LMFDB - Embedded newform 3696.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3696,2,Mod(1,3696)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.36147$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.93800 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.93800 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -3.93800 q^{13} -3.93800 q^{15} +4.72294 q^{17} -4.78493 q^{19} +1.00000 q^{21} -2.72294 q^{23} +10.5079 q^{25} +1.00000 q^{27} +7.93800 q^{29} -1.15307 q^{31} -1.00000 q^{33} -3.93800 q^{35} -5.50787 q^{37} -3.93800 q^{39} +0.430132 q^{41} -6.72294 q^{43} -3.93800 q^{45} +8.78493 q^{47} +1.00000 q^{49} +4.72294 q^{51} -3.15307 q^{53} +3.93800 q^{55} -4.78493 q^{57} +15.0911 q^{59} +6.00000 q^{61} +1.00000 q^{63} +15.5079 q^{65} -3.21507 q^{67} -2.72294 q^{69} -13.4459 q^{71} +11.9380 q^{73} +10.5079 q^{75} -1.00000 q^{77} +5.44588 q^{79} +1.00000 q^{81} +2.84693 q^{83} -18.5989 q^{85} +7.93800 q^{87} +12.3061 q^{89} -3.93800 q^{91} -1.15307 q^{93} +18.8431 q^{95} +11.1531 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 3 q^{11} - 12 q^{19} + 3 q^{21} + 6 q^{23} + 15 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + 6 q^{41} - 6 q^{43} + 24 q^{47} + 3 q^{49} - 12 q^{57} + 24 q^{59} + 18 q^{61} + 3 q^{63} + 30 q^{65} - 12 q^{67} + 6 q^{69} - 12 q^{71} + 24 q^{73} + 15 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 18 q^{83} - 18 q^{85} + 12 q^{87} + 18 q^{89} + 6 q^{93} - 12 q^{95} + 24 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 3 * q^11 - 12 * q^19 + 3 * q^21 + 6 * q^23 + 15 * q^25 + 3 * q^27 + 12 * q^29 + 6 * q^31 - 3 * q^33 + 6 * q^41 - 6 * q^43 + 24 * q^47 + 3 * q^49 - 12 * q^57 + 24 * q^59 + 18 * q^61 + 3 * q^63 + 30 * q^65 - 12 * q^67 + 6 * q^69 - 12 * q^71 + 24 * q^73 + 15 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 18 * q^83 - 18 * q^85 + 12 * q^87 + 18 * q^89 + 6 * q^93 - 12 * q^95 + 24 * q^97 - 3 * 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
$5$	−3.93800	−1.76113	−0.880564	−	0.473927i	$-0.842836\pi$
$5$	−3.93800	−1.76113	−0.880564	+	0.473927i	$0.842836\pi$
$6$	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
$13$	−3.93800	−1.09221	−0.546103	−	0.837718i	$-0.683889\pi$
$13$	−3.93800	−1.09221	−0.546103	+	0.837718i	$0.683889\pi$
$14$	0	0
$15$	−3.93800	−1.01679
$16$	0	0
$17$	4.72294	1.14548	0.572740	−	0.819737i	$-0.305880\pi$
$17$	4.72294	1.14548	0.572740	+	0.819737i	$0.305880\pi$
$18$	0	0
$19$	−4.78493	−1.09774	−0.548870	−	0.835908i	$-0.684942\pi$
