LMFDB - Embedded newform 8712.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$69.5656702409$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 792)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{5} -4.82843 q^{7} +O(q^{10})$	$q-2.82843 q^{5} -4.82843 q^{7} +0.828427 q^{13} -2.00000 q^{17} -2.00000 q^{19} +2.82843 q^{23} +3.00000 q^{25} -7.65685 q^{29} -1.65685 q^{31} +13.6569 q^{35} -3.65685 q^{37} -11.6569 q^{41} -11.6569 q^{43} +4.48528 q^{47} +16.3137 q^{49} +1.17157 q^{53} -9.65685 q^{59} -8.82843 q^{61} -2.34315 q^{65} -11.3137 q^{67} -12.4853 q^{71} -9.31371 q^{73} +0.828427 q^{79} -1.65685 q^{83} +5.65685 q^{85} +16.0000 q^{89} -4.00000 q^{91} +5.65685 q^{95} -17.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^7	$ 2 q - 4 q^{7} - 4 q^{13} - 4 q^{17} - 4 q^{19} + 6 q^{25} - 4 q^{29} + 8 q^{31} + 16 q^{35} + 4 q^{37} - 12 q^{41} - 12 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} - 8 q^{59} - 12 q^{61} - 16 q^{65} - 8 q^{71} + 4 q^{73} - 4 q^{79} + 8 q^{83} + 32 q^{89} - 8 q^{91} - 12 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 4 * q^13 - 4 * q^17 - 4 * q^19 + 6 * q^25 - 4 * q^29 + 8 * q^31 + 16 * q^35 + 4 * q^37 - 12 * q^41 - 12 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 - 8 * q^59 - 12 * q^61 - 16 * q^65 - 8 * q^71 + 4 * q^73 - 4 * q^79 + 8 * q^83 + 32 * q^89 - 8 * q^91 - 12 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	−1.65685	−0.297580	−0.148790	−	0.988869i	$-0.547538\pi$
$31$	−1.65685	−0.297580	−0.148790	+	0.988869i	$0.547538\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13.6569	2.30843
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.6569	−1.82049	−0.910247	−	0.414065i	$-0.864109\pi$
$41$	−11.6569	−1.82049	−0.910247	+	0.414065i	$0.864109\pi$
$42$	0	0
$43$	−11.6569	−1.77765	−0.888827	−	0.458243i	$-0.848479\pi$
$43$	−11.6569	−1.77765	−0.888827	+	0.458243i	$0.848479\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.48528	0.654246	0.327123	−	0.944982i	$-0.393921\pi$
$47$	4.48528	0.654246	0.327123	+	0.944982i	$0.393921\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.17157	0.160928	0.0804640	−	0.996758i	$-0.474360\pi$
$53$	1.17157	0.160928	0.0804640	+	0.996758i	$0.474360\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.65685	−1.25722	−0.628608	−	0.777723i	$-0.716375\pi$
$59$	−9.65685	−1.25722	−0.628608	+	0.777723i	$0.716375\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.34315	−0.290631
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.4853	−1.48173	−0.740865	−	0.671654i	$-0.765584\pi$
$71$	−12.4853	−1.48173	−0.740865	+	0.671654i	$0.765584\pi$
$72$	0	0
$73$	−9.31371	−1.09009	−0.545044	−	0.838408i	$-0.683487\pi$
$73$	−9.31371	−1.09009	−0.545044	+	0.838408i	$0.683487\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.828427	0.0932053	0.0466027	−	0.998914i	$-0.485161\pi$
$79$	0.828427	0.0932053	0.0466027	+	0.998914i	$0.485161\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.65685	−0.181863	−0.0909317	−	0.995857i	$-0.528984\pi$
$83$	−1.65685	−0.181863	−0.0909317	+	0.995857i	$0.528984\pi$
$84$	0	0
$85$	5.65685	0.613572
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.65685	0.580381
$96$	0	0
$97$	−17.3137	−1.75794	−0.878970	−	0.476876i	$-0.841769\pi$
$97$	−17.3137	−1.75794	−0.878970	+	0.476876i	$0.841769\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	16.8284	1.61187	0.805935	−	0.592003i	$-0.201663\pi$
$109$	16.8284	1.61187	0.805935	+	0.592003i	$0.201663\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−15.3137	−1.44059	−0.720296	−	0.693667i	$-0.755994\pi$
$113$	−15.3137	−1.44059	−0.720296	+	0.693667i	$0.755994\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.65685	0.885242
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	16.8284	1.49328	0.746641	−	0.665228i	$-0.231665\pi$
