LMFDB - Embedded newform 8112.2.a.cp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, 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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.69202 q^{5} +0.801938 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.69202 q^{5} +0.801938 q^{7} +1.00000 q^{9} +2.85086 q^{11} +3.69202 q^{15} +2.93900 q^{17} +2.44504 q^{19} +0.801938 q^{21} +7.78986 q^{23} +8.63102 q^{25} +1.00000 q^{27} +3.85086 q^{29} -2.34481 q^{31} +2.85086 q^{33} +2.96077 q^{35} +7.44504 q^{37} -0.850855 q^{41} +1.61596 q^{43} +3.69202 q^{45} -2.44504 q^{47} -6.35690 q^{49} +2.93900 q^{51} -9.96077 q^{53} +10.5254 q^{55} +2.44504 q^{57} +5.38404 q^{59} -13.2567 q^{61} +0.801938 q^{63} -14.3937 q^{67} +7.78986 q^{69} -8.12498 q^{71} -11.8877 q^{73} +8.63102 q^{75} +2.28621 q^{77} -5.40581 q^{79} +1.00000 q^{81} -7.04892 q^{83} +10.8509 q^{85} +3.85086 q^{87} +1.13169 q^{89} -2.34481 q^{93} +9.02715 q^{95} -5.94438 q^{97} +2.85086 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 6 q^{15} - q^{17} + 7 q^{19} - 2 q^{21} + 11 q^{25} + 3 q^{27} - 2 q^{29} + 16 q^{31} - 5 q^{33} - 4 q^{35} + 22 q^{37} + 11 q^{41} + 15 q^{43} + 6 q^{45} - 7 q^{47} - 15 q^{49} - q^{51} - 17 q^{53} - 3 q^{55} + 7 q^{57} + 6 q^{59} - 13 q^{61} - 2 q^{63} - 11 q^{67} + 6 q^{73} + 11 q^{75} + 15 q^{77} - 3 q^{79} + 3 q^{81} - 12 q^{83} + 19 q^{85} - 2 q^{87} + q^{89} + 16 q^{93} + 21 q^{95} - 5 q^{97} - 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9 - 5 * q^11 + 6 * q^15 - q^17 + 7 * q^19 - 2 * q^21 + 11 * q^25 + 3 * q^27 - 2 * q^29 + 16 * q^31 - 5 * q^33 - 4 * q^35 + 22 * q^37 + 11 * q^41 + 15 * q^43 + 6 * q^45 - 7 * q^47 - 15 * q^49 - q^51 - 17 * q^53 - 3 * q^55 + 7 * q^57 + 6 * q^59 - 13 * q^61 - 2 * q^63 - 11 * q^67 + 6 * q^73 + 11 * q^75 + 15 * q^77 - 3 * q^79 + 3 * q^81 - 12 * q^83 + 19 * q^85 - 2 * q^87 + q^89 + 16 * q^93 + 21 * q^95 - 5 * q^97 - 5 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.69202	1.65112	0.825561	−	0.564313i	$-0.190859\pi$
$5$	3.69202	1.65112	0.825561	+	0.564313i	$0.190859\pi$
$6$	0	0
$7$	0.801938	0.303104	0.151552	−	0.988449i	$-0.451573\pi$
$7$	0.801938	0.303104	0.151552	+	0.988449i	$0.451573\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.85086	0.859565	0.429783	−	0.902932i	$-0.358590\pi$
$11$	2.85086	0.859565	0.429783	+	0.902932i	$0.358590\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.69202	0.953276
$16$	0	0
$17$	2.93900	0.712812	0.356406	−	0.934331i	$-0.384002\pi$
$17$	2.93900	0.712812	0.356406	+	0.934331i	$0.384002\pi$
$18$	0	0
$19$	2.44504	0.560931	0.280466	−	0.959864i	$-0.409511\pi$
$19$	2.44504	0.560931	0.280466	+	0.959864i	$0.409511\pi$
$20$	0	0
$21$	0.801938	0.174997
$22$	0	0
$23$	7.78986	1.62430	0.812149	−	0.583451i	$-0.198298\pi$
$23$	7.78986	1.62430	0.812149	+	0.583451i	$0.198298\pi$
$24$	0	0
$25$	8.63102	1.72620
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.85086	0.715086	0.357543	−	0.933897i	$-0.383615\pi$
$29$	3.85086	0.715086	0.357543	+	0.933897i	$0.383615\pi$
$30$	0	0
$31$	−2.34481	−0.421141	−0.210571	−	0.977579i	$-0.567532\pi$
$31$	−2.34481	−0.421141	−0.210571	+	0.977579i	$0.567532\pi$
$32$	0	0
$33$	2.85086	0.496270
$34$	0	0
$35$	2.96077	0.500462
$36$	0	0
$37$	7.44504	1.22396	0.611979	−	0.790874i	$-0.290374\pi$
$37$	7.44504	1.22396	0.611979	+	0.790874i	$0.290374\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.850855	−0.132881	−0.0664406	−	0.997790i	$-0.521164\pi$
$41$	−0.850855	−0.132881	−0.0664406	+	0.997790i	$0.521164\pi$
$42$	0	0
$43$	1.61596	0.246431	0.123216	−	0.992380i	$-0.460679\pi$
$43$	1.61596	0.246431	0.123216	+	0.992380i	$0.460679\pi$
$44$	0	0
$45$	3.69202	0.550374
$46$	0	0
$47$	−2.44504	−0.356646	−0.178323	−	0.983972i	$-0.557067\pi$
$47$	−2.44504	−0.356646	−0.178323	+	0.983972i	$0.557067\pi$
$48$	0	0
$49$	−6.35690	−0.908128
$50$	0	0
$51$	2.93900	0.411542
$52$	0	0
$53$	−9.96077	−1.36822	−0.684109	−	0.729380i	$-0.739809\pi$
$53$	−9.96077	−1.36822	−0.684109	+	0.729380i	$0.739809\pi$
$54$	0	0
$55$	10.5254	1.41925
$56$	0	0
$57$	2.44504	0.323854
$58$	0	0
$59$	5.38404	0.700943	0.350471	−	0.936573i	$-0.386021\pi$
$59$	5.38404	0.700943	0.350471	+	0.936573i	$0.386021\pi$
$60$	0	0
$61$	−13.2567	−1.69734	−0.848671	−	0.528921i	$-0.822597\pi$
$61$	−13.2567	−1.69734	−0.848671	+	0.528921i	$0.822597\pi$
$62$	0	0
$63$	0.801938	0.101035
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.3937	−1.75847	−0.879237	−	0.476384i	$-0.841947\pi$
$67$	−14.3937	−1.75847	−0.879237	+	0.476384i	$0.841947\pi$
$68$	0	0
$69$	7.78986	0.937788
$70$	0	0
$71$	−8.12498	−0.964258	−0.482129	−	0.876100i	$-0.660136\pi$
$71$	−8.12498	−0.964258	−0.482129	+	0.876100i	$0.660136\pi$
$72$	0	0
$73$	−11.8877	−1.39135	−0.695674	−	0.718357i	$-0.744894\pi$