$19$	−4.78493	−1.09774	−0.548870	+	0.835908i	$0.684942\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.72294	−0.567772	−0.283886	−	0.958858i	$-0.591624\pi$
$23$	−2.72294	−0.567772	−0.283886	+	0.958858i	$0.591624\pi$
$24$	0	0
$25$	10.5079	2.10157
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.93800	1.47405	0.737025	−	0.675865i	$-0.236230\pi$
$29$	7.93800	1.47405	0.737025	+	0.675865i	$0.236230\pi$
$30$	0	0
$31$	−1.15307	−0.207097	−0.103549	−	0.994624i	$-0.533020\pi$
$31$	−1.15307	−0.207097	−0.103549	+	0.994624i	$0.533020\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−3.93800	−0.665644
$36$	0	0
$37$	−5.50787	−0.905489	−0.452744	−	0.891640i	$-0.649555\pi$
$37$	−5.50787	−0.905489	−0.452744	+	0.891640i	$0.649555\pi$
$38$	0	0
$39$	−3.93800	−0.630585
$40$	0	0
$41$	0.430132	0.0671753	0.0335877	−	0.999436i	$-0.489307\pi$
$41$	0.430132	0.0671753	0.0335877	+	0.999436i	$0.489307\pi$
$42$	0	0
$43$	−6.72294	−1.02524	−0.512619	−	0.858616i	$-0.671325\pi$
$43$	−6.72294	−1.02524	−0.512619	+	0.858616i	$0.671325\pi$
$44$	0	0
$45$	−3.93800	−0.587043
$46$	0	0
$47$	8.78493	1.28141	0.640707	−	0.767785i	$-0.278641\pi$
$47$	8.78493	1.28141	0.640707	+	0.767785i	$0.278641\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.72294	0.661344
$52$	0	0
$53$	−3.15307	−0.433107	−0.216554	−	0.976271i	$-0.569482\pi$
$53$	−3.15307	−0.433107	−0.216554	+	0.976271i	$0.569482\pi$
$54$	0	0
$55$	3.93800	0.531000
$56$	0	0
$57$	−4.78493	−0.633780
$58$	0	0
$59$	15.0911	1.96469	0.982345	−	0.187077i	$-0.0599015\pi$
$59$	15.0911	1.96469	0.982345	+	0.187077i	$0.0599015\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	15.5079	1.92351
$66$	0	0
$67$	−3.21507	−0.392783	−0.196391	−	0.980526i	$-0.562922\pi$
$67$	−3.21507	−0.392783	−0.196391	+	0.980526i	$0.562922\pi$
$68$	0	0
$69$	−2.72294	−0.327803
$70$	0	0
$71$	−13.4459	−1.59573	−0.797866	−	0.602835i	$-0.794038\pi$
$71$	−13.4459	−1.59573	−0.797866	+	0.602835i	$0.794038\pi$
$72$	0	0
$73$	11.9380	1.39724	0.698619	−	0.715494i	$-0.253798\pi$
$73$	11.9380	1.39724	0.698619	+	0.715494i	$0.253798\pi$
$74$	0	0
$75$	10.5079	1.21334
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.44588	0.612709	0.306354	−	0.951918i	$-0.400891\pi$
$79$	5.44588	0.612709	0.306354	+	0.951918i	$0.400891\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.84693	0.312491	0.156246	−	0.987718i	$-0.450061\pi$
$83$	2.84693	0.312491	0.156246	+	0.987718i	$0.450061\pi$
$84$	0	0
$85$	−18.5989	−2.01734
$86$	0	0
$87$	7.93800	0.851043
$88$	0	0
$89$	12.3061	1.30445	0.652224	−	0.758026i	$-0.273836\pi$
$89$	12.3061	1.30445	0.652224	+	0.758026i	$0.273836\pi$
$90$	0	0
$91$	−3.93800	−0.412815
