LMFDB - Embedded newform 882.3.c.f.685.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,3,Mod(685,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,8,0,0,0,0,0,0,36,0,0,0,0,16,0,0,0,0,0,24,60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.f.685.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +6.63103i q^{5} -2.82843 q^{8} -9.37769i q^{10} +4.75736 q^{11} +15.2913i q^{13} +4.00000 q^{16} -3.76127i q^{17} +4.18154i q^{19} +13.2621i q^{20} -6.72792 q^{22} +27.7279 q^{23} -18.9706 q^{25} -21.6251i q^{26} -3.51472 q^{29} +48.8667i q^{31} -5.65685 q^{32} +5.31925i q^{34} -2.94113 q^{37} -5.91359i q^{38} -18.7554i q^{40} +27.9590i q^{41} -10.4853 q^{43} +9.51472 q^{44} -39.2132 q^{46} -52.6790i q^{47} +26.8284 q^{50} +30.5826i q^{52} -55.9706 q^{53} +31.5462i q^{55} +4.97056 q^{58} +38.7206i q^{59} -90.5080i q^{61} -69.1080i q^{62} +8.00000 q^{64} -101.397 q^{65} -34.6396 q^{67} -7.52255i q^{68} -36.4264 q^{71} -52.6069i q^{73} +4.15938 q^{74} +8.36308i q^{76} -33.7868 q^{79} +26.5241i q^{80} -39.5400i q^{82} +127.577i q^{83} +24.9411 q^{85} +14.8284 q^{86} -13.4558 q^{88} +50.3314i q^{89} +55.4558 q^{92} +74.4993i q^{94} -27.7279 q^{95} +101.792i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 36 q^{11} + 16 q^{16} + 24 q^{22} + 60 q^{23} - 8 q^{25} - 48 q^{29} + 124 q^{37} - 8 q^{43} + 72 q^{44} - 72 q^{46} + 96 q^{50} - 156 q^{53} - 48 q^{58} + 32 q^{64} - 168 q^{65} + 116 q^{67}+ \cdots - 60 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 + 36 * q^11 + 16 * q^16 + 24 * q^22 + 60 * q^23 - 8 * q^25 - 48 * q^29 + 124 * q^37 - 8 * q^43 + 72 * q^44 - 72 * q^46 + 96 * q^50 - 156 * q^53 - 48 * q^58 + 32 * q^64 - 168 * q^65 + 116 * q^67 + 24 * q^71 + 192 * q^74 - 220 * q^79 - 36 * q^85 + 48 * q^86 + 48 * q^88 + 120 * q^92 - 60 * q^95

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
$5$	6.63103i	1.32621i	0.748528	+	0.663103i	$0.230761\pi$
$5$	6.63103i	1.32621i	−0.748528	+	0.663103i	$0.769239\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.82843	−0.353553
$9$	0	0
$10$	− 9.37769i	− 0.937769i
$11$	4.75736	0.432487	0.216244	−	0.976339i	$-0.430619\pi$
$11$	4.75736	0.432487	0.216244	+	0.976339i	$0.430619\pi$
$12$	0	0
$13$	15.2913i	1.17625i	0.808769	+	0.588126i	$0.200134\pi$
$13$	15.2913i	1.17625i	−0.808769	+	0.588126i	$0.799866\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 3.76127i	− 0.221251i	−0.993862	−	0.110626i	$-0.964715\pi$
$17$	− 3.76127i	− 0.221251i	0.993862	−	0.110626i	$-0.0352855\pi$
$18$	0	0
$19$	4.18154i	0.220081i	0.993927	+	0.110041i	$0.0350981\pi$
$19$	4.18154i	0.220081i	−0.993927	+	0.110041i	$0.964902\pi$
$20$	13.2621i	0.663103i
$21$	0	0
$22$	−6.72792	−0.305815
$23$	27.7279	1.20556	0.602781	−	0.797907i	$-0.294059\pi$
$23$	27.7279	1.20556	0.602781	+	0.797907i	$0.294059\pi$
$24$	0	0
$25$	−18.9706	−0.758823
$26$	− 21.6251i	− 0.831736i
$27$	0	0
$28$	0	0
$29$	−3.51472	−0.121197	−0.0605986	−	0.998162i	$-0.519301\pi$
$29$	−3.51472	−0.121197	−0.0605986	+	0.998162i	$0.519301\pi$
$30$	0	0
$31$	48.8667i	1.57635i	0.615454	+	0.788173i	$0.288973\pi$
$31$	48.8667i	1.57635i	−0.615454	+	0.788173i	$0.711027\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	5.31925i	0.156448i
$35$	0	0
$36$	0	0
$37$	−2.94113	−0.0794899	−0.0397449	−	0.999210i	$-0.512655\pi$
$37$	−2.94113	−0.0794899	−0.0397449	+	0.999210i	$0.512655\pi$
$38$	− 5.91359i	− 0.155621i
$39$	0	0
$40$	− 18.7554i	− 0.468885i
$41$	27.9590i	0.681927i	0.940077	+	0.340963i	$0.110753\pi$
$41$	27.9590i	0.681927i	−0.940077	+	0.340963i	$0.889247\pi$
$42$	0	0
$43$	−10.4853	−0.243844	−0.121922	−	0.992540i	$-0.538906\pi$
$43$	−10.4853	−0.243844	−0.121922	+	0.992540i	$0.538906\pi$
$44$	9.51472	0.216244
$45$	0	0
$46$	−39.2132	−0.852461
$47$	− 52.6790i	− 1.12083i	−0.828212	−	0.560415i	$-0.810642\pi$
$47$	− 52.6790i	− 1.12083i	0.828212	−	0.560415i	$-0.189358\pi$
$48$	0	0
$49$	0	0
$50$	26.8284	0.536569
$51$	0	0
$52$	30.5826i	0.588126i
$53$	−55.9706	−1.05605	−0.528024	−	0.849229i	$-0.677067\pi$
$53$	−55.9706	−1.05605	−0.528024	+	0.849229i	$0.677067\pi$
$54$	0	0
$55$	31.5462i	0.573567i
$56$	0	0
$57$	0	0
$58$	4.97056	0.0856994
$59$	38.7206i	0.656281i	0.944629	+	0.328141i	$0.106422\pi$
$59$	38.7206i	0.656281i	−0.944629	+	0.328141i	$0.893578\pi$
$60$	0	0
$61$	− 90.5080i	− 1.48374i	−0.670545	−	0.741869i	$-0.733940\pi$
$61$	− 90.5080i	− 1.48374i	0.670545	−	0.741869i	$-0.266060\pi$
$62$	− 69.1080i	− 1.11464i
$63$	0	0
$64$	8.00000	0.125000
$65$	−101.397	−1.55995
$66$	0	0
$67$	−34.6396	−0.517009	−0.258505	−	0.966010i	$-0.583230\pi$
$67$	−34.6396	−0.517009	−0.258505	+	0.966010i	$0.583230\pi$
$68$	− 7.52255i	− 0.110626i
$69$	0	0
$70$	0	0
$71$	−36.4264	−0.513048	−0.256524	−	0.966538i	$-0.582577\pi$
$71$	−36.4264	−0.513048	−0.256524	+	0.966538i	$0.582577\pi$
$72$	0	0
$73$	− 52.6069i	− 0.720642i	−0.932828	−	0.360321i	$-0.882667\pi$
$73$	− 52.6069i	− 0.720642i	0.932828	−	0.360321i	$-0.117333\pi$
$74$	4.15938	0.0562078
$75$	0	0
$76$	8.36308i	0.110041i
$77$	0	0
$78$	0	0
$79$	−33.7868	−0.427681	−0.213840	−	0.976869i	$-0.568597\pi$
$79$	−33.7868	−0.427681	−0.213840	+	0.976869i	$0.568597\pi$