$73$	−11.8877	−1.39135	−0.695674	+	0.718357i	$0.744894\pi$
$74$	0	0
$75$	8.63102	0.996625
$76$	0	0
$77$	2.28621	0.260538
$78$	0	0
$79$	−5.40581	−0.608202	−0.304101	−	0.952640i	$-0.598356\pi$
$79$	−5.40581	−0.608202	−0.304101	+	0.952640i	$0.598356\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.04892	−0.773719	−0.386860	−	0.922139i	$-0.626440\pi$
$83$	−7.04892	−0.773719	−0.386860	+	0.922139i	$0.626440\pi$
$84$	0	0
$85$	10.8509	1.17694
$86$	0	0
$87$	3.85086	0.412855
$88$	0	0
$89$	1.13169	0.119959	0.0599793	−	0.998200i	$-0.480897\pi$
$89$	1.13169	0.119959	0.0599793	+	0.998200i	$0.480897\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.34481	−0.243146
$94$	0	0
$95$	9.02715	0.926166
$96$	0	0
$97$	−5.94438	−0.603560	−0.301780	−	0.953378i	$-0.597581\pi$
$97$	−5.94438	−0.603560	−0.301780	+	0.953378i	$0.597581\pi$
$98$	0	0
$99$	2.85086	0.286522
$100$	0	0
$101$	4.62565	0.460269	0.230134	−	0.973159i	$-0.426083\pi$
$101$	4.62565	0.460269	0.230134	+	0.973159i	$0.426083\pi$
$102$	0	0
$103$	−1.20775	−0.119003	−0.0595016	−	0.998228i	$-0.518951\pi$
$103$	−1.20775	−0.119003	−0.0595016	+	0.998228i	$0.518951\pi$
$104$	0	0
$105$	2.96077	0.288942
$106$	0	0
$107$	9.52111	0.920440	0.460220	−	0.887805i	$-0.347771\pi$
$107$	9.52111	0.920440	0.460220	+	0.887805i	$0.347771\pi$
$108$	0	0
$109$	−1.78448	−0.170922	−0.0854611	−	0.996342i	$-0.527236\pi$
$109$	−1.78448	−0.170922	−0.0854611	+	0.996342i	$0.527236\pi$
$110$	0	0
$111$	7.44504	0.706652
$112$	0	0
$113$	4.95108	0.465759	0.232879	−	0.972506i	$-0.425185\pi$
$113$	4.95108	0.465759	0.232879	+	0.972506i	$0.425185\pi$
$114$	0	0
$115$	28.7603	2.68191
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.35690	0.216056
$120$	0	0
$121$	−2.87263	−0.261148
$122$	0	0
$123$	−0.850855	−0.0767190
$124$	0	0
$125$	13.4058	1.19905
$126$	0	0
$127$	−5.67025	−0.503153	−0.251577	−	0.967837i	$-0.580949\pi$
$127$	−5.67025	−0.503153	−0.251577	+	0.967837i	$0.580949\pi$
$128$	0	0
$129$	1.61596	0.142277
$130$	0	0
$131$	−18.2228	−1.59213	−0.796067	−	0.605208i	$-0.793090\pi$
$131$	−18.2228	−1.59213	−0.796067	+	0.605208i	$0.793090\pi$
$132$	0	0
$133$	1.96077	0.170020
$134$	0	0
$135$	3.69202	0.317759
$136$	0	0
$137$	9.45042	0.807404	0.403702	−	0.914891i	$-0.367723\pi$
$137$	9.45042	0.807404	0.403702	+	0.914891i	$0.367723\pi$
$138$	0	0
$139$	−4.01507	−0.340553	−0.170277	−	0.985396i	$-0.554466\pi$
$139$	−4.01507	−0.340553	−0.170277	+	0.985396i	$0.554466\pi$
$140$	0	0
$141$	−2.44504	−0.205910
$142$	0	0
$143$	0	0
$144$	0	0
$145$	14.2174	1.18069
$146$	0	0
$147$	−6.35690	−0.524308
$148$	0	0
$149$	19.4058	1.58979	0.794893	−	0.606750i	$-0.207527\pi$
$149$	19.4058	1.58979	0.794893	+	0.606750i	$0.207527\pi$
$150$	0	0
$151$	12.3623	1.00603	0.503014	−	0.864278i	$-0.332225\pi$
$151$	12.3623	1.00603	0.503014	+	0.864278i	$0.332225\pi$
$152$	0	0
$153$	2.93900	0.237604
$154$	0	0
$155$	−8.65710	−0.695355
$156$	0	0
$157$	−18.6775	−1.49063	−0.745315	−	0.666712i	$-0.767701\pi$
$157$	−18.6775	−1.49063	−0.745315	+	0.666712i	$0.767701\pi$
$158$	0	0
$159$	−9.96077	−0.789941
$160$	0	0
$161$	6.24698	0.492331
$162$	0	0
$163$	−12.3394	−0.966499	−0.483250	−	0.875483i	$-0.660544\pi$
$163$	−12.3394	−0.966499	−0.483250	+	0.875483i	$0.660544\pi$
$164$	0	0
$165$	10.5254	0.819403
$166$	0	0
$167$	−11.4940	−0.889429	−0.444715	−	0.895672i	$-0.646695\pi$
$167$	−11.4940	−0.889429	−0.444715	+	0.895672i	$0.646695\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	2.44504	0.186977
$172$	0	0
$173$	12.1142	0.921028	0.460514	−	0.887653i	$-0.347665\pi$
$173$	12.1142	0.921028	0.460514	+	0.887653i	$0.347665\pi$
$174$	0	0
$175$	6.92154	0.523219
$176$	0	0
$177$	5.38404	0.404689
$178$	0	0
$179$	0.538565	0.0402542	0.0201271	−	0.999797i	$-0.493593\pi$
$179$	0.538565	0.0402542	0.0201271	+	0.999797i	$0.493593\pi$
$180$	0	0
$181$	−23.2838	−1.73067	−0.865336	−	0.501192i	$-0.832895\pi$
$181$	−23.2838	−1.73067	−0.865336	+	0.501192i	$0.832895\pi$
$182$	0	0
$183$	−13.2567	−0.979961
$184$	0	0
$185$	27.4873	2.02090
$186$	0	0
$187$	8.37867	0.612709
$188$	0	0
$189$	0.801938	0.0583324
$190$	0	0
$191$	−16.7657	−1.21312	−0.606561	−	0.795037i	$-0.707452\pi$
$191$	−16.7657	−1.21312	−0.606561	+	0.795037i	$0.707452\pi$
$192$	0	0
$193$	25.7439	1.85309	0.926544	−	0.376186i	$-0.122765\pi$
$193$	25.7439	1.85309	0.926544	+	0.376186i	$0.122765\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.4209	1.52617	0.763087	−	0.646296i	$-0.223683\pi$
$197$	21.4209	1.52617	0.763087	+	0.646296i	$0.223683\pi$
$198$	0	0
$199$	−3.52781	−0.250080	−0.125040	−	0.992152i	$-0.539906\pi$
$199$	−3.52781	−0.250080	−0.125040	+	0.992152i	$0.539906\pi$
$200$	0	0
$201$	−14.3937	−1.01526
$202$	0	0
$203$	3.08815	0.216745
$204$	0	0
$205$	−3.14138	−0.219403
$206$	0	0
$207$	7.78986	0.541432