$92$	0	0
$93$	−1.15307	−0.119568
$94$	0	0
$95$	18.8431	1.93326
$96$	0	0
$97$	11.1531	1.13242	0.566211	−	0.824260i	$-0.308409\pi$
$97$	11.1531	1.13242	0.566211	+	0.824260i	$0.308409\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	16.4750	1.63932	0.819659	−	0.572851i	$-0.194163\pi$
$101$	16.4750	1.63932	0.819659	+	0.572851i	$0.194163\pi$
$102$	0	0
$103$	1.56987	0.154684	0.0773418	−	0.997005i	$-0.475357\pi$
$103$	1.56987	0.154684	0.0773418	+	0.997005i	$0.475357\pi$
$104$	0	0
$105$	−3.93800	−0.384310
$106$	0	0
$107$	4.78493	0.462577	0.231289	−	0.972885i	$-0.425706\pi$
$107$	4.78493	0.462577	0.231289	+	0.972885i	$0.425706\pi$
$108$	0	0
$109$	−15.0291	−1.43952	−0.719762	−	0.694221i	$-0.755749\pi$
$109$	−15.0291	−1.43952	−0.719762	+	0.694221i	$0.755749\pi$
$110$	0	0
$111$	−5.50787	−0.522784
$112$	0	0
$113$	7.44588	0.700449	0.350225	−	0.936666i	$-0.386105\pi$
$113$	7.44588	0.700449	0.350225	+	0.936666i	$0.386105\pi$
$114$	0	0
$115$	10.7229	0.999919
$116$	0	0
$117$	−3.93800	−0.364069
$118$	0	0
$119$	4.72294	0.432951
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.430132	0.0387837
$124$	0	0
$125$	−21.6900	−1.94001
$126$	0	0
$127$	−12.2928	−1.09081	−0.545405	−	0.838173i	$-0.683624\pi$
$127$	−12.2928	−1.09081	−0.545405	+	0.838173i	$0.683624\pi$
$128$	0	0
$129$	−6.72294	−0.591922
$130$	0	0
$131$	−7.13974	−0.623802	−0.311901	−	0.950115i	$-0.600966\pi$
$131$	−7.13974	−0.623802	−0.311901	+	0.950115i	$0.600966\pi$
$132$	0	0
$133$	−4.78493	−0.414906
$134$	0	0
$135$	−3.93800	−0.338929
$136$	0	0
$137$	1.58320	0.135262	0.0676310	−	0.997710i	$-0.478456\pi$
$137$	1.58320	0.135262	0.0676310	+	0.997710i	$0.478456\pi$
$138$	0	0
$139$	17.1979	1.45871	0.729353	−	0.684138i	$-0.239821\pi$
$139$	17.1979	1.45871	0.729353	+	0.684138i	$0.239821\pi$
$140$	0	0
$141$	8.78493	0.739825
$142$	0	0
$143$	3.93800	0.329312
$144$	0	0
$145$	−31.2599	−2.59599
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	15.9380	1.30569	0.652846	−	0.757491i	$-0.273575\pi$
$149$	15.9380	1.30569	0.652846	+	0.757491i	$0.273575\pi$
$150$	0	0
$151$	−6.59894	−0.537014	−0.268507	−	0.963278i	$-0.586530\pi$
$151$	−6.59894	−0.537014	−0.268507	+	0.963278i	$0.586530\pi$
$152$	0	0
$153$	4.72294	0.381827
$154$	0	0
$155$	4.54079	0.364725
$156$	0	0
$157$	7.32188	0.584350	0.292175	−	0.956365i	$-0.405621\pi$
$157$	7.32188	0.584350	0.292175	+	0.956365i	$0.405621\pi$
$158$	0	0
$159$	−3.15307	−0.250055
$160$	0	0
$161$	−2.72294	−0.214598
$162$	0	0
$163$	−14.1068	−1.10493	−0.552466	−	0.833536i	$-0.686313\pi$
$163$	−14.1068	−1.10493	−0.552466	+	0.833536i	$0.686313\pi$
