LMFDB - Embedded newform 3696.2.a.bc.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3696,2,Mod(1,3696)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3696.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3696, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3696.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,-2,0,2,0,-2,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	3696.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{13} +3.46410 q^{15} +3.46410 q^{17} -5.46410 q^{19} +1.00000 q^{21} -6.92820 q^{23} +7.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -5.46410 q^{31} +1.00000 q^{33} +3.46410 q^{35} -4.92820 q^{37} -2.00000 q^{39} +3.46410 q^{41} +10.9282 q^{43} -3.46410 q^{45} -9.46410 q^{47} +1.00000 q^{49} -3.46410 q^{51} -0.928203 q^{53} +3.46410 q^{55} +5.46410 q^{57} -6.92820 q^{59} +2.00000 q^{61} -1.00000 q^{63} -6.92820 q^{65} -14.9282 q^{67} +6.92820 q^{69} +12.0000 q^{71} -0.535898 q^{73} -7.00000 q^{75} +1.00000 q^{77} +10.9282 q^{79} +1.00000 q^{81} -4.39230 q^{83} -12.0000 q^{85} +6.00000 q^{87} -0.928203 q^{89} -2.00000 q^{91} +5.46410 q^{93} +18.9282 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} - 4 q^{19} + 2 q^{21} + 14 q^{25} - 2 q^{27} - 12 q^{29} - 4 q^{31} + 2 q^{33} + 4 q^{37} - 4 q^{39} + 8 q^{43} - 12 q^{47} + 2 q^{49} + 12 q^{53}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 - 4 * q^19 + 2 * q^21 + 14 * q^25 - 2 * q^27 - 12 * q^29 - 4 * q^31 + 2 * q^33 + 4 * q^37 - 4 * q^39 + 8 * q^43 - 12 * q^47 + 2 * q^49 + 12 * q^53 + 4 * q^57 + 4 * q^61 - 2 * q^63 - 16 * q^67 + 24 * q^71 - 8 * q^73 - 14 * q^75 + 2 * q^77 + 8 * q^79 + 2 * q^81 + 12 * q^83 - 24 * q^85 + 12 * q^87 + 12 * q^89 - 4 * q^91 + 4 * q^93 + 24 * q^95 + 4 * q^97 - 2 * q^99

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
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−5.46410	−0.981382	−0.490691	−	0.871334i	$-0.663256\pi$
$31$	−5.46410	−0.981382	−0.490691	+	0.871334i	$0.663256\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	3.46410	0.585540
$36$	0	0
$37$	−4.92820	−0.810192	−0.405096	−	0.914274i	$-0.632762\pi$
$37$	−4.92820	−0.810192	−0.405096	+	0.914274i	$0.632762\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	10.9282	1.66654	0.833268	−	0.552870i	$-0.186467\pi$
$43$	10.9282	1.66654	0.833268	+	0.552870i	$0.186467\pi$
$44$	0	0
$45$	−3.46410	−0.516398
$46$	0	0
$47$	−9.46410	−1.38048	−0.690241	−	0.723580i	$-0.742495\pi$
$47$	−9.46410	−1.38048	−0.690241	+	0.723580i	$0.742495\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.46410	−0.485071
$52$	0	0
$53$	−0.928203	−0.127499	−0.0637493	−	0.997966i	$-0.520306\pi$
$53$	−0.928203	−0.127499	−0.0637493	+	0.997966i	$0.520306\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	0	0
$57$	5.46410	0.723738
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−6.92820	−0.859338
$66$	0	0
$67$	−14.9282	−1.82377	−0.911885	−	0.410445i	$-0.865373\pi$
$67$	−14.9282	−1.82377	−0.911885	+	0.410445i	$0.865373\pi$
$68$	0	0
$69$	6.92820	0.834058
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−0.535898	−0.0627222	−0.0313611	−	0.999508i	$-0.509984\pi$
$73$	−0.535898	−0.0627222	−0.0313611	+	0.999508i	$0.509984\pi$
$74$	0	0
$75$	−7.00000	−0.808290
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.39230	−0.482118	−0.241059	−	0.970510i	$-0.577495\pi$
$83$	−4.39230	−0.482118	−0.241059	+	0.970510i	$0.577495\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	5.46410	0.566601
$94$	0	0
$95$	18.9282	1.94199
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−19.8564	−1.97579	−0.987893	−	0.155136i	$-0.950418\pi$
$101$	−19.8564	−1.97579	−0.987893	+	0.155136i	$0.950418\pi$
$102$	0	0
$103$	13.4641	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$103$	13.4641	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$104$	0	0
$105$	−3.46410	−0.338062
$106$	0	0
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	15.8564	1.51877	0.759384	−	0.650643i	$-0.225500\pi$