$208$	0	0
$209$	6.97046	0.482157
$210$	0	0
$211$	−1.21552	−0.0836799	−0.0418399	−	0.999124i	$-0.513322\pi$
$211$	−1.21552	−0.0836799	−0.0418399	+	0.999124i	$0.513322\pi$
$212$	0	0
$213$	−8.12498	−0.556715
$214$	0	0
$215$	5.96615	0.406888
$216$	0	0
$217$	−1.88040	−0.127650
$218$	0	0
$219$	−11.8877	−0.803296
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.3884	1.16441	0.582205	−	0.813042i	$-0.302190\pi$
$223$	17.3884	1.16441	0.582205	+	0.813042i	$0.302190\pi$
$224$	0	0
$225$	8.63102	0.575402
$226$	0	0
$227$	−17.4155	−1.15591	−0.577954	−	0.816070i	$-0.696149\pi$
$227$	−17.4155	−1.15591	−0.577954	+	0.816070i	$0.696149\pi$
$228$	0	0
$229$	18.7603	1.23972	0.619858	−	0.784714i	$-0.287190\pi$
$229$	18.7603	1.23972	0.619858	+	0.784714i	$0.287190\pi$
$230$	0	0
$231$	2.28621	0.150421
$232$	0	0
$233$	3.95108	0.258844	0.129422	−	0.991590i	$-0.458688\pi$
$233$	3.95108	0.258844	0.129422	+	0.991590i	$0.458688\pi$
$234$	0	0
$235$	−9.02715	−0.588866
$236$	0	0
$237$	−5.40581	−0.351145
$238$	0	0
$239$	−0.818331	−0.0529334	−0.0264667	−	0.999650i	$-0.508426\pi$
$239$	−0.818331	−0.0529334	−0.0264667	+	0.999650i	$0.508426\pi$
$240$	0	0
$241$	6.03252	0.388589	0.194295	−	0.980943i	$-0.437758\pi$
$241$	6.03252	0.388589	0.194295	+	0.980943i	$0.437758\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−23.4698	−1.49943
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−7.04892	−0.446707
$250$	0	0
$251$	26.8799	1.69665	0.848323	−	0.529479i	$-0.177613\pi$
$251$	26.8799	1.69665	0.848323	+	0.529479i	$0.177613\pi$
$252$	0	0
$253$	22.2078	1.39619
$254$	0	0
$255$	10.8509	0.679507
$256$	0	0
$257$	−9.05323	−0.564725	−0.282362	−	0.959308i	$-0.591118\pi$
$257$	−9.05323	−0.564725	−0.282362	+	0.959308i	$0.591118\pi$
$258$	0	0
$259$	5.97046	0.370986
$260$	0	0
$261$	3.85086	0.238362
$262$	0	0
$263$	−23.1511	−1.42756	−0.713778	−	0.700372i	$-0.753017\pi$
$263$	−23.1511	−1.42756	−0.713778	+	0.700372i	$0.753017\pi$
$264$	0	0
$265$	−36.7754	−2.25909
$266$	0	0
$267$	1.13169	0.0692581
$268$	0	0
$269$	−2.42088	−0.147604	−0.0738018	−	0.997273i	$-0.523513\pi$
$269$	−2.42088	−0.147604	−0.0738018	+	0.997273i	$0.523513\pi$
$270$	0	0
$271$	21.4450	1.30269	0.651347	−	0.758780i	$-0.274204\pi$
$271$	21.4450	1.30269	0.651347	+	0.758780i	$0.274204\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.6058	1.48379
$276$	0	0
$277$	−14.8073	−0.889685	−0.444843	−	0.895609i	$-0.646740\pi$
$277$	−14.8073	−0.889685	−0.444843	+	0.895609i	$0.646740\pi$
$278$	0	0
$279$	−2.34481	−0.140380
$280$	0	0
$281$	14.5036	0.865215	0.432608	−	0.901582i	$-0.357594\pi$
$281$	14.5036	0.865215	0.432608	+	0.901582i	$0.357594\pi$
$282$	0	0
$283$	−25.6722	−1.52605	−0.763026	−	0.646368i	$-0.776287\pi$
$283$	−25.6722	−1.52605	−0.763026	+	0.646368i	$0.776287\pi$
$284$	0	0
$285$	9.02715	0.534722
$286$	0	0
$287$	−0.682333	−0.0402768
$288$	0	0
$289$	−8.36227	−0.491898
$290$	0	0
$291$	−5.94438	−0.348466
$292$	0	0
$293$	−26.5230	−1.54949	−0.774746	−	0.632273i	$-0.782122\pi$
$293$	−26.5230	−1.54949	−0.774746	+	0.632273i	$0.782122\pi$
$294$	0	0
$295$	19.8780	1.15734
$296$	0	0
$297$	2.85086	0.165423
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.29590	0.0746943
$302$	0	0
$303$	4.62565	0.265736
$304$	0	0
$305$	−48.9439	−2.80252
$306$	0	0
$307$	8.24698	0.470680	0.235340	−	0.971913i	$-0.424380\pi$
$307$	8.24698	0.470680	0.235340	+	0.971913i	$0.424380\pi$
$308$	0	0
$309$	−1.20775	−0.0687066
$310$	0	0
$311$	14.4179	0.817564	0.408782	−	0.912632i	$-0.365954\pi$
$311$	14.4179	0.817564	0.408782	+	0.912632i	$0.365954\pi$
$312$	0	0
$313$	14.2338	0.804544	0.402272	−	0.915520i	$-0.368221\pi$
$313$	14.2338	0.804544	0.402272	+	0.915520i	$0.368221\pi$
$314$	0	0
$315$	2.96077	0.166821
$316$	0	0
$317$	−6.84415	−0.384406	−0.192203	−	0.981355i	$-0.561563\pi$
$317$	−6.84415	−0.384406	−0.192203	+	0.981355i	$0.561563\pi$
$318$	0	0
$319$	10.9782	0.614663
$320$	0	0
$321$	9.52111	0.531416
$322$	0	0
$323$	7.18598	0.399839
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.78448	−0.0986819
$328$	0	0
$329$	−1.96077	−0.108101
$330$	0	0
$331$	−9.44265	−0.519015	−0.259507	−	0.965741i	$-0.583560\pi$
$331$	−9.44265	−0.519015	−0.259507	+	0.965741i	$0.583560\pi$
$332$	0	0
$333$	7.44504	0.407986
$334$	0	0
$335$	−53.1420	−2.90346
$336$	0	0
$337$	2.64310	0.143979	0.0719895	−	0.997405i	$-0.477065\pi$
$337$	2.64310	0.143979	0.0719895	+	0.997405i	$0.477065\pi$
$338$	0	0
$339$	4.95108	0.268906
$340$	0	0
$341$	−6.68473	−0.361998
$342$	0	0
$343$	−10.7114	−0.578361
$344$	0	0
$345$	28.7603	1.54840
$346$	0	0
$347$	10.1588	0.545355	0.272677	−	0.962106i	$-0.412091\pi$
$347$	10.1588	0.545355	0.272677	+	0.962106i	$0.412091\pi$
$348$	0	0
$349$	10.4397	0.558822	0.279411	−	0.960172i	$-0.409861\pi$