$164$	0	0
$165$	3.93800	0.306573
$166$	0	0
$167$	0.416799	0.0322528	0.0161264	−	0.999870i	$-0.494867\pi$
$167$	0.416799	0.0322528	0.0161264	+	0.999870i	$0.494867\pi$
$168$	0	0
$169$	2.50787	0.192913
$170$	0	0
$171$	−4.78493	−0.365913
$172$	0	0
$173$	−8.43013	−0.640931	−0.320466	−	0.947260i	$-0.603839\pi$
$173$	−8.43013	−0.640931	−0.320466	+	0.947260i	$0.603839\pi$
$174$	0	0
$175$	10.5079	0.794320
$176$	0	0
$177$	15.0911	1.13431
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	20.5989	1.53111	0.765554	−	0.643372i	$-0.222465\pi$
$181$	20.5989	1.53111	0.765554	+	0.643372i	$0.222465\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	21.6900	1.59468
$186$	0	0
$187$	−4.72294	−0.345375
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	14.5989	1.05634	0.528171	−	0.849138i	$-0.322878\pi$
$191$	14.5989	1.05634	0.528171	+	0.849138i	$0.322878\pi$
$192$	0	0
$193$	13.4592	0.968815	0.484408	−	0.874842i	$-0.339035\pi$
$193$	13.4592	0.968815	0.484408	+	0.874842i	$0.339035\pi$
$194$	0	0
$195$	15.5079	1.11054
$196$	0	0
$197$	11.4459	0.815485	0.407742	−	0.913097i	$-0.366316\pi$
$197$	11.4459	0.815485	0.407742	+	0.913097i	$0.366316\pi$
$198$	0	0
$199$	13.0291	0.923607	0.461803	−	0.886982i	$-0.347203\pi$
$199$	13.0291	0.923607	0.461803	+	0.886982i	$0.347203\pi$
$200$	0	0
$201$	−3.21507	−0.226773
$202$	0	0
$203$	7.93800	0.557139
$204$	0	0
$205$	−1.69386	−0.118304
$206$	0	0
$207$	−2.72294	−0.189257
$208$	0	0
$209$	4.78493	0.330981
$210$	0	0
$211$	−2.43013	−0.167297	−0.0836486	−	0.996495i	$-0.526657\pi$
$211$	−2.43013	−0.167297	−0.0836486	+	0.996495i	$0.526657\pi$
$212$	0	0
$213$	−13.4459	−0.921296
$214$	0	0
$215$	26.4750	1.80558
$216$	0	0
$217$	−1.15307	−0.0782755
$218$	0	0
$219$	11.9380	0.806696
$220$	0	0
$221$	−18.5989	−1.25110
$222$	0	0
$223$	−22.7678	−1.52464	−0.762321	−	0.647199i	$-0.775940\pi$
$223$	−22.7678	−1.52464	−0.762321	+	0.647199i	$0.775940\pi$
$224$	0	0
$225$	10.5079	0.700525
$226$	0	0
$227$	15.8760	1.05373	0.526864	−	0.849950i	$-0.323368\pi$
$227$	15.8760	1.05373	0.526864	+	0.849950i	$0.323368\pi$
$228$	0	0
$229$	−3.15307	−0.208361	−0.104180	−	0.994558i	$-0.533222\pi$
$229$	−3.15307	−0.208361	−0.104180	+	0.994558i	$0.533222\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	13.1397	0.860813	0.430406	−	0.902635i	$-0.358370\pi$
$233$	13.1397	0.860813	0.430406	+	0.902635i	$0.358370\pi$
$234$	0	0
$235$	−34.5951	−2.25674
$236$	0	0
$237$	5.44588	0.353748
$238$	0	0
$239$	10.1068	0.653756	0.326878	−	0.945067i	$-0.394003\pi$
$239$	10.1068	0.653756	0.326878	+	0.945067i	$0.394003\pi$
$240$	0	0