$80$	26.5241i	0.331552i
$81$	0	0
$82$	− 39.5400i	− 0.482195i
$83$	127.577i	1.53708i	0.639803	+	0.768539i	$0.279016\pi$
$83$	127.577i	1.53708i	−0.639803	+	0.768539i	$0.720984\pi$
$84$	0	0
$85$	24.9411	0.293425
$86$	14.8284	0.172424
$87$	0	0
$88$	−13.4558	−0.152907
$89$	50.3314i	0.565522i	0.959190	+	0.282761i	$0.0912503\pi$
$89$	50.3314i	0.565522i	−0.959190	+	0.282761i	$0.908750\pi$
$90$	0	0
$91$	0	0
$92$	55.4558	0.602781
$93$	0	0
$94$	74.4993i	0.792546i
$95$	−27.7279	−0.291873
$96$	0	0
$97$	101.792i	1.04940i	0.851287	+	0.524700i	$0.175823\pi$
$97$	101.792i	1.04940i	−0.851287	+	0.524700i	$0.824177\pi$
$98$	0	0
$99$	0	0
$100$	−37.9411	−0.379411
$101$	− 59.6793i	− 0.590884i	−0.955361	−	0.295442i	$-0.904533\pi$
$101$	− 59.6793i	− 0.590884i	0.955361	−	0.295442i	$-0.0954669\pi$
$102$	0	0
$103$	120.178i	1.16678i	0.812193	+	0.583388i	$0.198273\pi$
$103$	120.178i	1.16678i	−0.812193	+	0.583388i	$0.801727\pi$
$104$	− 43.2503i	− 0.415868i
$105$	0	0
$106$	79.1543	0.746739
$107$	113.610	1.06178	0.530889	−	0.847442i	$-0.321858\pi$
$107$	113.610	1.06178	0.530889	+	0.847442i	$0.321858\pi$
$108$	0	0
$109$	−145.309	−1.33311	−0.666553	−	0.745457i	$-0.732231\pi$
$109$	−145.309	−1.33311	−0.666553	+	0.745457i	$0.732231\pi$
$110$	− 44.6131i	− 0.405573i
$111$	0	0
$112$	0	0
$113$	−34.5442	−0.305700	−0.152850	−	0.988249i	$-0.548845\pi$
$113$	−34.5442	−0.305700	−0.152850	+	0.988249i	$0.548845\pi$
$114$	0	0
$115$	183.865i	1.59882i
$116$	−7.02944	−0.0605986
$117$	0	0
$118$	− 54.7592i	− 0.464061i
$119$	0	0
$120$	0	0
$121$	−98.3675	−0.812955
$122$	127.998i	1.04916i
$123$	0	0
$124$	97.7334i	0.788173i
$125$	39.9814i	0.319851i
$126$	0	0
$127$	−247.338	−1.94754	−0.973772	−	0.227526i	$-0.926936\pi$
$127$	−247.338	−1.94754	−0.973772	+	0.227526i	$0.926936\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	143.397	1.10305
$131$	147.645i	1.12706i	0.826096	+	0.563529i	$0.190557\pi$
$131$	147.645i	1.12706i	−0.826096	+	0.563529i	$0.809443\pi$
$132$	0	0
$133$	0	0
$134$	48.9878	0.365581
$135$	0	0
$136$	10.6385i	0.0782242i
$137$	32.5736	0.237763	0.118882	−	0.992908i	$-0.462069\pi$
$137$	32.5736	0.237763	0.118882	+	0.992908i	$0.462069\pi$
$138$	0	0
$139$	68.5857i	0.493422i	0.969089	+	0.246711i	$0.0793499\pi$
$139$	68.5857i	0.493422i	−0.969089	+	0.246711i	$0.920650\pi$
$140$	0	0
$141$	0	0
$142$	51.5147	0.362780
$143$	72.7461i	0.508714i
$144$	0	0
$145$	− 23.3062i	− 0.160732i
$146$	74.3973i	0.509571i
$147$	0	0
$148$	−5.88225	−0.0397449
$149$	−92.3970	−0.620114	−0.310057	−	0.950718i	$-0.600348\pi$
$149$	−92.3970	−0.620114	−0.310057	+	0.950718i	$0.600348\pi$
$150$	0	0
$151$	−91.7868	−0.607860	−0.303930	−	0.952694i	$-0.598299\pi$
$151$	−91.7868	−0.607860	−0.303930	+	0.952694i	$0.598299\pi$
$152$	− 11.8272i	− 0.0778104i
$153$	0	0
$154$	0	0
$155$	−324.037	−2.09056
$156$	0	0
$157$	− 8.45631i	− 0.0538618i	−0.999637	−	0.0269309i	$-0.991427\pi$
$157$	− 8.45631i	− 0.0538618i	0.999637	−	0.0269309i	$-0.00857341\pi$
$158$	47.7817	0.302416
$159$	0	0
$160$	− 37.5108i	− 0.234442i
$161$	0	0
$162$	0	0
$163$	221.978	1.36183	0.680913	−	0.732364i	$-0.261583\pi$
$163$	221.978	1.36183	0.680913	+	0.732364i	$0.261583\pi$
$164$	55.9180i	0.340963i
$165$	0	0
$166$	− 180.422i	− 1.08688i
$167$	− 168.841i	− 1.01102i	−0.862820	−	0.505511i	$-0.831304\pi$
$167$	− 168.841i	− 1.01102i	0.862820	−	0.505511i	$-0.168696\pi$
$168$	0	0
$169$	−64.8234	−0.383570
$170$	−35.2721	−0.207483
$171$	0	0
$172$	−20.9706	−0.121922
$173$	164.341i	0.949947i	0.880000	+	0.474974i	$0.157542\pi$
$173$	164.341i	0.949947i	−0.880000	+	0.474974i	$0.842458\pi$
$174$	0	0
$175$	0	0
$176$	19.0294	0.108122
$177$	0	0
$178$	− 71.1794i	− 0.399884i
$179$	−185.184	−1.03455	−0.517273	−	0.855820i	$-0.673053\pi$
$179$	−185.184	−1.03455	−0.517273	+	0.855820i	$0.673053\pi$
$180$	0	0
$181$	− 155.086i	− 0.856830i	−0.903582	−	0.428415i	$-0.859072\pi$
$181$	− 155.086i	− 0.856830i	0.903582	−	0.428415i	$-0.140928\pi$
$182$	0	0
$183$	0	0
$184$	−78.4264	−0.426230
$185$	− 19.5027i	− 0.105420i
$186$	0	0
$187$	− 17.8937i	− 0.0956884i
$188$	− 105.358i	− 0.560415i
$189$	0	0
$190$	39.2132	0.206385
$191$	−248.095	−1.29893	−0.649465	−	0.760392i	$-0.725007\pi$
$191$	−248.095	−1.29893	−0.649465	+	0.760392i	$0.725007\pi$
$192$	0	0
$193$	154.338	0.799679	0.399840	−	0.916585i	$-0.369066\pi$
$193$	154.338	0.799679	0.399840	+	0.916585i	$0.369066\pi$
$194$	− 143.955i	− 0.742038i
$195$	0	0
$196$	0	0
$197$	181.103	0.919303	0.459651	−	0.888099i	$-0.347974\pi$
$197$	181.103	0.919303	0.459651	+	0.888099i	$0.347974\pi$
$198$	0	0
$199$	− 348.707i	− 1.75229i	−0.482043	−	0.876147i	$-0.660105\pi$
$199$	− 348.707i	− 1.75229i	0.482043	−	0.876147i	$-0.339895\pi$
$200$	53.6569	0.268284
$201$	0	0
$202$	84.3992i	0.417818i
$203$	0	0
$204$	0	0
$205$	−185.397	−0.904375
$206$	− 169.957i	− 0.825035i
$207$	0	0
$208$	61.1651i	0.294063i
$209$	19.8931i	0.0951823i
$210$	0	0
$211$	364.073	1.72547	0.862733	−	0.505660i	$-0.168751\pi$
$211$	364.073	1.72547	0.862733	+	0.505660i	$0.168751\pi$
$212$	−111.941	−0.528024
$213$	0	0
$214$	−160.669	−0.750790
$215$	− 69.5282i	− 0.323387i
$216$	0	0
