LMFDB - Embedded newform 882.3.c.e.685.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(685,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.2
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.e.685.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +4.46088i q^{5} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +4.46088i q^{5} -2.82843 q^{8} -6.30864i q^{10} -2.82843 q^{11} -18.6089i q^{13} +4.00000 q^{16} +12.0376i q^{17} +13.8854i q^{19} +8.92177i q^{20} +4.00000 q^{22} +36.4853 q^{23} +5.10051 q^{25} +26.3170i q^{26} +12.4437 q^{29} -15.1535i q^{31} -5.65685 q^{32} -17.0238i q^{34} +45.6985 q^{37} -19.6369i q^{38} -12.6173i q^{40} +52.6565i q^{41} -71.5980 q^{43} -5.65685 q^{44} -51.5980 q^{46} +84.7343i q^{47} -7.21320 q^{50} -37.2178i q^{52} -104.225 q^{53} -12.6173i q^{55} -17.5980 q^{58} -24.1203i q^{59} -13.5365i q^{61} +21.4303i q^{62} +8.00000 q^{64} +83.0122 q^{65} +4.00000 q^{67} +24.0753i q^{68} +92.2010 q^{71} +85.4678i q^{73} -64.6274 q^{74} +27.7708i q^{76} +3.59798 q^{79} +17.8435i q^{80} -74.4675i q^{82} +80.3410i q^{83} -53.6985 q^{85} +101.255 q^{86} +8.00000 q^{88} +111.925i q^{89} +72.9706 q^{92} -119.832i q^{94} -61.9411 q^{95} -23.0791i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 16 q^{16} + 16 q^{22} + 112 q^{23} + 60 q^{25} + 112 q^{29} + 64 q^{37} - 128 q^{43} - 48 q^{46} + 56 q^{50} - 168 q^{53} + 88 q^{58} + 32 q^{64} + 168 q^{65} + 16 q^{67} + 448 q^{71} - 168 q^{74} - 144 q^{79} - 96 q^{85} + 224 q^{86} + 32 q^{88} + 224 q^{92} - 112 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 + 16 * q^16 + 16 * q^22 + 112 * q^23 + 60 * q^25 + 112 * q^29 + 64 * q^37 - 128 * q^43 - 48 * q^46 + 56 * q^50 - 168 * q^53 + 88 * q^58 + 32 * q^64 + 168 * q^65 + 16 * q^67 + 448 * q^71 - 168 * q^74 - 144 * q^79 - 96 * q^85 + 224 * q^86 + 32 * q^88 + 224 * q^92 - 112 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	4.46088i	0.892177i	0.894989	+	0.446088i	$0.147183\pi$
$5$	4.46088i	0.892177i	−0.894989	+	0.446088i	$0.852817\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.82843	−0.353553
$9$	0	0
$10$	− 6.30864i	− 0.630864i
$11$	−2.82843	−0.257130	−0.128565	−	0.991701i	$-0.541037\pi$
$11$	−2.82843	−0.257130	−0.128565	+	0.991701i	$0.541037\pi$
$12$	0	0
$13$	− 18.6089i	− 1.43145i	−0.698380	−	0.715727i	$-0.746095\pi$
$13$	− 18.6089i	− 1.43145i	0.698380	−	0.715727i	$-0.253905\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	12.0376i	0.708096i	0.935227	+	0.354048i	$0.115195\pi$
$17$	12.0376i	0.708096i	−0.935227	+	0.354048i	$0.884805\pi$
$18$	0	0
$19$	13.8854i	0.730810i	0.930849	+	0.365405i	$0.119069\pi$
$19$	13.8854i	0.730810i	−0.930849	+	0.365405i	$0.880931\pi$
$20$	8.92177i	0.446088i
$21$	0	0
$22$	4.00000	0.181818
$23$	36.4853	1.58632	0.793158	−	0.609016i	$-0.208435\pi$
$23$	36.4853	1.58632	0.793158	+	0.609016i	$0.208435\pi$
$24$	0	0
$25$	5.10051	0.204020
$26$	26.3170i	1.01219i
$27$	0	0
$28$	0	0
$29$	12.4437	0.429091	0.214546	−	0.976714i	$-0.431173\pi$
$29$	12.4437	0.429091	0.214546	+	0.976714i	$0.431173\pi$
$30$	0	0
$31$	− 15.1535i	− 0.488822i	−0.969672	−	0.244411i	$-0.921405\pi$
$31$	− 15.1535i	− 0.488822i	0.969672	−	0.244411i	$-0.0785946\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	− 17.0238i	− 0.500699i
$35$	0	0
$36$	0	0
$37$	45.6985	1.23509	0.617547	−	0.786534i	$-0.288126\pi$
$37$	45.6985	1.23509	0.617547	+	0.786534i	$0.288126\pi$
$38$	− 19.6369i	− 0.516761i
$39$	0	0
$40$	− 12.6173i	− 0.315432i
$41$	52.6565i	1.28430i	0.766577	+	0.642152i	$0.221958\pi$
$41$	52.6565i	1.28430i	−0.766577	+	0.642152i	$0.778042\pi$
$42$	0	0
$43$	−71.5980	−1.66507	−0.832535	−	0.553973i	$-0.813111\pi$
$43$	−71.5980	−1.66507	−0.832535	+	0.553973i	$0.813111\pi$
$44$	−5.65685	−0.128565
$45$	0	0
$46$	−51.5980	−1.12170
$47$	84.7343i	1.80286i	0.432928	+	0.901429i	$0.357480\pi$
$47$	84.7343i	1.80286i	−0.432928	+	0.901429i	$0.642520\pi$
$48$	0	0
$49$	0	0
$50$	−7.21320	−0.144264
$51$	0	0
$52$	− 37.2178i	− 0.715727i
$53$	−104.225	−1.96652	−0.983258	−	0.182216i	$-0.941673\pi$
$53$	−104.225	−1.96652	−0.983258	+	0.182216i	$0.941673\pi$
$54$	0	0
$55$	− 12.6173i	− 0.229405i
$56$	0	0
$57$	0	0
$58$	−17.5980	−0.303413
$59$	− 24.1203i	− 0.408819i	−0.978885	−	0.204410i	$-0.934473\pi$
$59$	− 24.1203i	− 0.408819i	0.978885	−	0.204410i	$-0.0655274\pi$
$60$	0	0
$61$	− 13.5365i	− 0.221910i	−0.993825	−	0.110955i	$-0.964609\pi$
$61$	− 13.5365i	− 0.221910i	0.993825	−	0.110955i	$-0.0353909\pi$
$62$	21.4303i	0.345650i
$63$	0	0
$64$	8.00000	0.125000
$65$	83.0122	1.27711
$66$	0	0
$67$	4.00000	0.0597015	0.0298507	−	0.999554i	$-0.490497\pi$
$67$	4.00000	0.0597015	0.0298507	+	0.999554i	$0.490497\pi$
$68$	24.0753i	0.354048i
$69$	0	0
$70$	0	0
$71$	92.2010	1.29861	0.649303	−	0.760530i	$-0.275061\pi$
$71$	92.2010	1.29861	0.649303	+	0.760530i	$0.275061\pi$
$72$	0	0
$73$	85.4678i	1.17079i	0.810748	+	0.585396i	$0.199061\pi$
$73$	85.4678i	1.17079i	−0.810748	+	0.585396i	$0.800939\pi$
$74$	−64.6274	−0.873343
$75$	0	0