$241$	2.36814	0.152545	0.0762725	−	0.997087i	$-0.475698\pi$
$241$	2.36814	0.152545	0.0762725	+	0.997087i	$0.475698\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.93800	−0.251590
$246$	0	0
$247$	18.8431	1.19896
$248$	0	0
$249$	2.84693	0.180417
$250$	0	0
$251$	−10.2308	−0.645763	−0.322881	−	0.946439i	$-0.604652\pi$
$251$	−10.2308	−0.645763	−0.322881	+	0.946439i	$0.604652\pi$
$252$	0	0
$253$	2.72294	0.171190
$254$	0	0
$255$	−18.5989	−1.16471
$256$	0	0
$257$	−11.0777	−0.691010	−0.345505	−	0.938417i	$-0.612292\pi$
$257$	−11.0777	−0.691010	−0.345505	+	0.938417i	$0.612292\pi$
$258$	0	0
$259$	−5.50787	−0.342242
$260$	0	0
$261$	7.93800	0.491350
$262$	0	0
$263$	8.78493	0.541702	0.270851	−	0.962621i	$-0.412695\pi$
$263$	8.78493	0.541702	0.270851	+	0.962621i	$0.412695\pi$
$264$	0	0
$265$	12.4168	0.762758
$266$	0	0
$267$	12.3061	0.753123
$268$	0	0
$269$	5.75201	0.350706	0.175353	−	0.984506i	$-0.443893\pi$
$269$	5.75201	0.350706	0.175353	+	0.984506i	$0.443893\pi$
$270$	0	0
$271$	−0.784934	−0.0476813	−0.0238407	−	0.999716i	$-0.507589\pi$
$271$	−0.784934	−0.0476813	−0.0238407	+	0.999716i	$0.507589\pi$
$272$	0	0
$273$	−3.93800	−0.238339
$274$	0	0
$275$	−10.5079	−0.633648
$276$	0	0
$277$	−11.5699	−0.695166	−0.347583	−	0.937649i	$-0.612998\pi$
$277$	−11.5699	−0.695166	−0.347583	+	0.937649i	$0.612998\pi$
$278$	0	0
$279$	−1.15307	−0.0690325
$280$	0	0
$281$	−25.2599	−1.50688	−0.753439	−	0.657518i	$-0.771607\pi$
$281$	−25.2599	−1.50688	−0.753439	+	0.657518i	$0.771607\pi$
$282$	0	0
$283$	10.9671	0.651925	0.325963	−	0.945383i	$-0.394312\pi$
$283$	10.9671	0.651925	0.325963	+	0.945383i	$0.394312\pi$
$284$	0	0
$285$	18.8431	1.11617
$286$	0	0
$287$	0.430132	0.0253899
$288$	0	0
$289$	5.30614	0.312126
$290$	0	0
$291$	11.1531	0.653805
$292$	0	0
$293$	14.7363	0.860902	0.430451	−	0.902614i	$-0.358355\pi$
$293$	14.7363	0.860902	0.430451	+	0.902614i	$0.358355\pi$
$294$	0	0
$295$	−59.4287	−3.46007
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	10.7229	0.620123
$300$	0	0
$301$	−6.72294	−0.387504
$302$	0	0
$303$	16.4750	0.946461
$304$	0	0
$305$	−23.6280	−1.35294
$306$	0	0
$307$	−0.860264	−0.0490979	−0.0245489	−	0.999699i	$-0.507815\pi$
$307$	−0.860264	−0.0490979	−0.0245489	+	0.999699i	$0.507815\pi$
$308$	0	0
$309$	1.56987	0.0893067
$310$	0	0
$311$	−26.8918	−1.52489	−0.762446	−	0.647052i	$-0.776002\pi$
$311$	−26.8918	−1.52489	−0.762446	+	0.647052i	$0.776002\pi$
$312$	0	0
$313$	10.1688	0.574775	0.287388	−	0.957814i	$-0.407213\pi$
$313$	10.1688	0.574775	0.287388	+	0.957814i	$0.407213\pi$
$314$	0	0
$315$	−3.93800	−0.221881
$316$	0	0