$217$	0	0
$218$	205.497	0.942649
$219$	0	0
$220$	63.0924i	0.286784i
$221$	57.5147	0.260248
$222$	0	0
$223$	123.231i	0.552603i	0.961071	+	0.276302i	$0.0891089\pi$
$223$	123.231i	0.552603i	−0.961071	+	0.276302i	$0.910891\pi$
$224$	0	0
$225$	0	0
$226$	48.8528	0.216163
$227$	− 76.3756i	− 0.336456i	−0.985748	−	0.168228i	$-0.946195\pi$
$227$	− 76.3756i	− 0.336456i	0.985748	−	0.168228i	$-0.0538045\pi$
$228$	0	0
$229$	357.286i	1.56020i	0.625654	+	0.780101i	$0.284832\pi$
$229$	357.286i	1.56020i	−0.625654	+	0.780101i	$0.715168\pi$
$230$	− 260.024i	− 1.13054i
$231$	0	0
$232$	9.94113	0.0428497
$233$	273.073	1.17199	0.585994	−	0.810315i	$-0.300704\pi$
$233$	273.073	1.17199	0.585994	+	0.810315i	$0.300704\pi$
$234$	0	0
$235$	349.316	1.48645
$236$	77.4412i	0.328141i
$237$	0	0
$238$	0	0
$239$	265.103	1.10922	0.554608	−	0.832112i	$-0.312868\pi$
$239$	265.103	1.10922	0.554608	+	0.832112i	$0.312868\pi$
$240$	0	0
$241$	87.6383i	0.363644i	0.983331	+	0.181822i	$0.0581995\pi$
$241$	87.6383i	0.363644i	−0.983331	+	0.181822i	$0.941800\pi$
$242$	139.113	0.574846
$243$	0	0
$244$	− 181.016i	− 0.741869i
$245$	0	0
$246$	0	0
$247$	−63.9411	−0.258871
$248$	− 138.216i	− 0.557322i
$249$	0	0
$250$	− 56.5422i	− 0.226169i
$251$	495.655i	1.97472i	0.158491	+	0.987360i	$0.449337\pi$
$251$	495.655i	1.97472i	−0.158491	+	0.987360i	$0.550663\pi$
$252$	0	0
$253$	131.912	0.521390
$254$	349.789	1.37712
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 400.536i	− 1.55851i	−0.626709	−	0.779254i	$-0.715598\pi$
$257$	− 400.536i	− 1.55851i	0.626709	−	0.779254i	$-0.284402\pi$
$258$	0	0
$259$	0	0
$260$	−202.794	−0.779977
$261$	0	0
$262$	− 208.801i	− 0.796950i
$263$	32.3452	0.122986	0.0614928	−	0.998108i	$-0.480414\pi$
$263$	32.3452	0.122986	0.0614928	+	0.998108i	$0.480414\pi$
$264$	0	0
$265$	− 371.142i	− 1.40054i
$266$	0	0
$267$	0	0
$268$	−69.2792	−0.258505
$269$	− 306.963i	− 1.14113i	−0.821253	−	0.570564i	$-0.806725\pi$
$269$	− 306.963i	− 1.14113i	0.821253	−	0.570564i	$-0.193275\pi$
$270$	0	0
$271$	− 75.9852i	− 0.280388i	−0.990124	−	0.140194i	$-0.955227\pi$
$271$	− 75.9852i	− 0.280388i	0.990124	−	0.140194i	$-0.0447726\pi$
$272$	− 15.0451i	− 0.0553129i
$273$	0	0
$274$	−46.0660	−0.168124
$275$	−90.2498	−0.328181
$276$	0	0
$277$	278.411	1.00509	0.502547	−	0.864550i	$-0.332396\pi$
$277$	278.411	1.00509	0.502547	+	0.864550i	$0.332396\pi$
$278$	− 96.9948i	− 0.348902i
$279$	0	0
$280$	0	0
$281$	−394.690	−1.40459	−0.702296	−	0.711885i	$-0.747842\pi$
$281$	−394.690	−1.40459	−0.702296	+	0.711885i	$0.747842\pi$
$282$	0	0
$283$	− 146.396i	− 0.517301i	−0.965971	−	0.258650i	$-0.916722\pi$
$283$	− 146.396i	− 0.517301i	0.965971	−	0.258650i	$-0.0832778\pi$
$284$	−72.8528	−0.256524
$285$	0	0
$286$	− 102.879i	− 0.359715i
$287$	0	0
$288$	0	0
$289$	274.853	0.951048
$290$	32.9600i	0.113655i
$291$	0	0
$292$	− 105.214i	− 0.360321i
$293$	− 299.678i	− 1.02279i	−0.859345	−	0.511396i	$-0.829128\pi$
$293$	− 299.678i	− 1.02279i	0.859345	−	0.511396i	$-0.170872\pi$
$294$	0	0
$295$	−256.757	−0.870364
$296$	8.31876	0.0281039
$297$	0	0
$298$	130.669	0.438487
$299$	423.996i	1.41805i
$300$	0	0
$301$	0	0
$302$	129.806	0.429822
$303$	0	0
$304$	16.7262i	0.0550203i
$305$	600.161	1.96774
$306$	0	0
$307$	− 20.9886i	− 0.0683666i	−0.999416	−	0.0341833i	$-0.989117\pi$
$307$	− 20.9886i	− 0.0683666i	0.999416	−	0.0341833i	$-0.0108830\pi$
$308$	0	0
$309$	0	0
$310$	458.257	1.47825
$311$	182.039i	0.585336i	0.956214	+	0.292668i	$0.0945430\pi$
$311$	182.039i	0.585336i	−0.956214	+	0.292668i	$0.905457\pi$
$312$	0	0
$313$	− 97.9286i	− 0.312871i	−0.987688	−	0.156435i	$-0.950000\pi$
$313$	− 97.9286i	− 0.312871i	0.987688	−	0.156435i	$-0.0500003\pi$
$314$	11.9590i	0.0380861i
$315$	0	0
$316$	−67.5736	−0.213840
$317$	481.971	1.52041	0.760206	−	0.649682i	$-0.225098\pi$
$317$	481.971	1.52041	0.760206	+	0.649682i	$0.225098\pi$
$318$	0	0
$319$	−16.7208	−0.0524162
$320$	53.0482i	0.165776i
$321$	0	0
$322$	0	0
$323$	15.7279	0.0486933
$324$	0	0
$325$	− 290.084i	− 0.892567i
$326$	−313.924	−0.962957
$327$	0	0
$328$	− 79.0800i	− 0.241098i
$329$	0	0
$330$	0	0
$331$	225.007	0.679780	0.339890	−	0.940465i	$-0.389610\pi$
$331$	225.007	0.679780	0.339890	+	0.940465i	$0.389610\pi$
$332$	255.155i	0.768539i
$333$	0	0
$334$	238.777i	0.714901i
$335$	− 229.696i	− 0.685661i
$336$	0	0
$337$	−264.368	−0.784473	−0.392237	−	0.919864i	$-0.628299\pi$
$337$	−264.368	−0.784473	−0.392237	+	0.919864i	$0.628299\pi$
$338$	91.6741	0.271225
$339$	0	0
$340$	49.8823	0.146713
$341$	232.476i	0.681749i
$342$	0	0
$343$	0	0
$344$	29.6569	0.0862118
$345$	0	0
$346$	− 232.413i	− 0.671714i
$347$	191.257	0.551173	0.275586	−	0.961276i	$-0.411128\pi$
$347$	191.257	0.551173	0.275586	+	0.961276i	$0.411128\pi$
$348$	0	0
$349$	135.448i	0.388104i	0.980991	+	0.194052i	$0.0621630\pi$
$349$	135.448i	0.388104i	−0.980991	+	0.194052i	$0.937837\pi$
$350$	0	0
$351$	0	0
$352$	−26.9117	−0.0764537
$353$	− 348.490i	− 0.987225i	−0.869682	−	0.493612i	$-0.835676\pi$
$353$	− 348.490i	− 0.987225i	0.869682	−	0.493612i	$-0.164324\pi$
$354$	0	0
$355$	− 241.545i	− 0.680407i
$356$	100.663i	0.282761i
$357$	0	0