$76$	27.7708i	0.365405i
$77$	0	0
$78$	0	0
$79$	3.59798	0.0455440	0.0227720	−	0.999741i	$-0.492751\pi$
$79$	3.59798	0.0455440	0.0227720	+	0.999741i	$0.492751\pi$
$80$	17.8435i	0.223044i
$81$	0	0
$82$	− 74.4675i	− 0.908140i
$83$	80.3410i	0.967964i	0.875078	+	0.483982i	$0.160810\pi$
$83$	80.3410i	0.967964i	−0.875078	+	0.483982i	$0.839190\pi$
$84$	0	0
$85$	−53.6985	−0.631747
$86$	101.255	1.17738
$87$	0	0
$88$	8.00000	0.0909091
$89$	111.925i	1.25759i	0.777572	+	0.628794i	$0.216451\pi$
$89$	111.925i	1.25759i	−0.777572	+	0.628794i	$0.783549\pi$
$90$	0	0
$91$	0	0
$92$	72.9706	0.793158
$93$	0	0
$94$	− 119.832i	− 1.27481i
$95$	−61.9411	−0.652012
$96$	0	0
$97$	− 23.0791i	− 0.237929i	−0.992899	−	0.118965i	$-0.962043\pi$
$97$	− 23.0791i	− 0.237929i	0.992899	−	0.118965i	$-0.0379575\pi$
$98$	0	0
$99$	0	0
$100$	10.2010	0.102010
$101$	− 82.5602i	− 0.817427i	−0.912663	−	0.408714i	$-0.865977\pi$
$101$	− 82.5602i	− 0.817427i	0.912663	−	0.408714i	$-0.134023\pi$
$102$	0	0
$103$	162.757i	1.58016i	0.613003	+	0.790081i	$0.289961\pi$
$103$	162.757i	1.58016i	−0.613003	+	0.790081i	$0.710039\pi$
$104$	52.6339i	0.506096i
$105$	0	0
$106$	147.397	1.39054
$107$	13.8579	0.129513	0.0647564	−	0.997901i	$-0.479373\pi$
$107$	13.8579	0.129513	0.0647564	+	0.997901i	$0.479373\pi$
$108$	0	0
$109$	136.693	1.25407	0.627034	−	0.778992i	$-0.284269\pi$
$109$	136.693	1.25407	0.627034	+	0.778992i	$0.284269\pi$
$110$	17.8435i	0.162214i
$111$	0	0
$112$	0	0
$113$	−132.794	−1.17517	−0.587584	−	0.809163i	$-0.699921\pi$
$113$	−132.794	−1.17517	−0.587584	+	0.809163i	$0.699921\pi$
$114$	0	0
$115$	162.757i	1.41528i
$116$	24.8873	0.214546
$117$	0	0
$118$	34.1113i	0.289079i
$119$	0	0
$120$	0	0
$121$	−113.000	−0.933884
$122$	19.1435i	0.156914i
$123$	0	0
$124$	− 30.3070i	− 0.244411i
$125$	134.275i	1.07420i
$126$	0	0
$127$	150.392	1.18419	0.592094	−	0.805869i	$-0.298301\pi$
$127$	150.392	1.18419	0.592094	+	0.805869i	$0.298301\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	−117.397	−0.903054
$131$	217.709i	1.66190i	0.556345	+	0.830951i	$0.312203\pi$
$131$	217.709i	1.66190i	−0.556345	+	0.830951i	$0.687797\pi$
$132$	0	0
$133$	0	0
$134$	−5.65685	−0.0422153
$135$	0	0
$136$	− 34.0476i	− 0.250350i
$137$	−52.0416	−0.379866	−0.189933	−	0.981797i	$-0.560827\pi$
$137$	−52.0416	−0.379866	−0.189933	+	0.981797i	$0.560827\pi$
$138$	0	0
$139$	143.799i	1.03452i	0.855827	+	0.517262i	$0.173049\pi$
$139$	143.799i	1.03452i	−0.855827	+	0.517262i	$0.826951\pi$
$140$	0	0
$141$	0	0
$142$	−130.392	−0.918253
$143$	52.6339i	0.368069i
$144$	0	0
$145$	55.5097i	0.382825i
$146$	− 120.870i	− 0.827875i
$147$	0	0
$148$	91.3970	0.617547
$149$	19.0883	0.128109	0.0640547	−	0.997946i	$-0.479597\pi$
$149$	19.0883	0.128109	0.0640547	+	0.997946i	$0.479597\pi$
$150$	0	0
$151$	154.392	1.02246	0.511232	−	0.859443i	$-0.329189\pi$
$151$	154.392	1.02246	0.511232	+	0.859443i	$0.329189\pi$
$152$	− 39.2738i	− 0.258380i
$153$	0	0
$154$	0	0
$155$	67.5980	0.436116
$156$	0	0
$157$	117.677i	0.749535i	0.927119	+	0.374767i	$0.122277\pi$
$157$	117.677i	0.749535i	−0.927119	+	0.374767i	$0.877723\pi$
$158$	−5.08831	−0.0322045
$159$	0	0
$160$	− 25.2346i	− 0.157716i
$161$	0	0
$162$	0	0
$163$	185.990	1.14104	0.570521	−	0.821283i	$-0.306741\pi$
$163$	185.990	1.14104	0.570521	+	0.821283i	$0.306741\pi$
$164$	105.313i	0.642152i
$165$	0	0
$166$	− 113.619i	− 0.684454i
$167$	− 206.142i	− 1.23439i	−0.786812	−	0.617193i	$-0.788270\pi$
$167$	− 206.142i	− 1.23439i	0.786812	−	0.617193i	$-0.211730\pi$
$168$	0	0
$169$	−177.291	−1.04906
$170$	75.9411	0.446713
$171$	0	0
$172$	−143.196	−0.832535
$173$	− 206.075i	− 1.19118i	−0.803287	−	0.595592i	$-0.796917\pi$
$173$	− 206.075i	− 1.19118i	0.803287	−	0.595592i	$-0.203083\pi$
$174$	0	0
$175$	0	0
$176$	−11.3137	−0.0642824
$177$	0	0
$178$	− 158.286i	− 0.889250i
$179$	160.083	0.894320	0.447160	−	0.894454i	$-0.352435\pi$
$179$	160.083	0.894320	0.447160	+	0.894454i	$0.352435\pi$
$180$	0	0
$181$	99.9235i	0.552064i	0.961149	+	0.276032i	$0.0890195\pi$
$181$	99.9235i	0.552064i	−0.961149	+	0.276032i	$0.910980\pi$
$182$	0	0
$183$	0	0
$184$	−103.196	−0.560848
$185$	203.856i	1.10192i
$186$	0	0
$187$	− 34.0476i	− 0.182073i
$188$	169.469i	0.901429i
$189$	0	0
$190$	87.5980	0.461042
$191$	−69.5736	−0.364260	−0.182130	−	0.983274i	$-0.558299\pi$
$191$	−69.5736	−0.364260	−0.182130	+	0.983274i	$0.558299\pi$
$192$	0	0
$193$	135.196	0.700497	0.350249	−	0.936657i	$-0.386097\pi$
$193$	135.196	0.700497	0.350249	+	0.936657i	$0.386097\pi$
$194$	32.6388i	0.168241i
$195$	0	0
$196$	0	0
$197$	−21.4903	−0.109088	−0.0545440	−	0.998511i	$-0.517371\pi$
$197$	−21.4903	−0.109088	−0.0545440	+	0.998511i	$0.517371\pi$
$198$	0	0
$199$	259.827i	1.30566i	0.757503	+	0.652831i	$0.226419\pi$
$199$	259.827i	1.30566i	−0.757503	+	0.652831i	$0.773581\pi$
$200$	−14.4264	−0.0721320
$201$	0	0
$202$	116.758i	0.578008i
$203$	0	0
$204$	0	0
$205$	−234.894	−1.14583
$206$	− 230.173i	− 1.11734i
$207$	0	0
$208$	− 74.4356i	− 0.357864i
$209$	− 39.2738i	− 0.187913i
$210$	0	0
