LMFDB - Embedded newform 2800.2.g.f.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(449,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$, degree $1$, not 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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2800.449
Dual form			2800.2.g.f.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +6.00000 q^{19} -2.00000 q^{21} -3.00000i q^{23} -4.00000i q^{27} +3.00000 q^{29} +2.00000i q^{33} -9.00000i q^{37} +8.00000 q^{39} +2.00000 q^{41} -9.00000i q^{43} -6.00000i q^{47} -1.00000 q^{49} +6.00000i q^{53} -12.0000i q^{57} +8.00000 q^{59} -10.0000 q^{61} +1.00000i q^{63} -1.00000i q^{67} -6.00000 q^{69} +7.00000 q^{71} -2.00000i q^{73} +1.00000i q^{77} -9.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -6.00000i q^{87} +4.00000 q^{89} +4.00000 q^{91} -16.0000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 2 q^{11} + 12 q^{19} - 4 q^{21} + 6 q^{29} + 16 q^{39} + 4 q^{41} - 2 q^{49} + 16 q^{59} - 20 q^{61} - 12 q^{69} + 14 q^{71} - 18 q^{79} - 22 q^{81} + 8 q^{89} + 8 q^{91} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 + 12 * q^19 - 4 * q^21 + 6 * q^29 + 16 * q^39 + 4 * q^41 - 2 * q^49 + 16 * q^59 - 20 * q^61 - 12 * q^69 + 14 * q^71 - 18 * q^79 - 22 * q^81 + 8 * q^89 + 8 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.00000i	− 1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$	− 9.00000i	− 1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 12.0000i	− 1.58944i
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.00000i	− 0.122169i	−0.998133	−	0.0610847i	$-0.980544\pi$
$67$	− 1.00000i	− 0.122169i	0.998133	−	0.0610847i	$-0.0194560\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.00000i	0.113961i
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 16.0000i	− 1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$	− 16.0000i	− 1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.0000i	1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$	20.0000i	1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	−18.0000	−1.70848
$112$	0	0
$113$	− 9.00000i	− 0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$	− 9.00000i	− 0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	− 4.00000i	− 0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.00000i	− 0.0887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$	− 1.00000i	− 0.0887357i	0.999015	−	0.0443678i	$-0.0141274\pi$
$128$	0	0
$129$	−18.0000	−1.58481
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	− 6.00000i	− 0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.0000i	1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$	22.0000i	1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 16.0000i	− 1.20263i
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	20.0000i	1.47844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	− 1.00000i	− 0.0719816i	−0.999352	−	0.0359908i	$-0.988541\pi$
$193$	− 1.00000i	− 0.0719816i	0.999352	−	0.0359908i	$-0.0114587\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.00000i	0.208514i
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	− 14.0000i	− 0.959264i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 22.0000i	− 1.46019i	−0.683345	−	0.730096i	$-0.739475\pi$
$227$	− 22.0000i	− 1.46019i	0.683345	−	0.730096i	$-0.260525\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	− 19.0000i	− 1.24473i	−0.782727	−	0.622366i	$-0.786172\pi$
$233$	− 19.0000i	− 1.24473i	0.782727	−	0.622366i	$-0.213828\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	18.0000i	1.16923i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	24.0000i	1.52708i
$248$	0	0
$249$	−24.0000	−1.52094
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	3.00000i	0.188608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	15.0000i	0.924940i	0.886635	+	0.462470i	$0.153037\pi$
$263$	15.0000i	0.924940i	−0.886635	+	0.462470i	$0.846963\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 8.00000i	− 0.489592i
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	− 8.00000i	− 0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.00000	−0.417585	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000	−0.417585	−0.208792	+	0.977960i	$0.566953\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.00000i	− 0.118056i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	−32.0000	−1.87587
$292$	0	0
$293$	8.00000i	0.467365i	0.972313	+	0.233682i	$0.0750776\pi$
$293$	8.00000i	0.467365i	−0.972313	+	0.233682i	$0.924922\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000i	0.232104i
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	28.0000i	1.60856i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.0000i	1.85346i	0.375722	+	0.926732i	$0.377395\pi$
$317$	33.0000i	1.85346i	−0.375722	+	0.926732i	$0.622605\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	40.0000	2.23258
