LMFDB - Embedded newform 882.3.c.b.685.1

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,-24,0,0,0,0,16,0,0,0,0,0,0,48] 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 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -8.36308i q^{5} -2.82843 q^{8} +11.8272i q^{10} -6.00000 q^{11} +17.8639i q^{13} +4.00000 q^{16} -18.7554i q^{17} -17.0233i q^{19} -16.7262i q^{20} +8.48528 q^{22} -13.4558 q^{23} -44.9411 q^{25} -25.2633i q^{26} -33.9411 q^{29} -14.7479i q^{31} -5.65685 q^{32} +26.5241i q^{34} +5.97056 q^{37} +24.0746i q^{38} +23.6544i q^{40} +35.2354i q^{41} +15.4853 q^{43} -12.0000 q^{44} +19.0294 q^{46} +33.2061i q^{47} +63.5563 q^{50} +35.7277i q^{52} +34.5442 q^{53} +50.1785i q^{55} +48.0000 q^{58} +27.3647i q^{59} -40.3805i q^{61} +20.8567i q^{62} +8.00000 q^{64} +149.397 q^{65} -114.397 q^{67} -37.5108i q^{68} -18.6030 q^{71} +117.032i q^{73} -8.44365 q^{74} -34.0467i q^{76} -88.3381 q^{79} -33.4523i q^{80} -49.8303i q^{82} +75.7601i q^{83} -156.853 q^{85} -21.8995 q^{86} +16.9706 q^{88} +20.7846i q^{89} -26.9117 q^{92} -46.9606i q^{94} -142.368 q^{95} -30.5826i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 24 q^{11} + 16 q^{16} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 28 q^{43} - 48 q^{44} + 144 q^{46} + 192 q^{50} + 240 q^{53} + 192 q^{58} + 32 q^{64} + 360 q^{65} - 220 q^{67} - 312 q^{71} - 96 q^{74}+ \cdots - 264 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 - 24 * q^11 + 16 * q^16 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 28 * q^43 - 48 * q^44 + 144 * q^46 + 192 * q^50 + 240 * q^53 + 192 * q^58 + 32 * q^64 + 360 * q^65 - 220 * q^67 - 312 * q^71 - 96 * q^74 + 20 * q^79 - 288 * q^85 - 48 * q^86 + 96 * q^92 - 264 * 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
$5$	− 8.36308i	− 1.67262i	−0.548260	−	0.836308i	$-0.684709\pi$
$5$	− 8.36308i	− 1.67262i	0.548260	−	0.836308i	$-0.315291\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.82843	−0.353553
$9$	0	0
$10$	11.8272i	1.18272i
$11$	−6.00000	−0.545455	−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000	−0.545455	−0.272727	+	0.962091i	$0.587926\pi$
$12$	0	0
$13$	17.8639i	1.37414i	0.726590	+	0.687072i	$0.241104\pi$
$13$	17.8639i	1.37414i	−0.726590	+	0.687072i	$0.758896\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 18.7554i	− 1.10326i	−0.834090	−	0.551629i	$-0.814006\pi$
$17$	− 18.7554i	− 1.10326i	0.834090	−	0.551629i	$-0.185994\pi$
$18$	0	0
$19$	− 17.0233i	− 0.895965i	−0.894042	−	0.447983i	$-0.852143\pi$
$19$	− 17.0233i	− 0.895965i	0.894042	−	0.447983i	$-0.147857\pi$
$20$	− 16.7262i	− 0.836308i
$21$	0	0
$22$	8.48528	0.385695
$23$	−13.4558	−0.585037	−0.292518	−	0.956260i	$-0.594493\pi$
$23$	−13.4558	−0.585037	−0.292518	+	0.956260i	$0.594493\pi$
$24$	0	0
$25$	−44.9411	−1.79765
$26$	− 25.2633i	− 0.971666i
$27$	0	0
$28$	0	0
$29$	−33.9411	−1.17038	−0.585192	−	0.810895i	$-0.698981\pi$
$29$	−33.9411	−1.17038	−0.585192	+	0.810895i	$0.698981\pi$
$30$	0	0
$31$	− 14.7479i	− 0.475740i	−0.971297	−	0.237870i	$-0.923551\pi$
$31$	− 14.7479i	− 0.475740i	0.971297	−	0.237870i	$-0.0764491\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	26.5241i	0.780121i
$35$	0	0
$36$	0	0
$37$	5.97056	0.161367	0.0806833	−	0.996740i	$-0.474290\pi$
$37$	5.97056	0.161367	0.0806833	+	0.996740i	$0.474290\pi$
$38$	24.0746i	0.633543i
$39$	0	0
$40$	23.6544i	0.591359i
$41$	35.2354i	0.859399i	0.902972	+	0.429700i	$0.141381\pi$
$41$	35.2354i	0.859399i	−0.902972	+	0.429700i	$0.858619\pi$
$42$	0	0
$43$	15.4853	0.360123	0.180061	−	0.983655i	$-0.442370\pi$
$43$	15.4853	0.360123	0.180061	+	0.983655i	$0.442370\pi$
$44$	−12.0000	−0.272727
$45$	0	0
$46$	19.0294	0.413683
$47$	33.2061i	0.706514i	0.935526	+	0.353257i	$0.114926\pi$
$47$	33.2061i	0.706514i	−0.935526	+	0.353257i	$0.885074\pi$
$48$	0	0
$49$	0	0
$50$	63.5563	1.27113
$51$	0	0
$52$	35.7277i	0.687072i
$53$	34.5442	0.651777	0.325888	−	0.945408i	$-0.394337\pi$
$53$	34.5442	0.651777	0.325888	+	0.945408i	$0.394337\pi$
$54$	0	0
$55$	50.1785i	0.912336i
$56$	0	0
$57$	0	0
$58$	48.0000	0.827586
$59$	27.3647i	0.463808i	0.972739	+	0.231904i	$0.0744955\pi$
$59$	27.3647i	0.463808i	−0.972739	+	0.231904i	$0.925505\pi$
$60$	0	0
$61$	− 40.3805i	− 0.661976i	−0.943635	−	0.330988i	$-0.892618\pi$
$61$	− 40.3805i	− 0.661976i	0.943635	−	0.330988i	$-0.107382\pi$
$62$	20.8567i	0.336399i
$63$	0	0
$64$	8.00000	0.125000
$65$	149.397	2.29841
$66$	0	0
$67$	−114.397	−1.70742	−0.853709	−	0.520751i	$-0.825652\pi$
$67$	−114.397	−1.70742	−0.853709	+	0.520751i	$0.825652\pi$
$68$	− 37.5108i	− 0.551629i
$69$	0	0
$70$	0	0
$71$	−18.6030	−0.262015	−0.131007	−	0.991381i	$-0.541821\pi$
$71$	−18.6030	−0.262015	−0.131007	+	0.991381i	$0.541821\pi$
$72$	0	0
$73$	117.032i	1.60318i	0.597874	+	0.801590i	$0.296012\pi$
$73$	117.032i	1.60318i	−0.597874	+	0.801590i	$0.703988\pi$
$74$	−8.44365	−0.114103
$75$	0	0
$76$	− 34.0467i	− 0.447983i
$77$	0	0
$78$	0	0
$79$	−88.3381	−1.11820	−0.559102	−	0.829099i	$-0.688854\pi$
$79$	−88.3381	−1.11820	−0.559102	+	0.829099i	$0.688854\pi$
$80$	− 33.4523i	− 0.418154i