$211$	−251.598	−1.19241	−0.596204	−	0.802833i	$-0.703325\pi$
$211$	−251.598	−1.19241	−0.596204	+	0.802833i	$0.703325\pi$
$212$	−208.451	−0.983258
$213$	0	0
$214$	−19.5980	−0.0915793
$215$	− 319.390i	− 1.48554i
$216$	0	0
$217$	0	0
$218$	−193.314	−0.886760
$219$	0	0
$220$	− 25.2346i	− 0.114703i
$221$	224.007	1.01361
$222$	0	0
$223$	− 29.0389i	− 0.130219i	−0.997878	−	0.0651096i	$-0.979260\pi$
$223$	− 29.0389i	− 0.130219i	0.997878	−	0.0651096i	$-0.0207397\pi$
$224$	0	0
$225$	0	0
$226$	187.799	0.830969
$227$	− 275.543i	− 1.21385i	−0.794761	−	0.606923i	$-0.792404\pi$
$227$	− 275.543i	− 1.21385i	0.794761	−	0.606923i	$-0.207596\pi$
$228$	0	0
$229$	− 41.8457i	− 0.182733i	−0.995817	−	0.0913663i	$-0.970877\pi$
$229$	− 41.8457i	− 0.182733i	0.995817	−	0.0913663i	$-0.0291234\pi$
$230$	− 230.173i	− 1.00075i
$231$	0	0
$232$	−35.1960	−0.151707
$233$	−217.213	−0.932246	−0.466123	−	0.884720i	$-0.654349\pi$
$233$	−217.213	−0.932246	−0.466123	+	0.884720i	$0.654349\pi$
$234$	0	0
$235$	−377.990	−1.60847
$236$	− 48.2406i	− 0.204410i
$237$	0	0
$238$	0	0
$239$	−68.7208	−0.287535	−0.143767	−	0.989612i	$-0.545922\pi$
$239$	−68.7208	−0.287535	−0.143767	+	0.989612i	$0.545922\pi$
$240$	0	0
$241$	18.0386i	0.0748489i	0.999299	+	0.0374244i	$0.0119154\pi$
$241$	18.0386i	0.0748489i	−0.999299	+	0.0374244i	$0.988085\pi$
$242$	159.806	0.660356
$243$	0	0
$244$	− 27.0730i	− 0.110955i
$245$	0	0
$246$	0	0
$247$	258.392	1.04612
$248$	42.8605i	0.172825i
$249$	0	0
$250$	− 189.893i	− 0.759573i
$251$	− 166.869i	− 0.664815i	−0.943136	−	0.332408i	$-0.892139\pi$
$251$	− 166.869i	− 0.664815i	0.943136	−	0.332408i	$-0.107861\pi$
$252$	0	0
$253$	−103.196	−0.407889
$254$	−212.686	−0.837348
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 305.873i	− 1.19017i	−0.803664	−	0.595083i	$-0.797119\pi$
$257$	− 305.873i	− 1.19017i	0.803664	−	0.595083i	$-0.202881\pi$
$258$	0	0
$259$	0	0
$260$	166.024	0.638555
$261$	0	0
$262$	− 307.887i	− 1.17514i
$263$	9.62237	0.0365869	0.0182935	−	0.999833i	$-0.494177\pi$
$263$	9.62237	0.0365869	0.0182935	+	0.999833i	$0.494177\pi$
$264$	0	0
$265$	− 464.938i	− 1.75448i
$266$	0	0
$267$	0	0
$268$	8.00000	0.0298507
$269$	99.6422i	0.370417i	0.982699	+	0.185209i	$0.0592961\pi$
$269$	99.6422i	0.370417i	−0.982699	+	0.185209i	$0.940704\pi$
$270$	0	0
$271$	504.628i	1.86210i	0.364898	+	0.931048i	$0.381104\pi$
$271$	504.628i	1.86210i	−0.364898	+	0.931048i	$0.618896\pi$
$272$	48.1505i	0.177024i
$273$	0	0
$274$	73.5980	0.268606
$275$	−14.4264	−0.0524597
$276$	0	0
$277$	314.000	1.13357	0.566787	−	0.823864i	$-0.308186\pi$
$277$	314.000	1.13357	0.566787	+	0.823864i	$0.308186\pi$
$278$	− 203.362i	− 0.731519i
$279$	0	0
$280$	0	0
$281$	155.002	0.551609	0.275804	−	0.961214i	$-0.411056\pi$
$281$	155.002	0.551609	0.275804	+	0.961214i	$0.411056\pi$
$282$	0	0
$283$	− 276.503i	− 0.977044i	−0.872552	−	0.488522i	$-0.837536\pi$
$283$	− 276.503i	− 0.977044i	0.872552	−	0.488522i	$-0.162464\pi$
$284$	184.402	0.649303
$285$	0	0
$286$	− 74.4356i	− 0.260264i
$287$	0	0
$288$	0	0
$289$	144.095	0.498600
$290$	− 78.5026i	− 0.270698i
$291$	0	0
$292$	170.936i	0.585396i
$293$	− 485.027i	− 1.65538i	−0.561185	−	0.827690i	$-0.689654\pi$
$293$	− 485.027i	− 1.65538i	0.561185	−	0.827690i	$-0.310346\pi$
$294$	0	0
$295$	107.598	0.364739
$296$	−129.255	−0.436672
$297$	0	0
$298$	−26.9949	−0.0905871
$299$	− 678.951i	− 2.27074i
$300$	0	0
$301$	0	0
$302$	−218.343	−0.722991
$303$	0	0
$304$	55.5416i	0.182702i
$305$	60.3848	0.197983
$306$	0	0
$307$	126.364i	0.411609i	0.978593	+	0.205805i	$0.0659811\pi$
$307$	126.364i	0.411609i	−0.978593	+	0.205805i	$0.934019\pi$
$308$	0	0
$309$	0	0
$310$	−95.5980	−0.308381
$311$	421.968i	1.35681i	0.734688	+	0.678405i	$0.237329\pi$
$311$	421.968i	1.35681i	−0.734688	+	0.678405i	$0.762671\pi$
$312$	0	0
$313$	− 431.650i	− 1.37907i	−0.724251	−	0.689536i	$-0.757814\pi$
$313$	− 431.650i	− 1.37907i	0.724251	−	0.689536i	$-0.242186\pi$
$314$	− 166.420i	− 0.530001i
$315$	0	0
$316$	7.19596	0.0227720
$317$	−256.804	−0.810107	−0.405054	−	0.914293i	$-0.632747\pi$
$317$	−256.804	−0.810107	−0.405054	+	0.914293i	$0.632747\pi$
$318$	0	0
$319$	−35.1960	−0.110332
$320$	35.6871i	0.111522i
$321$	0	0
$322$	0	0
$323$	−167.147	−0.517484
$324$	0	0
$325$	− 94.9148i	− 0.292046i
$326$	−263.029	−0.806839
$327$	0	0
$328$	− 148.935i	− 0.454070i
$329$	0	0
$330$	0	0
$331$	155.598	0.470085	0.235042	−	0.971985i	$-0.424477\pi$
$331$	155.598	0.470085	0.235042	+	0.971985i	$0.424477\pi$
$332$	160.682i	0.483982i
$333$	0	0
$334$	291.529i	0.872843i
$335$	17.8435i	0.0532643i
$336$	0	0
$337$	39.7086	0.117830	0.0589148	−	0.998263i	$-0.481236\pi$
$337$	39.7086	0.117830	0.0589148	+	0.998263i	$0.481236\pi$
$338$	250.728	0.741799
$339$	0	0
$340$	−107.397	−0.315873
$341$	42.8605i	0.125691i
$342$	0	0
$343$	0	0
$344$	202.510	0.588691
$345$	0	0
$346$	291.434i	0.842294i
$347$	649.671	1.87225	0.936126	−	0.351666i	$-0.114385\pi$
$347$	649.671	1.87225	0.936126	+	0.351666i	$0.114385\pi$
$348$	0	0