$358$	261.889	0.731535
$359$	−304.831	−0.849110	−0.424555	−	0.905402i	$-0.639569\pi$
$359$	−304.831	−0.849110	−0.424555	+	0.905402i	$0.639569\pi$
$360$	0	0
$361$	343.515	0.951564
$362$	219.325i	0.605870i
$363$	0	0
$364$	0	0
$365$	348.838	0.955720
$366$	0	0
$367$	95.0042i	0.258867i	0.991588	+	0.129434i	$0.0413159\pi$
$367$	95.0042i	0.258867i	−0.991588	+	0.129434i	$0.958684\pi$
$368$	110.912	0.301390
$369$	0	0
$370$	27.5810i	0.0745432i
$371$	0	0
$372$	0	0
$373$	253.558	0.679781	0.339891	−	0.940465i	$-0.389610\pi$
$373$	253.558	0.679781	0.339891	+	0.940465i	$0.389610\pi$
$374$	25.3056i	0.0676619i
$375$	0	0
$376$	148.999i	0.396273i
$377$	− 53.7446i	− 0.142559i
$378$	0	0
$379$	508.250	1.34103	0.670514	−	0.741897i	$-0.266074\pi$
$379$	508.250	1.34103	0.670514	+	0.741897i	$0.266074\pi$
$380$	−55.4558	−0.145936
$381$	0	0
$382$	350.860	0.918482
$383$	− 477.761i	− 1.24742i	−0.781656	−	0.623709i	$-0.785625\pi$
$383$	− 477.761i	− 1.24742i	0.781656	−	0.623709i	$-0.214375\pi$
$384$	0	0
$385$	0	0
$386$	−218.267	−0.565459
$387$	0	0
$388$	203.584i	0.524700i
$389$	−170.220	−0.437584	−0.218792	−	0.975771i	$-0.570212\pi$
$389$	−170.220	−0.437584	−0.218792	+	0.975771i	$0.570212\pi$
$390$	0	0
$391$	− 104.292i	− 0.266732i
$392$	0	0
$393$	0	0
$394$	−256.118	−0.650045
$395$	− 224.041i	− 0.567193i
$396$	0	0
$397$	244.550i	0.615995i	0.951387	+	0.307997i	$0.0996588\pi$
$397$	244.550i	0.615995i	−0.951387	+	0.307997i	$0.900341\pi$
$398$	493.146i	1.23906i
$399$	0	0
$400$	−75.8823	−0.189706
$401$	417.573	1.04133	0.520664	−	0.853762i	$-0.325684\pi$
$401$	417.573	1.04133	0.520664	+	0.853762i	$0.325684\pi$
$402$	0	0
$403$	−747.235	−1.85418
$404$	− 119.359i	− 0.295442i
$405$	0	0
$406$	0	0
$407$	−13.9920	−0.0343784
$408$	0	0
$409$	− 308.212i	− 0.753574i	−0.926300	−	0.376787i	$-0.877029\pi$
$409$	− 308.212i	− 0.753574i	0.926300	−	0.376787i	$-0.122971\pi$
$410$	262.191	0.639490
$411$	0	0
$412$	240.356i	0.583388i
$413$	0	0
$414$	0	0
$415$	−845.970	−2.03848
$416$	− 86.5006i	− 0.207934i
$417$	0	0
$418$	− 28.1331i	− 0.0673040i
$419$	103.142i	0.246163i	0.992397	+	0.123081i	$0.0392776\pi$
$419$	103.142i	0.246163i	−0.992397	+	0.123081i	$0.960722\pi$
$420$	0	0
$421$	−165.220	−0.392447	−0.196224	−	0.980559i	$-0.562868\pi$
$421$	−165.220	−0.392447	−0.196224	+	0.980559i	$0.562868\pi$
$422$	−514.877	−1.22009
$423$	0	0
$424$	158.309	0.373369
$425$	71.3535i	0.167891i
$426$	0	0
$427$	0	0
$428$	227.220	0.530889
$429$	0	0
$430$	98.3277i	0.228669i
$431$	−594.536	−1.37943	−0.689717	−	0.724079i	$-0.742265\pi$
$431$	−594.536	−1.37943	−0.689717	+	0.724079i	$0.742265\pi$
$432$	0	0
$433$	− 40.6267i	− 0.0938261i	−0.998899	−	0.0469131i	$-0.985062\pi$
$433$	− 40.6267i	− 0.0938261i	0.998899	−	0.0469131i	$-0.0149384\pi$
$434$	0	0
$435$	0	0
$436$	−290.617	−0.666553
$437$	115.945i	0.265321i
$438$	0	0
$439$	146.600i	0.333941i	0.985962	+	0.166971i	$0.0533985\pi$
$439$	146.600i	0.333941i	−0.985962	+	0.166971i	$0.946602\pi$
$440$	− 89.2261i	− 0.202787i
$441$	0	0
$442$	−81.3381	−0.184023
$443$	−107.360	−0.242349	−0.121174	−	0.992631i	$-0.538666\pi$
$443$	−107.360	−0.242349	−0.121174	+	0.992631i	$0.538666\pi$
$444$	0	0
$445$	−333.749	−0.749999
$446$	− 174.274i	− 0.390750i
$447$	0	0
$448$	0	0
$449$	−135.161	−0.301028	−0.150514	−	0.988608i	$-0.548093\pi$
$449$	−135.161	−0.301028	−0.150514	+	0.988608i	$0.548093\pi$
$450$	0	0
$451$	133.011i	0.294925i
$452$	−69.0883	−0.152850
$453$	0	0
$454$	108.011i	0.237910i
$455$	0	0
$456$	0	0
$457$	159.735	0.349530	0.174765	−	0.984610i	$-0.444083\pi$
$457$	159.735	0.349530	0.174765	+	0.984610i	$0.444083\pi$
$458$	− 505.279i	− 1.10323i
$459$	0	0
$460$	367.729i	0.799412i
$461$	− 310.250i	− 0.672993i	−0.941685	−	0.336497i	$-0.890758\pi$
$461$	− 310.250i	− 0.672993i	0.941685	−	0.336497i	$-0.109242\pi$
$462$	0	0
$463$	−326.014	−0.704135	−0.352067	−	0.935975i	$-0.614521\pi$
$463$	−326.014	−0.704135	−0.352067	+	0.935975i	$0.614521\pi$
$464$	−14.0589	−0.0302993
$465$	0	0
$466$	−386.184	−0.828721
$467$	− 595.558i	− 1.27529i	−0.770332	−	0.637643i	$-0.779910\pi$
$467$	− 595.558i	− 1.27529i	0.770332	−	0.637643i	$-0.220090\pi$
$468$	0	0
$469$	0	0
$470$	−494.007	−1.05108
$471$	0	0
$472$	− 109.518i	− 0.232030i
$473$	−49.8823	−0.105459
$474$	0	0
$475$	− 79.3262i	− 0.167002i
$476$	0	0
$477$	0	0
$478$	−374.912	−0.784334
$479$	506.680i	1.05779i	0.848688	+	0.528894i	$0.177393\pi$
$479$	506.680i	1.05779i	−0.848688	+	0.528894i	$0.822607\pi$
$480$	0	0
$481$	− 44.9736i	− 0.0935002i
$482$	− 123.939i	− 0.257135i
$483$	0	0
$484$	−196.735	−0.406477
$485$	−674.985	−1.39172
$486$	0	0
$487$	211.302	0.433884	0.216942	−	0.976184i	$-0.430392\pi$
$487$	211.302	0.433884	0.216942	+	0.976184i	$0.430392\pi$
$488$	255.995i	0.524581i
$489$	0	0
$490$	0	0
$491$	784.161	1.59707	0.798534	−	0.601949i	$-0.205609\pi$
$491$	784.161	1.59707	0.798534	+	0.601949i	$0.205609\pi$
$492$	0	0
$493$	13.2198i	0.0268151i
$494$	90.4264	0.183049
$495$	0	0
$496$	195.467i	0.394086i
$497$	0	0
$498$	0	0
$499$	−171.492	−0.343672	−0.171836	−	0.985126i	$-0.554970\pi$
$499$	−171.492	−0.343672	−0.171836	+	0.985126i	$0.554970\pi$