$81$	0	0
$82$	− 49.8303i	− 0.607687i
$83$	75.7601i	0.912772i	0.889782	+	0.456386i	$0.150856\pi$
$83$	75.7601i	0.912772i	−0.889782	+	0.456386i	$0.849144\pi$
$84$	0	0
$85$	−156.853	−1.84533
$86$	−21.8995	−0.254645
$87$	0	0
$88$	16.9706	0.192847
$89$	20.7846i	0.233535i	0.993159	+	0.116767i	$0.0372532\pi$
$89$	20.7846i	0.233535i	−0.993159	+	0.116767i	$0.962747\pi$
$90$	0	0
$91$	0	0
$92$	−26.9117	−0.292518
$93$	0	0
$94$	− 46.9606i	− 0.499581i
$95$	−142.368	−1.49861
$96$	0	0
$97$	− 30.5826i	− 0.315284i	−0.987496	−	0.157642i	$-0.949611\pi$
$97$	− 30.5826i	− 0.315284i	0.987496	−	0.157642i	$-0.0503892\pi$
$98$	0	0
$99$	0	0
$100$	−89.8823	−0.898823
$101$	127.968i	1.26701i	0.773739	+	0.633504i	$0.218384\pi$
$101$	127.968i	1.26701i	−0.773739	+	0.633504i	$0.781616\pi$
$102$	0	0
$103$	− 80.9563i	− 0.785983i	−0.919542	−	0.392992i	$-0.871440\pi$
$103$	− 80.9563i	− 0.785983i	0.919542	−	0.392992i	$-0.128560\pi$
$104$	− 50.5266i	− 0.485833i
$105$	0	0
$106$	−48.8528	−0.460876
$107$	−169.456	−1.58370	−0.791850	−	0.610716i	$-0.790882\pi$
$107$	−169.456	−1.58370	−0.791850	+	0.610716i	$0.790882\pi$
$108$	0	0
$109$	178.941	1.64166	0.820831	−	0.571171i	$-0.193511\pi$
$109$	178.941	1.64166	0.820831	+	0.571171i	$0.193511\pi$
$110$	− 70.9631i	− 0.645119i
$111$	0	0
$112$	0	0
$113$	17.3970	0.153955	0.0769777	−	0.997033i	$-0.475473\pi$
$113$	17.3970	0.153955	0.0769777	+	0.997033i	$0.475473\pi$
$114$	0	0
$115$	112.532i	0.978542i
$116$	−67.8823	−0.585192
$117$	0	0
$118$	− 38.6995i	− 0.327962i
$119$	0	0
$120$	0	0
$121$	−85.0000	−0.702479
$122$	57.1067i	0.468088i
$123$	0	0
$124$	− 29.4959i	− 0.237870i
$125$	166.769i	1.33415i
$126$	0	0
$127$	167.426	1.31832	0.659159	−	0.752004i	$-0.270912\pi$
$127$	167.426	1.31832	0.659159	+	0.752004i	$0.270912\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	−211.279	−1.62522
$131$	1.78304i	0.0136110i	0.999977	+	0.00680549i	$0.00216627\pi$
$131$	1.78304i	0.0136110i	−0.999977	+	0.00680549i	$0.997834\pi$
$132$	0	0
$133$	0	0
$134$	161.782	1.20733
$135$	0	0
$136$	53.0482i	0.390061i
$137$	−100.971	−0.737011	−0.368506	−	0.929625i	$-0.620130\pi$
$137$	−100.971	−0.737011	−0.368506	+	0.929625i	$0.620130\pi$
$138$	0	0
$139$	140.542i	1.01110i	0.862799	+	0.505548i	$0.168710\pi$
$139$	140.542i	1.01110i	−0.862799	+	0.505548i	$0.831290\pi$
$140$	0	0
$141$	0	0
$142$	26.3087	0.185272
$143$	− 107.183i	− 0.749533i
$144$	0	0
$145$	283.852i	1.95760i
$146$	− 165.508i	− 1.13362i
$147$	0	0
$148$	11.9411	0.0806833
$149$	−182.912	−1.22760	−0.613798	−	0.789463i	$-0.710359\pi$
$149$	−182.912	−1.22760	−0.613798	+	0.789463i	$0.710359\pi$
$150$	0	0
$151$	−288.794	−1.91254	−0.956271	−	0.292481i	$-0.905519\pi$
$151$	−288.794	−1.91254	−0.956271	+	0.292481i	$0.905519\pi$
$152$	48.1493i	0.316771i
$153$	0	0
$154$	0	0
$155$	−123.338	−0.795730
$156$	0	0
$157$	− 187.061i	− 1.19147i	−0.803179	−	0.595737i	$-0.796860\pi$
$157$	− 187.061i	− 1.19147i	0.803179	−	0.595737i	$-0.203140\pi$
$158$	124.929	0.790689
$159$	0	0
$160$	47.3087i	0.295680i
$161$	0	0
$162$	0	0
$163$	16.0589	0.0985207	0.0492604	−	0.998786i	$-0.484314\pi$
$163$	16.0589	0.0985207	0.0492604	+	0.998786i	$0.484314\pi$
$164$	70.4707i	0.429700i
$165$	0	0
$166$	− 107.141i	− 0.645427i
$167$	− 176.117i	− 1.05459i	−0.849681	−	0.527297i	$-0.823206\pi$
$167$	− 176.117i	− 1.05459i	0.849681	−	0.527297i	$-0.176794\pi$
$168$	0	0
$169$	−150.118	−0.888271
$170$	221.823	1.30484
$171$	0	0
$172$	30.9706	0.180061
$173$	231.152i	1.33614i	0.744098	+	0.668070i	$0.232879\pi$
$173$	231.152i	1.33614i	−0.744098	+	0.668070i	$0.767121\pi$
$174$	0	0
$175$	0	0
$176$	−24.0000	−0.136364
$177$	0	0
$178$	− 29.3939i	− 0.165134i
$179$	85.2792	0.476420	0.238210	−	0.971214i	$-0.423439\pi$
$179$	85.2792	0.476420	0.238210	+	0.971214i	$0.423439\pi$
$180$	0	0
$181$	5.58655i	0.0308649i	0.999881	+	0.0154325i	$0.00491250\pi$
$181$	5.58655i	0.0308649i	−0.999881	+	0.0154325i	$0.995087\pi$
$182$	0	0
$183$	0	0
$184$	38.0589	0.206842
$185$	− 49.9323i	− 0.269904i
$186$	0	0
$187$	112.532i	0.601777i
$188$	66.4123i	0.353257i
$189$	0	0
$190$	201.338	1.05967
$191$	−185.397	−0.970665	−0.485332	−	0.874330i	$-0.661301\pi$
$191$	−185.397	−0.970665	−0.485332	+	0.874330i	$0.661301\pi$
$192$	0	0
$193$	227.794	1.18028	0.590140	−	0.807301i	$-0.299073\pi$
$193$	227.794	1.18028	0.590140	+	0.807301i	$0.299073\pi$
$194$	43.2503i	0.222940i
$195$	0	0
$196$	0	0
$197$	−123.161	−0.625185	−0.312593	−	0.949887i	$-0.601197\pi$
$197$	−123.161	−0.625185	−0.312593	+	0.949887i	$0.601197\pi$
$198$	0	0
$199$	− 6.23188i	− 0.0313160i	−0.999877	−	0.0156580i	$-0.995016\pi$
$199$	− 6.23188i	− 0.0313160i	0.999877	−	0.0156580i	$-0.00498430\pi$
$200$	127.113	0.635563
$201$	0	0
$202$	− 180.974i	− 0.895910i
$203$	0	0
$204$	0	0
$205$	294.676	1.43744
$206$	114.489i	0.555774i
$207$	0	0
$208$	71.4555i	0.343536i
$209$	102.140i	0.488708i
$210$	0	0
$211$	−124.912	−0.591999	−0.295999	−	0.955188i	$-0.595653\pi$
$211$	−124.912	−0.591999	−0.295999	+	0.955188i	$0.595653\pi$
$212$	69.0883	0.325888
$213$	0	0
$214$	239.647	1.11984
$215$	− 129.505i	− 0.602347i
$216$	0	0