$349$	10.4397	0.558822	0.279411	+	0.960172i	$0.409861\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.2911	0.973538	0.486769	−	0.873531i	$-0.338175\pi$
$353$	18.2911	0.973538	0.486769	+	0.873531i	$0.338175\pi$
$354$	0	0
$355$	−29.9976	−1.59211
$356$	0	0
$357$	2.35690	0.124740
$358$	0	0
$359$	15.2731	0.806081	0.403041	−	0.915182i	$-0.367953\pi$
$359$	15.2731	0.806081	0.403041	+	0.915182i	$0.367953\pi$
$360$	0	0
$361$	−13.0218	−0.685356
$362$	0	0
$363$	−2.87263	−0.150774
$364$	0	0
$365$	−43.8896	−2.29729
$366$	0	0
$367$	22.2717	1.16258	0.581288	−	0.813698i	$-0.302549\pi$
$367$	22.2717	1.16258	0.581288	+	0.813698i	$0.302549\pi$
$368$	0	0
$369$	−0.850855	−0.0442937
$370$	0	0
$371$	−7.98792	−0.414712
$372$	0	0
$373$	4.12631	0.213652	0.106826	−	0.994278i	$-0.465931\pi$
$373$	4.12631	0.213652	0.106826	+	0.994278i	$0.465931\pi$
$374$	0	0
$375$	13.4058	0.692273
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.7071	0.549986	0.274993	−	0.961446i	$-0.411324\pi$
$379$	10.7071	0.549986	0.274993	+	0.961446i	$0.411324\pi$
$380$	0	0
$381$	−5.67025	−0.290496
$382$	0	0
$383$	−6.52648	−0.333488	−0.166744	−	0.986000i	$-0.553325\pi$
$383$	−6.52648	−0.333488	−0.166744	+	0.986000i	$0.553325\pi$
$384$	0	0
$385$	8.44073	0.430179
$386$	0	0
$387$	1.61596	0.0821437
$388$	0	0
$389$	−11.7922	−0.597891	−0.298945	−	0.954270i	$-0.596635\pi$
$389$	−11.7922	−0.597891	−0.298945	+	0.954270i	$0.596635\pi$
$390$	0	0
$391$	22.8944	1.15782
$392$	0	0
$393$	−18.2228	−0.919219
$394$	0	0
$395$	−19.9584	−1.00422
$396$	0	0
$397$	−12.5429	−0.629509	−0.314754	−	0.949173i	$-0.601922\pi$
$397$	−12.5429	−0.629509	−0.314754	+	0.949173i	$0.601922\pi$
$398$	0	0
$399$	1.96077	0.0981613
$400$	0	0
$401$	17.8702	0.892397	0.446198	−	0.894934i	$-0.352778\pi$
$401$	17.8702	0.892397	0.446198	+	0.894934i	$0.352778\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.69202	0.183458
$406$	0	0
$407$	21.2247	1.05207
$408$	0	0
$409$	−27.9119	−1.38015	−0.690076	−	0.723737i	$-0.742423\pi$
$409$	−27.9119	−1.38015	−0.690076	+	0.723737i	$0.742423\pi$
$410$	0	0
$411$	9.45042	0.466155
$412$	0	0
$413$	4.31767	0.212459
$414$	0	0
$415$	−26.0248	−1.27750
$416$	0	0
$417$	−4.01507	−0.196619
$418$	0	0
$419$	16.4034	0.801360	0.400680	−	0.916218i	$-0.368774\pi$
$419$	16.4034	0.801360	0.400680	+	0.916218i	$0.368774\pi$
$420$	0	0
$421$	−3.03684	−0.148006	−0.0740032	−	0.997258i	$-0.523577\pi$
$421$	−3.03684	−0.148006	−0.0740032	+	0.997258i	$0.523577\pi$
$422$	0	0
$423$	−2.44504	−0.118882
$424$	0	0
$425$	25.3666	1.23046
$426$	0	0
$427$	−10.6310	−0.514471
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.33811	0.160791	0.0803955	−	0.996763i	$-0.474382\pi$
$431$	3.33811	0.160791	0.0803955	+	0.996763i	$0.474382\pi$
$432$	0	0
$433$	11.9028	0.572010	0.286005	−	0.958228i	$-0.407673\pi$
$433$	11.9028	0.572010	0.286005	+	0.958228i	$0.407673\pi$
$434$	0	0
$435$	14.2174	0.681674
$436$	0	0
$437$	19.0465	0.911119
$438$	0	0
$439$	−3.71810	−0.177455	−0.0887277	−	0.996056i	$-0.528280\pi$
$439$	−3.71810	−0.177455	−0.0887277	+	0.996056i	$0.528280\pi$
$440$	0	0
$441$	−6.35690	−0.302709
$442$	0	0
$443$	−1.45712	−0.0692300	−0.0346150	−	0.999401i	$-0.511021\pi$
$443$	−1.45712	−0.0692300	−0.0346150	+	0.999401i	$0.511021\pi$
$444$	0	0
$445$	4.17821	0.198066
$446$	0	0
$447$	19.4058	0.917863
$448$	0	0
$449$	−12.1274	−0.572326	−0.286163	−	0.958181i	$-0.592380\pi$
$449$	−12.1274	−0.572326	−0.286163	+	0.958181i	$0.592380\pi$
$450$	0	0
$451$	−2.42566	−0.114220
$452$	0	0
$453$	12.3623	0.580830
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.44803	−0.161292	−0.0806459	−	0.996743i	$-0.525698\pi$
$457$	−3.44803	−0.161292	−0.0806459	+	0.996743i	$0.525698\pi$
$458$	0	0
$459$	2.93900	0.137181
$460$	0	0
$461$	−6.75600	−0.314658	−0.157329	−	0.987546i	$-0.550288\pi$
$461$	−6.75600	−0.314658	−0.157329	+	0.987546i	$0.550288\pi$
$462$	0	0
$463$	7.45175	0.346312	0.173156	−	0.984894i	$-0.444604\pi$
$463$	7.45175	0.346312	0.173156	+	0.984894i	$0.444604\pi$
$464$	0	0
$465$	−8.65710	−0.401464
$466$	0	0
$467$	−32.6098	−1.50900	−0.754502	−	0.656298i	$-0.772121\pi$
$467$	−32.6098	−1.50900	−0.754502	+	0.656298i	$0.772121\pi$
$468$	0	0
$469$	−11.5429	−0.533001
$470$	0	0
$471$	−18.6775	−0.860616
$472$	0	0
$473$	4.60686	0.211824
$474$	0	0
$475$	21.1032	0.968282
$476$	0	0
$477$	−9.96077	−0.456072
$478$	0	0
$479$	−2.82908	−0.129264	−0.0646321	−	0.997909i	$-0.520587\pi$
$479$	−2.82908	−0.129264	−0.0646321	+	0.997909i	$0.520587\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	6.24698	0.284247
$484$	0	0
$485$	−21.9468	−0.996552
$486$	0	0
$487$	41.2935	1.87119	0.935594	−	0.353079i	$-0.114865\pi$
$487$	41.2935	1.87119	0.935594	+	0.353079i	$0.114865\pi$
$488$	0	0
$489$	−12.3394	−0.558009
$490$	0	0
$491$	34.6698	1.56463	0.782313	−	0.622886i	$-0.214040\pi$