$109$	15.8564	1.51877	0.759384	+	0.650643i	$0.225500\pi$
$110$	0	0
$111$	4.92820	0.467764
$112$	0	0
$113$	7.85641	0.739069	0.369534	−	0.929217i	$-0.379517\pi$
$113$	7.85641	0.739069	0.369534	+	0.929217i	$0.379517\pi$
$114$	0	0
$115$	24.0000	2.23801
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.46410	−0.312348
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	10.9282	0.969721	0.484861	−	0.874591i	$-0.338870\pi$
$127$	10.9282	0.969721	0.484861	+	0.874591i	$0.338870\pi$
$128$	0	0
$129$	−10.9282	−0.962175
$130$	0	0
$131$	4.39230	0.383757	0.191879	−	0.981419i	$-0.438542\pi$
$131$	4.39230	0.383757	0.191879	+	0.981419i	$0.438542\pi$
$132$	0	0
$133$	5.46410	0.473798
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	7.85641	0.671218	0.335609	−	0.942001i	$-0.391058\pi$
$137$	7.85641	0.671218	0.335609	+	0.942001i	$0.391058\pi$
$138$	0	0
$139$	13.4641	1.14201	0.571005	−	0.820947i	$-0.306554\pi$
$139$	13.4641	1.14201	0.571005	+	0.820947i	$0.306554\pi$
$140$	0	0
$141$	9.46410	0.797021
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	20.7846	1.72607
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	3.46410	0.280056
$154$	0	0
$155$	18.9282	1.52035
$156$	0	0
$157$	−2.39230	−0.190927	−0.0954634	−	0.995433i	$-0.530433\pi$
$157$	−2.39230	−0.190927	−0.0954634	+	0.995433i	$0.530433\pi$
$158$	0	0
$159$	0.928203	0.0736113
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−3.46410	−0.269680
$166$	0	0
$167$	−18.9282	−1.46471	−0.732354	−	0.680924i	$-0.761578\pi$
$167$	−18.9282	−1.46471	−0.732354	+	0.680924i	$0.761578\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−5.46410	−0.417850
$172$	0	0
$173$	−0.928203	−0.0705700	−0.0352850	−	0.999377i	$-0.511234\pi$
$173$	−0.928203	−0.0705700	−0.0352850	+	0.999377i	$0.511234\pi$
$174$	0	0
$175$	−7.00000	−0.529150
$176$	0	0
$177$	6.92820	0.520756
$178$	0	0
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	6.39230	0.475136	0.237568	−	0.971371i	$-0.423650\pi$
$181$	6.39230	0.475136	0.237568	+	0.971371i	$0.423650\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	17.0718	1.25514
$186$	0	0
$187$	−3.46410	−0.253320
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	0	0
$195$	6.92820	0.496139
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−0.392305	−0.0278098	−0.0139049	−	0.999903i	$-0.504426\pi$
$199$	−0.392305	−0.0278098	−0.0139049	+	0.999903i	$0.504426\pi$
$200$	0	0
$201$	14.9282	1.05295
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	−6.92820	−0.481543
$208$	0	0
$209$	5.46410	0.377960
$210$	0	0
$211$	−16.7846	−1.15550	−0.577750	−	0.816214i	$-0.696069\pi$
$211$	−16.7846	−1.15550	−0.577750	+	0.816214i	$0.696069\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	−37.8564	−2.58179
$216$	0	0
$217$	5.46410	0.370927
$218$	0	0
$219$	0.535898	0.0362127
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	8.39230	0.561990	0.280995	−	0.959709i	$-0.409336\pi$
$223$	8.39230	0.561990	0.280995	+	0.959709i	$0.409336\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	4.39230	0.291528	0.145764	−	0.989319i	$-0.453436\pi$
$227$	4.39230	0.291528	0.145764	+	0.989319i	$0.453436\pi$
$228$	0	0
$229$	−21.3205	−1.40890	−0.704449	−	0.709754i	$-0.748806\pi$
$229$	−21.3205	−1.40890	−0.704449	+	0.709754i	$0.748806\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	0	0
$235$	32.7846	2.13863
$236$	0	0
$237$	−10.9282	−0.709863
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	9.60770	0.618886	0.309443	−	0.950918i	$-0.399857\pi$
$241$	9.60770	0.618886	0.309443	+	0.950918i	$0.399857\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.46410	−0.221313
$246$	0	0
$247$	−10.9282	−0.695345
$248$	0	0
$249$	4.39230	0.278351
$250$	0	0
$251$	−30.9282	−1.95217	−0.976085	−	0.217387i	$-0.930247\pi$
$251$	−30.9282	−1.95217	−0.976085	+	0.217387i	$0.930247\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	0	0
$255$	12.0000	0.751469
$256$	0	0