$127$	16.8284	1.49328	0.746641	+	0.665228i	$0.231665\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.9706	−1.83221	−0.916103	−	0.400942i	$-0.868683\pi$
$131$	−20.9706	−1.83221	−0.916103	+	0.400942i	$0.868683\pi$
$132$	0	0
$133$	9.65685	0.837355
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.6569	1.50853	0.754263	−	0.656572i	$-0.227994\pi$
$137$	17.6569	1.50853	0.754263	+	0.656572i	$0.227994\pi$
$138$	0	0
$139$	−13.3137	−1.12925	−0.564627	−	0.825346i	$-0.690980\pi$
$139$	−13.3137	−1.12925	−0.564627	+	0.825346i	$0.690980\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	21.6569	1.79850
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.34315	0.355804	0.177902	−	0.984048i	$-0.443069\pi$
$149$	4.34315	0.355804	0.177902	+	0.984048i	$0.443069\pi$
$150$	0	0
$151$	−2.48528	−0.202249	−0.101125	−	0.994874i	$-0.532244\pi$
$151$	−2.48528	−0.202249	−0.101125	+	0.994874i	$0.532244\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.68629	0.376412
$156$	0	0
$157$	3.65685	0.291849	0.145924	−	0.989296i	$-0.453384\pi$
$157$	3.65685	0.291849	0.145924	+	0.989296i	$0.453384\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.6569	−1.07631
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.34315	0.634318	0.317159	−	0.948372i	$-0.397271\pi$
$173$	8.34315	0.634318	0.317159	+	0.948372i	$0.397271\pi$
$174$	0	0
$175$	−14.4853	−1.09498
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.3431	−0.773083	−0.386542	−	0.922272i	$-0.626330\pi$
$179$	−10.3431	−0.773083	−0.386542	+	0.922272i	$0.626330\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.3431	0.760443
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.17157	0.0847720	0.0423860	−	0.999101i	$-0.486504\pi$
$191$	1.17157	0.0847720	0.0423860	+	0.999101i	$0.486504\pi$
$192$	0	0
$193$	−15.6569	−1.12701	−0.563503	−	0.826114i	$-0.690546\pi$
$193$	−15.6569	−1.12701	−0.563503	+	0.826114i	$0.690546\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	0	0
$199$	−14.3431	−1.01676	−0.508379	−	0.861133i	$-0.669755\pi$
$199$	−14.3431	−1.01676	−0.508379	+	0.861133i	$0.669755\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	36.9706	2.59482
$204$	0	0
$205$	32.9706	2.30276
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.34315	−0.298994	−0.149497	−	0.988762i	$-0.547766\pi$
$211$	−4.34315	−0.298994	−0.149497	+	0.988762i	$0.547766\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	32.9706	2.24857
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.65685	−0.111452
$222$	0	0
$223$	−11.3137	−0.757622	−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137	−0.757622	−0.378811	+	0.925474i	$0.623667\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.6274	1.23635	0.618173	−	0.786042i	$-0.287873\pi$
$227$	18.6274	1.23635	0.618173	+	0.786042i	$0.287873\pi$
$228$	0	0
$229$	14.9706	0.989283	0.494641	−	0.869097i	$-0.335299\pi$
$229$	14.9706	0.989283	0.494641	+	0.869097i	$0.335299\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.97056	−0.456657	−0.228328	−	0.973584i	$-0.573326\pi$
$233$	−6.97056	−0.456657	−0.228328	+	0.973584i	$0.573326\pi$
$234$	0	0
$235$	−12.6863	−0.827562
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.6274	1.46365	0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274	1.46365	0.731823	+	0.681495i	$0.238670\pi$
$240$	0	0
$241$	7.65685	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$241$	7.65685	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−46.1421	−2.94791
$246$	0	0
$247$	−1.65685	−0.105423
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.6569	1.36697	0.683484	−	0.729965i	$-0.260464\pi$
$251$	21.6569	1.36697	0.683484	+	0.729965i	$0.260464\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.3137	1.20476	0.602378	−	0.798211i	$-0.294220\pi$
$257$	19.3137	1.20476	0.602378	+	0.798211i	$0.294220\pi$
$258$	0	0
$259$	17.6569	1.09714