$491$	34.6698	1.56463	0.782313	+	0.622886i	$0.214040\pi$
$492$	0	0
$493$	11.3177	0.509722
$494$	0	0
$495$	10.5254	0.473082
$496$	0	0
$497$	−6.51573	−0.292270
$498$	0	0
$499$	−17.9409	−0.803146	−0.401573	−	0.915827i	$-0.631536\pi$
$499$	−17.9409	−0.803146	−0.401573	+	0.915827i	$0.631536\pi$
$500$	0	0
$501$	−11.4940	−0.513512
$502$	0	0
$503$	26.1812	1.16736	0.583681	−	0.811983i	$-0.301612\pi$
$503$	26.1812	1.16736	0.583681	+	0.811983i	$0.301612\pi$
$504$	0	0
$505$	17.0780	0.759960
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.50604	−0.244051	−0.122025	−	0.992527i	$-0.538939\pi$
$509$	−5.50604	−0.244051	−0.122025	+	0.992527i	$0.538939\pi$
$510$	0	0
$511$	−9.53319	−0.421723
$512$	0	0
$513$	2.44504	0.107951
$514$	0	0
$515$	−4.45904	−0.196489
$516$	0	0
$517$	−6.97046	−0.306560
$518$	0	0
$519$	12.1142	0.531756
$520$	0	0
$521$	−26.7211	−1.17067	−0.585336	−	0.810791i	$-0.699037\pi$
$521$	−26.7211	−1.17067	−0.585336	+	0.810791i	$0.699037\pi$
$522$	0	0
$523$	−36.5230	−1.59704	−0.798520	−	0.601968i	$-0.794383\pi$
$523$	−36.5230	−1.59704	−0.798520	+	0.601968i	$0.794383\pi$
$524$	0	0
$525$	6.92154	0.302081
$526$	0	0
$527$	−6.89141	−0.300195
$528$	0	0
$529$	37.6819	1.63834
$530$	0	0
$531$	5.38404	0.233648
$532$	0	0
$533$	0	0
$534$	0	0
$535$	35.1521	1.51976
$536$	0	0
$537$	0.538565	0.0232408
$538$	0	0
$539$	−18.1226	−0.780595
$540$	0	0
$541$	18.4655	0.793893	0.396947	−	0.917842i	$-0.370070\pi$
$541$	18.4655	0.793893	0.396947	+	0.917842i	$0.370070\pi$
$542$	0	0
$543$	−23.2838	−0.999204
$544$	0	0
$545$	−6.58834	−0.282213
$546$	0	0
$547$	−39.8471	−1.70374	−0.851870	−	0.523753i	$-0.824532\pi$
$547$	−39.8471	−1.70374	−0.851870	+	0.523753i	$0.824532\pi$
$548$	0	0
$549$	−13.2567	−0.565781
$550$	0	0
$551$	9.41550	0.401114
$552$	0	0
$553$	−4.33513	−0.184348
$554$	0	0
$555$	27.4873	1.16677
$556$	0	0
$557$	9.20477	0.390018	0.195009	−	0.980801i	$-0.437526\pi$
$557$	9.20477	0.390018	0.195009	+	0.980801i	$0.437526\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.37867	0.353748
$562$	0	0
$563$	0.975246	0.0411017	0.0205509	−	0.999789i	$-0.493458\pi$
$563$	0.975246	0.0411017	0.0205509	+	0.999789i	$0.493458\pi$
$564$	0	0
$565$	18.2795	0.769024
$566$	0	0
$567$	0.801938	0.0336782
$568$	0	0
$569$	−16.8944	−0.708250	−0.354125	−	0.935198i	$-0.615221\pi$
$569$	−16.8944	−0.708250	−0.354125	+	0.935198i	$0.615221\pi$
$570$	0	0
$571$	44.3226	1.85484	0.927421	−	0.374019i	$-0.122021\pi$
$571$	44.3226	1.85484	0.927421	+	0.374019i	$0.122021\pi$
$572$	0	0
$573$	−16.7657	−0.700397
$574$	0	0
$575$	67.2344	2.80387
$576$	0	0
$577$	3.56704	0.148498	0.0742489	−	0.997240i	$-0.476344\pi$
$577$	3.56704	0.148498	0.0742489	+	0.997240i	$0.476344\pi$
$578$	0	0
$579$	25.7439	1.06988
$580$	0	0
$581$	−5.65279	−0.234517
$582$	0	0
$583$	−28.3967	−1.17607
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.1172	0.665229	0.332614	−	0.943063i	$-0.392069\pi$
$587$	16.1172	0.665229	0.332614	+	0.943063i	$0.392069\pi$
$588$	0	0
$589$	−5.73317	−0.236231
$590$	0	0
$591$	21.4209	0.881137
$592$	0	0
$593$	−42.8611	−1.76010	−0.880048	−	0.474885i	$-0.842490\pi$
$593$	−42.8611	−1.76010	−0.880048	+	0.474885i	$0.842490\pi$
$594$	0	0
$595$	8.70171	0.356735
$596$	0	0
$597$	−3.52781	−0.144384
$598$	0	0
$599$	−40.9420	−1.67284	−0.836422	−	0.548086i	$-0.815357\pi$
$599$	−40.9420	−1.67284	−0.836422	+	0.548086i	$0.815357\pi$
$600$	0	0
$601$	1.18705	0.0484206	0.0242103	−	0.999707i	$-0.492293\pi$
$601$	1.18705	0.0484206	0.0242103	+	0.999707i	$0.492293\pi$
$602$	0	0
$603$	−14.3937	−0.586158
$604$	0	0
$605$	−10.6058	−0.431187
$606$	0	0
$607$	−19.9922	−0.811460	−0.405730	−	0.913993i	$-0.632983\pi$
$607$	−19.9922	−0.811460	−0.405730	+	0.913993i	$0.632983\pi$
$608$	0	0
$609$	3.08815	0.125138
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−33.3618	−1.34747	−0.673735	−	0.738973i	$-0.735311\pi$
$613$	−33.3618	−1.34747	−0.673735	+	0.738973i	$0.735311\pi$
$614$	0	0
$615$	−3.14138	−0.126672
$616$	0	0
$617$	11.6233	0.467935	0.233967	−	0.972244i	$-0.424829\pi$
$617$	11.6233	0.467935	0.233967	+	0.972244i	$0.424829\pi$
$618$	0	0
$619$	16.5381	0.664722	0.332361	−	0.943152i	$-0.392155\pi$
$619$	16.5381	0.664722	0.332361	+	0.943152i	$0.392155\pi$
$620$	0	0
$621$	7.78986	0.312596
$622$	0	0
$623$	0.907542	0.0363599
$624$	0	0
$625$	6.33944	0.253577
$626$	0	0
$627$	6.97046	0.278373
$628$	0	0
$629$	21.8810	0.872452
$630$	0	0
$631$	36.4416	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$631$	36.4416	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$632$	0	0
$633$	−1.21552	−0.0483126
$634$	0	0
$635$	−20.9347	−0.830768
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.12498	−0.321419
$640$	0	0
$641$	27.2067	1.07460	0.537300	−	0.843391i	$-0.319444\pi$