$217$	0	0
$218$	−253.061	−1.16083
$219$	0	0
$220$	100.357i	0.456168i
$221$	335.044	1.51603
$222$	0	0
$223$	228.631i	1.02525i	0.858613	+	0.512625i	$0.171327\pi$
$223$	228.631i	1.02525i	−0.858613	+	0.512625i	$0.828673\pi$
$224$	0	0
$225$	0	0
$226$	−24.6030	−0.108863
$227$	− 169.537i	− 0.746859i	−0.927659	−	0.373430i	$-0.878182\pi$
$227$	− 169.537i	− 0.746859i	0.927659	−	0.373430i	$-0.121818\pi$
$228$	0	0
$229$	34.6920i	0.151493i	0.997127	+	0.0757467i	$0.0241340\pi$
$229$	34.6920i	0.151493i	−0.997127	+	0.0757467i	$0.975866\pi$
$230$	− 159.145i	− 0.691934i
$231$	0	0
$232$	96.0000	0.413793
$233$	254.485	1.09221	0.546106	−	0.837716i	$-0.316110\pi$
$233$	254.485	1.09221	0.546106	+	0.837716i	$0.316110\pi$
$234$	0	0
$235$	277.706	1.18173
$236$	54.7293i	0.231904i
$237$	0	0
$238$	0	0
$239$	−197.147	−0.824884	−0.412442	−	0.910984i	$-0.635324\pi$
$239$	−197.147	−0.824884	−0.412442	+	0.910984i	$0.635324\pi$
$240$	0	0
$241$	− 88.4701i	− 0.367096i	−0.983011	−	0.183548i	$-0.941242\pi$
$241$	− 88.4701i	− 0.367096i	0.983011	−	0.183548i	$-0.0587582\pi$
$242$	120.208	0.496728
$243$	0	0
$244$	− 80.7611i	− 0.330988i
$245$	0	0
$246$	0	0
$247$	304.103	1.23118
$248$	41.7134i	0.168199i
$249$	0	0
$250$	− 235.847i	− 0.943389i
$251$	− 215.903i	− 0.860172i	−0.902788	−	0.430086i	$-0.858483\pi$
$251$	− 215.903i	− 0.860172i	0.902788	−	0.430086i	$-0.141517\pi$
$252$	0	0
$253$	80.7351	0.319111
$254$	−236.777	−0.932192
$255$	0	0
$256$	16.0000	0.0625000
$257$	4.30463i	0.0167495i	0.999965	+	0.00837477i	$0.00266580\pi$
$257$	4.30463i	0.0167495i	−0.999965	+	0.00837477i	$0.997334\pi$
$258$	0	0
$259$	0	0
$260$	298.794	1.14921
$261$	0	0
$262$	− 2.52160i	− 0.00962441i
$263$	−282.676	−1.07481	−0.537407	−	0.843323i	$-0.680596\pi$
$263$	−282.676	−1.07481	−0.537407	+	0.843323i	$0.680596\pi$
$264$	0	0
$265$	− 288.896i	− 1.09017i
$266$	0	0
$267$	0	0
$268$	−228.794	−0.853709
$269$	− 381.934i	− 1.41983i	−0.704288	−	0.709914i	$-0.748734\pi$
$269$	− 381.934i	− 1.41983i	0.704288	−	0.709914i	$-0.251266\pi$
$270$	0	0
$271$	− 84.3271i	− 0.311170i	−0.987822	−	0.155585i	$-0.950274\pi$
$271$	− 84.3271i	− 0.311170i	0.987822	−	0.155585i	$-0.0497263\pi$
$272$	− 75.0215i	− 0.275814i
$273$	0	0
$274$	142.794	0.521146
$275$	269.647	0.980534
$276$	0	0
$277$	−137.118	−0.495010	−0.247505	−	0.968887i	$-0.579611\pi$
$277$	−137.118	−0.495010	−0.247505	+	0.968887i	$0.579611\pi$
$278$	− 198.757i	− 0.714953i
$279$	0	0
$280$	0	0
$281$	325.103	1.15695	0.578474	−	0.815701i	$-0.303648\pi$
$281$	325.103	1.15695	0.578474	+	0.815701i	$0.303648\pi$
$282$	0	0
$283$	− 194.575i	− 0.687545i	−0.939053	−	0.343773i	$-0.888295\pi$
$283$	− 194.575i	− 0.687545i	0.939053	−	0.343773i	$-0.111705\pi$
$284$	−37.2061	−0.131007
$285$	0	0
$286$	151.580i	0.530000i
$287$	0	0
$288$	0	0
$289$	−62.7645	−0.217178
$290$	− 401.428i	− 1.38423i
$291$	0	0
$292$	234.064i	0.801590i
$293$	− 239.702i	− 0.818095i	−0.912513	−	0.409048i	$-0.865861\pi$
$293$	− 239.702i	− 0.818095i	0.912513	−	0.409048i	$-0.134139\pi$
$294$	0	0
$295$	228.853	0.775772
$296$	−16.8873	−0.0570517
$297$	0	0
$298$	258.676	0.868041
$299$	− 240.373i	− 0.803924i
$300$	0	0
$301$	0	0
$302$	408.416	1.35237
$303$	0	0
$304$	− 68.0933i	− 0.223991i
$305$	−337.706	−1.10723
$306$	0	0
$307$	− 540.272i	− 1.75984i	−0.475120	−	0.879921i	$-0.657595\pi$
$307$	− 540.272i	− 1.75984i	0.475120	−	0.879921i	$-0.342405\pi$
$308$	0	0
$309$	0	0
$310$	174.426	0.562666
$311$	404.196i	1.29966i	0.760078	+	0.649832i	$0.225161\pi$
$311$	404.196i	1.29966i	−0.760078	+	0.649832i	$0.774839\pi$
$312$	0	0
$313$	− 131.296i	− 0.419477i	−0.977757	−	0.209739i	$-0.932739\pi$
$313$	− 131.296i	− 0.419477i	0.977757	−	0.209739i	$-0.0672613\pi$
$314$	264.545i	0.842500i
$315$	0	0
$316$	−176.676	−0.559102
$317$	93.9411	0.296344	0.148172	−	0.988962i	$-0.452661\pi$
$317$	93.9411	0.296344	0.148172	+	0.988962i	$0.452661\pi$
$318$	0	0
$319$	203.647	0.638391
$320$	− 66.9046i	− 0.209077i
$321$	0	0
$322$	0	0
$323$	−319.279	−0.988481
$324$	0	0
$325$	− 802.822i	− 2.47022i
$326$	−22.7107	−0.0696647
$327$	0	0
$328$	− 99.6607i	− 0.303843i
$329$	0	0
$330$	0	0
$331$	−261.368	−0.789630	−0.394815	−	0.918761i	$-0.629191\pi$
$331$	−261.368	−0.789630	−0.394815	+	0.918761i	$0.629191\pi$
$332$	151.520i	0.456386i
$333$	0	0
$334$	249.067i	0.745710i
$335$	956.711i	2.85585i
$336$	0	0
$337$	136.265	0.404347	0.202173	−	0.979350i	$-0.435200\pi$
$337$	136.265	0.404347	0.202173	+	0.979350i	$0.435200\pi$
$338$	212.299	0.628102
$339$	0	0
$340$	−313.706	−0.922664
$341$	88.4876i	0.259494i
$342$	0	0
$343$	0	0
$344$	−43.7990	−0.127323
$345$	0	0
$346$	− 326.899i	− 0.944794i
$347$	322.191	0.928504	0.464252	−	0.885703i	$-0.346323\pi$
$347$	322.191	0.928504	0.464252	+	0.885703i	$0.346323\pi$
$348$	0	0
$349$	− 346.495i	− 0.992821i	−0.868088	−	0.496411i	$-0.834651\pi$
$349$	− 346.495i	− 0.992821i	0.868088	−	0.496411i	$-0.165349\pi$
$350$	0	0
$351$	0	0
$352$	33.9411	0.0964237
$353$	− 620.591i	− 1.75805i	−0.476777	−	0.879025i	$-0.658195\pi$
$353$	− 620.591i	− 1.75805i	0.476777	−	0.879025i	$-0.341805\pi$
$354$	0	0
