.00	0.689560
$237$	0	0
$238$	1652.00	0.449930
$239$	1761.00	0.476609	0.238305	−	0.971190i	$-0.423408\pi$
$239$	1761.00	0.476609	0.238305	+	0.971190i	$0.423408\pi$
$240$	0	0
$241$	4951.00	1.32333	0.661664	−	0.749801i	$-0.269851\pi$
$241$	4951.00	1.32333	0.661664	+	0.749801i	$0.269851\pi$
$242$	−242.000	−0.0642824
$243$	0	0
$244$	2232.00	0.585611
$245$	−539.000	−0.140553
$246$	0	0
$247$	525.000	0.135243
$248$	5136.00	1.31507
$249$	0	0
$250$	−2838.00	−0.717964
$251$	3553.00	0.893480	0.446740	−	0.894664i	$-0.352585\pi$
$251$	3553.00	0.893480	0.446740	+	0.894664i	$0.352585\pi$
$252$	0	0
$253$	−748.000	−0.185875
$254$	−2260.00	−0.558287
$255$	0	0
$256$	−4352.00	−1.06250
$257$	4837.00	1.17402	0.587011	−	0.809579i	$-0.300304\pi$
$257$	4837.00	1.17402	0.587011	+	0.809579i	$0.300304\pi$
$258$	0	0
$259$	−231.000	−0.0554195
$260$	−220.000	−0.0524762
$261$	0	0
$262$	4416.00	1.04130
$263$	5951.00	1.39526	0.697632	−	0.716456i	$-0.254237\pi$
$263$	5951.00	1.39526	0.697632	+	0.716456i	$0.254237\pi$
$264$	0	0
$265$	264.000	0.0611977
$266$	−1470.00	−0.338840
$267$	0	0
$268$	−692.000	−0.157726
$269$	−6854.00	−1.55352	−0.776758	−	0.629799i	$-0.783137\pi$
$269$	−6854.00	−1.55352	−0.776758	+	0.629799i	$0.783137\pi$
$270$	0	0
$271$	−4429.00	−0.992777	−0.496389	−	0.868100i	$-0.665341\pi$
$271$	−4429.00	−0.992777	−0.496389	+	0.868100i	$0.665341\pi$
$272$	−1888.00	−0.420871
$273$	0	0
$274$	−3316.00	−0.731120
$275$	44.0000	0.00964836
$276$	0	0
$277$	−1700.00	−0.368748	−0.184374	−	0.982856i	$-0.559026\pi$
$277$	−1700.00	−0.368748	−0.184374	+	0.982856i	$0.559026\pi$
$278$	4240.00	0.914742
$279$	0	0
$280$	1848.00	0.394425
$281$	−4161.00	−0.883361	−0.441681	−	0.897172i	$-0.645618\pi$
$281$	−4161.00	−0.883361	−0.441681	+	0.897172i	$0.645618\pi$
$282$	0	0
$283$	1903.00	0.399723	0.199861	−	0.979824i	$-0.435951\pi$
$283$	1903.00	0.399723	0.199861	+	0.979824i	$0.435951\pi$
$284$	672.000	0.140408
$285$	0	0
$286$	−110.000	−0.0227428
$287$	−2632.00	−0.541331
$288$	0	0
$289$	9011.00	1.83411
$290$	4290.00	0.868681
$291$	0	0
$292$	−3892.00	−0.780007
$293$	8352.00	1.66529	0.832643	−	0.553809i	$-0.186826\pi$
$293$	8352.00	1.66529	0.832643	+	0.553809i	$0.186826\pi$
$294$	0	0
$295$	6875.00	1.35687
$296$	792.000	0.155520
$297$	0	0
$298$	−46.0000	−0.00894197
$299$	−340.000	−0.0657616
$300$	0	0
$301$	1176.00	0.225194
$302$	3620.00	0.689760
$303$	0	0
$304$	1680.00	0.316956
$305$	6138.00	1.15233
$306$	0	0
$307$	8900.00	1.65456	0.827280	−	0.561790i	$-0.189887\pi$
$307$	8900.00	1.65456	0.827280	+	0.561790i	$0.189887\pi$
$308$	−308.000	−0.0569803
$309$	0	0
$310$	4708.00	0.862569
$311$	−5976.00	−1.08961	−0.544804	−	0.838564i	$-0.683396\pi$
$311$	−5976.00	−1.08961	−0.544804	+	0.838564i	$0.683396\pi$
$312$	0	0
$313$	−2854.00	−0.515392	−0.257696	−	0.966226i	$-0.582963\pi$
$313$	−2854.00	−0.515392	−0.257696	+	0.966226i	$0.582963\pi$
$314$	1736.00	0.312000
$315$	0	0
$316$	4288.00	0.763351
$317$	−3152.00	−0.558467	−0.279233	−	0.960223i	$-0.590080\pi$
$317$	−3152.00	−0.558467	−0.279233	+	0.960223i	$0.590080\pi$
$318$	0	0
$319$	−2145.00	−0.376479
$320$	−4928.00	−0.860886
$321$	0	0
$322$	952.000	0.164761
$323$	−12390.0	−2.13436
$324$	0	0
$325$	20.0000	0.00341354
$326$	4534.00	0.770292
$327$	0	0
$328$	9024.00	1.51911
$329$	427.000	0.0715540
$330$	0	0
$331$	10432.0	1.73231	0.866155	−	0.499776i	$-0.166584\pi$