$500$	79.9628i	0.159926i
$501$	0	0
$502$	− 700.962i	− 1.39634i
$503$	20.0883i	0.0399370i	0.999801	+	0.0199685i	$0.00635659\pi$
$503$	20.0883i	0.0399370i	−0.999801	+	0.0199685i	$0.993643\pi$
$504$	0	0
$505$	395.735	0.783634
$506$	−186.551	−0.368678
$507$	0	0
$508$	−494.676	−0.973772
$509$	476.764i	0.936668i	0.883551	+	0.468334i	$0.155146\pi$
$509$	476.764i	0.936668i	−0.883551	+	0.468334i	$0.844854\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	566.444i	1.10203i
$515$	−796.904	−1.54739
$516$	0	0
$517$	− 250.613i	− 0.484744i
$518$	0	0
$519$	0	0
$520$	286.794	0.551527
$521$	854.274i	1.63968i	0.572592	+	0.819841i	$0.305938\pi$
$521$	854.274i	1.63968i	−0.572592	+	0.819841i	$0.694062\pi$
$522$	0	0
$523$	− 593.002i	− 1.13385i	−0.823771	−	0.566923i	$-0.808134\pi$
$523$	− 593.002i	− 1.13385i	0.823771	−	0.566923i	$-0.191866\pi$
$524$	295.289i	0.563529i
$525$	0	0
$526$	−45.7431	−0.0869640
$527$	183.801	0.348769
$528$	0	0
$529$	239.838	0.453379
$530$	524.875i	0.990330i
$531$	0	0
$532$	0	0
$533$	−427.529	−0.802118
$534$	0	0
$535$	753.352i	1.40814i
$536$	97.9756	0.182790
$537$	0	0
$538$	434.112i	0.806899i
$539$	0	0
$540$	0	0
$541$	855.191	1.58076	0.790380	−	0.612617i	$-0.209883\pi$
$541$	855.191	1.58076	0.790380	+	0.612617i	$0.209883\pi$
$542$	107.459i	0.198264i
$543$	0	0
$544$	21.2770i	0.0391121i
$545$	− 963.546i	− 1.76797i
$546$	0	0
$547$	415.897	0.760323	0.380161	−	0.924920i	$-0.375868\pi$
$547$	415.897	0.760323	0.380161	+	0.924920i	$0.375868\pi$
$548$	65.1472	0.118882
$549$	0	0
$550$	127.632	0.232059
$551$	− 14.6969i	− 0.0266732i
$552$	0	0
$553$	0	0
$554$	−393.733	−0.710709
$555$	0	0
$556$	137.171i	0.246711i
$557$	584.220	1.04887	0.524435	−	0.851451i	$-0.324277\pi$
$557$	584.220	1.04887	0.524435	+	0.851451i	$0.324277\pi$
$558$	0	0
$559$	− 160.333i	− 0.286822i
$560$	0	0
$561$	0	0
$562$	558.177	0.993197
$563$	911.147i	1.61838i	0.587548	+	0.809189i	$0.300093\pi$
$563$	911.147i	1.61838i	−0.587548	+	0.809189i	$0.699907\pi$
$564$	0	0
$565$	− 229.063i	− 0.405422i
$566$	207.035i	0.365787i
$567$	0	0
$568$	103.029	0.181390
$569$	−699.999	−1.23023	−0.615113	−	0.788439i	$-0.710890\pi$
$569$	−699.999	−1.23023	−0.615113	+	0.788439i	$0.710890\pi$
$570$	0	0
$571$	−562.463	−0.985049	−0.492525	−	0.870299i	$-0.663926\pi$
$571$	−562.463	−0.985049	−0.492525	+	0.870299i	$0.663926\pi$
$572$	145.492i	0.254357i
$573$	0	0
$574$	0	0
$575$	−526.014	−0.914807
$576$	0	0
$577$	661.659i	1.14672i	0.819302	+	0.573362i	$0.194361\pi$
$577$	661.659i	1.14672i	−0.819302	+	0.573362i	$0.805639\pi$
$578$	−388.701	−0.672492
$579$	0	0
$580$	− 46.6124i	− 0.0803662i
$581$	0	0
$582$	0	0
$583$	−266.272	−0.456727
$584$	148.795i	0.254785i
$585$	0	0
$586$	423.809i	0.723224i
$587$	823.029i	1.40209i	0.713116	+	0.701046i	$0.247283\pi$
$587$	823.029i	1.40209i	−0.713116	+	0.701046i	$0.752717\pi$
$588$	0	0
$589$	−204.338	−0.346924
$590$	363.110	0.615440
$591$	0	0
$592$	−11.7645	−0.0198725
$593$	622.256i	1.04934i	0.851307	+	0.524668i	$0.175811\pi$
$593$	622.256i	1.04934i	−0.851307	+	0.524668i	$0.824189\pi$
$594$	0	0
$595$	0	0
$596$	−184.794	−0.310057
$597$	0	0
$598$	− 599.620i	− 1.00271i
$599$	512.845	0.856168	0.428084	−	0.903739i	$-0.359189\pi$
$599$	512.845	0.856168	0.428084	+	0.903739i	$0.359189\pi$
$600$	0	0
$601$	680.160i	1.13171i	0.824504	+	0.565857i	$0.191454\pi$
$601$	680.160i	1.13171i	−0.824504	+	0.565857i	$0.808546\pi$
$602$	0	0
$603$	0	0
$604$	−183.574	−0.303930
$605$	− 652.278i	− 1.07815i
$606$	0	0
$607$	38.7381i	0.0638189i	0.999491	+	0.0319095i	$0.0101588\pi$
$607$	38.7381i	0.0638189i	−0.999491	+	0.0319095i	$0.989841\pi$
$608$	− 23.6544i	− 0.0389052i
$609$	0	0
$610$	−848.756	−1.39140
$611$	805.529	1.31838
$612$	0	0
$613$	401.103	0.654329	0.327164	−	0.944967i	$-0.393907\pi$
$613$	401.103	0.654329	0.327164	+	0.944967i	$0.393907\pi$
$614$	29.6823i	0.0483425i
$615$	0	0
$616$	0	0
$617$	959.044	1.55437	0.777183	−	0.629275i	$-0.216648\pi$
$617$	959.044	1.55437	0.777183	+	0.629275i	$0.216648\pi$
$618$	0	0
$619$	1004.53i	1.62283i	0.584469	+	0.811416i	$0.301303\pi$
$619$	1004.53i	1.62283i	−0.584469	+	0.811416i	$0.698697\pi$
$620$	−648.073	−1.04528
$621$	0	0
$622$	− 257.443i	− 0.413895i
$623$	0	0
$624$	0	0
$625$	−739.382	−1.18301
$626$	138.492i	0.221233i
$627$	0	0
$628$	− 16.9126i	− 0.0269309i
$629$	11.0624i	0.0175873i
$630$	0	0
$631$	−386.514	−0.612542	−0.306271	−	0.951944i	$-0.599081\pi$
$631$	−386.514	−0.612542	−0.306271	+	0.951944i	$0.599081\pi$
$632$	95.5635	0.151208
$633$	0	0
$634$	−681.609	−1.07509
$635$	− 1640.11i	− 2.58284i
$636$	0	0
$637$	0	0
$638$	23.6468	0.0370639
$639$	0	0
$640$	− 75.0215i	− 0.117221i
$641$	992.147	1.54781	0.773906	−	0.633301i	$-0.218300\pi$
$641$	992.147	1.54781	0.773906	+	0.633301i	$0.218300\pi$
$642$	0	0
$643$	944.986i	1.46965i	0.678256	+	0.734826i	$0.262736\pi$
$643$	944.986i	1.46965i	−0.678256	+	0.734826i	$0.737264\pi$
$644$	0	0
$645$	0	0
$646$	−22.2426	−0.0344313
$647$	2.89088i	0.00446812i	0.999998	+	0.00223406i	$0.000711124\pi$
$647$	2.89088i	0.00446812i	−0.999998	+	0.00223406i	$0.999289\pi$
$648$	0	0
$649$	184.208i	0.283833i
$650$	410.241i	0.631140i
$651$	0	0