$355$	155.579i	0.438250i
$356$	41.5692i	0.116767i
$357$	0	0
$358$	−120.603	−0.336880
$359$	−20.2355	−0.0563663	−0.0281831	−	0.999603i	$-0.508972\pi$
$359$	−20.2355	−0.0563663	−0.0281831	+	0.999603i	$0.508972\pi$
$360$	0	0
$361$	71.2061	0.197247
$362$	− 7.90058i	− 0.0218248i
$363$	0	0
$364$	0	0
$365$	978.749	2.68151
$366$	0	0
$367$	311.574i	0.848975i	0.905434	+	0.424488i	$0.139546\pi$
$367$	311.574i	0.848975i	−0.905434	+	0.424488i	$0.860454\pi$
$368$	−53.8234	−0.146259
$369$	0	0
$370$	70.6149i	0.190851i
$371$	0	0
$372$	0	0
$373$	−681.382	−1.82676	−0.913380	−	0.407107i	$-0.866538\pi$
$373$	−681.382	−1.82676	−0.913380	+	0.407107i	$0.866538\pi$
$374$	− 159.145i	− 0.425521i
$375$	0	0
$376$	− 93.9211i	− 0.249790i
$377$	− 606.320i	− 1.60828i
$378$	0	0
$379$	−624.779	−1.64849	−0.824246	−	0.566231i	$-0.808401\pi$
$379$	−624.779	−1.64849	−0.824246	+	0.566231i	$0.808401\pi$
$380$	−284.735	−0.749303
$381$	0	0
$382$	262.191	0.686364
$383$	− 138.300i	− 0.361098i	−0.983566	−	0.180549i	$-0.942213\pi$
$383$	− 138.300i	− 0.361098i	0.983566	−	0.180549i	$-0.0577874\pi$
$384$	0	0
$385$	0	0
$386$	−322.149	−0.834584
$387$	0	0
$388$	− 61.1651i	− 0.157642i
$389$	563.574	1.44878	0.724388	−	0.689393i	$-0.242123\pi$
$389$	563.574	1.44878	0.724388	+	0.689393i	$0.242123\pi$
$390$	0	0
$391$	252.370i	0.645446i
$392$	0	0
$393$	0	0
$394$	174.177	0.442073
$395$	738.779i	1.87033i
$396$	0	0
$397$	− 453.338i	− 1.14191i	−0.820981	−	0.570955i	$-0.806573\pi$
$397$	− 453.338i	− 1.14191i	0.820981	−	0.570955i	$-0.193427\pi$
$398$	8.81321i	0.0221438i
$399$	0	0
$400$	−179.765	−0.449411
$401$	275.750	0.687656	0.343828	−	0.939033i	$-0.388276\pi$
$401$	275.750	0.687656	0.343828	+	0.939033i	$0.388276\pi$
$402$	0	0
$403$	263.455	0.653734
$404$	255.936i	0.633504i
$405$	0	0
$406$	0	0
$407$	−35.8234	−0.0880181
$408$	0	0
$409$	435.831i	1.06560i	0.846240	+	0.532801i	$0.178861\pi$
$409$	435.831i	1.06560i	−0.846240	+	0.532801i	$0.821139\pi$
$410$	−416.735	−1.01643
$411$	0	0
$412$	− 161.913i	− 0.392992i
$413$	0	0
$414$	0	0
$415$	633.588	1.52672
$416$	− 101.053i	− 0.242917i
$417$	0	0
$418$	− 144.448i	− 0.345569i
$419$	− 301.257i	− 0.718991i	−0.933147	−	0.359496i	$-0.882949\pi$
$419$	− 301.257i	− 0.718991i	0.933147	−	0.359496i	$-0.117051\pi$
$420$	0	0
$421$	−203.794	−0.484071	−0.242036	−	0.970267i	$-0.577815\pi$
$421$	−203.794	−0.484071	−0.242036	+	0.970267i	$0.577815\pi$
$422$	176.652	0.418606
$423$	0	0
$424$	−97.7056	−0.230438
$425$	842.888i	1.98327i
$426$	0	0
$427$	0	0
$428$	−338.912	−0.791850
$429$	0	0
$430$	183.147i	0.425924i
$431$	395.720	0.918144	0.459072	−	0.888399i	$-0.348182\pi$
$431$	395.720	0.918144	0.459072	+	0.888399i	$0.348182\pi$
$432$	0	0
$433$	44.2685i	0.102237i	0.998693	+	0.0511184i	$0.0162786\pi$
$433$	44.2685i	0.102237i	−0.998693	+	0.0511184i	$0.983721\pi$
$434$	0	0
$435$	0	0
$436$	357.882	0.820831
$437$	229.063i	0.524172i
$438$	0	0
$439$	397.862i	0.906291i	0.891437	+	0.453146i	$0.149698\pi$
$439$	397.862i	0.906291i	−0.891437	+	0.453146i	$0.850302\pi$
$440$	− 141.926i	− 0.322560i
$441$	0	0
$442$	−473.823	−1.07200
$443$	118.544	0.267594	0.133797	−	0.991009i	$-0.457283\pi$
$443$	118.544	0.267594	0.133797	+	0.991009i	$0.457283\pi$
$444$	0	0
$445$	173.823	0.390614
$446$	− 323.333i	− 0.724961i
$447$	0	0
$448$	0	0
$449$	−713.897	−1.58997	−0.794985	−	0.606629i	$-0.792521\pi$
$449$	−713.897	−1.58997	−0.794985	+	0.606629i	$0.792521\pi$
$450$	0	0
$451$	− 211.412i	− 0.468763i
$452$	34.7939	0.0769777
$453$	0	0
$454$	239.762i	0.528109i
$455$	0	0
$456$	0	0
$457$	−125.177	−0.273909	−0.136955	−	0.990577i	$-0.543731\pi$
$457$	−125.177	−0.273909	−0.136955	+	0.990577i	$0.543731\pi$
$458$	− 49.0619i	− 0.107122i
$459$	0	0
$460$	225.065i	0.489271i
$461$	655.767i	1.42249i	0.702945	+	0.711244i	$0.251868\pi$
$461$	655.767i	1.42249i	−0.702945	+	0.711244i	$0.748132\pi$
$462$	0	0
$463$	869.396	1.87775	0.938873	−	0.344265i	$-0.111872\pi$
$463$	869.396	1.87775	0.938873	+	0.344265i	$0.111872\pi$
$464$	−135.765	−0.292596
$465$	0	0
$466$	−359.897	−0.772310
$467$	267.372i	0.572532i	0.958150	+	0.286266i	$0.0924141\pi$
$467$	267.372i	0.572532i	−0.958150	+	0.286266i	$0.907586\pi$
$468$	0	0
$469$	0	0
$470$	−392.735	−0.835607
$471$	0	0
$472$	− 77.3989i	− 0.163981i
$473$	−92.9117	−0.196431
$474$	0	0
$475$	765.048i	1.61063i
$476$	0	0
$477$	0	0
$478$	278.808	0.583281
$479$	271.737i	0.567300i	0.958928	+	0.283650i	$0.0915454\pi$
$479$	271.737i	0.567300i	−0.958928	+	0.283650i	$0.908455\pi$
$480$	0	0
$481$	106.657i	0.221741i
$482$	125.116i	0.259576i
$483$	0	0
$484$	−170.000	−0.351240
$485$	−255.765	−0.527349
$486$	0	0
$487$	−561.514	−1.15301	−0.576503	−	0.817095i	$-0.695583\pi$
$487$	−561.514	−1.15301	−0.576503	+	0.817095i	$0.695583\pi$
$488$	114.213i	0.234044i
$489$	0	0
$490$	0	0
$491$	406.441	0.827781	0.413891	−	0.910327i	$-0.364170\pi$
$491$	406.441	0.827781	0.413891	+	0.910327i	$0.364170\pi$
$492$	0	0
$493$	636.579i	1.29124i
$494$	−430.066	−0.870579
$495$	0	0
$496$	− 58.9917i	− 0.118935i
$497$	0	0
$498$	0	0
$499$	−371.426	−0.744341	−0.372171	−	0.928164i	$-0.621386\pi$