$331$	10432.0	1.73231	0.866155	+	0.499776i	$0.166584\pi$
$332$	5832.00	0.964074
$333$	0	0
$334$	6048.00	0.990814
$335$	−1903.00	−0.310364
$336$	0	0
$337$	2036.00	0.329104	0.164552	−	0.986368i	$-0.447382\pi$
$337$	2036.00	0.329104	0.164552	+	0.986368i	$0.447382\pi$
$338$	4344.00	0.699061
$339$	0	0
$340$	5192.00	0.828164
$341$	−2354.00	−0.373831
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−4032.00	−0.631950
$345$	0	0
$346$	2544.00	0.395278
$347$	−4832.00	−0.747537	−0.373769	−	0.927522i	$-0.621935\pi$
$347$	−4832.00	−0.747537	−0.373769	+	0.927522i	$0.621935\pi$
$348$	0	0
$349$	4551.00	0.698021	0.349011	−	0.937119i	$-0.386518\pi$
$349$	4551.00	0.698021	0.349011	+	0.937119i	$0.386518\pi$
$350$	−56.0000	−0.00855236
$351$	0	0
$352$	1760.00	0.266501
$353$	−10785.0	−1.62614	−0.813070	−	0.582166i	$-0.802206\pi$
$353$	−10785.0	−1.62614	−0.813070	+	0.582166i	$0.802206\pi$
$354$	0	0
$355$	1848.00	0.276286
$356$	−792.000	−0.117910
$357$	0	0
$358$	−6456.00	−0.953101
$359$	6988.00	1.02733	0.513666	−	0.857990i	$-0.328287\pi$
$359$	6988.00	1.02733	0.513666	+	0.857990i	$0.328287\pi$
$360$	0	0
$361$	4166.00	0.607377
$362$	6904.00	1.00239
$363$	0	0
$364$	−140.000	−0.0201593
$365$	−10703.0	−1.53485
$366$	0	0
$367$	4342.00	0.617576	0.308788	−	0.951131i	$-0.400077\pi$
$367$	4342.00	0.617576	0.308788	+	0.951131i	$0.400077\pi$
$368$	−1088.00	−0.154119
$369$	0	0
$370$	726.000	0.102008
$371$	168.000	0.0235098
$372$	0	0
$373$	−12674.0	−1.75934	−0.879671	−	0.475582i	$-0.842237\pi$
$373$	−12674.0	−1.75934	−0.879671	+	0.475582i	$0.842237\pi$
$374$	2596.00	0.358920
$375$	0	0
$376$	−1464.00	−0.200798
$377$	−975.000	−0.133196
$378$	0	0
$379$	−13181.0	−1.78644	−0.893222	−	0.449615i	$-0.851561\pi$
$379$	−13181.0	−1.78644	−0.893222	+	0.449615i	$0.851561\pi$
$380$	−4620.00	−0.623687
$381$	0	0
$382$	4148.00	0.555576
$383$	1088.00	0.145155	0.0725773	−	0.997363i	$-0.476878\pi$
$383$	1088.00	0.145155	0.0725773	+	0.997363i	$0.476878\pi$
$384$	0	0
$385$	−847.000	−0.112122
$386$	−2408.00	−0.317523
$387$	0	0
$388$	1408.00	0.184228
$389$	−1356.00	−0.176740	−0.0883701	−	0.996088i	$-0.528166\pi$
$389$	−1356.00	−0.176740	−0.0883701	+	0.996088i	$0.528166\pi$
$390$	0	0
$391$	8024.00	1.03783
$392$	1176.00	0.151523
$393$	0	0
$394$	−4172.00	−0.533458
$395$	11792.0	1.50208
$396$	0	0
$397$	−8102.00	−1.02425	−0.512126	−	0.858911i	$-0.671142\pi$
$397$	−8102.00	−1.02425	−0.512126	+	0.858911i	$0.671142\pi$
$398$	−2688.00	−0.338536
$399$	0	0
$400$	64.0000	0.00800000
$401$	12716.0	1.58356	0.791779	−	0.610808i	$-0.209155\pi$
$401$	12716.0	1.58356	0.791779	+	0.610808i	$0.209155\pi$
$402$	0	0
$403$	−1070.00	−0.132259
$404$	2824.00	0.347770
$405$	0	0
$406$	2730.00	0.333713
$407$	−363.000	−0.0442094
$408$	0	0
$409$	−8766.00	−1.05978	−0.529891	−	0.848066i	$-0.677767\pi$
$409$	−8766.00	−1.05978	−0.529891	+	0.848066i	$0.677767\pi$
$410$	8272.00	0.996402
$411$	0	0
$412$	−4136.00	−0.494578
$413$	4375.00	0.521258
$414$	0	0
$415$	16038.0	1.89705
$416$	800.000	0.0942866
$417$	0	0
$418$	−2310.00	−0.270301
$419$	5729.00	0.667971	0.333986	−	0.942578i	$-0.391606\pi$
$419$	5729.00	0.667971	0.333986	+	0.942578i	$0.391606\pi$
$420$	0	0
$421$	8983.00	1.03992	0.519958	−	0.854192i	$-0.325948\pi$
$421$	8983.00	1.03992	0.519958	+	0.854192i	$0.325948\pi$
$422$	−2604.00	−0.300381
$423$	0	0
$424$	−576.000	−0.0659741
$425$	−472.000	−0.0538714