$317$	−27.4907	−1.54403	−0.772016	−	0.635604i	$-0.780751\pi$
$317$	−27.4907	−1.54403	−0.772016	+	0.635604i	$0.780751\pi$
$318$	0	0
$319$	−7.93800	−0.444443
$320$	0	0
$321$	4.78493	0.267069
$322$	0	0
$323$	−22.5989	−1.25744
$324$	0	0
$325$	−41.3800	−2.29535
$326$	0	0
$327$	−15.0291	−0.831110
$328$	0	0
$329$	8.78493	0.484329
$330$	0	0
$331$	17.1979	0.945281	0.472641	−	0.881255i	$-0.343301\pi$
$331$	17.1979	0.945281	0.472641	+	0.881255i	$0.343301\pi$
$332$	0	0
$333$	−5.50787	−0.301830
$334$	0	0
$335$	12.6609	0.691741
$336$	0	0
$337$	19.7387	1.07523	0.537617	−	0.843189i	$-0.319325\pi$
$337$	19.7387	1.07523	0.537617	+	0.843189i	$0.319325\pi$
$338$	0	0
$339$	7.44588	0.404404
$340$	0	0
$341$	1.15307	0.0624422
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	10.7229	0.577304
$346$	0	0
$347$	−16.6123	−0.891794	−0.445897	−	0.895084i	$-0.647115\pi$
$347$	−16.6123	−0.891794	−0.445897	+	0.895084i	$0.647115\pi$
$348$	0	0
$349$	6.95375	0.372226	0.186113	−	0.982528i	$-0.440411\pi$
$349$	6.95375	0.372226	0.186113	+	0.982528i	$0.440411\pi$
$350$	0	0
$351$	−3.93800	−0.210195
$352$	0	0
$353$	−3.22840	−0.171830	−0.0859152	−	0.996302i	$-0.527381\pi$
$353$	−3.22840	−0.171830	−0.0859152	+	0.996302i	$0.527381\pi$
$354$	0	0
$355$	52.9499	2.81029
$356$	0	0
$357$	4.72294	0.249964
$358$	0	0
$359$	−10.3061	−0.543937	−0.271969	−	0.962306i	$-0.587675\pi$
$359$	−10.3061	−0.543937	−0.271969	+	0.962306i	$0.587675\pi$
$360$	0	0
$361$	3.89559	0.205031
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−47.0119	−2.46072
$366$	0	0
$367$	11.8760	0.619923	0.309961	−	0.950749i	$-0.399684\pi$
$367$	11.8760	0.619923	0.309961	+	0.950749i	$0.399684\pi$
$368$	0	0
$369$	0.430132	0.0223918
$370$	0	0
$371$	−3.15307	−0.163699
$372$	0	0
$373$	16.7229	0.865881	0.432940	−	0.901423i	$-0.357476\pi$
$373$	16.7229	0.865881	0.432940	+	0.901423i	$0.357476\pi$
$374$	0	0
$375$	−21.6900	−1.12007
$376$	0	0
$377$	−31.2599	−1.60997
$378$	0	0
$379$	10.9671	0.563341	0.281671	−	0.959511i	$-0.409111\pi$
$379$	10.9671	0.563341	0.281671	+	0.959511i	$0.409111\pi$
$380$	0	0
$381$	−12.2928	−0.629780
$382$	0	0
$383$	−15.7520	−0.804890	−0.402445	−	0.915444i	$-0.631840\pi$
$383$	−15.7520	−0.804890	−0.402445	+	0.915444i	$0.631840\pi$
$384$	0	0
$385$	3.93800	0.200699
$386$	0	0
$387$	−6.72294	−0.341746
$388$	0	0
$389$	36.7678	1.86420	0.932100	−	0.362202i	$-0.117975\pi$
$389$	36.7678	1.86420	0.932100	+	0.362202i	$0.117975\pi$
$390$	0	0
$391$	−12.8603	−0.650371
$392$	0	0
$393$	−7.13974	−0.360152
$394$	0	0
$395$	−21.4459	−1.07906
$396$	0	0
$397$	11.8627	0.595371	0.297685	−	0.954664i	$-0.403785\pi$