$349$	− 616.789i	− 1.76730i	−0.468143	−	0.883652i	$-0.655077\pi$
$349$	− 616.789i	− 1.76730i	0.468143	−	0.883652i	$-0.344923\pi$
$350$	0	0
$351$	0	0
$352$	16.0000	0.0454545
$353$	− 530.755i	− 1.50356i	−0.659416	−	0.751778i	$-0.729197\pi$
$353$	− 530.755i	− 1.50356i	0.659416	−	0.751778i	$-0.270803\pi$
$354$	0	0
$355$	411.298i	1.15859i
$356$	223.851i	0.628794i
$357$	0	0
$358$	−226.392	−0.632380
$359$	−532.014	−1.48193	−0.740967	−	0.671541i	$-0.765633\pi$
$359$	−532.014	−1.48193	−0.740967	+	0.671541i	$0.765633\pi$
$360$	0	0
$361$	168.196	0.465917
$362$	− 141.313i	− 0.390368i
$363$	0	0
$364$	0	0
$365$	−381.262	−1.04455
$366$	0	0
$367$	− 391.072i	− 1.06559i	−0.846244	−	0.532796i	$-0.821141\pi$
$367$	− 391.072i	− 1.06559i	0.846244	−	0.532796i	$-0.178859\pi$
$368$	145.941	0.396579
$369$	0	0
$370$	− 288.295i	− 0.779177i
$371$	0	0
$372$	0	0
$373$	72.8040	0.195185	0.0975925	−	0.995226i	$-0.468886\pi$
$373$	72.8040	0.195185	0.0975925	+	0.995226i	$0.468886\pi$
$374$	48.1505i	0.128745i
$375$	0	0
$376$	− 239.665i	− 0.637406i
$377$	− 231.563i	− 0.614225i
$378$	0	0
$379$	−30.8141	−0.0813038	−0.0406519	−	0.999173i	$-0.512943\pi$
$379$	−30.8141	−0.0813038	−0.0406519	+	0.999173i	$0.512943\pi$
$380$	−123.882	−0.326006
$381$	0	0
$382$	98.3919	0.257570
$383$	446.269i	1.16519i	0.812762	+	0.582596i	$0.197963\pi$
$383$	446.269i	1.16519i	−0.812762	+	0.582596i	$0.802037\pi$
$384$	0	0
$385$	0	0
$386$	−191.196	−0.495326
$387$	0	0
$388$	− 46.1582i	− 0.118965i
$389$	260.208	0.668916	0.334458	−	0.942411i	$-0.391447\pi$
$389$	260.208	0.668916	0.334458	+	0.942411i	$0.391447\pi$
$390$	0	0
$391$	439.196i	1.12326i
$392$	0	0
$393$	0	0
$394$	30.3919	0.0771369
$395$	16.0502i	0.0406334i
$396$	0	0
$397$	− 609.847i	− 1.53614i	−0.640367	−	0.768069i	$-0.721218\pi$
$397$	− 609.847i	− 1.53614i	0.640367	−	0.768069i	$-0.278782\pi$
$398$	− 367.451i	− 0.923243i
$399$	0	0
$400$	20.4020	0.0510051
$401$	147.057	0.366725	0.183363	−	0.983045i	$-0.441302\pi$
$401$	147.057	0.366725	0.183363	+	0.983045i	$0.441302\pi$
$402$	0	0
$403$	−281.990	−0.699727
$404$	− 165.120i	− 0.408714i
$405$	0	0
$406$	0	0
$407$	−129.255	−0.317579
$408$	0	0
$409$	158.763i	0.388173i	0.980984	+	0.194087i	$0.0621743\pi$
$409$	158.763i	0.388173i	−0.980984	+	0.194087i	$0.937826\pi$
$410$	332.191	0.810222
$411$	0	0
$412$	325.513i	0.790081i
$413$	0	0
$414$	0	0
$415$	−358.392	−0.863595
$416$	105.268i	0.253048i
$417$	0	0
$418$	55.5416i	0.132875i
$419$	577.000i	1.37709i	0.725195	+	0.688544i	$0.241750\pi$
$419$	577.000i	1.37709i	−0.725195	+	0.688544i	$0.758250\pi$
$420$	0	0
$421$	−37.9798	−0.0902133	−0.0451066	−	0.998982i	$-0.514363\pi$
$421$	−37.9798	−0.0902133	−0.0451066	+	0.998982i	$0.514363\pi$
$422$	355.813	0.843159
$423$	0	0
$424$	294.794	0.695269
$425$	61.3980i	0.144466i
$426$	0	0
$427$	0	0
$428$	27.7157	0.0647564
$429$	0	0
$430$	451.686i	1.05043i
$431$	−785.151	−1.82170	−0.910848	−	0.412741i	$-0.864571\pi$
$431$	−785.151	−1.82170	−0.910848	+	0.412741i	$0.864571\pi$
$432$	0	0
$433$	582.519i	1.34531i	0.739957	+	0.672655i	$0.234846\pi$
$433$	582.519i	1.34531i	−0.739957	+	0.672655i	$0.765154\pi$
$434$	0	0
$435$	0	0
$436$	273.387	0.627034
$437$	506.612i	1.15930i
$438$	0	0
$439$	− 6.46796i	− 0.0147334i	−0.999973	−	0.00736669i	$-0.997655\pi$
$439$	− 6.46796i	− 0.0147334i	0.999973	−	0.00736669i	$-0.00234491\pi$
$440$	35.6871i	0.0811070i
$441$	0	0
$442$	−316.794	−0.716728
$443$	291.897	0.658909	0.329454	−	0.944172i	$-0.393135\pi$
$443$	291.897	0.658909	0.329454	+	0.944172i	$0.393135\pi$
$444$	0	0
$445$	−499.286	−1.12199
$446$	41.0672i	0.0920789i
$447$	0	0
$448$	0	0
$449$	457.480	1.01889	0.509443	−	0.860504i	$-0.329851\pi$
$449$	457.480	1.01889	0.509443	+	0.860504i	$0.329851\pi$
$450$	0	0
$451$	− 148.935i	− 0.330233i
$452$	−265.588	−0.587584
$453$	0	0
$454$	389.677i	0.858319i
$455$	0	0
$456$	0	0
$457$	−680.764	−1.48964	−0.744818	−	0.667268i	$-0.767464\pi$
$457$	−680.764	−1.48964	−0.744818	+	0.667268i	$0.767464\pi$
$458$	59.1788i	0.129211i
$459$	0	0
$460$	325.513i	0.707638i
$461$	156.401i	0.339265i	0.985507	+	0.169633i	$0.0542581\pi$
$461$	156.401i	0.339265i	−0.985507	+	0.169633i	$0.945742\pi$
$462$	0	0
$463$	370.774	0.800807	0.400404	−	0.916339i	$-0.368870\pi$
$463$	370.774	0.800807	0.400404	+	0.916339i	$0.368870\pi$
$464$	49.7746	0.107273
$465$	0	0
$466$	307.186	0.659197
$467$	− 40.1705i	− 0.0860182i	−0.999075	−	0.0430091i	$-0.986306\pi$
$467$	− 40.1705i	− 0.0860182i	0.999075	−	0.0430091i	$-0.0136944\pi$
$468$	0	0
$469$	0	0
$470$	534.558	1.13736
$471$	0	0
$472$	68.2226i	0.144539i
$473$	202.510	0.428139
$474$	0	0
$475$	70.8225i	0.149100i
$476$	0	0
$477$	0	0
$478$	97.1859	0.203318
$479$	− 213.136i	− 0.444960i	−0.974937	−	0.222480i	$-0.928585\pi$
$479$	− 213.136i	− 0.444960i	0.974937	−	0.222480i	$-0.0714152\pi$
$480$	0	0
$481$	− 850.399i	− 1.76798i
$482$	− 25.5104i	− 0.0529262i
$483$	0	0
$484$	−226.000	−0.466942
$485$	102.953	0.212275
$486$	0	0
$487$	−843.598	−1.73223	−0.866117	−	0.499841i	$-0.833392\pi$
$487$	−843.598	−1.73223	−0.866117	+	0.499841i	$0.833392\pi$