$426$	0	0
$427$	3906.00	0.442681
$428$	−2700.00	−0.304929
$429$	0	0
$430$	−3696.00	−0.414505
$431$	10239.0	1.14430	0.572152	−	0.820147i	$-0.306109\pi$
$431$	10239.0	1.14430	0.572152	+	0.820147i	$0.306109\pi$
$432$	0	0
$433$	−9932.00	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$433$	−9932.00	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$434$	2996.00	0.331365
$435$	0	0
$436$	4568.00	0.501760
$437$	−7140.00	−0.781585
$438$	0	0
$439$	−9893.00	−1.07555	−0.537776	−	0.843088i	$-0.680735\pi$
$439$	−9893.00	−1.07555	−0.537776	+	0.843088i	$0.680735\pi$
$440$	2904.00	0.314643
$441$	0	0
$442$	1180.00	0.126984
$443$	−15176.0	−1.62762	−0.813808	−	0.581134i	$-0.802609\pi$
$443$	−15176.0	−1.62762	−0.813808	+	0.581134i	$0.802609\pi$
$444$	0	0
$445$	−2178.00	−0.232016
$446$	1852.00	0.196625
$447$	0	0
$448$	−3136.00	−0.330719
$449$	−856.000	−0.0899714	−0.0449857	−	0.998988i	$-0.514324\pi$
$449$	−856.000	−0.0899714	−0.0449857	+	0.998988i	$0.514324\pi$
$450$	0	0
$451$	−4136.00	−0.431833
$452$	648.000	0.0674322
$453$	0	0
$454$	−8412.00	−0.869592
$455$	−385.000	−0.0396683
$456$	0	0
$457$	−8280.00	−0.847532	−0.423766	−	0.905772i	$-0.639292\pi$
$457$	−8280.00	−0.847532	−0.423766	+	0.905772i	$0.639292\pi$
$458$	3216.00	0.328109
$459$	0	0
$460$	2992.00	0.303267
$461$	−2088.00	−0.210950	−0.105475	−	0.994422i	$-0.533636\pi$
$461$	−2088.00	−0.210950	−0.105475	+	0.994422i	$0.533636\pi$
$462$	0	0
$463$	−19283.0	−1.93554	−0.967772	−	0.251827i	$-0.918968\pi$
$463$	−19283.0	−1.93554	−0.967772	+	0.251827i	$0.918968\pi$
$464$	−3120.00	−0.312160
$465$	0	0
$466$	12804.0	1.27282
$467$	−10797.0	−1.06986	−0.534931	−	0.844896i	$-0.679662\pi$
$467$	−10797.0	−1.06986	−0.534931	+	0.844896i	$0.679662\pi$
$468$	0	0
$469$	−1211.00	−0.119230
$470$	−1342.00	−0.131706
$471$	0	0
$472$	−15000.0	−1.46278
$473$	1848.00	0.179643
$474$	0	0
$475$	420.000	0.0405704
$476$	3304.00	0.318148
$477$	0	0
$478$	−3522.00	−0.337014
$479$	2594.00	0.247438	0.123719	−	0.992317i	$-0.460518\pi$
$479$	2594.00	0.247438	0.123719	+	0.992317i	$0.460518\pi$
$480$	0	0
$481$	−165.000	−0.0156411
$482$	−9902.00	−0.935734
$483$	0	0
$484$	−484.000	−0.0454545
$485$	3872.00	0.362512
$486$	0	0
$487$	12032.0	1.11955	0.559776	−	0.828644i	$-0.310887\pi$
$487$	12032.0	1.11955	0.559776	+	0.828644i	$0.310887\pi$
$488$	−13392.0	−1.24227
$489$	0	0
$490$	1078.00	0.0993859
$491$	−7949.00	−0.730618	−0.365309	−	0.930886i	$-0.619037\pi$
$491$	−7949.00	−0.730618	−0.365309	+	0.930886i	$0.619037\pi$
$492$	0	0
$493$	23010.0	2.10207
$494$	−1050.00	−0.0956310
$495$	0	0
$496$	−3424.00	−0.309964
$497$	1176.00	0.106138
$498$	0	0
$499$	−17053.0	−1.52985	−0.764927	−	0.644117i	$-0.777225\pi$
$499$	−17053.0	−1.52985	−0.764927	+	0.644117i	$0.777225\pi$
$500$	−5676.00	−0.507677
$501$	0	0
$502$	−7106.00	−0.631785
$503$	−6080.00	−0.538954	−0.269477	−	0.963007i	$-0.586851\pi$
$503$	−6080.00	−0.538954	−0.269477	+	0.963007i	$0.586851\pi$
$504$	0	0
$505$	7766.00	0.684322
$506$	1496.00	0.131433
$507$	0	0
$508$	−4520.00	−0.394769
$509$	−3510.00	−0.305654	−0.152827	−	0.988253i	$-0.548838\pi$
$509$	−3510.00	−0.305654	−0.152827	+	0.988253i	$0.548838\pi$
$510$	0	0
$511$	−6811.00	−0.589630
$512$	5632.00	0.486136
$513$	0	0
$514$	−9674.00	−0.830159
$515$	−11374.0	−0.973201
$516$	0	0
$517$	671.000	0.0570804
$518$	462.000	0.0391875
$519$	0	0
$520$	1320.00	0.111319