$397$	11.8627	0.595371	0.297685	+	0.954664i	$0.403785\pi$
$398$	0	0
$399$	−4.78493	−0.239546
$400$	0	0
$401$	3.40106	0.169841	0.0849203	−	0.996388i	$-0.472936\pi$
$401$	3.40106	0.169841	0.0849203	+	0.996388i	$0.472936\pi$
$402$	0	0
$403$	4.54079	0.226193
$404$	0	0
$405$	−3.93800	−0.195681
$406$	0	0
$407$	5.50787	0.273015
$408$	0	0
$409$	−28.0315	−1.38607	−0.693034	−	0.720905i	$-0.743726\pi$
$409$	−28.0315	−1.38607	−0.693034	+	0.720905i	$0.743726\pi$
$410$	0	0
$411$	1.58320	0.0780936
$412$	0	0
$413$	15.0911	0.742583
$414$	0	0
$415$	−11.2112	−0.550337
$416$	0	0
$417$	17.1979	0.842184
$418$	0	0
$419$	−28.2890	−1.38201	−0.691003	−	0.722852i	$-0.742831\pi$
$419$	−28.2890	−1.38201	−0.691003	+	0.722852i	$0.742831\pi$
$420$	0	0
$421$	−14.4921	−0.706303	−0.353152	−	0.935566i	$-0.614890\pi$
$421$	−14.4921	−0.706303	−0.353152	+	0.935566i	$0.614890\pi$
$422$	0	0
$423$	8.78493	0.427138
$424$	0	0
$425$	49.6280	2.40731
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	3.93800	0.190129
$430$	0	0
$431$	24.5369	1.18190	0.590952	−	0.806707i	$-0.298752\pi$
$431$	24.5369	1.18190	0.590952	+	0.806707i	$0.298752\pi$
$432$	0	0
$433$	−7.32188	−0.351867	−0.175934	−	0.984402i	$-0.556294\pi$
$433$	−7.32188	−0.351867	−0.175934	+	0.984402i	$0.556294\pi$
$434$	0	0
$435$	−31.2599	−1.49880
$436$	0	0
$437$	13.0291	0.623265
$438$	0	0
$439$	−16.5369	−0.789265	−0.394633	−	0.918839i	$-0.629128\pi$
$439$	−16.5369	−0.789265	−0.394633	+	0.918839i	$0.629128\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−25.1979	−1.19719	−0.598594	−	0.801053i	$-0.704274\pi$
$443$	−25.1979	−1.19719	−0.598594	+	0.801053i	$0.704274\pi$
$444$	0	0
$445$	−48.4616	−2.29730
$446$	0	0
$447$	15.9380	0.753842
$448$	0	0
$449$	−36.5198	−1.72347	−0.861737	−	0.507355i	$-0.830623\pi$
$449$	−36.5198	−1.72347	−0.861737	+	0.507355i	$0.830623\pi$
$450$	0	0
$451$	−0.430132	−0.0202541
$452$	0	0
$453$	−6.59894	−0.310045
$454$	0	0
$455$	15.5079	0.727020
$456$	0	0
$457$	3.15307	0.147494	0.0737472	−	0.997277i	$-0.476504\pi$
$457$	3.15307	0.147494	0.0737472	+	0.997277i	$0.476504\pi$
$458$	0	0
$459$	4.72294	0.220448
$460$	0	0
$461$	−24.7678	−1.15355	−0.576775	−	0.816903i	$-0.695689\pi$
$461$	−24.7678	−1.15355	−0.576775	+	0.816903i	$0.695689\pi$
$462$	0	0
$463$	35.4287	1.64651	0.823256	−	0.567671i	$-0.192155\pi$
$463$	35.4287	1.64651	0.823256	+	0.567671i	$0.192155\pi$
$464$	0	0
$465$	4.54079	0.210574
$466$	0	0
$467$	23.9247	1.10710	0.553551	−	0.832815i	$-0.313272\pi$
$467$	23.9247	1.10710	0.553551	+	0.832815i	$0.313272\pi$
$468$	0	0
$469$	−3.21507	−0.148458