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.3431	−1.13109	−0.565543	−	0.824719i	$-0.691334\pi$
$263$	−18.3431	−1.13109	−0.565543	+	0.824719i	$0.691334\pi$
$264$	0	0
$265$	−3.31371	−0.203559
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.4853	0.761241	0.380621	−	0.924731i	$-0.375710\pi$
$269$	12.4853	0.761241	0.380621	+	0.924731i	$0.375710\pi$
$270$	0	0
$271$	13.7990	0.838229	0.419114	−	0.907933i	$-0.362341\pi$
$271$	13.7990	0.838229	0.419114	+	0.907933i	$0.362341\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.4558	−1.16899	−0.584494	−	0.811398i	$-0.698707\pi$
$277$	−19.4558	−1.16899	−0.584494	+	0.811398i	$0.698707\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.3137	1.27147	0.635735	−	0.771908i	$-0.280697\pi$
$281$	21.3137	1.27147	0.635735	+	0.771908i	$0.280697\pi$
$282$	0	0
$283$	2.97056	0.176582	0.0882908	−	0.996095i	$-0.471860\pi$
$283$	2.97056	0.176582	0.0882908	+	0.996095i	$0.471860\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	56.2843	3.32236
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	27.3137	1.59027
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.34315	0.135508
$300$	0	0
$301$	56.2843	3.24417
$302$	0	0
$303$	0	0
$304$	0	0
$305$	24.9706	1.42981
$306$	0	0
$307$	9.31371	0.531561	0.265781	−	0.964034i	$-0.414370\pi$
$307$	9.31371	0.531561	0.265781	+	0.964034i	$0.414370\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.1716	−1.20053	−0.600265	−	0.799801i	$-0.704938\pi$
$311$	−21.1716	−1.20053	−0.600265	+	0.799801i	$0.704938\pi$
$312$	0	0
$313$	2.68629	0.151838	0.0759191	−	0.997114i	$-0.475811\pi$
$313$	2.68629	0.151838	0.0759191	+	0.997114i	$0.475811\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.17157	−0.290464	−0.145232	−	0.989398i	$-0.546393\pi$
$317$	−5.17157	−0.290464	−0.145232	+	0.989398i	$0.546393\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	2.48528	0.137859
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−21.6569	−1.19398
$330$	0	0
$331$	26.6274	1.46358	0.731788	−	0.681533i	$-0.238686\pi$
$331$	26.6274	1.46358	0.731788	+	0.681533i	$0.238686\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	32.0000	1.74835
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−44.9706	−2.42818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.9706	−1.12576	−0.562879	−	0.826539i	$-0.690306\pi$
$347$	−20.9706	−1.12576	−0.562879	+	0.826539i	$0.690306\pi$
$348$	0	0
$349$	−25.7990	−1.38099	−0.690494	−	0.723338i	$-0.742607\pi$
$349$	−25.7990	−1.38099	−0.690494	+	0.723338i	$0.742607\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	35.3137	1.87426
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.3431	1.37886
$366$	0	0
$367$	−19.3137	−1.00817	−0.504084	−	0.863655i	$-0.668170\pi$
$367$	−19.3137	−1.00817	−0.504084	+	0.863655i	$0.668170\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.65685	−0.293689
$372$	0	0
$373$	8.14214	0.421584	0.210792	−	0.977531i	$-0.432396\pi$
$373$	8.14214	0.421584	0.210792	+	0.977531i	$0.432396\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.34315	−0.326689
$378$	0	0
$379$	−23.3137	−1.19754	−0.598772	−	0.800919i	$-0.704345\pi$
$379$	−23.3137	−1.19754	−0.598772	+	0.800919i	$0.704345\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.5147	0.792765	0.396383	−	0.918085i	$-0.370265\pi$
$383$	15.5147	0.792765	0.396383	+	0.918085i	$0.370265\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.8284	−1.36026	−0.680128	−	0.733094i	$-0.738076\pi$
$389$	−26.8284	−1.36026	−0.680128	+	0.733094i	$0.738076\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.34315	−0.117896
$396$	0	0
$397$	29.3137	1.47121	0.735606	−	0.677409i	$-0.236897\pi$
$397$	29.3137	1.47121	0.735606	+	0.677409i	$0.236897\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.3431	0.716263	0.358131	−	0.933671i	$-0.383414\pi$
$401$	14.3431	0.716263	0.358131	+	0.933671i	$0.383414\pi$