$521$	5781.00	0.486123	0.243062	−	0.970011i	$-0.421848\pi$
$521$	5781.00	0.486123	0.243062	+	0.970011i	$0.421848\pi$
$522$	0	0
$523$	−11959.0	−0.999867	−0.499934	−	0.866064i	$-0.666642\pi$
$523$	−11959.0	−0.999867	−0.499934	+	0.866064i	$0.666642\pi$
$524$	8832.00	0.736312
$525$	0	0
$526$	−11902.0	−0.986600
$527$	25252.0	2.08728
$528$	0	0
$529$	−7543.00	−0.619956
$530$	−528.000	−0.0432733
$531$	0	0
$532$	−2940.00	−0.239596
$533$	−1880.00	−0.152780
$534$	0	0
$535$	−7425.00	−0.600020
$536$	4152.00	0.334588
$537$	0	0
$538$	13708.0	1.09850
$539$	−539.000	−0.0430730
$540$	0	0
$541$	5692.00	0.452344	0.226172	−	0.974087i	$-0.427379\pi$
$541$	5692.00	0.452344	0.226172	+	0.974087i	$0.427379\pi$
$542$	8858.00	0.701999
$543$	0	0
$544$	−18880.0	−1.48800
$545$	12562.0	0.987334
$546$	0	0
$547$	−5620.00	−0.439294	−0.219647	−	0.975579i	$-0.570491\pi$
$547$	−5620.00	−0.439294	−0.219647	+	0.975579i	$0.570491\pi$
$548$	−6632.00	−0.516980
$549$	0	0
$550$	−88.0000	−0.00682242
$551$	−20475.0	−1.58306
$552$	0	0
$553$	7504.00	0.577039
$554$	3400.00	0.260744
$555$	0	0
$556$	8480.00	0.646820
$557$	16639.0	1.26574	0.632870	−	0.774258i	$-0.281877\pi$
$557$	16639.0	1.26574	0.632870	+	0.774258i	$0.281877\pi$
$558$	0	0
$559$	840.000	0.0635567
$560$	−1232.00	−0.0929670
$561$	0	0
$562$	8322.00	0.624631
$563$	−20168.0	−1.50973	−0.754867	−	0.655878i	$-0.772298\pi$
$563$	−20168.0	−1.50973	−0.754867	+	0.655878i	$0.772298\pi$
$564$	0	0
$565$	1782.00	0.132689
$566$	−3806.00	−0.282647
$567$	0	0
$568$	−4032.00	−0.297850
$569$	378.000	0.0278499	0.0139249	−	0.999903i	$-0.495567\pi$
$569$	378.000	0.0278499	0.0139249	+	0.999903i	$0.495567\pi$
$570$	0	0
$571$	3212.00	0.235408	0.117704	−	0.993049i	$-0.462447\pi$
$571$	3212.00	0.235408	0.117704	+	0.993049i	$0.462447\pi$
$572$	−220.000	−0.0160816
$573$	0	0
$574$	5264.00	0.382779
$575$	−272.000	−0.0197273
$576$	0	0
$577$	1086.00	0.0783549	0.0391774	−	0.999232i	$-0.487526\pi$
$577$	1086.00	0.0783549	0.0391774	+	0.999232i	$0.487526\pi$
$578$	−18022.0	−1.29691
$579$	0	0
$580$	8580.00	0.614250
$581$	10206.0	0.728771
$582$	0	0
$583$	264.000	0.0187543
$584$	23352.0	1.65464
$585$	0	0
$586$	−16704.0	−1.17754
$587$	−8267.00	−0.581287	−0.290644	−	0.956831i	$-0.593869\pi$
$587$	−8267.00	−0.581287	−0.290644	+	0.956831i	$0.593869\pi$
$588$	0	0
$589$	−22470.0	−1.57192
$590$	−13750.0	−0.959455
$591$	0	0
$592$	−528.000	−0.0366565
$593$	−6700.00	−0.463973	−0.231987	−	0.972719i	$-0.574523\pi$
$593$	−6700.00	−0.463973	−0.231987	+	0.972719i	$0.574523\pi$
$594$	0	0
$595$	9086.00	0.626033
$596$	−92.0000	−0.00632293
$597$	0	0
$598$	680.000	0.0465004
$599$	−10496.0	−0.715951	−0.357976	−	0.933731i	$-0.616533\pi$
$599$	−10496.0	−0.715951	−0.357976	+	0.933731i	$0.616533\pi$
$600$	0	0
$601$	−10855.0	−0.736747	−0.368373	−	0.929678i	$-0.620085\pi$
$601$	−10855.0	−0.736747	−0.368373	+	0.929678i	$0.620085\pi$
$602$	−2352.00	−0.159236
$603$	0	0
$604$	7240.00	0.487734
$605$	−1331.00	−0.0894427
$606$	0	0
$607$	−11605.0	−0.776001	−0.388000	−	0.921659i	$-0.626834\pi$
$607$	−11605.0	−0.776001	−0.388000	+	0.921659i	$0.626834\pi$
$608$	16800.0	1.12061
$609$	0	0
$610$	−12276.0	−0.814821
$611$	305.000	0.0201947
$612$	0	0
$613$	11354.0	0.748097	0.374049	−	0.927409i	$-0.377969\pi$
$613$	11354.0	0.748097	0.374049	+	0.927409i	$0.377969\pi$
$614$	−17800.0	−1.16995
$615$	0	0