$257$	12.9282	0.806439	0.403220	−	0.915103i	$-0.367891\pi$
$257$	12.9282	0.806439	0.403220	+	0.915103i	$0.367891\pi$
$258$	0	0
$259$	4.92820	0.306224
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	5.07180	0.312740	0.156370	−	0.987699i	$-0.450021\pi$
$263$	5.07180	0.312740	0.156370	+	0.987699i	$0.450021\pi$
$264$	0	0
$265$	3.21539	0.197520
$266$	0	0
$267$	0.928203	0.0568051
$268$	0	0
$269$	24.2487	1.47847	0.739235	−	0.673448i	$-0.235187\pi$
$269$	24.2487	1.47847	0.739235	+	0.673448i	$0.235187\pi$
$270$	0	0
$271$	24.7846	1.50556	0.752779	−	0.658273i	$-0.228713\pi$
$271$	24.7846	1.50556	0.752779	+	0.658273i	$0.228713\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	−7.00000	−0.422116
$276$	0	0
$277$	−11.8564	−0.712382	−0.356191	−	0.934413i	$-0.615925\pi$
$277$	−11.8564	−0.712382	−0.356191	+	0.934413i	$0.615925\pi$
$278$	0	0
$279$	−5.46410	−0.327127
$280$	0	0
$281$	−7.85641	−0.468674	−0.234337	−	0.972155i	$-0.575292\pi$
$281$	−7.85641	−0.468674	−0.234337	+	0.972155i	$0.575292\pi$
$282$	0	0
$283$	13.4641	0.800358	0.400179	−	0.916437i	$-0.368948\pi$
$283$	13.4641	0.800358	0.400179	+	0.916437i	$0.368948\pi$
$284$	0	0
$285$	−18.9282	−1.12121
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−24.9282	−1.45632	−0.728161	−	0.685407i	$-0.759625\pi$
$293$	−24.9282	−1.45632	−0.728161	+	0.685407i	$0.759625\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−13.8564	−0.801337
$300$	0	0
$301$	−10.9282	−0.629891
$302$	0	0
$303$	19.8564	1.14072
$304$	0	0
$305$	−6.92820	−0.396708
$306$	0	0
$307$	−10.5359	−0.601315	−0.300658	−	0.953732i	$-0.597206\pi$
$307$	−10.5359	−0.601315	−0.300658	+	0.953732i	$0.597206\pi$
$308$	0	0
$309$	−13.4641	−0.765946
$310$	0	0
$311$	−28.3923	−1.60998	−0.804990	−	0.593288i	$-0.797829\pi$
$311$	−28.3923	−1.60998	−0.804990	+	0.593288i	$0.797829\pi$
$312$	0	0
$313$	−16.9282	−0.956839	−0.478419	−	0.878132i	$-0.658790\pi$
$313$	−16.9282	−0.956839	−0.478419	+	0.878132i	$0.658790\pi$
$314$	0	0
$315$	3.46410	0.195180
$316$	0	0
$317$	12.9282	0.726120	0.363060	−	0.931766i	$-0.381732\pi$
$317$	12.9282	0.726120	0.363060	+	0.931766i	$0.381732\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−6.92820	−0.386695
$322$	0	0
$323$	−18.9282	−1.05319
$324$	0	0
$325$	14.0000	0.776580
$326$	0	0
$327$	−15.8564	−0.876861
$328$	0	0
$329$	9.46410	0.521773
$330$	0	0
$331$	9.07180	0.498631	0.249316	−	0.968422i	$-0.419794\pi$
$331$	9.07180	0.498631	0.249316	+	0.968422i	$0.419794\pi$
$332$	0	0
$333$	−4.92820	−0.270064
$334$	0	0
$335$	51.7128	2.82537
$336$	0	0
$337$	20.9282	1.14003	0.570016	−	0.821634i	$-0.306937\pi$
$337$	20.9282	1.14003	0.570016	+	0.821634i	$0.306937\pi$
$338$	0	0
$339$	−7.85641	−0.426701
$340$	0	0
$341$	5.46410	0.295898
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−24.0000	−1.29212
$346$	0	0
$347$	20.7846	1.11578	0.557888	−	0.829916i	$-0.311612\pi$
$347$	20.7846	1.11578	0.557888	+	0.829916i	$0.311612\pi$
$348$	0	0
$349$	10.7846	0.577287	0.288643	−	0.957437i	$-0.406796\pi$
$349$	10.7846	0.577287	0.288643	+	0.957437i	$0.406796\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	12.9282	0.688099	0.344049	−	0.938952i	$-0.388201\pi$
$353$	12.9282	0.688099	0.344049	+	0.938952i	$0.388201\pi$
$354$	0	0
$355$	−41.5692	−2.20627
$356$	0	0
$357$	3.46410	0.183340
$358$	0	0
$359$	5.07180	0.267679	0.133840	−	0.991003i	$-0.457269\pi$
$359$	5.07180	0.267679	0.133840	+	0.991003i	$0.457269\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	1.85641	0.0971688
$366$	0	0
$367$	−19.3205	−1.00852	−0.504261	−	0.863551i	$-0.668235\pi$
$367$	−19.3205	−1.00852	−0.504261	+	0.863551i	$0.668235\pi$
$368$	0	0
$369$	3.46410	0.180334
$370$	0	0
$371$	0.928203	0.0481899
$372$	0	0
$373$	29.7128	1.53847	0.769236	−	0.638965i	$-0.220637\pi$
$373$	29.7128	1.53847	0.769236	+	0.638965i	$0.220637\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	17.8564	0.917222	0.458611	−	0.888637i	$-0.348347\pi$
$379$	17.8564	0.917222	0.458611	+	0.888637i	$0.348347\pi$