$402$	0	0
$403$	−1.37258	−0.0683732
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	23.6569	1.16976	0.584878	−	0.811121i	$-0.301142\pi$
$409$	23.6569	1.16976	0.584878	+	0.811121i	$0.301142\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	46.6274	2.29439
$414$	0	0
$415$	4.68629	0.230041
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.2843	−0.990951	−0.495476	−	0.868622i	$-0.665006\pi$
$419$	−20.2843	−0.990951	−0.495476	+	0.868622i	$0.665006\pi$
$420$	0	0
$421$	4.34315	0.211672	0.105836	−	0.994384i	$-0.466248\pi$
$421$	4.34315	0.211672	0.105836	+	0.994384i	$0.466248\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	42.6274	2.06289
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.6569	1.04317	0.521587	−	0.853198i	$-0.325340\pi$
$431$	21.6569	1.04317	0.521587	+	0.853198i	$0.325340\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.65685	−0.270604
$438$	0	0
$439$	8.14214	0.388603	0.194301	−	0.980942i	$-0.437756\pi$
$439$	8.14214	0.388603	0.194301	+	0.980942i	$0.437756\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.9706	−0.616250	−0.308125	−	0.951346i	$-0.599702\pi$
$443$	−12.9706	−0.616250	−0.308125	+	0.951346i	$0.599702\pi$
$444$	0	0
$445$	−45.2548	−2.14528
$446$	0	0
$447$	0	0
$448$	0	0
$449$	29.6569	1.39959	0.699797	−	0.714342i	$-0.253274\pi$
$449$	29.6569	1.39959	0.699797	+	0.714342i	$0.253274\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11.3137	0.530395
$456$	0	0
$457$	−16.3431	−0.764500	−0.382250	−	0.924059i	$-0.624851\pi$
$457$	−16.3431	−0.764500	−0.382250	+	0.924059i	$0.624851\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.3137	−0.806380	−0.403190	−	0.915116i	$-0.632099\pi$
$461$	−17.3137	−0.806380	−0.403190	+	0.915116i	$0.632099\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.3137	−1.07883	−0.539415	−	0.842040i	$-0.681355\pi$
$467$	−23.3137	−1.07883	−0.539415	+	0.842040i	$0.681355\pi$
$468$	0	0
$469$	54.6274	2.52246
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.97056	0.409875	0.204938	−	0.978775i	$-0.434301\pi$
$479$	8.97056	0.409875	0.204938	+	0.978775i	$0.434301\pi$
$480$	0	0
$481$	−3.02944	−0.138130
$482$	0	0
$483$	0	0
$484$	0	0
$485$	48.9706	2.22364
$486$	0	0
$487$	1.65685	0.0750792	0.0375396	−	0.999295i	$-0.488048\pi$
$487$	1.65685	0.0750792	0.0375396	+	0.999295i	$0.488048\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.6274	0.840644	0.420322	−	0.907375i	$-0.361917\pi$
$491$	18.6274	0.840644	0.420322	+	0.907375i	$0.361917\pi$
$492$	0	0
$493$	15.3137	0.689695
$494$	0	0
$495$	0	0
$496$	0	0
$497$	60.2843	2.70412
$498$	0	0
$499$	−20.6863	−0.926046	−0.463023	−	0.886346i	$-0.653235\pi$
$499$	−20.6863	−0.926046	−0.463023	+	0.886346i	$0.653235\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.31371	−0.147751	−0.0738755	−	0.997267i	$-0.523537\pi$
$503$	−3.31371	−0.147751	−0.0738755	+	0.997267i	$0.523537\pi$
$504$	0	0
$505$	−39.5980	−1.76209
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.201010	−0.00890962	−0.00445481	−	0.999990i	$-0.501418\pi$
$509$	−0.201010	−0.00890962	−0.00445481	+	0.999990i	$0.501418\pi$
$510$	0	0
$511$	44.9706	1.98938
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.97056	−0.217764	−0.108882	−	0.994055i	$-0.534727\pi$
$521$	−4.97056	−0.217764	−0.108882	+	0.994055i	$0.534727\pi$
$522$	0	0
$523$	−22.9706	−1.00443	−0.502216	−	0.864742i	$-0.667482\pi$
$523$	−22.9706	−1.00443	−0.502216	+	0.864742i	$0.667482\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.31371	0.144347
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.65685	−0.418285
$534$	0	0
$535$	−11.3137	−0.489134
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24.8284	1.06746	0.533729	−	0.845656i	$-0.320790\pi$
$541$	24.8284	1.06746	0.533729	+	0.845656i	$0.320790\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−47.5980	−2.03887
$546$	0	0