$616$	1848.00	0.120873
$617$	−2994.00	−0.195355	−0.0976774	−	0.995218i	$-0.531141\pi$
$617$	−2994.00	−0.195355	−0.0976774	+	0.995218i	$0.531141\pi$
$618$	0	0
$619$	13082.0	0.849451	0.424725	−	0.905322i	$-0.360371\pi$
$619$	13082.0	0.849451	0.424725	+	0.905322i	$0.360371\pi$
$620$	9416.00	0.609928
$621$	0	0
$622$	11952.0	0.770469
$623$	−1386.00	−0.0891315
$624$	0	0
$625$	−15109.0	−0.966976
$626$	5708.00	0.364437
$627$	0	0
$628$	3472.00	0.220618
$629$	3894.00	0.246843
$630$	0	0
$631$	14212.0	0.896626	0.448313	−	0.893877i	$-0.352025\pi$
$631$	14212.0	0.896626	0.448313	+	0.893877i	$0.352025\pi$
$632$	−25728.0	−1.61931
$633$	0	0
$634$	6304.00	0.394896
$635$	−12430.0	−0.776802
$636$	0	0
$637$	−245.000	−0.0152390
$638$	4290.00	0.266211
$639$	0	0
$640$	−4224.00	−0.260888
$641$	−30.0000	−0.00184856	−0.000924281	−	1.00000i	$-0.500294\pi$
$641$	−30.0000	−0.00184856	−0.000924281		1.00000i	$0.500294\pi$
$642$	0	0
$643$	−28216.0	−1.73053	−0.865265	−	0.501315i	$-0.832850\pi$
$643$	−28216.0	−1.73053	−0.865265	+	0.501315i	$0.832850\pi$
$644$	1904.00	0.116503
$645$	0	0
$646$	24780.0	1.50922
$647$	−15991.0	−0.971671	−0.485835	−	0.874050i	$-0.661485\pi$
$647$	−15991.0	−0.971671	−0.485835	+	0.874050i	$0.661485\pi$
$648$	0	0
$649$	6875.00	0.415820
$650$	−40.0000	−0.00241374
$651$	0	0
$652$	9068.00	0.544678
$653$	20752.0	1.24363	0.621814	−	0.783165i	$-0.286396\pi$
$653$	20752.0	1.24363	0.621814	+	0.783165i	$0.286396\pi$
$654$	0	0
$655$	24288.0	1.44887
$656$	−6016.00	−0.358057
$657$	0	0
$658$	−854.000	−0.0505963
$659$	10447.0	0.617538	0.308769	−	0.951137i	$-0.400083\pi$
$659$	10447.0	0.617538	0.308769	+	0.951137i	$0.400083\pi$
$660$	0	0
$661$	−17564.0	−1.03353	−0.516763	−	0.856129i	$-0.672863\pi$
$661$	−17564.0	−1.03353	−0.516763	+	0.856129i	$0.672863\pi$
$662$	−20864.0	−1.22493
$663$	0	0
$664$	−34992.0	−2.04511
$665$	−8085.00	−0.471463
$666$	0	0
$667$	13260.0	0.769759
$668$	12096.0	0.700611
$669$	0	0
$670$	3806.00	0.219461
$671$	6138.00	0.353137
$672$	0	0
$673$	−21088.0	−1.20785	−0.603925	−	0.797041i	$-0.706397\pi$
$673$	−21088.0	−1.20785	−0.603925	+	0.797041i	$0.706397\pi$
$674$	−4072.00	−0.232712
$675$	0	0
$676$	8688.00	0.494310
$677$	−16470.0	−0.934998	−0.467499	−	0.883994i	$-0.654845\pi$
$677$	−16470.0	−0.934998	−0.467499	+	0.883994i	$0.654845\pi$
$678$	0	0
$679$	2464.00	0.139263
$680$	−31152.0	−1.75680
$681$	0	0
$682$	4708.00	0.264338
$683$	−13030.0	−0.729984	−0.364992	−	0.931011i	$-0.618928\pi$
$683$	−13030.0	−0.729984	−0.364992	+	0.931011i	$0.618928\pi$
$684$	0	0
$685$	−18238.0	−1.01728
$686$	686.000	0.0381802
$687$	0	0
$688$	2688.00	0.148952
$689$	120.000	0.00663518
$690$	0	0
$691$	−18886.0	−1.03974	−0.519868	−	0.854247i	$-0.674019\pi$
$691$	−18886.0	−1.03974	−0.519868	+	0.854247i	$0.674019\pi$
$692$	5088.00	0.279504
$693$	0	0
$694$	9664.00	0.528589
$695$	23320.0	1.27277
$696$	0	0
$697$	44368.0	2.41113
$698$	−9102.00	−0.493576
$699$	0	0
$700$	−112.000	−0.00604743
$701$	29966.0	1.61455	0.807275	−	0.590175i	$-0.200941\pi$
$701$	29966.0	1.61455	0.807275	+	0.590175i	$0.200941\pi$
$702$	0	0
$703$	−3465.00	−0.185896
$704$	−4928.00	−0.263822
$705$	0	0
$706$	21570.0	1.14986
$707$	4942.00	0.262890
$708$	0	0
$709$	−22597.0	−1.19696	−0.598482	−	0.801136i	$-0.704229\pi$
$709$	−22597.0	−1.19696	−0.598482	+	0.801136i	$0.704229\pi$
$710$	−3696.00	−0.195364
$711$	0	0
$712$	4752.00	0.250125