$380$	0	0
$381$	−10.9282	−0.559869
$382$	0	0
$383$	−14.5359	−0.742750	−0.371375	−	0.928483i	$-0.621114\pi$
$383$	−14.5359	−0.742750	−0.371375	+	0.928483i	$0.621114\pi$
$384$	0	0
$385$	−3.46410	−0.176547
$386$	0	0
$387$	10.9282	0.555512
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	−4.39230	−0.221562
$394$	0	0
$395$	−37.8564	−1.90476
$396$	0	0
$397$	−2.39230	−0.120066	−0.0600332	−	0.998196i	$-0.519121\pi$
$397$	−2.39230	−0.120066	−0.0600332	+	0.998196i	$0.519121\pi$
$398$	0	0
$399$	−5.46410	−0.273547
$400$	0	0
$401$	4.14359	0.206921	0.103461	−	0.994634i	$-0.467008\pi$
$401$	4.14359	0.206921	0.103461	+	0.994634i	$0.467008\pi$
$402$	0	0
$403$	−10.9282	−0.544373
$404$	0	0
$405$	−3.46410	−0.172133
$406$	0	0
$407$	4.92820	0.244282
$408$	0	0
$409$	−9.32051	−0.460869	−0.230435	−	0.973088i	$-0.574015\pi$
$409$	−9.32051	−0.460869	−0.230435	+	0.973088i	$0.574015\pi$
$410$	0	0
$411$	−7.85641	−0.387528
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	15.2154	0.746894
$416$	0	0
$417$	−13.4641	−0.659340
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−37.7128	−1.83801	−0.919005	−	0.394246i	$-0.871006\pi$
$421$	−37.7128	−1.83801	−0.919005	+	0.394246i	$0.871006\pi$
$422$	0	0
$423$	−9.46410	−0.460160
$424$	0	0
$425$	24.2487	1.17624
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	18.9282	0.911739	0.455870	−	0.890047i	$-0.349328\pi$
$431$	18.9282	0.911739	0.455870	+	0.890047i	$0.349328\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−20.7846	−0.996546
$436$	0	0
$437$	37.8564	1.81092
$438$	0	0
$439$	24.7846	1.18290	0.591452	−	0.806340i	$-0.298555\pi$
$439$	24.7846	1.18290	0.591452	+	0.806340i	$0.298555\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	0	0
$445$	3.21539	0.152424
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	−19.8564	−0.937082	−0.468541	−	0.883442i	$-0.655220\pi$
$449$	−19.8564	−0.937082	−0.468541	+	0.883442i	$0.655220\pi$
$450$	0	0
$451$	−3.46410	−0.163118
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	15.8564	0.741731	0.370866	−	0.928687i	$-0.379061\pi$
$457$	15.8564	0.741731	0.370866	+	0.928687i	$0.379061\pi$
$458$	0	0
$459$	−3.46410	−0.161690
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	−18.9282	−0.877774
$466$	0	0
$467$	15.7128	0.727102	0.363551	−	0.931574i	$-0.381564\pi$
$467$	15.7128	0.727102	0.363551	+	0.931574i	$0.381564\pi$
$468$	0	0
$469$	14.9282	0.689320
$470$	0	0
$471$	2.39230	0.110232
$472$	0	0
$473$	−10.9282	−0.502479
$474$	0	0
$475$	−38.2487	−1.75497
$476$	0	0
$477$	−0.928203	−0.0424995
$478$	0	0
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	−9.85641	−0.449413
$482$	0	0
$483$	−6.92820	−0.315244
$484$	0	0
$485$	−6.92820	−0.314594
$486$	0	0
$487$	−40.7846	−1.84813	−0.924064	−	0.382239i	$-0.875153\pi$
$487$	−40.7846	−1.84813	−0.924064	+	0.382239i	$0.875153\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	1.85641	0.0837785	0.0418892	−	0.999122i	$-0.486662\pi$
$491$	1.85641	0.0837785	0.0418892	+	0.999122i	$0.486662\pi$
$492$	0	0
$493$	−20.7846	−0.936092
$494$	0	0
$495$	3.46410	0.155700
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	9.07180	0.406109	0.203055	−	0.979167i	$-0.434913\pi$
$499$	9.07180	0.406109	0.203055	+	0.979167i	$0.434913\pi$
$500$	0	0
$501$	18.9282	0.845650
$502$	0	0
$503$	−18.9282	−0.843967	−0.421983	−	0.906604i	$-0.638666\pi$
$503$	−18.9282	−0.843967	−0.421983	+	0.906604i	$0.638666\pi$
$504$	0	0
$505$	68.7846	3.06087
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−41.3205	−1.83150	−0.915750	−	0.401749i	$-0.868402\pi$
$509$	−41.3205	−1.83150	−0.915750	+	0.401749i	$0.868402\pi$
$510$	0	0
$511$	0.535898	0.0237067
$512$	0	0
$513$	5.46410	0.241246
$514$	0	0
$515$	−46.6410	−2.05525
$516$	0	0
$517$	9.46410	0.416231
$518$	0	0
$519$	0.928203	0.0407436
$520$	0	0
$521$	36.9282	1.61785	0.808927	−	0.587909i	$-0.200049\pi$
$521$	36.9282	1.61785	0.808927	+	0.587909i	$0.200049\pi$
$522$	0	0