$547$	−25.3137	−1.08234	−0.541168	−	0.840914i	$-0.682018\pi$
$547$	−25.3137	−1.08234	−0.541168	+	0.840914i	$0.682018\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.3137	0.652386
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.31371	0.225149	0.112575	−	0.993643i	$-0.464090\pi$
$557$	5.31371	0.225149	0.112575	+	0.993643i	$0.464090\pi$
$558$	0	0
$559$	−9.65685	−0.408441
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.6863	0.703243	0.351622	−	0.936142i	$-0.385630\pi$
$563$	16.6863	0.703243	0.351622	+	0.936142i	$0.385630\pi$
$564$	0	0
$565$	43.3137	1.82222
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.3137	−0.558140	−0.279070	−	0.960271i	$-0.590026\pi$
$569$	−13.3137	−0.558140	−0.279070	+	0.960271i	$0.590026\pi$
$570$	0	0
$571$	16.6274	0.695836	0.347918	−	0.937525i	$-0.386889\pi$
$571$	16.6274	0.695836	0.347918	+	0.937525i	$0.386889\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	16.6274	0.692208	0.346104	−	0.938196i	$-0.387504\pi$
$577$	16.6274	0.692208	0.346104	+	0.938196i	$0.387504\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.65685	0.398581	0.199291	−	0.979940i	$-0.436136\pi$
$587$	9.65685	0.398581	0.199291	+	0.979940i	$0.436136\pi$
$588$	0	0
$589$	3.31371	0.136539
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−27.3137	−1.11975
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.4853	−1.00044	−0.500221	−	0.865898i	$-0.666748\pi$
$599$	−24.4853	−1.00044	−0.500221	+	0.865898i	$0.666748\pi$
$600$	0	0
$601$	−5.31371	−0.216751	−0.108375	−	0.994110i	$-0.534565\pi$
$601$	−5.31371	−0.216751	−0.108375	+	0.994110i	$0.534565\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.1716	−1.26522	−0.632608	−	0.774473i	$-0.718015\pi$
$607$	−31.1716	−1.26522	−0.632608	+	0.774473i	$0.718015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.71573	0.150322
$612$	0	0
$613$	−0.828427	−0.0334599	−0.0167299	−	0.999860i	$-0.505326\pi$
$613$	−0.828427	−0.0334599	−0.0167299	+	0.999860i	$0.505326\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.68629	0.349697	0.174848	−	0.984595i	$-0.444056\pi$
$617$	8.68629	0.349697	0.174848	+	0.984595i	$0.444056\pi$
$618$	0	0
$619$	−23.3137	−0.937057	−0.468529	−	0.883448i	$-0.655216\pi$
$619$	−23.3137	−0.937057	−0.468529	+	0.883448i	$0.655216\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−77.2548	−3.09515
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.31371	0.291617
$630$	0	0
$631$	−44.9706	−1.79025	−0.895125	−	0.445815i	$-0.852914\pi$
$631$	−44.9706	−1.79025	−0.895125	+	0.445815i	$0.852914\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−47.5980	−1.88887
$636$	0	0
$637$	13.5147	0.535473
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.2843	1.11716	0.558581	−	0.829450i	$-0.311346\pi$
$641$	28.2843	1.11716	0.558581	+	0.829450i	$0.311346\pi$
$642$	0	0
$643$	10.6274	0.419104	0.209552	−	0.977797i	$-0.432799\pi$
$643$	10.6274	0.419104	0.209552	+	0.977797i	$0.432799\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.7990	−1.09289	−0.546446	−	0.837495i	$-0.684019\pi$
$647$	−27.7990	−1.09289	−0.546446	+	0.837495i	$0.684019\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.1127	−1.37407	−0.687033	−	0.726626i	$-0.741087\pi$
$653$	−35.1127	−1.37407	−0.687033	+	0.726626i	$0.741087\pi$
$654$	0	0
$655$	59.3137	2.31758
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.3137	−0.908173	−0.454087	−	0.890958i	$-0.650034\pi$
$659$	−23.3137	−0.908173	−0.454087	+	0.890958i	$0.650034\pi$
$660$	0	0
$661$	0.343146	0.0133468	0.00667341	−	0.999978i	$-0.497876\pi$
$661$	0.343146	0.0133468	0.00667341	+	0.999978i	$0.497876\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−27.3137	−1.05918
$666$	0	0
$667$	−21.6569	−0.838557
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.6569	0.757716	0.378858	−	0.925455i	$-0.376317\pi$