$713$	14552.0	0.764343
$714$	0	0
$715$	−605.000	−0.0316444
$716$	−12912.0	−0.673944
$717$	0	0
$718$	−13976.0	−0.726434
$719$	20229.0	1.04926	0.524628	−	0.851332i	$-0.324205\pi$
$719$	20229.0	1.04926	0.524628	+	0.851332i	$0.324205\pi$
$720$	0	0
$721$	−7238.00	−0.373866
$722$	−8332.00	−0.429481
$723$	0	0
$724$	13808.0	0.708799
$725$	−780.000	−0.0399565
$726$	0	0
$727$	−15792.0	−0.805630	−0.402815	−	0.915281i	$-0.631968\pi$
$727$	−15792.0	−0.805630	−0.402815	+	0.915281i	$0.631968\pi$
$728$	840.000	0.0427644
$729$	0	0
$730$	21406.0	1.08530
$731$	−19824.0	−1.00303
$732$	0	0
$733$	17370.0	0.875274	0.437637	−	0.899152i	$-0.355816\pi$
$733$	17370.0	0.875274	0.437637	+	0.899152i	$0.355816\pi$
$734$	−8684.00	−0.436692
$735$	0	0
$736$	−10880.0	−0.544894
$737$	−1903.00	−0.0951125
$738$	0	0
$739$	36824.0	1.83301	0.916504	−	0.400026i	$-0.130999\pi$
$739$	36824.0	1.83301	0.916504	+	0.400026i	$0.130999\pi$
$740$	1452.00	0.0721305
$741$	0	0
$742$	−336.000	−0.0166239
$743$	10327.0	0.509907	0.254953	−	0.966953i	$-0.417940\pi$
$743$	10327.0	0.509907	0.254953	+	0.966953i	$0.417940\pi$
$744$	0	0
$745$	−253.000	−0.0124419
$746$	25348.0	1.24404
$747$	0	0
$748$	5192.00	0.253795
$749$	−4725.00	−0.230504
$750$	0	0
$751$	−22955.0	−1.11537	−0.557683	−	0.830054i	$-0.688310\pi$
$751$	−22955.0	−1.11537	−0.557683	+	0.830054i	$0.688310\pi$
$752$	976.000	0.0473285
$753$	0	0
$754$	1950.00	0.0941841
$755$	19910.0	0.959734
$756$	0	0
$757$	−28591.0	−1.37273	−0.686366	−	0.727257i	$-0.740795\pi$
$757$	−28591.0	−1.37273	−0.686366	+	0.727257i	$0.740795\pi$
$758$	26362.0	1.26321
$759$	0	0
$760$	27720.0	1.32304
$761$	−1920.00	−0.0914585	−0.0457293	−	0.998954i	$-0.514561\pi$
$761$	−1920.00	−0.0914585	−0.0457293	+	0.998954i	$0.514561\pi$
$762$	0	0
$763$	7994.00	0.379295
$764$	8296.00	0.392852
$765$	0	0
$766$	−2176.00	−0.102640
$767$	3125.00	0.147115
$768$	0	0
$769$	−34393.0	−1.61280	−0.806400	−	0.591370i	$-0.798587\pi$
$769$	−34393.0	−1.61280	−0.806400	+	0.591370i	$0.798587\pi$
$770$	1694.00	0.0792825
$771$	0	0
$772$	−4816.00	−0.224523
$773$	2481.00	0.115440	0.0577202	−	0.998333i	$-0.481617\pi$
$773$	2481.00	0.115440	0.0577202	+	0.998333i	$0.481617\pi$
$774$	0	0
$775$	−856.000	−0.0396754
$776$	−8448.00	−0.390806
$777$	0	0
$778$	2712.00	0.124974
$779$	−39480.0	−1.81581
$780$	0	0
$781$	1848.00	0.0846692
$782$	−16048.0	−0.733856
$783$	0	0
$784$	−784.000	−0.0357143
$785$	9548.00	0.434118
$786$	0	0
$787$	−14255.0	−0.645662	−0.322831	−	0.946457i	$-0.604635\pi$
$787$	−14255.0	−0.645662	−0.322831	+	0.946457i	$0.604635\pi$
$788$	−8344.00	−0.377212
$789$	0	0
$790$	−23584.0	−1.06213
$791$	1134.00	0.0509740
$792$	0	0
$793$	2790.00	0.124938
$794$	16204.0	0.724255
$795$	0	0
$796$	−5376.00	−0.239381
$797$	−9649.00	−0.428840	−0.214420	−	0.976742i	$-0.568786\pi$
$797$	−9649.00	−0.428840	−0.214420	+	0.976742i	$0.568786\pi$
$798$	0	0
$799$	−7198.00	−0.318707
$800$	640.000	0.0282843
$801$	0	0
$802$	−25432.0	−1.11974
$803$	−10703.0	−0.470362
$804$	0	0
$805$	5236.00	0.229248
$806$	2140.00	0.0935214
$807$	0	0
$808$	−16944.0	−0.737732
$809$	−6285.00	−0.273138	−0.136569	−	0.990631i	$-0.543608\pi$
$809$	−6285.00	−0.273138	−0.136569	+	0.990631i	$0.543608\pi$
$810$	0	0
$811$	−14567.0	−0.630723	−0.315362	−	0.948972i	$-0.602126\pi$
$811$	−14567.0	−0.630723	−0.315362	+	0.948972i	$0.602126\pi$
$812$	5460.00	0.235971
$813$	0	0