$652$	443.955	0.680913
$653$	323.059	0.494730	0.247365	−	0.968922i	$-0.420435\pi$
$653$	323.059	0.494730	0.247365	+	0.968922i	$0.420435\pi$
$654$	0	0
$655$	−979.036	−1.49471
$656$	111.836i	0.170482i
$657$	0	0
$658$	0	0
$659$	−295.955	−0.449098	−0.224549	−	0.974463i	$-0.572091\pi$
$659$	−295.955	−0.449098	−0.224549	+	0.974463i	$0.572091\pi$
$660$	0	0
$661$	− 20.7511i	− 0.0313935i	−0.999877	−	0.0156968i	$-0.995003\pi$
$661$	− 20.7511i	− 0.0313935i	0.999877	−	0.0156968i	$-0.00499664\pi$
$662$	−318.208	−0.480677
$663$	0	0
$664$	− 360.843i	− 0.543439i
$665$	0	0
$666$	0	0
$667$	−97.4558	−0.146111
$668$	− 337.681i	− 0.505511i
$669$	0	0
$670$	324.840i	0.484835i
$671$	− 430.579i	− 0.641698i
$672$	0	0
$673$	627.044	0.931714	0.465857	−	0.884860i	$-0.345746\pi$
$673$	627.044	0.931714	0.465857	+	0.884860i	$0.345746\pi$
$674$	373.872	0.554706
$675$	0	0
$676$	−129.647	−0.191785
$677$	109.246i	0.161368i	0.996740	+	0.0806838i	$0.0257104\pi$
$677$	109.246i	0.161368i	−0.996740	+	0.0806838i	$0.974290\pi$
$678$	0	0
$679$	0	0
$680$	−70.5442	−0.103741
$681$	0	0
$682$	− 328.771i	− 0.482069i
$683$	−793.566	−1.16188	−0.580941	−	0.813946i	$-0.697315\pi$
$683$	−793.566	−1.16188	−0.580941	+	0.813946i	$0.697315\pi$
$684$	0	0
$685$	215.996i	0.315323i
$686$	0	0
$687$	0	0
$688$	−41.9411	−0.0609609
$689$	− 855.862i	− 1.24218i
$690$	0	0
$691$	183.889i	0.266121i	0.991108	+	0.133060i	$0.0424804\pi$
$691$	183.889i	0.266121i	−0.991108	+	0.133060i	$0.957520\pi$
$692$	328.682i	0.474974i
$693$	0	0
$694$	−270.478	−0.389738
$695$	−454.794	−0.654380
$696$	0	0
$697$	105.161	0.150877
$698$	− 191.553i	− 0.274431i
$699$	0	0
$700$	0	0
$701$	1043.82	1.48905	0.744525	−	0.667595i	$-0.232676\pi$
$701$	1043.82	1.48905	0.744525	+	0.667595i	$0.232676\pi$
$702$	0	0
$703$	− 12.2984i	− 0.0174942i
$704$	38.0589	0.0540609
$705$	0	0
$706$	492.840i	0.698073i
$707$	0	0
$708$	0	0
$709$	−980.558	−1.38301	−0.691507	−	0.722369i	$-0.743053\pi$
$709$	−980.558	−1.38301	−0.691507	+	0.722369i	$0.743053\pi$
$710$	341.596i	0.481121i
$711$	0	0
$712$	− 142.359i	− 0.199942i
$713$	1354.97i	1.90038i
$714$	0	0
$715$	−482.382	−0.674660
$716$	−370.368	−0.517273
$717$	0	0
$718$	431.095	0.600411
$719$	778.484i	1.08273i	0.840787	+	0.541366i	$0.182093\pi$
$719$	778.484i	1.08273i	−0.840787	+	0.541366i	$0.817907\pi$
$720$	0	0
$721$	0	0
$722$	−485.803	−0.672858
$723$	0	0
$724$	− 310.173i	− 0.428415i
$725$	66.6762	0.0919672
$726$	0	0
$727$	− 735.255i	− 1.01135i	−0.862723	−	0.505677i	$-0.831243\pi$
$727$	− 735.255i	− 1.01135i	0.862723	−	0.505677i	$-0.168757\pi$
$728$	0	0
$729$	0	0
$730$	−493.331	−0.675796
$731$	39.4380i	0.0539508i
$732$	0	0
$733$	− 478.860i	− 0.653288i	−0.945147	−	0.326644i	$-0.894082\pi$
$733$	− 478.860i	− 0.653288i	0.945147	−	0.326644i	$-0.105918\pi$
$734$	− 134.356i	− 0.183047i
$735$	0	0
$736$	−156.853	−0.213115
$737$	−164.793	−0.223600
$738$	0	0
$739$	−19.9045	−0.0269344	−0.0134672	−	0.999909i	$-0.504287\pi$
$739$	−19.9045	−0.0269344	−0.0134672	+	0.999909i	$0.504287\pi$
$740$	− 39.0054i	− 0.0527100i
$741$	0	0
$742$	0	0
$743$	−43.3095	−0.0582901	−0.0291450	−	0.999575i	$-0.509278\pi$
$743$	−43.3095	−0.0582901	−0.0291450	+	0.999575i	$0.509278\pi$
$744$	0	0
$745$	− 612.687i	− 0.822399i
$746$	−358.586	−0.480678
$747$	0	0
$748$	− 35.7875i	− 0.0478442i
$749$	0	0
$750$	0	0
$751$	225.330	0.300040	0.150020	−	0.988683i	$-0.452066\pi$
$751$	225.330	0.300040	0.150020	+	0.988683i	$0.452066\pi$
$752$	− 210.716i	− 0.280207i
$753$	0	0
$754$	76.0063i	0.100804i
$755$	− 608.641i	− 0.806147i
$756$	0	0
$757$	935.779	1.23617	0.618084	−	0.786112i	$-0.287909\pi$
$757$	935.779	1.23617	0.618084	+	0.786112i	$0.287909\pi$
$758$	−718.774	−0.948250
$759$	0	0
$760$	78.4264	0.103193
$761$	− 1402.71i	− 1.84325i	−0.388081	−	0.921625i	$-0.626862\pi$
$761$	− 1402.71i	− 1.84325i	0.388081	−	0.921625i	$-0.373138\pi$
$762$	0	0
$763$	0	0
$764$	−496.191	−0.649465
$765$	0	0
$766$	675.656i	0.882058i
$767$	−592.087	−0.771952
$768$	0	0
$769$	1.72330i	0.00224097i	0.999999	+	0.00112048i	$0.000356661\pi$
$769$	1.72330i	0.00224097i	−0.999999	+	0.00112048i	$0.999643\pi$
$770$	0	0
$771$	0	0
$772$	308.676	0.399840
$773$	224.258i	0.290113i	0.989423	+	0.145057i	$0.0463364\pi$
$773$	224.258i	0.290113i	−0.989423	+	0.145057i	$0.953664\pi$
$774$	0	0
$775$	− 927.029i	− 1.19617i
$776$	− 287.911i	− 0.371019i
$777$	0	0
$778$	240.728	0.309419
$779$	−116.912	−0.150079
$780$	0	0
$781$	−173.294	−0.221887
$782$	147.492i	0.188608i
$783$	0	0
$784$	0	0
$785$	56.0740	0.0714319
$786$	0	0
$787$	− 70.2034i	− 0.0892038i	−0.999005	−	0.0446019i	$-0.985798\pi$
$787$	− 70.2034i	− 0.0892038i	0.999005	−	0.0446019i	$-0.0142019\pi$
$788$	362.205	0.459651
$789$	0	0
$790$	316.842i	0.401066i
$791$	0	0
$792$	0	0
$793$	1383.98	1.74525
$794$	− 345.846i	− 0.435574i
$795$	0	0
$796$	− 697.413i	− 0.876147i
$797$	− 1305.38i	− 1.63787i	−0.573889	−	0.818933i	$-0.694566\pi$
$797$	− 1305.38i	− 1.63787i	0.573889	−	0.818933i	$-0.305434\pi$
$798$	0	0
$799$	−198.140	−0.247985
$800$	107.314	0.134142
$801$	0	0
$802$	−590.537	−0.736330
$803$	− 250.270i	− 0.311668i
$804$	0	0
$805$	0	0
$806$	1056.75	1.31110