$488$	38.2870i	0.0784570i
$489$	0	0
$490$	0	0
$491$	−614.621	−1.25177	−0.625887	−	0.779913i	$-0.715263\pi$
$491$	−614.621	−1.25177	−0.625887	+	0.779913i	$0.715263\pi$
$492$	0	0
$493$	149.792i	0.303838i
$494$	−365.421	−0.739719
$495$	0	0
$496$	− 60.6140i	− 0.122206i
$497$	0	0
$498$	0	0
$499$	659.980	1.32260	0.661302	−	0.750119i	$-0.270004\pi$
$499$	659.980	1.32260	0.661302	+	0.750119i	$0.270004\pi$
$500$	268.550i	0.537100i
$501$	0	0
$502$	235.988i	0.470095i
$503$	− 301.276i	− 0.598959i	−0.954103	−	0.299480i	$-0.903187\pi$
$503$	− 301.276i	− 0.598959i	0.954103	−	0.299480i	$-0.0968130\pi$
$504$	0	0
$505$	368.291	0.729290
$506$	145.941	0.288421
$507$	0	0
$508$	300.784	0.592094
$509$	866.644i	1.70264i	0.524646	+	0.851320i	$0.324198\pi$
$509$	866.644i	1.70264i	−0.524646	+	0.851320i	$0.675802\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	432.569i	0.841574i
$515$	−726.039	−1.40978
$516$	0	0
$517$	− 239.665i	− 0.463568i
$518$	0	0
$519$	0	0
$520$	−234.794	−0.451527
$521$	− 623.830i	− 1.19737i	−0.800984	−	0.598685i	$-0.795690\pi$
$521$	− 623.830i	− 1.19737i	0.800984	−	0.598685i	$-0.204310\pi$
$522$	0	0
$523$	770.604i	1.47343i	0.676203	+	0.736715i	$0.263624\pi$
$523$	770.604i	1.47343i	−0.676203	+	0.736715i	$0.736376\pi$
$524$	435.418i	0.830951i
$525$	0	0
$526$	−13.6081	−0.0258709
$527$	182.412	0.346133
$528$	0	0
$529$	802.176	1.51640
$530$	657.521i	1.24061i
$531$	0	0
$532$	0	0
$533$	979.879	1.83842
$534$	0	0
$535$	61.8183i	0.115548i
$536$	−11.3137	−0.0211077
$537$	0	0
$538$	− 140.915i	− 0.261924i
$539$	0	0
$540$	0	0
$541$	−1033.57	−1.91048	−0.955238	−	0.295838i	$-0.904401\pi$
$541$	−1033.57	−1.91048	−0.955238	+	0.295838i	$0.904401\pi$
$542$	− 713.651i	− 1.31670i
$543$	0	0
$544$	− 68.0951i	− 0.125175i
$545$	609.774i	1.11885i
$546$	0	0
$547$	923.176	1.68771	0.843854	−	0.536574i	$-0.180282\pi$
$547$	923.176	1.68771	0.843854	+	0.536574i	$0.180282\pi$
$548$	−104.083	−0.189933
$549$	0	0
$550$	20.4020	0.0370946
$551$	172.785i	0.313584i
$552$	0	0
$553$	0	0
$554$	−444.063	−0.801558
$555$	0	0
$556$	287.598i	0.517262i
$557$	−381.255	−0.684479	−0.342240	−	0.939613i	$-0.611185\pi$
$557$	−381.255	−0.684479	−0.342240	+	0.939613i	$0.611185\pi$
$558$	0	0
$559$	1332.36i	2.38347i
$560$	0	0
$561$	0	0
$562$	−219.206	−0.390046
$563$	214.659i	0.381277i	0.981660	+	0.190638i	$0.0610557\pi$
$563$	214.659i	0.381277i	−0.981660	+	0.190638i	$0.938944\pi$
$564$	0	0
$565$	− 592.378i	− 1.04846i
$566$	391.035i	0.690874i
$567$	0	0
$568$	−260.784	−0.459126
$569$	131.252	0.230671	0.115336	−	0.993327i	$-0.463206\pi$
$569$	131.252	0.230671	0.115336	+	0.993327i	$0.463206\pi$
$570$	0	0
$571$	−101.186	−0.177208	−0.0886041	−	0.996067i	$-0.528241\pi$
$571$	−101.186	−0.177208	−0.0886041	+	0.996067i	$0.528241\pi$
$572$	105.268i	0.184035i
$573$	0	0
$574$	0	0
$575$	186.093	0.323641
$576$	0	0
$577$	− 920.331i	− 1.59503i	−0.603301	−	0.797514i	$-0.706148\pi$
$577$	− 920.331i	− 1.59503i	0.603301	−	0.797514i	$-0.293852\pi$
$578$	−203.782	−0.352564
$579$	0	0
$580$	111.019i	0.191413i
$581$	0	0
$582$	0	0
$583$	294.794	0.505650
$584$	− 241.739i	− 0.413937i
$585$	0	0
$586$	685.931i	1.17053i
$587$	595.384i	1.01428i	0.861863	+	0.507141i	$0.169298\pi$
$587$	595.384i	1.01428i	−0.861863	+	0.507141i	$0.830702\pi$
$588$	0	0
$589$	210.412	0.357236
$590$	−152.167	−0.257909
$591$	0	0
$592$	182.794	0.308774
$593$	441.134i	0.743903i	0.928252	+	0.371951i	$0.121311\pi$
$593$	441.134i	0.743903i	−0.928252	+	0.371951i	$0.878689\pi$
$594$	0	0
$595$	0	0
$596$	38.1766	0.0640547
$597$	0	0
$598$	960.182i	1.60566i
$599$	−250.053	−0.417451	−0.208725	−	0.977974i	$-0.566931\pi$
$599$	−250.053	−0.417451	−0.208725	+	0.977974i	$0.566931\pi$
$600$	0	0
$601$	− 593.457i	− 0.987449i	−0.869618	−	0.493725i	$-0.835635\pi$
$601$	− 593.457i	− 0.987449i	0.869618	−	0.493725i	$-0.164365\pi$
$602$	0	0
$603$	0	0
$604$	308.784	0.511232
$605$	− 504.080i	− 0.833190i
$606$	0	0
$607$	− 338.449i	− 0.557577i	−0.960353	−	0.278788i	$-0.910067\pi$
$607$	− 338.449i	− 0.557577i	0.960353	−	0.278788i	$-0.0899328\pi$
$608$	− 78.5476i	− 0.129190i
$609$	0	0
$610$	−85.3970	−0.139995
$611$	1576.81	2.58071
$612$	0	0
$613$	805.477	1.31399	0.656996	−	0.753894i	$-0.271827\pi$
$613$	805.477	1.31399	0.656996	+	0.753894i	$0.271827\pi$
$614$	− 178.706i	− 0.291052i
$615$	0	0
$616$	0	0
$617$	−18.1148	−0.0293595	−0.0146797	−	0.999892i	$-0.504673\pi$
$617$	−18.1148	−0.0293595	−0.0146797	+	0.999892i	$0.504673\pi$
$618$	0	0
$619$	− 448.137i	− 0.723969i	−0.932184	−	0.361984i	$-0.882099\pi$
$619$	− 448.137i	− 0.723969i	0.932184	−	0.361984i	$-0.117901\pi$
$620$	135.196	0.218058
$621$	0	0
$622$	− 596.753i	− 0.959410i
$623$	0	0
$624$	0	0
$625$	−471.472	−0.754356
$626$	610.445i	0.975152i
$627$	0	0
$628$	235.354i	0.374767i
$629$	550.101i	0.874565i
$630$	0	0
$631$	696.764	1.10422	0.552111	−	0.833771i	$-0.313823\pi$
$631$	696.764	1.10422	0.552111	+	0.833771i	$0.313823\pi$
$632$	−10.1766	−0.0161023
$633$	0	0
$634$	363.176	0.572832