$523$	−24.3923	−1.06660	−0.533301	−	0.845926i	$-0.679048\pi$
$523$	−24.3923	−1.06660	−0.533301	+	0.845926i	$0.679048\pi$
$524$	0	0
$525$	7.00000	0.305505
$526$	0	0
$527$	−18.9282	−0.824525
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	−6.92820	−0.300658
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−20.7846	−0.896922
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−25.7128	−1.10548	−0.552740	−	0.833354i	$-0.686418\pi$
$541$	−25.7128	−1.10548	−0.552740	+	0.833354i	$0.686418\pi$
$542$	0	0
$543$	−6.39230	−0.274320
$544$	0	0
$545$	−54.9282	−2.35287
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	32.7846	1.39667
$552$	0	0
$553$	−10.9282	−0.464714
$554$	0	0
$555$	−17.0718	−0.724657
$556$	0	0
$557$	21.7128	0.920001	0.460001	−	0.887919i	$-0.347849\pi$
$557$	21.7128	0.920001	0.460001	+	0.887919i	$0.347849\pi$
$558$	0	0
$559$	21.8564	0.924427
$560$	0	0
$561$	3.46410	0.146254
$562$	0	0
$563$	18.2487	0.769091	0.384546	−	0.923106i	$-0.374358\pi$
$563$	18.2487	0.769091	0.384546	+	0.923106i	$0.374358\pi$
$564$	0	0
$565$	−27.2154	−1.14496
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−31.8564	−1.33549	−0.667745	−	0.744390i	$-0.732740\pi$
$569$	−31.8564	−1.33549	−0.667745	+	0.744390i	$0.732740\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	20.7846	0.868290
$574$	0	0
$575$	−48.4974	−2.02248
$576$	0	0
$577$	−30.7846	−1.28158	−0.640790	−	0.767716i	$-0.721393\pi$
$577$	−30.7846	−1.28158	−0.640790	+	0.767716i	$0.721393\pi$
$578$	0	0
$579$	−26.0000	−1.08052
$580$	0	0
$581$	4.39230	0.182224
$582$	0	0
$583$	0.928203	0.0384422
$584$	0	0
$585$	−6.92820	−0.286446
$586$	0	0
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	0	0
$589$	29.8564	1.23021
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	−20.5359	−0.843308	−0.421654	−	0.906757i	$-0.638550\pi$
$593$	−20.5359	−0.843308	−0.421654	+	0.906757i	$0.638550\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	0.392305	0.0160560
$598$	0	0
$599$	−1.85641	−0.0758507	−0.0379254	−	0.999281i	$-0.512075\pi$
$599$	−1.85641	−0.0758507	−0.0379254	+	0.999281i	$0.512075\pi$
$600$	0	0
$601$	18.3923	0.750238	0.375119	−	0.926977i	$-0.377602\pi$
$601$	18.3923	0.750238	0.375119	+	0.926977i	$0.377602\pi$
$602$	0	0
$603$	−14.9282	−0.607923
$604$	0	0
$605$	−3.46410	−0.140836
$606$	0	0
$607$	19.7128	0.800118	0.400059	−	0.916489i	$-0.368990\pi$
$607$	19.7128	0.800118	0.400059	+	0.916489i	$0.368990\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−18.9282	−0.765753
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−19.8564	−0.799389	−0.399694	−	0.916648i	$-0.630884\pi$
$617$	−19.8564	−0.799389	−0.399694	+	0.916648i	$0.630884\pi$
$618$	0	0
$619$	31.7128	1.27465	0.637323	−	0.770597i	$-0.280042\pi$
$619$	31.7128	1.27465	0.637323	+	0.770597i	$0.280042\pi$
$620$	0	0
$621$	6.92820	0.278019
$622$	0	0
$623$	0.928203	0.0371877
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−5.46410	−0.218215
$628$	0	0
$629$	−17.0718	−0.680697
$630$	0	0
$631$	0.784610	0.0312348	0.0156174	−	0.999878i	$-0.495029\pi$
$631$	0.784610	0.0312348	0.0156174	+	0.999878i	$0.495029\pi$
$632$	0	0
$633$	16.7846	0.667128
$634$	0	0
$635$	−37.8564	−1.50229
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	31.8564	1.25825	0.629126	−	0.777303i	$-0.283413\pi$
$641$	31.8564	1.25825	0.629126	+	0.777303i	$0.283413\pi$
$642$	0	0
$643$	−14.9282	−0.588711	−0.294355	−	0.955696i	$-0.595105\pi$
$643$	−14.9282	−0.588711	−0.294355	+	0.955696i	$0.595105\pi$
$644$	0	0
$645$	37.8564	1.49059
$646$	0	0
$647$	−0.679492	−0.0267136	−0.0133568	−	0.999911i	$-0.504252\pi$
$647$	−0.679492	−0.0267136	−0.0133568	+	0.999911i	$0.504252\pi$
$648$	0	0
$649$	6.92820	0.271956
$650$	0	0
$651$	−5.46410	−0.214155
$652$	0	0
$653$	−33.7128	−1.31928	−0.659642	−	0.751580i	$-0.729292\pi$
$653$	−33.7128	−1.31928	−0.659642	+	0.751580i	$0.729292\pi$
$654$	0	0
$655$	−15.2154	−0.594514
$656$	0	0
$657$	−0.535898	−0.0209074
$658$	0	0