$814$	726.000	0.0312608
$815$	24937.0	1.07179
$816$	0	0
$817$	17640.0	0.755380
$818$	17532.0	0.749379
$819$	0	0
$820$	16544.0	0.704563
$821$	35833.0	1.52324	0.761620	−	0.648024i	$-0.224404\pi$
$821$	35833.0	1.52324	0.761620	+	0.648024i	$0.224404\pi$
$822$	0	0
$823$	16917.0	0.716512	0.358256	−	0.933623i	$-0.383371\pi$
$823$	16917.0	0.716512	0.358256	+	0.933623i	$0.383371\pi$
$824$	24816.0	1.04916
$825$	0	0
$826$	−8750.00	−0.368585
$827$	11243.0	0.472742	0.236371	−	0.971663i	$-0.424042\pi$
$827$	11243.0	0.472742	0.236371	+	0.971663i	$0.424042\pi$
$828$	0	0
$829$	−5312.00	−0.222549	−0.111275	−	0.993790i	$-0.535493\pi$
$829$	−5312.00	−0.222549	−0.111275	+	0.993790i	$0.535493\pi$
$830$	−32076.0	−1.34141
$831$	0	0
$832$	−2240.00	−0.0933390
$833$	5782.00	0.240498
$834$	0	0
$835$	33264.0	1.37862
$836$	−4620.00	−0.191132
$837$	0	0
$838$	−11458.0	−0.472327
$839$	−44429.0	−1.82820	−0.914100	−	0.405489i	$-0.867101\pi$
$839$	−44429.0	−1.82820	−0.914100	+	0.405489i	$0.867101\pi$
$840$	0	0
$841$	13636.0	0.559105
$842$	−17966.0	−0.735332
$843$	0	0
$844$	−5208.00	−0.212401
$845$	23892.0	0.972674
$846$	0	0
$847$	−847.000	−0.0343604
$848$	384.000	0.0155503
$849$	0	0
$850$	944.000	0.0380929
$851$	2244.00	0.0903917
$852$	0	0
$853$	35678.0	1.43211	0.716056	−	0.698043i	$-0.245946\pi$
$853$	35678.0	1.43211	0.716056	+	0.698043i	$0.245946\pi$
$854$	−7812.00	−0.313022
$855$	0	0
$856$	16200.0	0.646851
$857$	−20830.0	−0.830267	−0.415134	−	0.909760i	$-0.636265\pi$
$857$	−20830.0	−0.830267	−0.415134	+	0.909760i	$0.636265\pi$
$858$	0	0
$859$	−30336.0	−1.20495	−0.602474	−	0.798138i	$-0.705818\pi$
$859$	−30336.0	−1.20495	−0.602474	+	0.798138i	$0.705818\pi$
$860$	−7392.00	−0.293099
$861$	0	0
$862$	−20478.0	−0.809146
$863$	28716.0	1.13268	0.566341	−	0.824171i	$-0.308359\pi$
$863$	28716.0	1.13268	0.566341	+	0.824171i	$0.308359\pi$
$864$	0	0
$865$	13992.0	0.549991
$866$	19864.0	0.779453
$867$	0	0
$868$	5992.00	0.234311
$869$	11792.0	0.460318
$870$	0	0
$871$	−865.000	−0.0336503
$872$	−27408.0	−1.06439
$873$	0	0
$874$	14280.0	0.552664
$875$	−9933.00	−0.383768
$876$	0	0
$877$	39408.0	1.51735	0.758674	−	0.651471i	$-0.225848\pi$
$877$	39408.0	1.51735	0.758674	+	0.651471i	$0.225848\pi$
$878$	19786.0	0.760530
$879$	0	0
$880$	−1936.00	−0.0741620
$881$	4373.00	0.167231	0.0836153	−	0.996498i	$-0.473353\pi$
$881$	4373.00	0.167231	0.0836153	+	0.996498i	$0.473353\pi$
$882$	0	0
$883$	36163.0	1.37824	0.689118	−	0.724649i	$-0.257998\pi$
$883$	36163.0	1.37824	0.689118	+	0.724649i	$0.257998\pi$
$884$	2360.00	0.0897912
$885$	0	0
$886$	30352.0	1.15090
$887$	−82.0000	−0.00310405	−0.00155202	−	0.999999i	$-0.500494\pi$
$887$	−82.0000	−0.00310405	−0.00155202	+	0.999999i	$0.500494\pi$
$888$	0	0
$889$	−7910.00	−0.298417
$890$	4356.00	0.164060
$891$	0	0
$892$	3704.00	0.139035
$893$	6405.00	0.240017
$894$	0	0
$895$	−35508.0	−1.32615
$896$	−2688.00	−0.100223
$897$	0	0
$898$	1712.00	0.0636194
$899$	41730.0	1.54814
$900$	0	0
$901$	−2832.00	−0.104714
$902$	8272.00	0.305352
$903$	0	0
$904$	−3888.00	−0.143045
$905$	37972.0	1.39473
$906$	0	0
$907$	−5580.00	−0.204279	−0.102139	−	0.994770i	$-0.532569\pi$
$907$	−5580.00	−0.204279	−0.102139	+	0.994770i	$0.532569\pi$
$908$	−16824.0	−0.614894
$909$	0	0
$910$	770.000	0.0280497
$911$	25050.0	0.911025	0.455512	−	0.890229i	$-0.349456\pi$
$911$	25050.0	0.911025	0.455512	+	0.890229i	$0.349456\pi$