$499$	−371.426	−0.744341	−0.372171	+	0.928164i	$0.621386\pi$
$500$	333.538i	0.667077i
$501$	0	0
$502$	305.333i	0.608234i
$503$	64.6292i	0.128488i	0.997934	+	0.0642438i	$0.0204635\pi$
$503$	64.6292i	0.128488i	−0.997934	+	0.0642438i	$0.979536\pi$
$504$	0	0
$505$	1070.21	2.11922
$506$	−114.177	−0.225646
$507$	0	0
$508$	334.853	0.659159
$509$	− 1006.77i	− 1.97794i	−0.148125	−	0.988969i	$-0.547324\pi$
$509$	− 1006.77i	− 1.97794i	0.148125	−	0.988969i	$-0.452676\pi$
$510$	0	0
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	− 6.08767i	− 0.0118437i
$515$	−677.044	−1.31465
$516$	0	0
$517$	− 199.237i	− 0.385371i
$518$	0	0
$519$	0	0
$520$	−422.558	−0.812612
$521$	− 372.153i	− 0.714306i	−0.934046	−	0.357153i	$-0.883747\pi$
$521$	− 372.153i	− 0.714306i	0.934046	−	0.357153i	$-0.116253\pi$
$522$	0	0
$523$	− 637.284i	− 1.21852i	−0.792972	−	0.609258i	$-0.791467\pi$
$523$	− 637.284i	− 1.21852i	0.792972	−	0.609258i	$-0.208533\pi$
$524$	3.56608i	0.00680549i
$525$	0	0
$526$	399.765	0.760009
$527$	−276.603	−0.524863
$528$	0	0
$529$	−347.940	−0.657732
$530$	408.560i	0.770868i
$531$	0	0
$532$	0	0
$533$	−629.440	−1.18094
$534$	0	0
$535$	1417.17i	2.64892i
$536$	323.563	0.603663
$537$	0	0
$538$	540.136i	1.00397i
$539$	0	0
$540$	0	0
$541$	220.823	0.408176	0.204088	−	0.978953i	$-0.434577\pi$
$541$	220.823	0.408176	0.204088	+	0.978953i	$0.434577\pi$
$542$	119.257i	0.220031i
$543$	0	0
$544$	106.096i	0.195030i
$545$	− 1496.50i	− 2.74587i
$546$	0	0
$547$	−160.676	−0.293741	−0.146870	−	0.989156i	$-0.546920\pi$
$547$	−160.676	−0.293741	−0.146870	+	0.989156i	$0.546920\pi$
$548$	−201.941	−0.368506
$549$	0	0
$550$	−381.338	−0.693342
$551$	577.791i	1.04862i
$552$	0	0
$553$	0	0
$554$	193.914	0.350025
$555$	0	0
$556$	281.085i	0.505548i
$557$	474.353	0.851622	0.425811	−	0.904812i	$-0.359989\pi$
$557$	474.353	0.851622	0.425811	+	0.904812i	$0.359989\pi$
$558$	0	0
$559$	276.627i	0.494860i
$560$	0	0
$561$	0	0
$562$	−459.765	−0.818086
$563$	− 496.868i	− 0.882537i	−0.897375	−	0.441269i	$-0.854529\pi$
$563$	− 496.868i	− 0.882537i	0.897375	−	0.441269i	$-0.145471\pi$
$564$	0	0
$565$	− 145.492i	− 0.257508i
$566$	275.171i	0.486168i
$567$	0	0
$568$	52.6173	0.0926361
$569$	−785.294	−1.38013	−0.690065	−	0.723748i	$-0.742418\pi$
$569$	−785.294	−1.38013	−0.690065	+	0.723748i	$0.742418\pi$
$570$	0	0
$571$	−715.043	−1.25226	−0.626132	−	0.779717i	$-0.715363\pi$
$571$	−715.043	−1.25226	−0.626132	+	0.779717i	$0.715363\pi$
$572$	− 214.366i	− 0.374766i
$573$	0	0
$574$	0	0
$575$	604.721	1.05169
$576$	0	0
$577$	− 772.630i	− 1.33905i	−0.742791	−	0.669524i	$-0.766498\pi$
$577$	− 772.630i	− 1.33905i	0.742791	−	0.669524i	$-0.233502\pi$
$578$	88.7624	0.153568
$579$	0	0
$580$	567.705i	0.978801i
$581$	0	0
$582$	0	0
$583$	−207.265	−0.355514
$584$	− 331.017i	− 0.566810i
$585$	0	0
$586$	338.990i	0.578481i
$587$	− 436.477i	− 0.743572i	−0.928318	−	0.371786i	$-0.878746\pi$
$587$	− 436.477i	− 0.743572i	0.928318	−	0.371786i	$-0.121254\pi$
$588$	0	0
$589$	−251.059	−0.426246
$590$	−323.647	−0.548554
$591$	0	0
$592$	23.8823	0.0403416
$593$	− 834.152i	− 1.40666i	−0.710861	−	0.703332i	$-0.751695\pi$
$593$	− 834.152i	− 1.40666i	0.710861	−	0.703332i	$-0.248305\pi$
$594$	0	0
$595$	0	0
$596$	−365.823	−0.613798
$597$	0	0
$598$	339.939i	0.568460i
$599$	−873.588	−1.45841	−0.729205	−	0.684295i	$-0.760110\pi$
$599$	−873.588	−1.45841	−0.729205	+	0.684295i	$0.760110\pi$
$600$	0	0
$601$	− 198.982i	− 0.331085i	−0.986203	−	0.165542i	$-0.947063\pi$
$601$	− 198.982i	− 0.331085i	0.986203	−	0.165542i	$-0.0529375\pi$
$602$	0	0
$603$	0	0
$604$	−577.588	−0.956271
$605$	710.862i	1.17498i
$606$	0	0
$607$	158.950i	0.261861i	0.991392	+	0.130930i	$0.0417964\pi$
$607$	158.950i	0.261861i	−0.991392	+	0.130930i	$0.958204\pi$
$608$	96.2985i	0.158386i
$609$	0	0
$610$	477.588	0.782931
$611$	−593.190	−0.970851
$612$	0	0
$613$	−714.735	−1.16596	−0.582981	−	0.812486i	$-0.698114\pi$
$613$	−714.735	−1.16596	−0.582981	+	0.812486i	$0.698114\pi$
$614$	764.059i	1.24440i
$615$	0	0
$616$	0	0
$617$	639.381	1.03627	0.518137	−	0.855298i	$-0.326626\pi$
$617$	639.381	1.03627	0.518137	+	0.855298i	$0.326626\pi$
$618$	0	0
$619$	− 172.025i	− 0.277908i	−0.990299	−	0.138954i	$-0.955626\pi$
$619$	− 172.025i	− 0.277908i	0.990299	−	0.138954i	$-0.0443740\pi$
$620$	−246.676	−0.397865
$621$	0	0
$622$	− 571.619i	− 0.919002i
$623$	0	0
$624$	0	0
$625$	271.177	0.433883
$626$	185.681i	0.296615i
$627$	0	0
$628$	− 374.123i	− 0.595737i
$629$	− 111.980i	− 0.178029i
$630$	0	0
$631$	−1141.06	−1.80833	−0.904166	−	0.427180i	$-0.859507\pi$
$631$	−1141.06	−1.80833	−0.904166	+	0.427180i	$0.859507\pi$
$632$	249.858	0.395345
$633$	0	0
$634$	−132.853	−0.209547
$635$	− 1400.20i	− 2.20504i
$636$	0	0
$637$	0	0
$638$	−288.000	−0.451411
$639$	0	0
$640$	94.6175i	0.147840i
$641$	−229.103	−0.357414	−0.178707	−	0.983902i	$-0.557191\pi$
$641$	−229.103	−0.357414	−0.178707	+	0.983902i	$0.557191\pi$
$642$	0	0
$643$	− 707.670i	− 1.10058i	−0.834975	−	0.550288i	$-0.814518\pi$
$643$	− 707.670i	− 1.10058i	0.834975	−	0.550288i	$-0.185482\pi$
$644$	0	0
$645$	0	0