$641$	27.2067	1.07460	0.537300	+	0.843391i	$0.319444\pi$
$642$	0	0
$643$	5.06962	0.199926	0.0999632	−	0.994991i	$-0.468127\pi$
$643$	5.06962	0.199926	0.0999632	+	0.994991i	$0.468127\pi$
$644$	0	0
$645$	5.96615	0.234917
$646$	0	0
$647$	−19.3207	−0.759573	−0.379787	−	0.925074i	$-0.624003\pi$
$647$	−19.3207	−0.759573	−0.379787	+	0.925074i	$0.624003\pi$
$648$	0	0
$649$	15.3491	0.602506
$650$	0	0
$651$	−1.88040	−0.0736985
$652$	0	0
$653$	35.2355	1.37887	0.689436	−	0.724347i	$-0.257859\pi$
$653$	35.2355	1.37887	0.689436	+	0.724347i	$0.257859\pi$
$654$	0	0
$655$	−67.2790	−2.62881
$656$	0	0
$657$	−11.8877	−0.463783
$658$	0	0
$659$	4.36168	0.169907	0.0849535	−	0.996385i	$-0.472926\pi$
$659$	4.36168	0.169907	0.0849535	+	0.996385i	$0.472926\pi$
$660$	0	0
$661$	−15.4709	−0.601747	−0.300873	−	0.953664i	$-0.597278\pi$
$661$	−15.4709	−0.601747	−0.300873	+	0.953664i	$0.597278\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.23921	0.280725
$666$	0	0
$667$	29.9976	1.16151
$668$	0	0
$669$	17.3884	0.672273
$670$	0	0
$671$	−37.7928	−1.45898
$672$	0	0
$673$	−11.7409	−0.452580	−0.226290	−	0.974060i	$-0.572660\pi$
$673$	−11.7409	−0.452580	−0.226290	+	0.974060i	$0.572660\pi$
$674$	0	0
$675$	8.63102	0.332208
$676$	0	0
$677$	3.44504	0.132404	0.0662019	−	0.997806i	$-0.478912\pi$
$677$	3.44504	0.132404	0.0662019	+	0.997806i	$0.478912\pi$
$678$	0	0
$679$	−4.76702	−0.182941
$680$	0	0
$681$	−17.4155	−0.667363
$682$	0	0
$683$	−20.4058	−0.780807	−0.390403	−	0.920644i	$-0.627664\pi$
$683$	−20.4058	−0.780807	−0.390403	+	0.920644i	$0.627664\pi$
$684$	0	0
$685$	34.8911	1.33312
$686$	0	0
$687$	18.7603	0.715751
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−27.8039	−1.05771	−0.528854	−	0.848713i	$-0.677378\pi$
$691$	−27.8039	−1.05771	−0.528854	+	0.848713i	$0.677378\pi$
$692$	0	0
$693$	2.28621	0.0868459
$694$	0	0
$695$	−14.8237	−0.562295
$696$	0	0
$697$	−2.50066	−0.0947194
$698$	0	0
$699$	3.95108	0.149444
$700$	0	0
$701$	11.9715	0.452158	0.226079	−	0.974109i	$-0.427409\pi$
$701$	11.9715	0.452158	0.226079	+	0.974109i	$0.427409\pi$
$702$	0	0
$703$	18.2034	0.686556
$704$	0	0
$705$	−9.02715	−0.339982
$706$	0	0
$707$	3.70948	0.139509
$708$	0	0
$709$	32.2664	1.21179	0.605894	−	0.795545i	$-0.292815\pi$
$709$	32.2664	1.21179	0.605894	+	0.795545i	$0.292815\pi$
$710$	0	0
$711$	−5.40581	−0.202734
$712$	0	0
$713$	−18.2658	−0.684058
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.818331	−0.0305611
$718$	0	0
$719$	−12.1086	−0.451574	−0.225787	−	0.974177i	$-0.572495\pi$
$719$	−12.1086	−0.451574	−0.225787	+	0.974177i	$0.572495\pi$
$720$	0	0
$721$	−0.968541	−0.0360704
$722$	0	0
$723$	6.03252	0.224352
$724$	0	0
$725$	33.2368	1.23438
$726$	0	0
$727$	16.6200	0.616402	0.308201	−	0.951321i	$-0.400273\pi$
$727$	16.6200	0.616402	0.308201	+	0.951321i	$0.400273\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.74930	0.175659
$732$	0	0
$733$	17.7912	0.657132	0.328566	−	0.944481i	$-0.393435\pi$
$733$	17.7912	0.657132	0.328566	+	0.944481i	$0.393435\pi$
$734$	0	0
$735$	−23.4698	−0.865696
$736$	0	0
$737$	−41.0344	−1.51152
$738$	0	0
$739$	27.3618	1.00652	0.503260	−	0.864135i	$-0.332134\pi$
$739$	27.3618	1.00652	0.503260	+	0.864135i	$0.332134\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.38596	−0.307651	−0.153826	−	0.988098i	$-0.549159\pi$
$743$	−8.38596	−0.307651	−0.153826	+	0.988098i	$0.549159\pi$
$744$	0	0
$745$	71.6467	2.62493
$746$	0	0
$747$	−7.04892	−0.257906
$748$	0	0
$749$	7.63533	0.278989
$750$	0	0
$751$	38.7778	1.41502	0.707511	−	0.706703i	$-0.249818\pi$
$751$	38.7778	1.41502	0.707511	+	0.706703i	$0.249818\pi$
$752$	0	0
$753$	26.8799	0.979559
$754$	0	0
$755$	45.6418	1.66107
$756$	0	0
$757$	12.9729	0.471506	0.235753	−	0.971813i	$-0.424244\pi$
$757$	12.9729	0.471506	0.235753	+	0.971813i	$0.424244\pi$
$758$	0	0
$759$	22.2078	0.806090
$760$	0	0
$761$	−5.15585	−0.186899	−0.0934497	−	0.995624i	$-0.529789\pi$
$761$	−5.15585	−0.186899	−0.0934497	+	0.995624i	$0.529789\pi$
$762$	0	0
$763$	−1.43104	−0.0518072
$764$	0	0
$765$	10.8509	0.392313
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−35.5013	−1.28021	−0.640104	−	0.768288i	$-0.721109\pi$
$769$	−35.5013	−1.28021	−0.640104	+	0.768288i	$0.721109\pi$
$770$	0	0
$771$	−9.05323	−0.326044
$772$	0	0
$773$	−6.15585	−0.221411	−0.110705	−	0.993853i	$-0.535311\pi$
$773$	−6.15585	−0.221411	−0.110705	+	0.993853i	$0.535311\pi$
$774$	0	0
$775$	−20.2381	−0.726976
$776$	0	0
$777$	5.97046	0.214189
$778$	0	0
$779$	−2.08038	−0.0745372
$780$	0	0
$781$	−23.1631	−0.828843
$782$	0	0
$783$	3.85086	0.137618
$784$	0	0
$785$	−68.9579	−2.46121
$786$	0	0
$787$	−14.2107	−0.506558	−0.253279	−	0.967393i	$-0.581509\pi$
$787$	−14.2107	−0.506558	−0.253279	+	0.967393i	$0.581509\pi$
$788$	0	0
$789$	−23.1511	−0.824200
$790$	0	0
$791$	3.97046	0.141173