$807$	0	0
$808$	168.798i	0.208909i
$809$	−762.765	−0.942849	−0.471424	−	0.881907i	$-0.656260\pi$
$809$	−762.765	−0.942849	−0.471424	+	0.881907i	$0.656260\pi$
$810$	0	0
$811$	1214.98i	1.49813i	0.662498	+	0.749064i	$0.269496\pi$
$811$	1214.98i	1.49813i	−0.662498	+	0.749064i	$0.730504\pi$
$812$	0	0
$813$	0	0
$814$	19.7877	0.0243092
$815$	1471.94i	1.80606i
$816$	0	0
$817$	− 43.8446i	− 0.0536654i
$818$	435.877i	0.532857i
$819$	0	0
$820$	−370.794	−0.452188
$821$	583.368	0.710557	0.355279	−	0.934760i	$-0.384386\pi$
$821$	583.368	0.710557	0.355279	+	0.934760i	$0.384386\pi$
$822$	0	0
$823$	1030.74	1.25242	0.626210	−	0.779655i	$-0.284605\pi$
$823$	1030.74	1.25242	0.626210	+	0.779655i	$0.284605\pi$
$824$	− 339.915i	− 0.412518i
$825$	0	0
$826$	0	0
$827$	−152.102	−0.183920	−0.0919599	−	0.995763i	$-0.529313\pi$
$827$	−152.102	−0.183920	−0.0919599	+	0.995763i	$0.529313\pi$
$828$	0	0
$829$	− 614.410i	− 0.741146i	−0.928803	−	0.370573i	$-0.879161\pi$
$829$	− 614.410i	− 0.741146i	0.928803	−	0.370573i	$-0.120839\pi$
$830$	1196.38	1.44142
$831$	0	0
$832$	122.330i	0.147032i
$833$	0	0
$834$	0	0
$835$	1119.59	1.34082
$836$	39.7862i	0.0475911i
$837$	0	0
$838$	− 145.865i	− 0.174063i
$839$	1546.14i	1.84284i	0.388568	+	0.921420i	$0.372970\pi$
$839$	1546.14i	1.84284i	−0.388568	+	0.921420i	$0.627030\pi$
$840$	0	0
$841$	−828.647	−0.985311
$842$	233.657	0.277502
$843$	0	0
$844$	728.146	0.862733
$845$	− 429.846i	− 0.508693i
$846$	0	0
$847$	0	0
$848$	−223.882	−0.264012
$849$	0	0
$850$	− 100.909i	− 0.118717i
$851$	−81.5513	−0.0958300
$852$	0	0
$853$	− 1235.15i	− 1.44800i	−0.689798	−	0.724002i	$-0.742301\pi$
$853$	− 1235.15i	− 1.44800i	0.689798	−	0.724002i	$-0.257699\pi$
$854$	0	0
$855$	0	0
$856$	−321.338	−0.375395
$857$	− 1100.68i	− 1.28434i	−0.766561	−	0.642172i	$-0.778034\pi$
$857$	− 1100.68i	− 1.28434i	0.766561	−	0.642172i	$-0.221966\pi$
$858$	0	0
$859$	591.771i	0.688906i	0.938803	+	0.344453i	$0.111936\pi$
$859$	591.771i	0.688906i	−0.938803	+	0.344453i	$0.888064\pi$
$860$	− 139.056i	− 0.161694i
$861$	0	0
$862$	840.801	0.975407
$863$	64.7271	0.0750024	0.0375012	−	0.999297i	$-0.488060\pi$
$863$	64.7271	0.0750024	0.0375012	+	0.999297i	$0.488060\pi$
$864$	0	0
$865$	−1089.75	−1.25983
$866$	57.4548i	0.0663451i
$867$	0	0
$868$	0	0
$869$	−160.736	−0.184967
$870$	0	0
$871$	− 529.684i	− 0.608133i
$872$	410.995	0.471324
$873$	0	0
$874$	− 163.972i	− 0.187611i
$875$	0	0
$876$	0	0
$877$	−304.192	−0.346855	−0.173427	−	0.984847i	$-0.555484\pi$
$877$	−304.192	−0.346855	−0.173427	+	0.984847i	$0.555484\pi$
$878$	− 207.324i	− 0.236132i
$879$	0	0
$880$	126.185i	0.143392i
$881$	− 863.732i	− 0.980400i	−0.871610	−	0.490200i	$-0.836924\pi$
$881$	− 863.732i	− 0.980400i	0.871610	−	0.490200i	$-0.163076\pi$
$882$	0	0
$883$	−567.456	−0.642645	−0.321323	−	0.946970i	$-0.604127\pi$
$883$	−567.456	−0.642645	−0.321323	+	0.946970i	$0.604127\pi$
$884$	115.029	0.130124
$885$	0	0
$886$	151.831	0.171366
$887$	889.761i	1.00311i	0.865125	+	0.501556i	$0.167239\pi$
$887$	889.761i	1.00311i	−0.865125	+	0.501556i	$0.832761\pi$
$888$	0	0
$889$	0	0
$890$	471.993	0.530329
$891$	0	0
$892$	246.461i	0.276302i
$893$	220.279	0.246673
$894$	0	0
$895$	− 1227.96i	− 1.37202i
$896$	0	0
$897$	0	0
$898$	191.147	0.212859
$899$	− 171.753i	− 0.191049i
$900$	0	0
$901$	210.521i	0.233652i
$902$	− 188.106i	− 0.208543i
$903$	0	0
$904$	97.7056	0.108081
$905$	1028.38	1.13633
$906$	0	0
$907$	373.978	0.412324	0.206162	−	0.978518i	$-0.433903\pi$
$907$	373.978	0.412324	0.206162	+	0.978518i	$0.433903\pi$
$908$	− 152.751i	− 0.168228i
$909$	0	0
$910$	0	0
$911$	−1133.75	−1.24451	−0.622256	−	0.782814i	$-0.713784\pi$
$911$	−1133.75	−1.24451	−0.622256	+	0.782814i	$0.713784\pi$
$912$	0	0
$913$	606.932i	0.664766i
$914$	−225.899	−0.247155
$915$	0	0
$916$	714.572i	0.780101i
$917$	0	0
$918$	0	0
$919$	456.302	0.496521	0.248260	−	0.968693i	$-0.420141\pi$
$919$	456.302	0.496521	0.248260	+	0.968693i	$0.420141\pi$
$920$	− 520.048i	− 0.565269i
$921$	0	0
$922$	438.759i	0.475878i
$923$	− 557.007i	− 0.603474i
$924$	0	0
$925$	55.7948	0.0603187
$926$	461.054	0.497898
$927$	0	0
$928$	19.8823	0.0214248
$929$	951.540i	1.02426i	0.858907	+	0.512131i	$0.171144\pi$
$929$	951.540i	1.02426i	−0.858907	+	0.512131i	$0.828856\pi$
$930$	0	0
$931$	0	0
$932$	546.146	0.585994
$933$	0	0
$934$	842.246i	0.901763i
$935$	118.654	0.126903
$936$	0	0
$937$	− 1295.71i	− 1.38283i	−0.722460	−	0.691413i	$-0.756989\pi$
$937$	− 1295.71i	− 1.38283i	0.722460	−	0.691413i	$-0.243011\pi$
$938$	0	0
$939$	0	0
$940$	698.632	0.743225
$941$	− 1356.89i	− 1.44197i	−0.692952	−	0.720984i	$-0.743690\pi$
$941$	− 1356.89i	− 1.44197i	0.692952	−	0.720984i	$-0.256310\pi$
$942$	0	0
$943$	775.245i	0.822105i
$944$	154.882i	0.164070i
$945$	0	0
$946$	70.5442	0.0745710
$947$	709.462	0.749168	0.374584	−	0.927193i	$-0.377786\pi$
$947$	709.462	0.749168	0.374584	+	0.927193i	$0.377786\pi$
$948$	0	0
$949$	804.426	0.847657
$950$	112.184i	0.118089i
$951$	0	0
$952$	0	0
$953$	−936.603	−0.982794	−0.491397	−	0.870936i	$-0.663514\pi$
$953$	−936.603	−0.982794	−0.491397	+	0.870936i	$0.663514\pi$
$954$	0	0