$646$	451.529	0.698961
$647$	1179.37i	1.82283i	0.411484	+	0.911417i	$0.365011\pi$
$647$	1179.37i	1.82283i	−0.411484	+	0.911417i	$0.634989\pi$
$648$	0	0
$649$	− 164.188i	− 0.252986i
$650$	1135.36i	1.74671i
$651$	0	0
$652$	32.1177	0.0492604
$653$	154.764	0.237004	0.118502	−	0.992954i	$-0.462191\pi$
$653$	154.764	0.237004	0.118502	+	0.992954i	$0.462191\pi$
$654$	0	0
$655$	14.9117	0.0227659
$656$	140.941i	0.214850i
$657$	0	0
$658$	0	0
$659$	−591.308	−0.897280	−0.448640	−	0.893712i	$-0.648092\pi$
$659$	−591.308	−0.897280	−0.448640	+	0.893712i	$0.648092\pi$
$660$	0	0
$661$	− 162.167i	− 0.245337i	−0.992448	−	0.122668i	$-0.960855\pi$
$661$	− 162.167i	− 0.245337i	0.992448	−	0.122668i	$-0.0391451\pi$
$662$	369.630	0.558353
$663$	0	0
$664$	− 214.282i	− 0.322714i
$665$	0	0
$666$	0	0
$667$	456.706	0.684717
$668$	− 352.234i	− 0.527297i
$669$	0	0
$670$	− 1352.99i	− 2.01939i
$671$	242.283i	0.361078i
$672$	0	0
$673$	42.3238	0.0628883	0.0314441	−	0.999506i	$-0.489989\pi$
$673$	42.3238	0.0628883	0.0314441	+	0.999506i	$0.489989\pi$
$674$	−192.708	−0.285916
$675$	0	0
$676$	−300.235	−0.444135
$677$	− 497.354i	− 0.734643i	−0.930094	−	0.367322i	$-0.880275\pi$
$677$	− 497.354i	− 0.734643i	0.930094	−	0.367322i	$-0.119725\pi$
$678$	0	0
$679$	0	0
$680$	443.647	0.652422
$681$	0	0
$682$	− 125.140i	− 0.183490i
$683$	−1216.16	−1.78062	−0.890308	−	0.455359i	$-0.849511\pi$
$683$	−1216.16	−1.78062	−0.890308	+	0.455359i	$0.849511\pi$
$684$	0	0
$685$	844.425i	1.23274i
$686$	0	0
$687$	0	0
$688$	61.9411	0.0900307
$689$	617.092i	0.895635i
$690$	0	0
$691$	− 1076.39i	− 1.55773i	−0.627191	−	0.778865i	$-0.715796\pi$
$691$	− 1076.39i	− 1.55773i	0.627191	−	0.778865i	$-0.284204\pi$
$692$	462.305i	0.668070i
$693$	0	0
$694$	−455.647	−0.656552
$695$	1175.37	1.69118
$696$	0	0
$697$	660.853	0.948139
$698$	490.017i	0.702031i
$699$	0	0
$700$	0	0
$701$	695.897	0.992720	0.496360	−	0.868117i	$-0.334670\pi$
$701$	695.897	0.992720	0.496360	+	0.868117i	$0.334670\pi$
$702$	0	0
$703$	− 101.639i	− 0.144579i
$704$	−48.0000	−0.0681818
$705$	0	0
$706$	877.649i	1.24313i
$707$	0	0
$708$	0	0
$709$	254.824	0.359414	0.179707	−	0.983720i	$-0.442485\pi$
$709$	254.824	0.359414	0.179707	+	0.983720i	$0.442485\pi$
$710$	− 220.021i	− 0.309889i
$711$	0	0
$712$	− 58.7878i	− 0.0825671i
$713$	198.446i	0.278325i
$714$	0	0
$715$	−896.382	−1.25368
$716$	170.558	0.238210
$717$	0	0
$718$	28.6173	0.0398570
$719$	1114.20i	1.54965i	0.632175	+	0.774826i	$0.282162\pi$
$719$	1114.20i	1.54965i	−0.632175	+	0.774826i	$0.717838\pi$
$720$	0	0
$721$	0	0
$722$	−100.701	−0.139474
$723$	0	0
$724$	11.1731i	0.0154325i
$725$	1525.35	2.10393
$726$	0	0
$727$	398.345i	0.547930i	0.961740	+	0.273965i	$0.0883353\pi$
$727$	398.345i	0.547930i	−0.961740	+	0.273965i	$0.911665\pi$
$728$	0	0
$729$	0	0
$730$	−1384.16	−1.89611
$731$	− 290.432i	− 0.397308i
$732$	0	0
$733$	945.139i	1.28941i	0.764431	+	0.644706i	$0.223020\pi$
$733$	945.139i	1.28941i	−0.764431	+	0.644706i	$0.776980\pi$
$734$	− 440.632i	− 0.600316i
$735$	0	0
$736$	76.1177	0.103421
$737$	686.382	0.931319
$738$	0	0
$739$	192.632	0.260666	0.130333	−	0.991470i	$-0.458395\pi$
$739$	192.632	0.260666	0.130333	+	0.991470i	$0.458395\pi$
$740$	− 99.8646i	− 0.134952i
$741$	0	0
$742$	0	0
$743$	911.616	1.22694	0.613470	−	0.789718i	$-0.289773\pi$
$743$	911.616	1.22694	0.613470	+	0.789718i	$0.289773\pi$
$744$	0	0
$745$	1529.71i	2.05330i
$746$	963.619	1.29172
$747$	0	0
$748$	225.065i	0.300889i
$749$	0	0
$750$	0	0
$751$	−391.662	−0.521521	−0.260760	−	0.965404i	$-0.583973\pi$
$751$	−391.662	−0.521521	−0.260760	+	0.965404i	$0.583973\pi$
$752$	132.825i	0.176628i
$753$	0	0
$754$	857.466i	1.13722i
$755$	2415.21i	3.19895i
$756$	0	0
$757$	−152.823	−0.201879	−0.100940	−	0.994893i	$-0.532185\pi$
$757$	−152.823	−0.201879	−0.100940	+	0.994893i	$0.532185\pi$
$758$	883.571	1.16566
$759$	0	0
$760$	402.676	0.529837
$761$	126.245i	0.165893i	0.996554	+	0.0829465i	$0.0264330\pi$
$761$	126.245i	0.165893i	−0.996554	+	0.0829465i	$0.973567\pi$
$762$	0	0
$763$	0	0
$764$	−370.794	−0.485332
$765$	0	0
$766$	195.586i	0.255335i
$767$	−488.839	−0.637338
$768$	0	0
$769$	369.148i	0.480037i	0.970768	+	0.240018i	$0.0771535\pi$
$769$	369.148i	0.480037i	−0.970768	+	0.240018i	$0.922847\pi$
$770$	0	0
$771$	0	0
$772$	455.588	0.590140
$773$	− 1403.71i	− 1.81592i	−0.419056	−	0.907961i	$-0.637639\pi$
$773$	− 1403.71i	− 1.81592i	0.419056	−	0.907961i	$-0.362361\pi$
$774$	0	0
$775$	662.788i	0.855211i
$776$	86.5006i	0.111470i
$777$	0	0
$778$	−797.013	−1.02444
$779$	599.823	0.769991
$780$	0	0
$781$	111.618	0.142917
$782$	− 356.904i	− 0.456400i
$783$	0	0
$784$	0	0
$785$	−1564.41	−1.99288
$786$	0	0
$787$	− 226.507i	− 0.287810i	−0.989591	−	0.143905i	$-0.954034\pi$
$787$	− 226.507i	− 0.287810i	0.989591	−	0.143905i	$-0.0459660\pi$
$788$	−246.323	−0.312593
$789$	0	0
$790$	− 1044.79i	− 1.32252i
$791$	0	0
$792$	0	0
$793$	721.352	0.909650
$794$	641.117i	0.807453i
$795$	0	0
$796$	− 12.4638i	− 0.0156580i
$797$	− 688.414i	− 0.863756i	−0.901932	−	0.431878i	$-0.857851\pi$
$797$	− 688.414i	− 0.863756i	0.901932	−	0.431878i	$-0.142149\pi$