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 22.0000i	− 1.21660i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	9.00000i	0.493197i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.0000i	0.912608i	0.889824	+	0.456304i	$0.150827\pi$
$347$	17.0000i	0.912608i	−0.889824	+	0.456304i	$0.849173\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	20.0000i	1.04973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	15.0000i	0.776671i	0.921518	+	0.388335i	$0.126950\pi$
$373$	15.0000i	0.776671i	−0.921518	+	0.388335i	$0.873050\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	0	0
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.00000i	0.457496i
$388$	0	0
$389$	35.0000	1.77457	0.887285	−	0.461221i	$-0.152589\pi$
$389$	35.0000	1.77457	0.887285	+	0.461221i	$0.152589\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 24.0000i	− 1.21064i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.0000i	− 1.70641i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	− 34.0000i	− 1.70641i	0.521575	−	0.853206i	$-0.325345\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	−39.0000	−1.94757	−0.973784	−	0.227477i	$-0.926952\pi$
$401$	−39.0000	−1.94757	−0.973784	+	0.227477i	$0.926952\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.00000i	0.446113i
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	44.0000	2.17036
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0000i	0.587643i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	23.0000	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$422$	0	0
$423$	6.00000i	0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.0000i	0.483934i
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 18.0000i	− 0.861057i
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 34.0000i	− 1.60814i
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	34.0000i	1.59746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11.0000i	− 0.514558i	−0.966337	−	0.257279i	$-0.917174\pi$
$457$	− 11.0000i	− 0.514558i	0.966337	−	0.257279i	$-0.0828260\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	0	0
$469$	−1.00000	−0.0461757
$470$	0	0
$471$	28.0000	1.29017
$472$	0	0
$473$	9.00000i	0.413820i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.0000i	0.770344i	0.922845	+	0.385172i	$0.125858\pi$
$487$	17.0000i	0.770344i	−0.922845	+	0.385172i	$0.874142\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	39.0000	1.76005	0.880023	−	0.474932i	$-0.157527\pi$
$491$	39.0000	1.76005	0.880023	+	0.474932i	$0.157527\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.00000i	− 0.313993i
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	0	0
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.00000i	0.266469i
$508$	0	0
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	− 24.0000i	− 1.05963i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.00000i	0.263880i
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	8.00000i	0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 32.0000i	− 1.38090i
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	0	0
$543$	− 44.0000i	− 1.88822i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.0000i	0.470326i	0.971956	+	0.235163i	$0.0755624\pi$
$547$	11.0000i	0.470326i	−0.971956	+	0.235163i	$0.924438\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	18.0000	0.766826
$552$	0	0
$553$	9.00000i	0.382719i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.0000i	1.39825i	0.714997	+	0.699127i	$0.246428\pi$
$557$	33.0000i	1.39825i	−0.714997	+	0.699127i	$0.753572\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.00000i	0.0842900i	0.999112	+	0.0421450i	$0.0134191\pi$
$563$	2.00000i	0.0842900i	−0.999112	+	0.0421450i	$0.986581\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	11.0000i	0.461957i
$568$	0	0
$569$	−37.0000	−1.55112	−0.775560	−	0.631273i	$-0.782533\pi$
$569$	−37.0000	−1.55112	−0.775560	+	0.631273i	$0.782533\pi$
$570$	0	0
$571$	−15.0000	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$571$	−15.0000	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$572$	0	0
$573$	40.0000i	1.67102i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	− 6.00000i	− 0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22.0000i	− 0.908037i	−0.890992	−	0.454019i	$-0.849990\pi$
$587$	− 22.0000i	− 0.908037i	0.890992	−	0.454019i	$-0.150010\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	44.0000i	1.80686i	0.428732	+	0.903432i	$0.358960\pi$
$593$	44.0000i	1.80686i	−0.428732	+	0.903432i	$0.641040\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	36.0000i	1.47338i
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	1.00000i	0.0407231i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	5.00000i	0.201948i	0.994889	+	0.100974i	$0.0321959\pi$