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−36.7754	−1.30429
$796$	0	0
$797$	11.9022	0.421596	0.210798	−	0.977530i	$-0.432394\pi$
$797$	11.9022	0.421596	0.210798	+	0.977530i	$0.432394\pi$
$798$	0	0
$799$	−7.18598	−0.254222
$800$	0	0
$801$	1.13169	0.0399862
$802$	0	0
$803$	−33.8901	−1.19596
$804$	0	0
$805$	23.0640	0.812899
$806$	0	0
$807$	−2.42088	−0.0852190
$808$	0	0
$809$	1.61596	0.0568140	0.0284070	−	0.999596i	$-0.490957\pi$
$809$	1.61596	0.0568140	0.0284070	+	0.999596i	$0.490957\pi$
$810$	0	0
$811$	51.9657	1.82476	0.912381	−	0.409342i	$-0.134242\pi$
$811$	51.9657	1.82476	0.912381	+	0.409342i	$0.134242\pi$
$812$	0	0
$813$	21.4450	0.752110
$814$	0	0
$815$	−45.5575	−1.59581
$816$	0	0
$817$	3.95108	0.138231
$818$	0	0
$819$	0	0
$820$	0	0
$821$	54.8327	1.91367	0.956837	−	0.290627i	$-0.0938638\pi$
$821$	54.8327	1.91367	0.956837	+	0.290627i	$0.0938638\pi$
$822$	0	0
$823$	46.8514	1.63314	0.816569	−	0.577247i	$-0.195873\pi$
$823$	46.8514	1.63314	0.816569	+	0.577247i	$0.195873\pi$
$824$	0	0
$825$	24.6058	0.856664
$826$	0	0
$827$	−21.2021	−0.737270	−0.368635	−	0.929574i	$-0.620175\pi$
$827$	−21.2021	−0.737270	−0.368635	+	0.929574i	$0.620175\pi$
$828$	0	0
$829$	18.2972	0.635489	0.317744	−	0.948176i	$-0.397075\pi$
$829$	18.2972	0.635489	0.317744	+	0.948176i	$0.397075\pi$
$830$	0	0
$831$	−14.8073	−0.513660
$832$	0	0
$833$	−18.6829	−0.647325
$834$	0	0
$835$	−42.4359	−1.46856
$836$	0	0
$837$	−2.34481	−0.0810486
$838$	0	0
$839$	−21.1414	−0.729881	−0.364941	−	0.931031i	$-0.618911\pi$
$839$	−21.1414	−0.729881	−0.364941	+	0.931031i	$0.618911\pi$
$840$	0	0
$841$	−14.1709	−0.488652
$842$	0	0
$843$	14.5036	0.499532
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.30367	−0.0791549
$848$	0	0
$849$	−25.6722	−0.881067
$850$	0	0
$851$	57.9958	1.98807
$852$	0	0
$853$	−7.13036	−0.244139	−0.122069	−	0.992522i	$-0.538953\pi$
$853$	−7.13036	−0.244139	−0.122069	+	0.992522i	$0.538953\pi$
$854$	0	0
$855$	9.02715	0.308722
$856$	0	0
$857$	−44.7741	−1.52945	−0.764726	−	0.644355i	$-0.777126\pi$
$857$	−44.7741	−1.52945	−0.764726	+	0.644355i	$0.777126\pi$
$858$	0	0
$859$	−57.3782	−1.95772	−0.978859	−	0.204535i	$-0.934432\pi$
$859$	−57.3782	−1.95772	−0.978859	+	0.204535i	$0.934432\pi$
$860$	0	0
$861$	−0.682333	−0.0232538
$862$	0	0
$863$	6.67563	0.227241	0.113621	−	0.993524i	$-0.463755\pi$
$863$	6.67563	0.227241	0.113621	+	0.993524i	$0.463755\pi$
$864$	0	0
$865$	44.7260	1.52073
$866$	0	0
$867$	−8.36227	−0.283998
$868$	0	0
$869$	−15.4112	−0.522789
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−5.94438	−0.201187
$874$	0	0
$875$	10.7506	0.363438
$876$	0	0
$877$	25.2983	0.854263	0.427131	−	0.904190i	$-0.359524\pi$
$877$	25.2983	0.854263	0.427131	+	0.904190i	$0.359524\pi$
$878$	0	0
$879$	−26.5230	−0.894599
$880$	0	0
$881$	−35.3787	−1.19194	−0.595969	−	0.803008i	$-0.703232\pi$
$881$	−35.3787	−1.19194	−0.595969	+	0.803008i	$0.703232\pi$
$882$	0	0
$883$	11.5851	0.389869	0.194935	−	0.980816i	$-0.437551\pi$
$883$	11.5851	0.389869	0.194935	+	0.980816i	$0.437551\pi$
$884$	0	0
$885$	19.8780	0.668192
$886$	0	0
$887$	27.2892	0.916281	0.458141	−	0.888880i	$-0.348516\pi$
$887$	27.2892	0.916281	0.458141	+	0.888880i	$0.348516\pi$
$888$	0	0
$889$	−4.54719	−0.152508
$890$	0	0
$891$	2.85086	0.0955072
$892$	0	0
$893$	−5.97823	−0.200054
$894$	0	0
$895$	1.98839	0.0664646
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.02954	−0.301152
$900$	0	0
$901$	−29.2747	−0.975282
$902$	0	0
$903$	1.29590	0.0431247
$904$	0	0
$905$	−85.9643	−2.85755
$906$	0	0
$907$	30.4219	1.01014	0.505072	−	0.863077i	$-0.331466\pi$
$907$	30.4219	1.01014	0.505072	+	0.863077i	$0.331466\pi$
$908$	0	0
$909$	4.62565	0.153423
$910$	0	0
$911$	−53.5719	−1.77492	−0.887459	−	0.460887i	$-0.847531\pi$
$911$	−53.5719	−1.77492	−0.887459	+	0.460887i	$0.847531\pi$
$912$	0	0
$913$	−20.0954	−0.665062
$914$	0	0
$915$	−48.9439	−1.61804
$916$	0	0
$917$	−14.6136	−0.482582
$918$	0	0
$919$	36.1672	1.19305	0.596523	−	0.802596i	$-0.296549\pi$
$919$	36.1672	1.19305	0.596523	+	0.802596i	$0.296549\pi$
$920$	0	0
$921$	8.24698	0.271747
$922$	0	0
$923$	0	0
$924$	0	0
$925$	64.2583	2.11280
$926$	0	0
$927$	−1.20775	−0.0396677
$928$	0	0
$929$	−13.3478	−0.437927	−0.218964	−	0.975733i	$-0.570268\pi$
$929$	−13.3478	−0.437927	−0.218964	+	0.975733i	$0.570268\pi$
$930$	0	0
$931$	−15.5429	−0.509397
$932$	0	0
$933$	14.4179	0.472021
$934$	0	0
$935$	30.9342	1.01166
$936$	0	0
$937$	38.6872	1.26386	0.631928	−	0.775027i	$-0.282264\pi$
$937$	38.6872	1.26386	0.631928	+	0.775027i	$0.282264\pi$
$938$	0	0
$939$	14.2338	0.464504
$940$	0	0
$941$	35.7275	1.16468	0.582342	−	0.812944i	$-0.302136\pi$
$941$	35.7275	1.16468	0.582342	+	0.812944i	$0.302136\pi$
$942$	0	0
$943$	−6.62804	−0.215839
$944$	0	0
$945$	2.96077	0.0963139