$659$	17.0718	0.665023	0.332511	−	0.943099i	$-0.392104\pi$
$659$	17.0718	0.665023	0.332511	+	0.943099i	$0.392104\pi$
$660$	0	0
$661$	44.2487	1.72108	0.860538	−	0.509387i	$-0.170128\pi$
$661$	44.2487	1.72108	0.860538	+	0.509387i	$0.170128\pi$
$662$	0	0
$663$	−6.92820	−0.269069
$664$	0	0
$665$	−18.9282	−0.734004
$666$	0	0
$667$	41.5692	1.60957
$668$	0	0
$669$	−8.39230	−0.324465
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	20.9282	0.806723	0.403361	−	0.915041i	$-0.367842\pi$
$673$	20.9282	0.806723	0.403361	+	0.915041i	$0.367842\pi$
$674$	0	0
$675$	−7.00000	−0.269430
$676$	0	0
$677$	2.78461	0.107021	0.0535106	−	0.998567i	$-0.482959\pi$
$677$	2.78461	0.107021	0.0535106	+	0.998567i	$0.482959\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−4.39230	−0.168313
$682$	0	0
$683$	−3.21539	−0.123033	−0.0615167	−	0.998106i	$-0.519594\pi$
$683$	−3.21539	−0.123033	−0.0615167	+	0.998106i	$0.519594\pi$
$684$	0	0
$685$	−27.2154	−1.03985
$686$	0	0
$687$	21.3205	0.813428
$688$	0	0
$689$	−1.85641	−0.0707235
$690$	0	0
$691$	33.0718	1.25811	0.629055	−	0.777361i	$-0.283442\pi$
$691$	33.0718	1.25811	0.629055	+	0.777361i	$0.283442\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	−46.6410	−1.76919
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	7.85641	0.297157
$700$	0	0
$701$	45.7128	1.72655	0.863275	−	0.504735i	$-0.168410\pi$
$701$	45.7128	1.72655	0.863275	+	0.504735i	$0.168410\pi$
$702$	0	0
$703$	26.9282	1.01562
$704$	0	0
$705$	−32.7846	−1.23474
$706$	0	0
$707$	19.8564	0.746777
$708$	0	0
$709$	46.7846	1.75703	0.878516	−	0.477712i	$-0.158534\pi$
$709$	46.7846	1.75703	0.878516	+	0.477712i	$0.158534\pi$
$710$	0	0
$711$	10.9282	0.409840
$712$	0	0
$713$	37.8564	1.41773
$714$	0	0
$715$	6.92820	0.259100
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	37.1769	1.38646	0.693232	−	0.720714i	$-0.256186\pi$
$719$	37.1769	1.38646	0.693232	+	0.720714i	$0.256186\pi$
$720$	0	0
$721$	−13.4641	−0.501429
$722$	0	0
$723$	−9.60770	−0.357314
$724$	0	0
$725$	−42.0000	−1.55984
$726$	0	0
$727$	−33.1769	−1.23046	−0.615232	−	0.788346i	$-0.710938\pi$
$727$	−33.1769	−1.23046	−0.615232	+	0.788346i	$0.710938\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	37.8564	1.40017
$732$	0	0
$733$	12.1436	0.448534	0.224267	−	0.974528i	$-0.428001\pi$
$733$	12.1436	0.448534	0.224267	+	0.974528i	$0.428001\pi$
$734$	0	0
$735$	3.46410	0.127775
$736$	0	0
$737$	14.9282	0.549887
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	10.9282	0.401458
$742$	0	0
$743$	41.5692	1.52503	0.762513	−	0.646972i	$-0.223965\pi$
$743$	41.5692	1.52503	0.762513	+	0.646972i	$0.223965\pi$
$744$	0	0
$745$	20.7846	0.761489
$746$	0	0
$747$	−4.39230	−0.160706
$748$	0	0
$749$	−6.92820	−0.253151
$750$	0	0
$751$	24.7846	0.904403	0.452202	−	0.891916i	$-0.350639\pi$
$751$	24.7846	0.904403	0.452202	+	0.891916i	$0.350639\pi$
$752$	0	0
$753$	30.9282	1.12709
$754$	0	0
$755$	−55.4256	−2.01715
$756$	0	0
$757$	22.7846	0.828121	0.414060	−	0.910249i	$-0.364110\pi$
$757$	22.7846	0.828121	0.414060	+	0.910249i	$0.364110\pi$
$758$	0	0
$759$	−6.92820	−0.251478
$760$	0	0
$761$	−25.6077	−0.928278	−0.464139	−	0.885762i	$-0.653636\pi$
$761$	−25.6077	−0.928278	−0.464139	+	0.885762i	$0.653636\pi$
$762$	0	0
$763$	−15.8564	−0.574040
$764$	0	0
$765$	−12.0000	−0.433861
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	28.5359	1.02903	0.514515	−	0.857481i	$-0.327972\pi$
$769$	28.5359	1.02903	0.514515	+	0.857481i	$0.327972\pi$
$770$	0	0
$771$	−12.9282	−0.465598
$772$	0	0
$773$	−32.5359	−1.17023	−0.585117	−	0.810949i	$-0.698952\pi$
$773$	−32.5359	−1.17023	−0.585117	+	0.810949i	$0.698952\pi$
$774$	0	0
$775$	−38.2487	−1.37393
$776$	0	0
$777$	−4.92820	−0.176798
$778$	0	0
$779$	−18.9282	−0.678173
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	8.28719	0.295782
$786$	0	0
$787$	−33.1769	−1.18263	−0.591315	−	0.806441i	$-0.701391\pi$
$787$	−33.1769	−1.18263	−0.591315	+	0.806441i	$0.701391\pi$
$788$	0	0
$789$	−5.07180	−0.180561