$613$	5.00000i	0.201948i	−0.994889	+	0.100974i	$0.967804\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.0000i	0.925945i	0.886373	+	0.462973i	$0.153217\pi$
$617$	23.0000i	0.925945i	−0.886373	+	0.462973i	$0.846783\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	−12.0000	−0.481543
$622$	0	0
$623$	− 4.00000i	− 0.160257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.0000i	0.479234i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	45.0000	1.79142	0.895711	−	0.444637i	$-0.146667\pi$
$631$	45.0000	1.79142	0.895711	+	0.444637i	$0.146667\pi$
$632$	0	0
$633$	8.00000i	0.317971i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	−7.00000	−0.276916
$640$	0	0
$641$	49.0000	1.93538	0.967692	−	0.252136i	$-0.0811330\pi$
$641$	49.0000	1.93538	0.967692	+	0.252136i	$0.0811330\pi$
$642$	0	0
$643$	− 8.00000i	− 0.315489i	−0.987480	−	0.157745i	$-0.949578\pi$
$643$	− 8.00000i	− 0.315489i	0.987480	−	0.157745i	$-0.0504223\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.0000i	1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$	42.0000i	1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000i	0.0780274i
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	0	0
$669$	−52.0000	−2.01044
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−44.0000	−1.68608
$682$	0	0
$683$	23.0000i	0.880071i	0.897980	+	0.440035i	$0.145034\pi$
$683$	23.0000i	0.880071i	−0.897980	+	0.440035i	$0.854966\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 52.0000i	− 1.98392i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	0	0
$693$	− 1.00000i	− 0.0379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−38.0000	−1.43729
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	− 54.0000i	− 2.03665i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0000i	0.526524i
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	9.00000	0.337526
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	32.0000i	1.19506i
$718$	0	0
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	− 36.0000i	− 1.33885i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.00000i	0.0368355i
$738$	0	0
$739$	−43.0000	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$739$	−43.0000	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$740$	0	0
$741$	48.0000	1.76332
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	20.0000	0.730784
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	40.0000i	1.45768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 17.0000i	− 0.617876i	−0.951082	−	0.308938i	$-0.900027\pi$
$757$	− 17.0000i	− 0.617876i	0.951082	−	0.308938i	$-0.0999735\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	32.0000	1.16000	0.580000	−	0.814617i	$-0.303053\pi$
$761$	32.0000	1.16000	0.580000	+	0.814617i	$0.303053\pi$
$762$	0	0
$763$	− 11.0000i	− 0.398227i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.0000i	1.15545i
$768$	0	0
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40.0000i	1.43870i	0.694648	+	0.719350i	$0.255560\pi$
$773$	40.0000i	1.43870i	−0.694648	+	0.719350i	$0.744440\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	18.0000i	0.645746i
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	−7.00000	−0.250480
$782$	0	0
$783$	− 12.0000i	− 0.428845i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	0	0
$789$	30.0000	1.06803
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.0000i	0.991811i	0.868377	+	0.495905i	$0.165164\pi$
$797$	28.0000i	0.991811i	−0.868377	+	0.495905i	$0.834836\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	2.00000i	0.0705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.00000i	0.281613i
$808$	0	0
$809$	−43.0000	−1.51180	−0.755900	−	0.654687i	$-0.772800\pi$
$809$	−43.0000	−1.51180	−0.755900	+	0.654687i	$0.772800\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 54.0000i	− 1.88922i
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	0	0
$823$	− 23.0000i	− 0.801730i	−0.916137	−	0.400865i	$-0.868710\pi$
$823$	− 23.0000i	− 0.801730i	0.916137	−	0.400865i	$-0.131290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.00000i	− 0.243414i	−0.992566	−	0.121707i	$-0.961163\pi$
$827$	− 7.00000i	− 0.243414i	0.992566	−	0.121707i	$-0.0388368\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	14.0000i	0.482186i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−27.0000	−0.925548
$852$	0	0
$853$	28.0000i	0.958702i	0.877623	+	0.479351i	$0.159128\pi$
$853$	28.0000i	0.958702i	−0.877623	+	0.479351i	$0.840872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.0000i	1.29806i	0.760765	+	0.649028i	$0.224824\pi$
$857$	38.0000i	1.29806i	−0.760765	+	0.649028i	$0.775176\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	7.00000i	0.238283i	0.992877	+	0.119141i	$0.0380142\pi$
$863$	7.00000i	0.238283i	−0.992877	+	0.119141i	$0.961986\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 34.0000i	− 1.15470i