$635$	670.881i	1.05651i
$636$	0	0
$637$	0	0
$638$	49.7746	0.0780166
$639$	0	0
$640$	− 50.4692i	− 0.0788581i
$641$	−985.394	−1.53728	−0.768638	−	0.639684i	$-0.779065\pi$
$641$	−985.394	−1.53728	−0.768638	+	0.639684i	$0.779065\pi$
$642$	0	0
$643$	− 675.943i	− 1.05123i	−0.850722	−	0.525616i	$-0.823835\pi$
$643$	− 675.943i	− 1.05123i	0.850722	−	0.525616i	$-0.176165\pi$
$644$	0	0
$645$	0	0
$646$	236.382	0.365916
$647$	116.745i	0.180440i	0.995922	+	0.0902199i	$0.0287570\pi$
$647$	116.745i	0.180440i	−0.995922	+	0.0902199i	$0.971243\pi$
$648$	0	0
$649$	68.2226i	0.105120i
$650$	134.230i	0.206507i
$651$	0	0
$652$	371.980	0.570521
$653$	941.248	1.44142	0.720710	−	0.693236i	$-0.243816\pi$
$653$	941.248	1.44142	0.720710	+	0.693236i	$0.243816\pi$
$654$	0	0
$655$	−971.176	−1.48271
$656$	210.626i	0.321076i
$657$	0	0
$658$	0	0
$659$	1159.61	1.75965	0.879827	−	0.475294i	$-0.157658\pi$
$659$	1159.61	1.75965	0.879827	+	0.475294i	$0.157658\pi$
$660$	0	0
$661$	− 740.805i	− 1.12073i	−0.828244	−	0.560367i	$-0.810660\pi$
$661$	− 740.805i	− 1.12073i	0.828244	−	0.560367i	$-0.189340\pi$
$662$	−220.049	−0.332400
$663$	0	0
$664$	− 227.239i	− 0.342227i
$665$	0	0
$666$	0	0
$667$	454.010	0.680675
$668$	− 412.285i	− 0.617193i
$669$	0	0
$670$	− 25.2346i	− 0.0376635i
$671$	38.2870i	0.0570596i
$672$	0	0
$673$	−573.075	−0.851523	−0.425762	−	0.904835i	$-0.639994\pi$
$673$	−573.075	−0.851523	−0.425762	+	0.904835i	$0.639994\pi$
$674$	−56.1564	−0.0833181
$675$	0	0
$676$	−354.583	−0.524531
$677$	− 136.629i	− 0.201816i	−0.994896	−	0.100908i	$-0.967825\pi$
$677$	− 136.629i	− 0.201816i	0.994896	−	0.100908i	$-0.0321747\pi$
$678$	0	0
$679$	0	0
$680$	151.882	0.223356
$681$	0	0
$682$	− 60.6140i	− 0.0888768i
$683$	266.455	0.390124	0.195062	−	0.980791i	$-0.437509\pi$
$683$	266.455	0.390124	0.195062	+	0.980791i	$0.437509\pi$
$684$	0	0
$685$	− 232.152i	− 0.338908i
$686$	0	0
$687$	0	0
$688$	−286.392	−0.416267
$689$	1939.52i	2.81498i
$690$	0	0
$691$	127.568i	0.184614i	0.995731	+	0.0923071i	$0.0294242\pi$
$691$	127.568i	0.184614i	−0.995731	+	0.0923071i	$0.970576\pi$
$692$	− 412.150i	− 0.595592i
$693$	0	0
$694$	−918.774	−1.32388
$695$	−641.470	−0.922979
$696$	0	0
$697$	−633.859	−0.909410
$698$	872.272i	1.24967i
$699$	0	0
$700$	0	0
$701$	−545.325	−0.777924	−0.388962	−	0.921254i	$-0.627166\pi$
$701$	−545.325	−0.777924	−0.388962	+	0.921254i	$0.627166\pi$
$702$	0	0
$703$	634.541i	0.902619i
$704$	−22.6274	−0.0321412
$705$	0	0
$706$	750.601i	1.06317i
$707$	0	0
$708$	0	0
$709$	−171.106	−0.241334	−0.120667	−	0.992693i	$-0.538503\pi$
$709$	−171.106	−0.241334	−0.120667	+	0.992693i	$0.538503\pi$
$710$	− 581.663i	− 0.819244i
$711$	0	0
$712$	− 316.573i	− 0.444625i
$713$	− 552.879i	− 0.775427i
$714$	0	0
$715$	−234.794	−0.328383
$716$	320.167	0.447160
$717$	0	0
$718$	752.382	1.04789
$719$	233.579i	0.324867i	0.986720	+	0.162433i	$0.0519342\pi$
$719$	233.579i	0.324867i	−0.986720	+	0.162433i	$0.948066\pi$
$720$	0	0
$721$	0	0
$722$	−237.865	−0.329453
$723$	0	0
$724$	199.847i	0.276032i
$725$	63.4689	0.0875433
$726$	0	0
$727$	912.568i	1.25525i	0.778515	+	0.627626i	$0.215973\pi$
$727$	912.568i	1.25525i	−0.778515	+	0.627626i	$0.784027\pi$
$728$	0	0
$729$	0	0
$730$	539.186	0.738611
$731$	− 861.870i	− 1.17903i
$732$	0	0
$733$	− 333.754i	− 0.455327i	−0.973740	−	0.227663i	$-0.926891\pi$
$733$	− 333.754i	− 0.455327i	0.973740	−	0.227663i	$-0.0731085\pi$
$734$	553.060i	0.753487i
$735$	0	0
$736$	−206.392	−0.280424
$737$	−11.3137	−0.0153510
$738$	0	0
$739$	−640.382	−0.866552	−0.433276	−	0.901261i	$-0.642642\pi$
$739$	−640.382	−0.866552	−0.433276	+	0.901261i	$0.642642\pi$
$740$	407.711i	0.550961i
$741$	0	0
$742$	0	0
$743$	−750.642	−1.01028	−0.505142	−	0.863036i	$-0.668560\pi$
$743$	−750.642	−1.01028	−0.505142	+	0.863036i	$0.668560\pi$
$744$	0	0
$745$	85.1508i	0.114296i
$746$	−102.960	−0.138017
$747$	0	0
$748$	− 68.0951i	− 0.0910363i
$749$	0	0
$750$	0	0
$751$	795.980	1.05989	0.529947	−	0.848031i	$-0.322212\pi$
$751$	795.980	1.05989	0.529947	+	0.848031i	$0.322212\pi$
$752$	338.937i	0.450714i
$753$	0	0
$754$	327.479i	0.434323i
$755$	688.725i	0.912218i
$756$	0	0
$757$	−145.678	−0.192442	−0.0962208	−	0.995360i	$-0.530675\pi$
$757$	−145.678	−0.192442	−0.0962208	+	0.995360i	$0.530675\pi$
$758$	43.5778	0.0574905
$759$	0	0
$760$	175.196	0.230521
$761$	866.153i	1.13818i	0.822276	+	0.569089i	$0.192704\pi$
$761$	866.153i	1.13818i	−0.822276	+	0.569089i	$0.807296\pi$
$762$	0	0
$763$	0	0
$764$	−139.147	−0.182130
$765$	0	0
$766$	− 631.119i	− 0.823916i
$767$	−448.853	−0.585206
$768$	0	0
$769$	− 1051.96i	− 1.36796i	−0.729502	−	0.683978i	$-0.760248\pi$
$769$	− 1051.96i	− 1.36796i	0.729502	−	0.683978i	$-0.239752\pi$
$770$	0	0
$771$	0	0
$772$	270.392	0.350249
$773$	− 855.796i	− 1.10711i	−0.832813	−	0.553555i	$-0.813271\pi$
$773$	− 855.796i	− 1.10711i	0.832813	−	0.553555i	$-0.186729\pi$
$774$	0	0
$775$	− 77.2905i	− 0.0997296i
$776$	65.2776i	0.0841206i
$777$	0	0
$778$	−367.990	−0.472995
$779$	−731.156	−0.938582
$780$	0	0
$781$	−260.784	−0.333910