$673$	19.6569	0.757716	0.378858	+	0.925455i	$0.376317\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.9411	−1.68879	−0.844397	−	0.535717i	$-0.820041\pi$
$677$	−43.9411	−1.68879	−0.844397	+	0.535717i	$0.820041\pi$
$678$	0	0
$679$	83.5980	3.20820
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.02944	0.268974	0.134487	−	0.990915i	$-0.457061\pi$
$683$	7.02944	0.268974	0.134487	+	0.990915i	$0.457061\pi$
$684$	0	0
$685$	−49.9411	−1.90815
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.970563	0.0369755
$690$	0	0
$691$	−21.9411	−0.834680	−0.417340	−	0.908750i	$-0.637038\pi$
$691$	−21.9411	−0.834680	−0.417340	+	0.908750i	$0.637038\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	37.6569	1.42841
$696$	0	0
$697$	23.3137	0.883070
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0.627417	0.0236972	0.0118486	−	0.999930i	$-0.496228\pi$
$701$	0.627417	0.0236972	0.0118486	+	0.999930i	$0.496228\pi$
$702$	0	0
$703$	7.31371	0.275842
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−67.5980	−2.54228
$708$	0	0
$709$	−28.3431	−1.06445	−0.532225	−	0.846603i	$-0.678644\pi$
$709$	−28.3431	−1.06445	−0.532225	+	0.846603i	$0.678644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.68629	−0.175503
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.4853	−0.763972	−0.381986	−	0.924168i	$-0.624760\pi$
$719$	−20.4853	−0.763972	−0.381986	+	0.924168i	$0.624760\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−22.9706	−0.853105
$726$	0	0
$727$	12.9706	0.481052	0.240526	−	0.970643i	$-0.422680\pi$
$727$	12.9706	0.481052	0.240526	+	0.970643i	$0.422680\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	23.3137	0.862289
$732$	0	0
$733$	−13.7990	−0.509677	−0.254839	−	0.966984i	$-0.582022\pi$
$733$	−13.7990	−0.509677	−0.254839	+	0.966984i	$0.582022\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.02944	0.0378685	0.0189342	−	0.999821i	$-0.493973\pi$
$739$	1.02944	0.0378685	0.0189342	+	0.999821i	$0.493973\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.9706	0.916081	0.458041	−	0.888931i	$-0.348551\pi$
$743$	24.9706	0.916081	0.458041	+	0.888931i	$0.348551\pi$
$744$	0	0
$745$	−12.2843	−0.450061
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19.3137	−0.705708
$750$	0	0
$751$	35.5980	1.29899	0.649494	−	0.760366i	$-0.274981\pi$
$751$	35.5980	1.29899	0.649494	+	0.760366i	$0.274981\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.02944	0.255827
$756$	0	0
$757$	−11.6569	−0.423676	−0.211838	−	0.977305i	$-0.567945\pi$
$757$	−11.6569	−0.423676	−0.211838	+	0.977305i	$0.567945\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.6274	0.457744	0.228872	−	0.973457i	$-0.426496\pi$
$761$	12.6274	0.457744	0.228872	+	0.973457i	$0.426496\pi$
$762$	0	0
$763$	−81.2548	−2.94162
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	16.3431	0.589349	0.294674	−	0.955598i	$-0.404789\pi$
$769$	16.3431	0.589349	0.294674	+	0.955598i	$0.404789\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36.4853	1.31228	0.656142	−	0.754637i	$-0.272187\pi$
$773$	36.4853	1.31228	0.656142	+	0.754637i	$0.272187\pi$
$774$	0	0
$775$	−4.97056	−0.178548
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.3137	0.835300
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.3431	−0.369163
$786$	0	0
$787$	20.3431	0.725155	0.362577	−	0.931954i	$-0.381897\pi$
$787$	20.3431	0.725155	0.362577	+	0.931954i	$0.381897\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	73.9411	2.62904
$792$	0	0
$793$	−7.31371	−0.259717
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.85786	0.0658089	0.0329045	−	0.999459i	$-0.489524\pi$
$797$	1.85786	0.0658089	0.0329045	+	0.999459i	$0.489524\pi$
$798$	0	0
$799$	−8.97056	−0.317356
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	38.6274	1.36144
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.5980	1.32187	0.660937	−	0.750441i	$-0.270159\pi$