$798$	0	0
$799$	622.794	0.779467
$800$	254.225	0.317782
$801$	0	0
$802$	−389.970	−0.486247
$803$	− 702.193i	− 0.874462i
$804$	0	0
$805$	0	0
$806$	−372.582	−0.462260
$807$	0	0
$808$	− 361.948i	− 0.447955i
$809$	−25.2792	−0.0312475	−0.0156237	−	0.999878i	$-0.504973\pi$
$809$	−25.2792	−0.0312475	−0.0156237	+	0.999878i	$0.504973\pi$
$810$	0	0
$811$	− 1527.62i	− 1.88362i	−0.336145	−	0.941810i	$-0.609123\pi$
$811$	− 1527.62i	− 1.88362i	0.336145	−	0.941810i	$-0.390877\pi$
$812$	0	0
$813$	0	0
$814$	50.6619	0.0622382
$815$	− 134.302i	− 0.164787i
$816$	0	0
$817$	− 263.611i	− 0.322657i
$818$	− 616.359i	− 0.753495i
$819$	0	0
$820$	589.352	0.718722
$821$	116.662	0.142097	0.0710487	−	0.997473i	$-0.477365\pi$
$821$	116.662	0.142097	0.0710487	+	0.997473i	$0.477365\pi$
$822$	0	0
$823$	125.911	0.152990	0.0764950	−	0.997070i	$-0.475627\pi$
$823$	125.911	0.152990	0.0764950	+	0.997070i	$0.475627\pi$
$824$	228.979i	0.277887i
$825$	0	0
$826$	0	0
$827$	1434.40	1.73446	0.867229	−	0.497910i	$-0.165899\pi$
$827$	1434.40	1.73446	0.867229	+	0.497910i	$0.165899\pi$
$828$	0	0
$829$	37.3228i	0.0450215i	0.999747	+	0.0225107i	$0.00716600\pi$
$829$	37.3228i	0.0450215i	−0.999747	+	0.0225107i	$0.992834\pi$
$830$	−896.029	−1.07955
$831$	0	0
$832$	142.911i	0.171768i
$833$	0	0
$834$	0	0
$835$	−1472.88	−1.76393
$836$	204.280i	0.244354i
$837$	0	0
$838$	426.042i	0.508404i
$839$	3.07370i	0.00366353i	0.999998	+	0.00183177i	$0.000583069\pi$
$839$	3.07370i	0.00366353i	−0.999998	+	0.00183177i	$0.999417\pi$
$840$	0	0
$841$	311.000	0.369798
$842$	288.208	0.342290
$843$	0	0
$844$	−249.823	−0.295999
$845$	1255.45i	1.48574i
$846$	0	0
$847$	0	0
$848$	138.177	0.162944
$849$	0	0
$850$	− 1192.02i	− 1.40238i
$851$	−80.3390	−0.0944054
$852$	0	0
$853$	155.257i	0.182013i	0.995850	+	0.0910063i	$0.0290083\pi$
$853$	155.257i	0.182013i	−0.995850	+	0.0910063i	$0.970992\pi$
$854$	0	0
$855$	0	0
$856$	479.294	0.559922
$857$	− 1603.86i	− 1.87148i	−0.352686	−	0.935742i	$-0.614731\pi$
$857$	− 1603.86i	− 1.87148i	0.352686	−	0.935742i	$-0.385269\pi$
$858$	0	0
$859$	629.735i	0.733103i	0.930398	+	0.366551i	$0.119462\pi$
$859$	629.735i	0.733103i	−0.930398	+	0.366551i	$0.880538\pi$
$860$	− 259.009i	− 0.301174i
$861$	0	0
$862$	−559.632	−0.649226
$863$	−1029.41	−1.19283	−0.596414	−	0.802677i	$-0.703408\pi$
$863$	−1029.41	−1.19283	−0.596414	+	0.802677i	$0.703408\pi$
$864$	0	0
$865$	1933.15	2.23485
$866$	− 62.6051i	− 0.0722923i
$867$	0	0
$868$	0	0
$869$	530.029	0.609929
$870$	0	0
$871$	− 2043.57i	− 2.34624i
$872$	−506.122	−0.580415
$873$	0	0
$874$	− 323.944i	− 0.370646i
$875$	0	0
$876$	0	0
$877$	648.441	0.739385	0.369693	−	0.929154i	$-0.379463\pi$
$877$	648.441	0.739385	0.369693	+	0.929154i	$0.379463\pi$
$878$	− 562.662i	− 0.640845i
$879$	0	0
$880$	200.714i	0.228084i
$881$	− 363.857i	− 0.413005i	−0.978446	−	0.206502i	$-0.933792\pi$
$881$	− 363.857i	− 0.413005i	0.978446	−	0.206502i	$-0.0662082\pi$
$882$	0	0
$883$	1536.16	1.73971	0.869853	−	0.493312i	$-0.164214\pi$
$883$	1536.16	1.73971	0.869853	+	0.493312i	$0.164214\pi$
$884$	670.087	0.758017
$885$	0	0
$886$	−167.647	−0.189218
$887$	− 1125.51i	− 1.26889i	−0.772966	−	0.634447i	$-0.781228\pi$
$887$	− 1125.51i	− 1.26889i	0.772966	−	0.634447i	$-0.218772\pi$
$888$	0	0
$889$	0	0
$890$	−245.823	−0.276206
$891$	0	0
$892$	457.261i	0.512625i
$893$	565.279	0.633011
$894$	0	0
$895$	− 713.197i	− 0.796868i
$896$	0	0
$897$	0	0
$898$	1009.60	1.12428
$899$	500.561i	0.556798i
$900$	0	0
$901$	− 647.889i	− 0.719078i
$902$	298.982i	0.331466i
$903$	0	0
$904$	−49.2061	−0.0544315
$905$	46.7208	0.0516252
$906$	0	0
$907$	234.897	0.258982	0.129491	−	0.991581i	$-0.458666\pi$
$907$	234.897	0.258982	0.129491	+	0.991581i	$0.458666\pi$
$908$	− 339.074i	− 0.373430i
$909$	0	0
$910$	0	0
$911$	−224.278	−0.246189	−0.123095	−	0.992395i	$-0.539282\pi$
$911$	−224.278	−0.246189	−0.123095	+	0.992395i	$0.539282\pi$
$912$	0	0
$913$	− 454.561i	− 0.497876i
$914$	177.026	0.193683
$915$	0	0
$916$	69.3840i	0.0757467i
$917$	0	0
$918$	0	0
$919$	−932.161	−1.01432	−0.507161	−	0.861851i	$-0.669305\pi$
$919$	−932.161	−1.01432	−0.507161	+	0.861851i	$0.669305\pi$
$920$	− 318.289i	− 0.345967i
$921$	0	0
$922$	− 927.394i	− 1.00585i
$923$	− 332.322i	− 0.360046i
$924$	0	0
$925$	−268.324	−0.290080
$926$	−1229.51	−1.32777
$927$	0	0
$928$	192.000	0.206897
$929$	714.055i	0.768628i	0.923203	+	0.384314i	$0.125562\pi$
$929$	714.055i	0.768628i	−0.923203	+	0.384314i	$0.874438\pi$
$930$	0	0
$931$	0	0
$932$	508.971	0.546106
$933$	0	0
$934$	− 378.122i	− 0.404841i
$935$	941.117	1.00654
$936$	0	0
$937$	1723.25i	1.83912i	0.392952	+	0.919559i	$0.371454\pi$
$937$	1723.25i	1.83912i	−0.392952	+	0.919559i	$0.628546\pi$
$938$	0	0
$939$	0	0
$940$	555.411	0.590863
$941$	964.761i	1.02525i	0.858612	+	0.512625i	$0.171327\pi$
$941$	964.761i	1.02525i	−0.858612	+	0.512625i	$0.828673\pi$
$942$	0	0
$943$	− 474.122i	− 0.502780i
$944$	109.459i	0.115952i
$945$	0	0
$946$	131.397	0.138897
$947$	−1451.76	−1.53301	−0.766506	−	0.642237i	$-0.778007\pi$
$947$	−1451.76	−1.53301	−0.766506	+	0.642237i	$0.778007\pi$