$790$	0	0
$791$	−7.85641	−0.279342
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	−3.21539	−0.114038
$796$	0	0
$797$	−32.5359	−1.15248	−0.576240	−	0.817280i	$-0.695481\pi$
$797$	−32.5359	−1.15248	−0.576240	+	0.817280i	$0.695481\pi$
$798$	0	0
$799$	−32.7846	−1.15984
$800$	0	0
$801$	−0.928203	−0.0327964
$802$	0	0
$803$	0.535898	0.0189114
$804$	0	0
$805$	−24.0000	−0.845889
$806$	0	0
$807$	−24.2487	−0.853595
$808$	0	0
$809$	11.0718	0.389264	0.194632	−	0.980876i	$-0.437649\pi$
$809$	11.0718	0.389264	0.194632	+	0.980876i	$0.437649\pi$
$810$	0	0
$811$	17.1769	0.603163	0.301582	−	0.953440i	$-0.402485\pi$
$811$	17.1769	0.603163	0.301582	+	0.953440i	$0.402485\pi$
$812$	0	0
$813$	−24.7846	−0.869234
$814$	0	0
$815$	−13.8564	−0.485369
$816$	0	0
$817$	−59.7128	−2.08909
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−40.7846	−1.42166	−0.710831	−	0.703363i	$-0.751681\pi$
$823$	−40.7846	−1.42166	−0.710831	+	0.703363i	$0.751681\pi$
$824$	0	0
$825$	7.00000	0.243709
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	44.2487	1.53682	0.768411	−	0.639957i	$-0.221048\pi$
$829$	44.2487	1.53682	0.768411	+	0.639957i	$0.221048\pi$
$830$	0	0
$831$	11.8564	0.411294
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	65.5692	2.26912
$836$	0	0
$837$	5.46410	0.188867
$838$	0	0
$839$	33.4641	1.15531	0.577655	−	0.816281i	$-0.303968\pi$
$839$	33.4641	1.15531	0.577655	+	0.816281i	$0.303968\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	7.85641	0.270589
$844$	0	0
$845$	31.1769	1.07252
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−13.4641	−0.462087
$850$	0	0
$851$	34.1436	1.17043
$852$	0	0
$853$	−49.7128	−1.70213	−0.851067	−	0.525057i	$-0.824044\pi$
$853$	−49.7128	−1.70213	−0.851067	+	0.525057i	$0.824044\pi$
$854$	0	0
$855$	18.9282	0.647331
$856$	0	0
$857$	−5.32051	−0.181745	−0.0908725	−	0.995863i	$-0.528966\pi$
$857$	−5.32051	−0.181745	−0.0908725	+	0.995863i	$0.528966\pi$
$858$	0	0
$859$	−52.7846	−1.80099	−0.900494	−	0.434869i	$-0.856795\pi$
$859$	−52.7846	−1.80099	−0.900494	+	0.434869i	$0.856795\pi$
$860$	0	0
$861$	3.46410	0.118056
$862$	0	0
$863$	−44.7846	−1.52449	−0.762243	−	0.647291i	$-0.775902\pi$
$863$	−44.7846	−1.52449	−0.762243	+	0.647291i	$0.775902\pi$
$864$	0	0
$865$	3.21539	0.109327
$866$	0	0
$867$	5.00000	0.169809
$868$	0	0
$869$	−10.9282	−0.370714
$870$	0	0
$871$	−29.8564	−1.01165
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	0	0
$879$	24.9282	0.840807
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	41.8564	1.40858	0.704290	−	0.709912i	$-0.251265\pi$
$883$	41.8564	1.40858	0.704290	+	0.709912i	$0.251265\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	−10.1436	−0.340589	−0.170294	−	0.985393i	$-0.554472\pi$
$887$	−10.1436	−0.340589	−0.170294	+	0.985393i	$0.554472\pi$
$888$	0	0
$889$	−10.9282	−0.366520
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	51.7128	1.73050
$894$	0	0
$895$	−72.0000	−2.40669
$896$	0	0
$897$	13.8564	0.462652
$898$	0	0
$899$	32.7846	1.09343
$900$	0	0
$901$	−3.21539	−0.107120
$902$	0	0
$903$	10.9282	0.363668
$904$	0	0
$905$	−22.1436	−0.736078
$906$	0	0
$907$	−14.9282	−0.495683	−0.247841	−	0.968801i	$-0.579721\pi$
$907$	−14.9282	−0.495683	−0.247841	+	0.968801i	$0.579721\pi$
$908$	0	0
$909$	−19.8564	−0.658595
$910$	0	0
$911$	48.4974	1.60679	0.803396	−	0.595446i	$-0.203024\pi$
$911$	48.4974	1.60679	0.803396	+	0.595446i	$0.203024\pi$
$912$	0	0
$913$	4.39230	0.145364
$914$	0	0
$915$	6.92820	0.229039
$916$	0	0
$917$	−4.39230	−0.145047
$918$	0	0
$919$	−21.8564	−0.720976	−0.360488	−	0.932764i	$-0.617390\pi$
$919$	−21.8564	−0.720976	−0.360488	+	0.932764i	$0.617390\pi$
$920$	0	0
$921$	10.5359	0.347170
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−34.4974	−1.13427
$926$	0	0
$927$	13.4641	0.442219
$928$	0	0
$929$	40.6410	1.33339	0.666694	−	0.745331i	$-0.267709\pi$
$929$	40.6410	1.33339	0.666694	+	0.745331i	$0.267709\pi$
$930$	0	0
$931$	−5.46410	−0.179079
$932$	0	0