$912$	0	0
$913$	16038.0	0.581359
$914$	16560.0	0.599296
$915$	0	0
$916$	6432.00	0.232008
$917$	15456.0	0.556600
$918$	0	0
$919$	−10316.0	−0.370287	−0.185143	−	0.982712i	$-0.559275\pi$
$919$	−10316.0	−0.370287	−0.185143	+	0.982712i	$0.559275\pi$
$920$	−17952.0	−0.643326
$921$	0	0
$922$	4176.00	0.149164
$923$	840.000	0.0299555
$924$	0	0
$925$	−132.000	−0.00469204
$926$	38566.0	1.36864
$927$	0	0
$928$	−31200.0	−1.10365
$929$	−12969.0	−0.458018	−0.229009	−	0.973424i	$-0.573549\pi$
$929$	−12969.0	−0.458018	−0.229009	+	0.973424i	$0.573549\pi$
$930$	0	0
$931$	−5145.00	−0.181118
$932$	25608.0	0.900019
$933$	0	0
$934$	21594.0	0.756507
$935$	14278.0	0.499402
$936$	0	0
$937$	−50986.0	−1.77763	−0.888815	−	0.458265i	$-0.848471\pi$
$937$	−50986.0	−1.77763	−0.888815	+	0.458265i	$0.848471\pi$
$938$	2422.00	0.0843082
$939$	0	0
$940$	−2684.00	−0.0931302
$941$	17172.0	0.594890	0.297445	−	0.954739i	$-0.403866\pi$
$941$	17172.0	0.594890	0.297445	+	0.954739i	$0.403866\pi$
$942$	0	0
$943$	25568.0	0.882936
$944$	10000.0	0.344780
$945$	0	0
$946$	−3696.00	−0.127027
$947$	−52482.0	−1.80088	−0.900441	−	0.434978i	$-0.856756\pi$
$947$	−52482.0	−1.80088	−0.900441	+	0.434978i	$0.856756\pi$
$948$	0	0
$949$	−4865.00	−0.166412
$950$	−840.000	−0.0286876
$951$	0	0
$952$	−19824.0	−0.674894
$953$	−22709.0	−0.771896	−0.385948	−	0.922521i	$-0.626126\pi$
$953$	−22709.0	−0.771896	−0.385948	+	0.922521i	$0.626126\pi$
$954$	0	0
$955$	22814.0	0.773030
$956$	−7044.00	−0.238305
$957$	0	0
$958$	−5188.00	−0.174965
$959$	−11606.0	−0.390800
$960$	0	0
$961$	16005.0	0.537243
$962$	330.000	0.0110599
$963$	0	0
$964$	−19804.0	−0.661664
$965$	−13244.0	−0.441803
$966$	0	0
$967$	49354.0	1.64128	0.820640	−	0.571446i	$-0.193617\pi$
$967$	49354.0	1.64128	0.820640	+	0.571446i	$0.193617\pi$
$968$	2904.00	0.0964237
$969$	0	0
$970$	−7744.00	−0.256335
$971$	−26683.0	−0.881873	−0.440936	−	0.897538i	$-0.645354\pi$
$971$	−26683.0	−0.881873	−0.440936	+	0.897538i	$0.645354\pi$
$972$	0	0
$973$	14840.0	0.488950
$974$	−24064.0	−0.791643
$975$	0	0
$976$	8928.00	0.292806
$977$	41856.0	1.37062	0.685308	−	0.728253i	$-0.259668\pi$
$977$	41856.0	1.37062	0.685308	+	0.728253i	$0.259668\pi$
$978$	0	0
$979$	−2178.00	−0.0711023
$980$	2156.00	0.0702764
$981$	0	0
$982$	15898.0	0.516625
$983$	15720.0	0.510061	0.255031	−	0.966933i	$-0.417914\pi$
$983$	15720.0	0.510061	0.255031	+	0.966933i	$0.417914\pi$
$984$	0	0
$985$	−22946.0	−0.742254
$986$	−46020.0	−1.48638
$987$	0	0
$988$	−2100.00	−0.0676214
$989$	−11424.0	−0.367302
$990$	0	0
$991$	−24851.0	−0.796587	−0.398294	−	0.917258i	$-0.630398\pi$
$991$	−24851.0	−0.796587	−0.398294	+	0.917258i	$0.630398\pi$
$992$	−34240.0	−1.09589
$993$	0	0
$994$	−2352.00	−0.0750512
$995$	−14784.0	−0.471040
$996$	0	0
$997$	42518.0	1.35061	0.675305	−	0.737538i	$-0.264012\pi$
$997$	42518.0	1.35061	0.675305	+	0.737538i	$0.264012\pi$
$998$	34106.0	1.08177
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.d.1.1		1
3.2	odd	2		231.4.a.c.1.1	✓	1
21.20	even	2		1617.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.c.1.1	✓	1	3.2	odd	2
693.4.a.d.1.1		1	1.1	even	1	trivial
1617.4.a.e.1.1		1	21.20	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	693.4.a.d.1.1

Level	$693$
Weight	$4$
Character	693.1
Self dual	yes
Analytic conductor	$40.888$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$