$946$	0	0
$947$	17.8436	0.579838	0.289919	−	0.957051i	$-0.406372\pi$
$947$	17.8436	0.579838	0.289919	+	0.957051i	$0.406372\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.84415	−0.221937
$952$	0	0
$953$	20.1691	0.653342	0.326671	−	0.945138i	$-0.394073\pi$
$953$	20.1691	0.653342	0.326671	+	0.945138i	$0.394073\pi$
$954$	0	0
$955$	−61.8993	−2.00301
$956$	0	0
$957$	10.9782	0.354876
$958$	0	0
$959$	7.57865	0.244727
$960$	0	0
$961$	−25.5018	−0.822640
$962$	0	0
$963$	9.52111	0.306813
$964$	0	0
$965$	95.0471	3.05967
$966$	0	0
$967$	32.7894	1.05444	0.527218	−	0.849730i	$-0.323235\pi$
$967$	32.7894	1.05444	0.527218	+	0.849730i	$0.323235\pi$
$968$	0	0
$969$	7.18598	0.230847
$970$	0	0
$971$	26.9845	0.865973	0.432986	−	0.901401i	$-0.357460\pi$
$971$	26.9845	0.865973	0.432986	+	0.901401i	$0.357460\pi$
$972$	0	0
$973$	−3.21983	−0.103223
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.9571	−0.542504	−0.271252	−	0.962508i	$-0.587438\pi$
$977$	−16.9571	−0.542504	−0.271252	+	0.962508i	$0.587438\pi$
$978$	0	0
$979$	3.22627	0.103112
$980$	0	0
$981$	−1.78448	−0.0569740
$982$	0	0
$983$	32.6631	1.04179	0.520895	−	0.853621i	$-0.325598\pi$
$983$	32.6631	1.04179	0.520895	+	0.853621i	$0.325598\pi$
$984$	0	0
$985$	79.0863	2.51990
$986$	0	0
$987$	−1.96077	−0.0624120
$988$	0	0
$989$	12.5881	0.400277
$990$	0	0
$991$	−7.64310	−0.242791	−0.121396	−	0.992604i	$-0.538737\pi$
$991$	−7.64310	−0.242791	−0.121396	+	0.992604i	$0.538737\pi$
$992$	0	0
$993$	−9.44265	−0.299653
$994$	0	0
$995$	−13.0248	−0.412912
$996$	0	0
$997$	36.1256	1.14411	0.572054	−	0.820216i	$-0.306147\pi$
$997$	36.1256	1.14411	0.572054	+	0.820216i	$0.306147\pi$
$998$	0	0
$999$	7.44504	0.235551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cp.1.3		3
4.3	odd	2		507.2.a.l.1.2	yes	3
12.11	even	2		1521.2.a.n.1.2		3
13.12	even	2		8112.2.a.cg.1.1		3
52.3	odd	6		507.2.e.i.22.2		6
52.7	even	12		507.2.j.i.361.5		12
52.11	even	12		507.2.j.i.316.2		12
52.15	even	12		507.2.j.i.316.5		12
52.19	even	12		507.2.j.i.361.2		12
52.23	odd	6		507.2.e.l.22.2		6
52.31	even	4		507.2.b.f.337.2		6
52.35	odd	6		507.2.e.i.484.2		6
52.43	odd	6		507.2.e.l.484.2		6
52.47	even	4		507.2.b.f.337.5		6
52.51	odd	2		507.2.a.i.1.2	✓	3
156.47	odd	4		1521.2.b.k.1351.2		6
156.83	odd	4		1521.2.b.k.1351.5		6
156.155	even	2		1521.2.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.2	✓	3	52.51	odd	2
507.2.a.l.1.2	yes	3	4.3	odd	2
507.2.b.f.337.2		6	52.31	even	4
507.2.b.f.337.5		6	52.47	even	4
507.2.e.i.22.2		6	52.3	odd	6
507.2.e.i.484.2		6	52.35	odd	6
507.2.e.l.22.2		6	52.23	odd	6
507.2.e.l.484.2		6	52.43	odd	6
507.2.j.i.316.2		12	52.11	even	12
507.2.j.i.316.5		12	52.15	even	12
507.2.j.i.361.2		12	52.19	even	12
507.2.j.i.361.5		12	52.7	even	12
1521.2.a.n.1.2		3	12.11	even	2
1521.2.a.s.1.2		3	156.155	even	2
1521.2.b.k.1351.2		6	156.47	odd	4
1521.2.b.k.1351.5		6	156.83	odd	4
8112.2.a.cg.1.1		3	13.12	even	2
8112.2.a.cp.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.69202 q^{5} +0.801938 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.69202 q^{5} +0.801938 q^{7} +1.00000 q^{9} +2.85086 q^{11} +3.69202 q^{15} +2.93900 q^{17} +2.44504 q^{19} +0.801938 q^{21} +7.78986 q^{23} +8.63102 q^{25} +1.00000 q^{27} +3.85086 q^{29} -2.34481 q^{31} +2.85086 q^{33} +2.96077 q^{35} +7.44504 q^{37} -0.850855 q^{41} +1.61596 q^{43} +3.69202 q^{45} -2.44504 q^{47} -6.35690 q^{49} +2.93900 q^{51} -9.96077 q^{53} +10.5254 q^{55} +2.44504 q^{57} +5.38404 q^{59} -13.2567 q^{61} +0.801938 q^{63} -14.3937 q^{67} +7.78986 q^{69} -8.12498 q^{71} -11.8877 q^{73} +8.63102 q^{75} +2.28621 q^{77} -5.40581 q^{79} +1.00000 q^{81} -7.04892 q^{83} +10.8509 q^{85} +3.85086 q^{87} +1.13169 q^{89} -2.34481 q^{93} +9.02715 q^{95} -5.94438 q^{97} +2.85086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 6 q^{15} - q^{17} + 7 q^{19} - 2 q^{21} + 11 q^{25} + 3 q^{27} - 2 q^{29} + 16 q^{31} - 5 q^{33} - 4 q^{35} + 22 q^{37} + 11 q^{41} + 15 q^{43} + 6 q^{45} - 7 q^{47} - 15 q^{49} - q^{51} - 17 q^{53} - 3 q^{55} + 7 q^{57} + 6 q^{59} - 13 q^{61} - 2 q^{63} - 11 q^{67} + 6 q^{73} + 11 q^{75} + 15 q^{77} - 3 q^{79} + 3 q^{81} - 12 q^{83} + 19 q^{85} - 2 q^{87} + q^{89} + 16 q^{93} + 21 q^{95} - 5 q^{97} - 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9 - 5 * q^11 + 6 * q^15 - q^17 + 7 * q^19 - 2 * q^21 + 11 * q^25 + 3 * q^27 - 2 * q^29 + 16 * q^31 - 5 * q^33 - 4 * q^35 + 22 * q^37 + 11 * q^41 + 15 * q^43 + 6 * q^45 - 7 * q^47 - 15 * q^49 - q^51 - 17 * q^53 - 3 * q^55 + 7 * q^57 + 6 * q^59 - 13 * q^61 - 2 * q^63 - 11 * q^67 + 6 * q^73 + 11 * q^75 + 15 * q^77 - 3 * q^79 + 3 * q^81 - 12 * q^83 + 19 * q^85 - 2 * q^87 + q^89 + 16 * q^93 + 21 * q^95 - 5 * q^97 - 5 * q^99

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