$782$	− 621.117i	− 0.794268i
$783$	0	0
$784$	0	0
$785$	−524.943	−0.668717
$786$	0	0
$787$	− 1405.91i	− 1.78642i	−0.449643	−	0.893208i	$-0.648449\pi$
$787$	− 1405.91i	− 1.78642i	0.449643	−	0.893208i	$-0.351551\pi$
$788$	−42.9807	−0.0545440
$789$	0	0
$790$	− 22.6984i	− 0.0287321i
$791$	0	0
$792$	0	0
$793$	−251.899	−0.317654
$794$	862.454i	1.08621i
$795$	0	0
$796$	519.654i	0.652831i
$797$	− 253.331i	− 0.317856i	−0.987290	−	0.158928i	$-0.949196\pi$
$797$	− 253.331i	− 0.317856i	0.987290	−	0.158928i	$-0.0508037\pi$
$798$	0	0
$799$	−1020.00	−1.27660
$800$	−28.8528	−0.0360660
$801$	0	0
$802$	−207.970	−0.259314
$803$	− 241.739i	− 0.301045i
$804$	0	0
$805$	0	0
$806$	398.794	0.494782
$807$	0	0
$808$	233.515i	0.289004i
$809$	739.882	0.914564	0.457282	−	0.889322i	$-0.348823\pi$
$809$	739.882	0.914564	0.457282	+	0.889322i	$0.348823\pi$
$810$	0	0
$811$	− 764.518i	− 0.942686i	−0.881950	−	0.471343i	$-0.843769\pi$
$811$	− 764.518i	− 0.942686i	0.881950	−	0.471343i	$-0.156231\pi$
$812$	0	0
$813$	0	0
$814$	182.794	0.224563
$815$	829.680i	1.01801i
$816$	0	0
$817$	− 994.166i	− 1.21685i
$818$	− 224.524i	− 0.274480i
$819$	0	0
$820$	−469.789	−0.572913
$821$	−119.902	−0.146044	−0.0730222	−	0.997330i	$-0.523264\pi$
$821$	−119.902	−0.146044	−0.0730222	+	0.997330i	$0.523264\pi$
$822$	0	0
$823$	−307.196	−0.373264	−0.186632	−	0.982430i	$-0.559757\pi$
$823$	−307.196	−0.373264	−0.186632	+	0.982430i	$0.559757\pi$
$824$	− 460.345i	− 0.558671i
$825$	0	0
$826$	0	0
$827$	514.504	0.622133	0.311066	−	0.950388i	$-0.399314\pi$
$827$	514.504	0.622133	0.311066	+	0.950388i	$0.399314\pi$
$828$	0	0
$829$	806.651i	0.973041i	0.873669	+	0.486520i	$0.161734\pi$
$829$	806.651i	0.973041i	−0.873669	+	0.486520i	$0.838266\pi$
$830$	506.843	0.610654
$831$	0	0
$832$	− 148.871i	− 0.178932i
$833$	0	0
$834$	0	0
$835$	919.578	1.10129
$836$	− 78.5476i	− 0.0939565i
$837$	0	0
$838$	− 816.001i	− 0.973748i
$839$	100.694i	0.120017i	0.998198	+	0.0600085i	$0.0191128\pi$
$839$	100.694i	0.120017i	−0.998198	+	0.0600085i	$0.980887\pi$
$840$	0	0
$841$	−686.156	−0.815881
$842$	53.7115	0.0637904
$843$	0	0
$844$	−503.196	−0.596204
$845$	− 790.877i	− 0.935949i
$846$	0	0
$847$	0	0
$848$	−416.902	−0.491629
$849$	0	0
$850$	− 86.8299i	− 0.102153i
$851$	1667.32	1.95925
$852$	0	0
$853$	137.712i	0.161444i	0.996737	+	0.0807219i	$0.0257226\pi$
$853$	137.712i	0.161444i	−0.996737	+	0.0807219i	$0.974277\pi$
$854$	0	0
$855$	0	0
$856$	−39.1960	−0.0457897
$857$	− 557.744i	− 0.650809i	−0.945575	−	0.325405i	$-0.894499\pi$
$857$	− 557.744i	− 0.650809i	0.945575	−	0.325405i	$-0.105501\pi$
$858$	0	0
$859$	1578.94i	1.83811i	0.394126	+	0.919056i	$0.371047\pi$
$859$	1578.94i	1.83811i	−0.394126	+	0.919056i	$0.628953\pi$
$860$	− 638.781i	− 0.742768i
$861$	0	0
$862$	1110.37	1.28813
$863$	525.249	0.608631	0.304316	−	0.952571i	$-0.401572\pi$
$863$	525.249	0.608631	0.304316	+	0.952571i	$0.401572\pi$
$864$	0	0
$865$	919.276	1.06275
$866$	− 823.806i	− 0.951277i
$867$	0	0
$868$	0	0
$869$	−10.1766	−0.0117107
$870$	0	0
$871$	− 74.4356i	− 0.0854600i
$872$	−386.627	−0.443380
$873$	0	0
$874$	− 716.458i	− 0.819746i
$875$	0	0
$876$	0	0
$877$	−34.2611	−0.0390663	−0.0195331	−	0.999809i	$-0.506218\pi$
$877$	−34.2611	−0.0390663	−0.0195331	+	0.999809i	$0.506218\pi$
$878$	9.14707i	0.0104181i
$879$	0	0
$880$	− 50.4692i	− 0.0573513i
$881$	− 366.173i	− 0.415634i	−0.978168	−	0.207817i	$-0.933364\pi$
$881$	− 366.173i	− 0.415634i	0.978168	−	0.207817i	$-0.0666358\pi$
$882$	0	0
$883$	493.186	0.558534	0.279267	−	0.960213i	$-0.409908\pi$
$883$	493.186	0.558534	0.279267	+	0.960213i	$0.409908\pi$
$884$	448.014	0.506803
$885$	0	0
$886$	−412.804	−0.465919
$887$	204.619i	0.230687i	0.993326	+	0.115344i	$0.0367969\pi$
$887$	204.619i	0.230687i	−0.993326	+	0.115344i	$0.963203\pi$
$888$	0	0
$889$	0	0
$890$	706.098	0.793368
$891$	0	0
$892$	− 58.0778i	− 0.0651096i
$893$	−1176.57	−1.31755
$894$	0	0
$895$	714.113i	0.797892i
$896$	0	0
$897$	0	0
$898$	−646.975	−0.720462
$899$	− 188.565i	− 0.209749i
$900$	0	0
$901$	− 1254.63i	− 1.39248i
$902$	210.626i	0.233510i
$903$	0	0
$904$	375.598	0.415484
$905$	−445.747	−0.492538
$906$	0	0
$907$	−136.764	−0.150787	−0.0753934	−	0.997154i	$-0.524021\pi$
$907$	−136.764	−0.150787	−0.0753934	+	0.997154i	$0.524021\pi$
$908$	− 551.086i	− 0.606923i
$909$	0	0
$910$	0	0
$911$	676.563	0.742659	0.371330	−	0.928501i	$-0.378902\pi$
$911$	676.563	0.742659	0.371330	+	0.928501i	$0.378902\pi$
$912$	0	0
$913$	− 227.239i	− 0.248892i
$914$	962.745	1.05333
$915$	0	0
$916$	− 83.6915i	− 0.0913663i
$917$	0	0
$918$	0	0
$919$	−1022.41	−1.11253	−0.556263	−	0.831006i	$-0.687765\pi$
$919$	−1022.41	−1.11253	−0.556263	+	0.831006i	$0.687765\pi$
$920$	− 460.345i	− 0.500375i
$921$	0	0
$922$	− 221.185i	− 0.239897i
$923$	− 1715.76i	− 1.85889i
$924$	0	0
$925$	233.085	0.251984
$926$	−524.353	−0.566256
$927$	0	0
$928$	−70.3919	−0.0758534
$929$	− 1797.60i	− 1.93499i	−0.252894	−	0.967494i	$-0.581382\pi$
$929$	− 1797.60i	− 1.93499i	0.252894	−	0.967494i	$-0.418618\pi$