$948$	0	0
$949$	−2090.65	−2.20300
$950$	− 1081.94i	− 1.13889i
$951$	0	0
$952$	0	0
$953$	−1147.43	−1.20401	−0.602007	−	0.798491i	$-0.705632\pi$
$953$	−1147.43	−1.20401	−0.602007	+	0.798491i	$0.705632\pi$
$954$	0	0
$955$	1550.49i	1.62355i
$956$	−394.294	−0.412442
$957$	0	0
$958$	− 384.294i	− 0.401142i
$959$	0	0
$960$	0	0
$961$	743.499	0.773672
$962$	− 150.836i	− 0.156794i
$963$	0	0
$964$	− 176.940i	− 0.183548i
$965$	− 1905.06i	− 1.97415i
$966$	0	0
$967$	412.190	0.426257	0.213128	−	0.977024i	$-0.431635\pi$
$967$	412.190	0.426257	0.213128	+	0.977024i	$0.431635\pi$
$968$	240.416	0.248364
$969$	0	0
$970$	361.706	0.372892
$971$	− 1004.12i	− 1.03411i	−0.855952	−	0.517056i	$-0.827028\pi$
$971$	− 1004.12i	− 1.03411i	0.855952	−	0.517056i	$-0.172972\pi$
$972$	0	0
$973$	0	0
$974$	794.101	0.815298
$975$	0	0
$976$	− 161.522i	− 0.165494i
$977$	1588.23	1.62562	0.812812	−	0.582527i	$-0.197936\pi$
$977$	1588.23	1.62562	0.812812	+	0.582527i	$0.197936\pi$
$978$	0	0
$979$	− 124.708i	− 0.127383i
$980$	0	0
$981$	0	0
$982$	−574.794	−0.585330
$983$	833.533i	0.847948i	0.905674	+	0.423974i	$0.139365\pi$
$983$	833.533i	0.847948i	−0.905674	+	0.423974i	$0.860635\pi$
$984$	0	0
$985$	1030.01i	1.04569i
$986$	− 900.259i	− 0.913041i
$987$	0	0
$988$	608.205	0.615592
$989$	−208.368	−0.210685
$990$	0	0
$991$	66.8965	0.0675041	0.0337520	−	0.999430i	$-0.489254\pi$
$991$	66.8965	0.0675041	0.0337520	+	0.999430i	$0.489254\pi$
$992$	83.4269i	0.0840997i
$993$	0	0
$994$	0	0
$995$	−52.1177	−0.0523796
$996$	0	0
$997$	− 1464.91i	− 1.46931i	−0.678439	−	0.734657i	$-0.737343\pi$
$997$	− 1464.91i	− 1.46931i	0.678439	−	0.734657i	$-0.262657\pi$
$998$	525.276	0.526329
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.b.685.1		4
3.2	odd	2		294.3.c.a.97.3		4
7.2	even	3		126.3.n.a.73.2		4
7.3	odd	6		126.3.n.a.19.2		4
7.4	even	3		882.3.n.e.19.2		4
7.5	odd	6		882.3.n.e.325.2		4
7.6	odd	2	inner	882.3.c.b.685.2		4
12.11	even	2		2352.3.f.e.97.4		4
21.2	odd	6		42.3.g.a.31.1	yes	4
21.5	even	6		294.3.g.a.31.1		4
21.11	odd	6		294.3.g.a.19.1		4
21.17	even	6		42.3.g.a.19.1	✓	4
21.20	even	2		294.3.c.a.97.4		4
28.3	even	6		1008.3.cg.h.145.1		4
28.23	odd	6		1008.3.cg.h.577.1		4
84.23	even	6		336.3.bh.e.241.2		4
84.59	odd	6		336.3.bh.e.145.2		4
84.83	odd	2		2352.3.f.e.97.1		4
105.2	even	12		1050.3.q.a.199.4		8
105.17	odd	12		1050.3.q.a.649.1		8
105.23	even	12		1050.3.q.a.199.1		8
105.38	odd	12		1050.3.q.a.649.4		8
105.44	odd	6		1050.3.p.a.451.2		4
105.59	even	6		1050.3.p.a.901.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.g.a.19.1	✓	4	21.17	even	6
42.3.g.a.31.1	yes	4	21.2	odd	6
126.3.n.a.19.2		4	7.3	odd	6
126.3.n.a.73.2		4	7.2	even	3
294.3.c.a.97.3		4	3.2	odd	2
294.3.c.a.97.4		4	21.20	even	2
294.3.g.a.19.1		4	21.11	odd	6
294.3.g.a.31.1		4	21.5	even	6
336.3.bh.e.145.2		4	84.59	odd	6
336.3.bh.e.241.2		4	84.23	even	6
882.3.c.b.685.1		4	1.1	even	1	trivial
882.3.c.b.685.2		4	7.6	odd	2	inner
882.3.n.e.19.2		4	7.4	even	3
882.3.n.e.325.2		4	7.5	odd	6
1008.3.cg.h.145.1		4	28.3	even	6
1008.3.cg.h.577.1		4	28.23	odd	6
1050.3.p.a.451.2		4	105.44	odd	6
1050.3.p.a.901.2		4	105.59	even	6
1050.3.q.a.199.1		8	105.23	even	12
1050.3.q.a.199.4		8	105.2	even	12
1050.3.q.a.649.1		8	105.17	odd	12
1050.3.q.a.649.4		8	105.38	odd	12
2352.3.f.e.97.1		4	84.83	odd	2
2352.3.f.e.97.4		4	12.11	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

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

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -8.36308i q^{5} -2.82843 q^{8} +11.8272i q^{10} -6.00000 q^{11} +17.8639i q^{13} +4.00000 q^{16} -18.7554i q^{17} -17.0233i q^{19} -16.7262i q^{20} +8.48528 q^{22} -13.4558 q^{23} -44.9411 q^{25} -25.2633i q^{26} -33.9411 q^{29} -14.7479i q^{31} -5.65685 q^{32} +26.5241i q^{34} +5.97056 q^{37} +24.0746i q^{38} +23.6544i q^{40} +35.2354i q^{41} +15.4853 q^{43} -12.0000 q^{44} +19.0294 q^{46} +33.2061i q^{47} +63.5563 q^{50} +35.7277i q^{52} +34.5442 q^{53} +50.1785i q^{55} +48.0000 q^{58} +27.3647i q^{59} -40.3805i q^{61} +20.8567i q^{62} +8.00000 q^{64} +149.397 q^{65} -114.397 q^{67} -37.5108i q^{68} -18.6030 q^{71} +117.032i q^{73} -8.44365 q^{74} -34.0467i q^{76} -88.3381 q^{79} -33.4523i q^{80} -49.8303i q^{82} +75.7601i q^{83} -156.853 q^{85} -21.8995 q^{86} +16.9706 q^{88} +20.7846i q^{89} -26.9117 q^{92} -46.9606i q^{94} -142.368 q^{95} -30.5826i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 24 q^{11} + 16 q^{16} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 28 q^{43} - 48 q^{44} + 144 q^{46} + 192 q^{50} + 240 q^{53} + 192 q^{58} + 32 q^{64} + 360 q^{65} - 220 q^{67} - 312 q^{71} - 96 q^{74}+ \cdots - 264 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 - 24 * q^11 + 16 * q^16 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 28 * q^43 - 48 * q^44 + 144 * q^46 + 192 * q^50 + 240 * q^53 + 192 * q^58 + 32 * q^64 + 360 * q^65 - 220 * q^67 - 312 * q^71 - 96 * q^74 + 20 * q^79 - 288 * q^85 - 48 * q^86 + 96 * q^92 - 264 * q^95

Introduction

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