$809$	37.5980	1.32187	0.660937	+	0.750441i	$0.270159\pi$
$810$	0	0
$811$	−23.6569	−0.830705	−0.415352	−	0.909661i	$-0.636342\pi$
$811$	−23.6569	−0.830705	−0.415352	+	0.909661i	$0.636342\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.3137	0.396302
$816$	0	0
$817$	23.3137	0.815643
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.37258	−0.117704	−0.0588520	−	0.998267i	$-0.518744\pi$
$821$	−3.37258	−0.117704	−0.0588520	+	0.998267i	$0.518744\pi$
$822$	0	0
$823$	12.6863	0.442216	0.221108	−	0.975249i	$-0.429033\pi$
$823$	12.6863	0.442216	0.221108	+	0.975249i	$0.429033\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.9706	0.729218	0.364609	−	0.931161i	$-0.381203\pi$
$827$	20.9706	0.729218	0.364609	+	0.931161i	$0.381203\pi$
$828$	0	0
$829$	−24.6274	−0.855346	−0.427673	−	0.903934i	$-0.640666\pi$
$829$	−24.6274	−0.855346	−0.427673	+	0.903934i	$0.640666\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−32.6274	−1.13047
$834$	0	0
$835$	−45.2548	−1.56611
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.8284	−0.373839	−0.186919	−	0.982375i	$-0.559850\pi$
$839$	−10.8284	−0.373839	−0.186919	+	0.982375i	$0.559850\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	0	0
$844$	0	0
$845$	34.8284	1.19813
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.3431	−0.354558
$852$	0	0
$853$	−44.1421	−1.51140	−0.755699	−	0.654919i	$-0.772703\pi$
$853$	−44.1421	−1.51140	−0.755699	+	0.654919i	$0.772703\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.9706	−0.648022	−0.324011	−	0.946053i	$-0.605031\pi$
$857$	−18.9706	−0.648022	−0.324011	+	0.946053i	$0.605031\pi$
$858$	0	0
$859$	−53.9411	−1.84045	−0.920224	−	0.391393i	$-0.871993\pi$
$859$	−53.9411	−1.84045	−0.920224	+	0.391393i	$0.871993\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.7696	−1.52397	−0.761987	−	0.647593i	$-0.775776\pi$
$863$	−44.7696	−1.52397	−0.761987	+	0.647593i	$0.775776\pi$
$864$	0	0
$865$	−23.5980	−0.802355
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−9.37258	−0.317578
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−27.3137	−0.923372
$876$	0	0
$877$	27.4558	0.927118	0.463559	−	0.886066i	$-0.346572\pi$
$877$	27.4558	0.927118	0.463559	+	0.886066i	$0.346572\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34.6274	1.16663	0.583314	−	0.812247i	$-0.301756\pi$
$881$	34.6274	1.16663	0.583314	+	0.812247i	$0.301756\pi$
$882$	0	0
$883$	51.3137	1.72684	0.863422	−	0.504483i	$-0.168317\pi$
$883$	51.3137	1.72684	0.863422	+	0.504483i	$0.168317\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.3431	1.15313	0.576565	−	0.817051i	$-0.304393\pi$
$887$	34.3431	1.15313	0.576565	+	0.817051i	$0.304393\pi$
$888$	0	0
$889$	−81.2548	−2.72520
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.97056	−0.300188
$894$	0	0
$895$	29.2548	0.977881
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.6863	0.423112
$900$	0	0
$901$	−2.34315	−0.0780615
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.9706	0.564121
$906$	0	0
$907$	−24.6863	−0.819695	−0.409847	−	0.912154i	$-0.634418\pi$
$907$	−24.6863	−0.819695	−0.409847	+	0.912154i	$0.634418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28.4853	−0.943759	−0.471880	−	0.881663i	$-0.656424\pi$
$911$	−28.4853	−0.943759	−0.471880	+	0.881663i	$0.656424\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	101.255	3.34373
$918$	0	0
$919$	0.544156	0.0179500	0.00897502	−	0.999960i	$-0.497143\pi$
$919$	0.544156	0.0179500	0.00897502	+	0.999960i	$0.497143\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.3431	−0.340449
$924$	0	0
$925$	−10.9706	−0.360710
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.34315	0.208112	0.104056	−	0.994571i	$-0.466818\pi$
$929$	6.34315	0.208112	0.104056	+	0.994571i	$0.466818\pi$
$930$	0	0
$931$	−32.6274	−1.06932