$930$	0	0
$931$	0	0
$932$	−434.426	−0.466123
$933$	0	0
$934$	56.8097i	0.0608240i
$935$	151.882	0.162441
$936$	0	0
$937$	− 494.548i	− 0.527800i	−0.964550	−	0.263900i	$-0.914991\pi$
$937$	− 494.548i	− 0.527800i	0.964550	−	0.263900i	$-0.0850088\pi$
$938$	0	0
$939$	0	0
$940$	−755.980	−0.804234
$941$	39.6996i	0.0421888i	0.999777	+	0.0210944i	$0.00671505\pi$
$941$	39.6996i	0.0421888i	−0.999777	+	0.0210944i	$0.993285\pi$
$942$	0	0
$943$	1921.19i	2.03731i
$944$	− 96.4813i	− 0.102205i
$945$	0	0
$946$	−286.392	−0.302740
$947$	−1517.96	−1.60291	−0.801455	−	0.598055i	$-0.795940\pi$
$947$	−1517.96	−1.60291	−0.801455	+	0.598055i	$0.795940\pi$
$948$	0	0
$949$	1590.46	1.67593
$950$	− 100.158i	− 0.105430i
$951$	0	0
$952$	0	0
$953$	−1512.57	−1.58717	−0.793583	−	0.608462i	$-0.791787\pi$
$953$	−1512.57	−1.58717	−0.793583	+	0.608462i	$0.791787\pi$
$954$	0	0
$955$	− 310.360i	− 0.324984i
$956$	−137.442	−0.143767
$957$	0	0
$958$	301.419i	0.314634i
$959$	0	0
$960$	0	0
$961$	731.372	0.761053
$962$	1202.65i	1.25015i
$963$	0	0
$964$	36.0772i	0.0374244i
$965$	603.094i	0.624967i
$966$	0	0
$967$	308.382	0.318906	0.159453	−	0.987206i	$-0.449027\pi$
$967$	308.382	0.318906	0.159453	+	0.987206i	$0.449027\pi$
$968$	319.612	0.330178
$969$	0	0
$970$	−145.598	−0.150101
$971$	− 683.164i	− 0.703568i	−0.936081	−	0.351784i	$-0.885575\pi$
$971$	− 683.164i	− 0.703568i	0.936081	−	0.351784i	$-0.114425\pi$
$972$	0	0
$973$	0	0
$974$	1193.03	1.22487
$975$	0	0
$976$	− 54.1460i	− 0.0554775i
$977$	1339.53	1.37106	0.685532	−	0.728042i	$-0.259570\pi$
$977$	1339.53	1.37106	0.685532	+	0.728042i	$0.259570\pi$
$978$	0	0
$979$	− 316.573i	− 0.323363i
$980$	0	0
$981$	0	0
$982$	869.206	0.885139
$983$	− 952.611i	− 0.969085i	−0.874768	−	0.484543i	$-0.838986\pi$
$983$	− 952.611i	− 0.969085i	0.874768	−	0.484543i	$-0.161014\pi$
$984$	0	0
$985$	− 95.8659i	− 0.0973258i
$986$	− 211.838i	− 0.214846i
$987$	0	0
$988$	516.784	0.523061
$989$	−2612.27	−2.64133
$990$	0	0
$991$	1307.92	1.31980	0.659899	−	0.751355i	$-0.270599\pi$
$991$	1307.92	1.31980	0.659899	+	0.751355i	$0.270599\pi$
$992$	85.7211i	0.0864124i
$993$	0	0
$994$	0	0
$995$	−1159.06	−1.16488
$996$	0	0
$997$	248.288i	0.249035i	0.992217	+	0.124518i	$0.0397383\pi$
$997$	248.288i	0.249035i	−0.992217	+	0.124518i	$0.960262\pi$
$998$	−933.352	−0.935223
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.e.685.2	yes	4
3.2	odd	2		882.3.c.d.685.3	✓	4
7.2	even	3		882.3.n.g.325.4		8
7.3	odd	6		882.3.n.g.19.4		8
7.4	even	3		882.3.n.g.19.3		8
7.5	odd	6		882.3.n.g.325.3		8
7.6	odd	2	inner	882.3.c.e.685.1	yes	4
21.2	odd	6		882.3.n.h.325.1		8
21.5	even	6		882.3.n.h.325.2		8
21.11	odd	6		882.3.n.h.19.2		8
21.17	even	6		882.3.n.h.19.1		8
21.20	even	2		882.3.c.d.685.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.3.c.d.685.3	✓	4	3.2	odd	2
882.3.c.d.685.4	yes	4	21.20	even	2
882.3.c.e.685.1	yes	4	7.6	odd	2	inner
882.3.c.e.685.2	yes	4	1.1	even	1	trivial
882.3.n.g.19.3		8	7.4	even	3
882.3.n.g.19.4		8	7.3	odd	6
882.3.n.g.325.3		8	7.5	odd	6
882.3.n.g.325.4		8	7.2	even	3
882.3.n.h.19.1		8	21.17	even	6
882.3.n.h.19.2		8	21.11	odd	6
882.3.n.h.325.1		8	21.2	odd	6
882.3.n.h.325.2		8	21.5	even	6

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +4.46088i q^{5} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +4.46088i q^{5} -2.82843 q^{8} -6.30864i q^{10} -2.82843 q^{11} -18.6089i q^{13} +4.00000 q^{16} +12.0376i q^{17} +13.8854i q^{19} +8.92177i q^{20} +4.00000 q^{22} +36.4853 q^{23} +5.10051 q^{25} +26.3170i q^{26} +12.4437 q^{29} -15.1535i q^{31} -5.65685 q^{32} -17.0238i q^{34} +45.6985 q^{37} -19.6369i q^{38} -12.6173i q^{40} +52.6565i q^{41} -71.5980 q^{43} -5.65685 q^{44} -51.5980 q^{46} +84.7343i q^{47} -7.21320 q^{50} -37.2178i q^{52} -104.225 q^{53} -12.6173i q^{55} -17.5980 q^{58} -24.1203i q^{59} -13.5365i q^{61} +21.4303i q^{62} +8.00000 q^{64} +83.0122 q^{65} +4.00000 q^{67} +24.0753i q^{68} +92.2010 q^{71} +85.4678i q^{73} -64.6274 q^{74} +27.7708i q^{76} +3.59798 q^{79} +17.8435i q^{80} -74.4675i q^{82} +80.3410i q^{83} -53.6985 q^{85} +101.255 q^{86} +8.00000 q^{88} +111.925i q^{89} +72.9706 q^{92} -119.832i q^{94} -61.9411 q^{95} -23.0791i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 16 q^{16} + 16 q^{22} + 112 q^{23} + 60 q^{25} + 112 q^{29} + 64 q^{37} - 128 q^{43} - 48 q^{46} + 56 q^{50} - 168 q^{53} + 88 q^{58} + 32 q^{64} + 168 q^{65} + 16 q^{67} + 448 q^{71} - 168 q^{74} - 144 q^{79} - 96 q^{85} + 224 q^{86} + 32 q^{88} + 224 q^{92} - 112 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 + 16 * q^16 + 16 * q^22 + 112 * q^23 + 60 * q^25 + 112 * q^29 + 64 * q^37 - 128 * q^43 - 48 * q^46 + 56 * q^50 - 168 * q^53 + 88 * q^58 + 32 * q^64 + 168 * q^65 + 16 * q^67 + 448 * q^71 - 168 * q^74 - 144 * q^79 - 96 * q^85 + 224 * q^86 + 32 * q^88 + 224 * q^92 - 112 * q^95

Introduction

L-functions

Modular forms

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Database

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