$955$	− 1645.13i	− 1.72265i
$956$	530.205	0.554608
$957$	0	0
$958$	− 716.554i	− 0.747969i
$959$	0	0
$960$	0	0
$961$	−1426.95	−1.48486
$962$	63.6023i	0.0661146i
$963$	0	0
$964$	175.277i	0.181822i
$965$	1023.42i	1.06054i
$966$	0	0
$967$	1374.37	1.42127	0.710635	−	0.703561i	$-0.248408\pi$
$967$	1374.37	1.42127	0.710635	+	0.703561i	$0.248408\pi$
$968$	278.225	0.287423
$969$	0	0
$970$	954.573	0.984096
$971$	31.4617i	0.0324014i	0.999869	+	0.0162007i	$0.00515706\pi$
$971$	31.4617i	0.0324014i	−0.999869	+	0.0162007i	$0.994843\pi$
$972$	0	0
$973$	0	0
$974$	−298.825	−0.306802
$975$	0	0
$976$	− 362.032i	− 0.370935i
$977$	541.897	0.554654	0.277327	−	0.960776i	$-0.410551\pi$
$977$	541.897	0.554654	0.277327	+	0.960776i	$0.410551\pi$
$978$	0	0
$979$	239.445i	0.244581i
$980$	0	0
$981$	0	0
$982$	−1108.97	−1.12930
$983$	22.4198i	0.0228075i	0.999935	+	0.0114038i	$0.00363001\pi$
$983$	22.4198i	0.0228075i	−0.999935	+	0.0114038i	$0.996370\pi$
$984$	0	0
$985$	1200.90i	1.21918i
$986$	− 18.6957i	− 0.0189611i
$987$	0	0
$988$	−127.882	−0.129435
$989$	−290.735	−0.293969
$990$	0	0
$991$	678.035	0.684193	0.342096	−	0.939665i	$-0.388863\pi$
$991$	678.035	0.684193	0.342096	+	0.939665i	$0.388863\pi$
$992$	− 276.432i	− 0.278661i
$993$	0	0
$994$	0	0
$995$	2312.28	2.32390
$996$	0	0
$997$	− 876.163i	− 0.878799i	−0.898292	−	0.439400i	$-0.855191\pi$
$997$	− 876.163i	− 0.878799i	0.898292	−	0.439400i	$-0.144809\pi$
$998$	242.527	0.243013
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.f.685.2		4
3.2	odd	2		98.3.b.b.97.3		4
7.2	even	3		126.3.n.c.73.2		4
7.3	odd	6		126.3.n.c.19.2		4
7.4	even	3		882.3.n.b.19.2		4
7.5	odd	6		882.3.n.b.325.2		4
7.6	odd	2	inner	882.3.c.f.685.1		4
12.11	even	2		784.3.c.e.97.3		4
21.2	odd	6		14.3.d.a.3.1	✓	4
21.5	even	6		98.3.d.a.31.1		4
21.11	odd	6		98.3.d.a.19.1		4
21.17	even	6		14.3.d.a.5.1	yes	4
21.20	even	2		98.3.b.b.97.4		4
28.3	even	6		1008.3.cg.l.145.2		4
28.23	odd	6		1008.3.cg.l.577.2		4
84.11	even	6		784.3.s.c.705.2		4
84.23	even	6		112.3.s.b.17.1		4
84.47	odd	6		784.3.s.c.129.2		4
84.59	odd	6		112.3.s.b.33.1		4
84.83	odd	2		784.3.c.e.97.2		4
105.2	even	12		350.3.i.a.199.3		8
105.17	odd	12		350.3.i.a.299.2		8
105.23	even	12		350.3.i.a.199.2		8
105.38	odd	12		350.3.i.a.299.3		8
105.44	odd	6		350.3.k.a.101.2		4
105.59	even	6		350.3.k.a.201.2		4
168.59	odd	6		448.3.s.c.257.2		4
168.101	even	6		448.3.s.d.257.1		4
168.107	even	6		448.3.s.c.129.2		4
168.149	odd	6		448.3.s.d.129.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.3.d.a.3.1	✓	4	21.2	odd	6
14.3.d.a.5.1	yes	4	21.17	even	6
98.3.b.b.97.3		4	3.2	odd	2
98.3.b.b.97.4		4	21.20	even	2
98.3.d.a.19.1		4	21.11	odd	6
98.3.d.a.31.1		4	21.5	even	6
112.3.s.b.17.1		4	84.23	even	6
112.3.s.b.33.1		4	84.59	odd	6
126.3.n.c.19.2		4	7.3	odd	6
126.3.n.c.73.2		4	7.2	even	3
350.3.i.a.199.2		8	105.23	even	12
350.3.i.a.199.3		8	105.2	even	12
350.3.i.a.299.2		8	105.17	odd	12
350.3.i.a.299.3		8	105.38	odd	12
350.3.k.a.101.2		4	105.44	odd	6
350.3.k.a.201.2		4	105.59	even	6
448.3.s.c.129.2		4	168.107	even	6
448.3.s.c.257.2		4	168.59	odd	6
448.3.s.d.129.1		4	168.149	odd	6
448.3.s.d.257.1		4	168.101	even	6
784.3.c.e.97.2		4	84.83	odd	2
784.3.c.e.97.3		4	12.11	even	2
784.3.s.c.129.2		4	84.47	odd	6
784.3.s.c.705.2		4	84.11	even	6
882.3.c.f.685.1		4	7.6	odd	2	inner
882.3.c.f.685.2		4	1.1	even	1	trivial
882.3.n.b.19.2		4	7.4	even	3
882.3.n.b.325.2		4	7.5	odd	6
1008.3.cg.l.145.2		4	28.3	even	6
1008.3.cg.l.577.2		4	28.23	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +6.63103i q^{5} -2.82843 q^{8} -9.37769i q^{10} +4.75736 q^{11} +15.2913i q^{13} +4.00000 q^{16} -3.76127i q^{17} +4.18154i q^{19} +13.2621i q^{20} -6.72792 q^{22} +27.7279 q^{23} -18.9706 q^{25} -21.6251i q^{26} -3.51472 q^{29} +48.8667i q^{31} -5.65685 q^{32} +5.31925i q^{34} -2.94113 q^{37} -5.91359i q^{38} -18.7554i q^{40} +27.9590i q^{41} -10.4853 q^{43} +9.51472 q^{44} -39.2132 q^{46} -52.6790i q^{47} +26.8284 q^{50} +30.5826i q^{52} -55.9706 q^{53} +31.5462i q^{55} +4.97056 q^{58} +38.7206i q^{59} -90.5080i q^{61} -69.1080i q^{62} +8.00000 q^{64} -101.397 q^{65} -34.6396 q^{67} -7.52255i q^{68} -36.4264 q^{71} -52.6069i q^{73} +4.15938 q^{74} +8.36308i q^{76} -33.7868 q^{79} +26.5241i q^{80} -39.5400i q^{82} +127.577i q^{83} +24.9411 q^{85} +14.8284 q^{86} -13.4558 q^{88} +50.3314i q^{89} +55.4558 q^{92} +74.4993i q^{94} -27.7279 q^{95} +101.792i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 36 q^{11} + 16 q^{16} + 24 q^{22} + 60 q^{23} - 8 q^{25} - 48 q^{29} + 124 q^{37} - 8 q^{43} + 72 q^{44} - 72 q^{46} + 96 q^{50} - 156 q^{53} - 48 q^{58} + 32 q^{64} - 168 q^{65} + 116 q^{67}+ \cdots - 60 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 + 36 * q^11 + 16 * q^16 + 24 * q^22 + 60 * q^23 - 8 * q^25 - 48 * q^29 + 124 * q^37 - 8 * q^43 + 72 * q^44 - 72 * q^46 + 96 * q^50 - 156 * q^53 - 48 * q^58 + 32 * q^64 - 168 * q^65 + 116 * q^67 + 24 * q^71 + 192 * q^74 - 220 * q^79 - 36 * q^85 + 48 * q^86 + 48 * q^88 + 120 * q^92 - 60 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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