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	47.9411	1.56617	0.783084	−	0.621916i	$-0.213645\pi$
$937$	47.9411	1.56617	0.783084	+	0.621916i	$0.213645\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.62742	0.150849	0.0754247	−	0.997151i	$-0.475969\pi$
$941$	4.62742	0.150849	0.0754247	+	0.997151i	$0.475969\pi$
$942$	0	0
$943$	−32.9706	−1.07367
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	0	0
$949$	−7.71573	−0.250463
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32.3431	1.04770	0.523849	−	0.851811i	$-0.324496\pi$
$953$	32.3431	1.04770	0.523849	+	0.851811i	$0.324496\pi$
$954$	0	0
$955$	−3.31371	−0.107229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−85.2548	−2.75302
$960$	0	0
$961$	−28.2548	−0.911446
$962$	0	0
$963$	0	0
$964$	0	0
$965$	44.2843	1.42556
$966$	0	0
$967$	−34.4853	−1.10897	−0.554486	−	0.832193i	$-0.687085\pi$
$967$	−34.4853	−1.10897	−0.554486	+	0.832193i	$0.687085\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.65685	0.0531710	0.0265855	−	0.999647i	$-0.491537\pi$
$971$	1.65685	0.0531710	0.0265855	+	0.999647i	$0.491537\pi$
$972$	0	0
$973$	64.2843	2.06086
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.6569	−0.564893	−0.282446	−	0.959283i	$-0.591146\pi$
$977$	−17.6569	−0.564893	−0.282446	+	0.959283i	$0.591146\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.85786	−0.314417	−0.157209	−	0.987565i	$-0.550249\pi$
$983$	−9.85786	−0.314417	−0.157209	+	0.987565i	$0.550249\pi$
$984$	0	0
$985$	−26.3431	−0.839362
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.9706	−1.04840
$990$	0	0
$991$	−40.2843	−1.27967	−0.639836	−	0.768511i	$-0.720998\pi$
$991$	−40.2843	−1.27967	−0.639836	+	0.768511i	$0.720998\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	40.5685	1.28611
$996$	0	0
$997$	12.8284	0.406280	0.203140	−	0.979150i	$-0.434885\pi$
$997$	12.8284	0.406280	0.203140	+	0.979150i	$0.434885\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bh.1.1		2
3.2	odd	2		8712.2.a.bi.1.2		2
11.10	odd	2		792.2.a.i.1.1	✓	2
33.32	even	2		792.2.a.j.1.2	yes	2
44.43	even	2		1584.2.a.v.1.1		2
88.21	odd	2		6336.2.a.cr.1.2		2
88.43	even	2		6336.2.a.co.1.2		2
132.131	odd	2		1584.2.a.u.1.2		2
264.131	odd	2		6336.2.a.cp.1.1		2
264.197	even	2		6336.2.a.cq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
792.2.a.i.1.1	✓	2	11.10	odd	2
792.2.a.j.1.2	yes	2	33.32	even	2
1584.2.a.u.1.2		2	132.131	odd	2
1584.2.a.v.1.1		2	44.43	even	2
6336.2.a.co.1.2		2	88.43	even	2
6336.2.a.cp.1.1		2	264.131	odd	2
6336.2.a.cq.1.1		2	264.197	even	2
6336.2.a.cr.1.2		2	88.21	odd	2
8712.2.a.bh.1.1		2	1.1	even	1	trivial
8712.2.a.bi.1.2		2	3.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{5} -4.82843 q^{7} +O(q^{10})\)	\(q-2.82843 q^{5} -4.82843 q^{7} +0.828427 q^{13} -2.00000 q^{17} -2.00000 q^{19} +2.82843 q^{23} +3.00000 q^{25} -7.65685 q^{29} -1.65685 q^{31} +13.6569 q^{35} -3.65685 q^{37} -11.6569 q^{41} -11.6569 q^{43} +4.48528 q^{47} +16.3137 q^{49} +1.17157 q^{53} -9.65685 q^{59} -8.82843 q^{61} -2.34315 q^{65} -11.3137 q^{67} -12.4853 q^{71} -9.31371 q^{73} +0.828427 q^{79} -1.65685 q^{83} +5.65685 q^{85} +16.0000 q^{89} -4.00000 q^{91} +5.65685 q^{95} -17.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^7	\( 2 q - 4 q^{7} - 4 q^{13} - 4 q^{17} - 4 q^{19} + 6 q^{25} - 4 q^{29} + 8 q^{31} + 16 q^{35} + 4 q^{37} - 12 q^{41} - 12 q^{43} - 8 q^{47} + 10 q^{49} + 8 q^{53} - 8 q^{59} - 12 q^{61} - 16 q^{65} - 8 q^{71} + 4 q^{73} - 4 q^{79} + 8 q^{83} + 32 q^{89} - 8 q^{91} - 12 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 4 * q^13 - 4 * q^17 - 4 * q^19 + 6 * q^25 - 4 * q^29 + 8 * q^31 + 16 * q^35 + 4 * q^37 - 12 * q^41 - 12 * q^43 - 8 * q^47 + 10 * q^49 + 8 * q^53 - 8 * q^59 - 12 * q^61 - 16 * q^65 - 8 * q^71 + 4 * q^73 - 4 * q^79 + 8 * q^83 + 32 * q^89 - 8 * q^91 - 12 * q^97

Introduction

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