$868$	0	0
$869$	9.00000	0.305304
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	16.0000i	0.541518i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.0000i	1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$	46.0000i	1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	25.0000i	0.841317i	0.907219	+	0.420658i	$0.138201\pi$
$883$	25.0000i	0.841317i	−0.907219	+	0.420658i	$0.861799\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.00000i	− 0.0671534i	−0.999436	−	0.0335767i	$-0.989310\pi$
$887$	− 2.00000i	− 0.0671534i	0.999436	−	0.0335767i	$-0.0106898\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	11.0000	0.368514
$892$	0	0
$893$	− 36.0000i	− 1.20469i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 24.0000i	− 0.801337i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	18.0000i	0.599002i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	−23.0000	−0.762024	−0.381012	−	0.924570i	$-0.624424\pi$
$911$	−23.0000	−0.762024	−0.381012	+	0.924570i	$0.624424\pi$
$912$	0	0
$913$	12.0000i	0.397142i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−29.0000	−0.956622	−0.478311	−	0.878191i	$-0.658751\pi$
$919$	−29.0000	−0.956622	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	28.0000i	0.921631i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.00000i	− 0.197066i
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 10.0000i	− 0.326686i	−0.986569	−	0.163343i	$-0.947772\pi$
$937$	− 10.0000i	− 0.326686i	0.986569	−	0.163343i	$-0.0522277\pi$
$938$	0	0
$939$	−44.0000	−1.43589
$940$	0	0
$941$	−56.0000	−1.82555	−0.912774	−	0.408465i	$-0.866064\pi$
$941$	−56.0000	−1.82555	−0.912774	+	0.408465i	$0.866064\pi$
$942$	0	0
$943$	− 6.00000i	− 0.195387i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.0000i	0.649913i	0.945729	+	0.324956i	$0.105350\pi$
$947$	20.0000i	0.649913i	−0.945729	+	0.324956i	$0.894650\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	66.0000	2.14020
$952$	0	0
$953$	11.0000i	0.356325i	0.984001	+	0.178162i	$0.0570153\pi$
$953$	11.0000i	0.356325i	−0.984001	+	0.178162i	$0.942985\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.00000i	0.193952i
$958$	0	0
$959$	22.0000	0.710417
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 20.0000i	− 0.644491i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.0000	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$972$	0	0
$973$	6.00000i	0.192351i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.00000i	0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$	3.00000i	0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	−11.0000	−0.351203
$982$	0	0
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0000i	0.381964i
$988$	0	0
$989$	−27.0000	−0.858550
$990$	0	0
$991$	−23.0000	−0.730619	−0.365310	−	0.930886i	$-0.619037\pi$
$991$	−23.0000	−0.730619	−0.365310	+	0.930886i	$0.619037\pi$
$992$	0	0
$993$	− 34.0000i	− 1.07896i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	0	0
$999$	−36.0000	−1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.g.f.449.1		2
4.3	odd	2		1400.2.g.c.449.2		2
5.2	odd	4		2800.2.a.f.1.1		1
5.3	odd	4		2800.2.a.bb.1.1		1
5.4	even	2	inner	2800.2.g.f.449.2		2
20.3	even	4		1400.2.a.c.1.1	✓	1
20.7	even	4		1400.2.a.l.1.1	yes	1
20.19	odd	2		1400.2.g.c.449.1		2
140.27	odd	4		9800.2.a.h.1.1		1
140.83	odd	4		9800.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.a.c.1.1	✓	1	20.3	even	4
1400.2.a.l.1.1	yes	1	20.7	even	4
1400.2.g.c.449.1		2	20.19	odd	2
1400.2.g.c.449.2		2	4.3	odd	2
2800.2.a.f.1.1		1	5.2	odd	4
2800.2.a.bb.1.1		1	5.3	odd	4
2800.2.g.f.449.1		2	1.1	even	1	trivial
2800.2.g.f.449.2		2	5.4	even	2	inner
9800.2.a.h.1.1		1	140.27	odd	4
9800.2.a.bk.1.1		1	140.83	odd	4

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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +6.00000 q^{19} -2.00000 q^{21} -3.00000i q^{23} -4.00000i q^{27} +3.00000 q^{29} +2.00000i q^{33} -9.00000i q^{37} +8.00000 q^{39} +2.00000 q^{41} -9.00000i q^{43} -6.00000i q^{47} -1.00000 q^{49} +6.00000i q^{53} -12.0000i q^{57} +8.00000 q^{59} -10.0000 q^{61} +1.00000i q^{63} -1.00000i q^{67} -6.00000 q^{69} +7.00000 q^{71} -2.00000i q^{73} +1.00000i q^{77} -9.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -6.00000i q^{87} +4.00000 q^{89} +4.00000 q^{91} -16.0000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 2 q^{11} + 12 q^{19} - 4 q^{21} + 6 q^{29} + 16 q^{39} + 4 q^{41} - 2 q^{49} + 16 q^{59} - 20 q^{61} - 12 q^{69} + 14 q^{71} - 18 q^{79} - 22 q^{81} + 8 q^{89} + 8 q^{91} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 + 12 * q^19 - 4 * q^21 + 6 * q^29 + 16 * q^39 + 4 * q^41 - 2 * q^49 + 16 * q^59 - 20 * q^61 - 12 * q^69 + 14 * q^71 - 18 * q^79 - 22 * q^81 + 8 * q^89 + 8 * q^91 + 2 * q^99

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