$933$	28.3923	0.929522
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	−14.3923	−0.470176	−0.235088	−	0.971974i	$-0.575538\pi$
$937$	−14.3923	−0.470176	−0.235088	+	0.971974i	$0.575538\pi$
$938$	0	0
$939$	16.9282	0.552431
$940$	0	0
$941$	−11.0718	−0.360930	−0.180465	−	0.983581i	$-0.557760\pi$
$941$	−11.0718	−0.360930	−0.180465	+	0.983581i	$0.557760\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	−3.46410	−0.112687
$946$	0	0
$947$	−49.8564	−1.62012	−0.810058	−	0.586350i	$-0.800564\pi$
$947$	−49.8564	−1.62012	−0.810058	+	0.586350i	$0.800564\pi$
$948$	0	0
$949$	−1.07180	−0.0347920
$950$	0	0
$951$	−12.9282	−0.419226
$952$	0	0
$953$	−26.7846	−0.867639	−0.433819	−	0.901000i	$-0.642834\pi$
$953$	−26.7846	−0.867639	−0.433819	+	0.901000i	$0.642834\pi$
$954$	0	0
$955$	72.0000	2.32987
$956$	0	0
$957$	−6.00000	−0.193952
$958$	0	0
$959$	−7.85641	−0.253697
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	6.92820	0.223258
$964$	0	0
$965$	−90.0666	−2.89935
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	18.9282	0.608061
$970$	0	0
$971$	15.7128	0.504248	0.252124	−	0.967695i	$-0.418871\pi$
$971$	15.7128	0.504248	0.252124	+	0.967695i	$0.418871\pi$
$972$	0	0
$973$	−13.4641	−0.431639
$974$	0	0
$975$	−14.0000	−0.448359
$976$	0	0
$977$	−47.5692	−1.52187	−0.760937	−	0.648826i	$-0.775260\pi$
$977$	−47.5692	−1.52187	−0.760937	+	0.648826i	$0.775260\pi$
$978$	0	0
$979$	0.928203	0.0296655
$980$	0	0
$981$	15.8564	0.506256
$982$	0	0
$983$	−23.3205	−0.743809	−0.371904	−	0.928271i	$-0.621295\pi$
$983$	−23.3205	−0.743809	−0.371904	+	0.928271i	$0.621295\pi$
$984$	0	0
$985$	−62.3538	−1.98676
$986$	0	0
$987$	−9.46410	−0.301246
$988$	0	0
$989$	−75.7128	−2.40753
$990$	0	0
$991$	2.14359	0.0680935	0.0340467	−	0.999420i	$-0.489160\pi$
$991$	2.14359	0.0680935	0.0340467	+	0.999420i	$0.489160\pi$
$992$	0	0
$993$	−9.07180	−0.287885
$994$	0	0
$995$	1.35898	0.0430827
$996$	0	0
$997$	24.6410	0.780389	0.390194	−	0.920732i	$-0.372408\pi$
$997$	24.6410	0.780389	0.390194	+	0.920732i	$0.372408\pi$
$998$	0	0
$999$	4.92820	0.155921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bc.1.1		2
4.3	odd	2		462.2.a.h.1.1	✓	2
12.11	even	2		1386.2.a.p.1.2		2
28.27	even	2		3234.2.a.x.1.2		2
44.43	even	2		5082.2.a.bu.1.1		2
84.83	odd	2		9702.2.a.dd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.h.1.1	✓	2	4.3	odd	2
1386.2.a.p.1.2		2	12.11	even	2
3234.2.a.x.1.2		2	28.27	even	2
3696.2.a.bc.1.1		2	1.1	even	1	trivial
5082.2.a.bu.1.1		2	44.43	even	2
9702.2.a.dd.1.1		2	84.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{13} +3.46410 q^{15} +3.46410 q^{17} -5.46410 q^{19} +1.00000 q^{21} -6.92820 q^{23} +7.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -5.46410 q^{31} +1.00000 q^{33} +3.46410 q^{35} -4.92820 q^{37} -2.00000 q^{39} +3.46410 q^{41} +10.9282 q^{43} -3.46410 q^{45} -9.46410 q^{47} +1.00000 q^{49} -3.46410 q^{51} -0.928203 q^{53} +3.46410 q^{55} +5.46410 q^{57} -6.92820 q^{59} +2.00000 q^{61} -1.00000 q^{63} -6.92820 q^{65} -14.9282 q^{67} +6.92820 q^{69} +12.0000 q^{71} -0.535898 q^{73} -7.00000 q^{75} +1.00000 q^{77} +10.9282 q^{79} +1.00000 q^{81} -4.39230 q^{83} -12.0000 q^{85} +6.00000 q^{87} -0.928203 q^{89} -2.00000 q^{91} +5.46410 q^{93} +18.9282 q^{95} +2.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} - 4 q^{19} + 2 q^{21} + 14 q^{25} - 2 q^{27} - 12 q^{29} - 4 q^{31} + 2 q^{33} + 4 q^{37} - 4 q^{39} + 8 q^{43} - 12 q^{47} + 2 q^{49} + 12 q^{53}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 - 4 * q^19 + 2 * q^21 + 14 * q^25 - 2 * q^27 - 12 * q^29 - 4 * q^31 + 2 * q^33 + 4 * q^37 - 4 * q^39 + 8 * q^43 - 12 * q^47 + 2 * q^49 + 12 * q^53 + 4 * q^57 + 4 * q^61 - 2 * q^63 - 16 * q^67 + 24 * q^71 - 8 * q^73 - 14 * q^75 + 2 * q^77 + 8 * q^79 + 2 * q^81 + 12 * q^83 - 24 * q^85 + 12 * q^87 + 12 * q^89 - 4 * q^91 + 4 * q^93 + 24 * q^95 + 4 * q^97 - 2 * q^99

Introduction

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