$470$	0	0
$471$	7.32188	0.337375
$472$	0	0
$473$	6.72294	0.309121
$474$	0	0
$475$	−50.2795	−2.30698
$476$	0	0
$477$	−3.15307	−0.144369
$478$	0	0
$479$	−4.54079	−0.207474	−0.103737	−	0.994605i	$-0.533080\pi$
$479$	−4.54079	−0.207474	−0.103737	+	0.994605i	$0.533080\pi$
$480$	0	0
$481$	21.6900	0.988980
$482$	0	0
$483$	−2.72294	−0.123898
$484$	0	0
$485$	−43.9208	−1.99434
$486$	0	0
$487$	−27.1397	−1.22982	−0.614909	−	0.788598i	$-0.710807\pi$
$487$	−27.1397	−1.22982	−0.614909	+	0.788598i	$0.710807\pi$
$488$	0	0
$489$	−14.1068	−0.637932
$490$	0	0
$491$	−3.46305	−0.156285	−0.0781427	−	0.996942i	$-0.524899\pi$
$491$	−3.46305	−0.156285	−0.0781427	+	0.996942i	$0.524899\pi$
$492$	0	0
$493$	37.4907	1.68850
$494$	0	0
$495$	3.93800	0.177000
$496$	0	0
$497$	−13.4459	−0.603130
$498$	0	0
$499$	18.9671	0.849083	0.424542	−	0.905408i	$-0.360435\pi$
$499$	18.9671	0.849083	0.424542	+	0.905408i	$0.360435\pi$
$500$	0	0
$501$	0.416799	0.0186212
$502$	0	0
$503$	−5.86267	−0.261404	−0.130702	−	0.991422i	$-0.541723\pi$
$503$	−5.86267	−0.261404	−0.130702	+	0.991422i	$0.541723\pi$
$504$	0	0
$505$	−64.8784	−2.88705
$506$	0	0
$507$	2.50787	0.111378
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	11.9380	0.528106
$512$	0	0
$513$	−4.78493	−0.211260
$514$	0	0
$515$	−6.18215	−0.272418
$516$	0	0
$517$	−8.78493	−0.386361
$518$	0	0
$519$	−8.43013	−0.370042
$520$	0	0
$521$	−2.24414	−0.0983177	−0.0491588	−	0.998791i	$-0.515654\pi$
$521$	−2.24414	−0.0983177	−0.0491588	+	0.998791i	$0.515654\pi$
$522$	0	0
$523$	17.9828	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$523$	17.9828	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$524$	0	0
$525$	10.5079	0.458601
$526$	0	0
$527$	−5.44588	−0.237226
$528$	0	0
$529$	−15.5856	−0.677635
$530$	0	0
$531$	15.0911	0.654897
$532$	0	0
$533$	−1.69386	−0.0733693
$534$	0	0
$535$	−18.8431	−0.814658
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	28.7678	1.23682	0.618411	−	0.785855i	$-0.287777\pi$
$541$	28.7678	1.23682	0.618411	+	0.785855i	$0.287777\pi$
$542$	0	0
$543$	20.5989	0.883985
$544$	0	0
$545$	59.1846	2.53519
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−37.9828	−1.61812
$552$	0	0
$553$	5.44588	0.231582
$554$	0	0
$555$	21.6900	0.920690
$556$	0	0
$557$	−13.5079	−0.572347	−0.286173	−	0.958178i	$-0.592383\pi$
$557$	−13.5079	−0.572347	−0.286173	+	0.958178i	$0.592383\pi$
$558$	0	0
$559$	26.4750	1.11977
$560$	0	0
$561$	−4.72294	−0.199403
$562$	0	0
$563$	−24.7811	−1.04440	−0.522199	−	0.852824i	$-0.674888\pi$
$563$	−24.7811	−1.04440	−0.522199	+	0.852824i	$0.674888\pi$
$564$	0	0
$565$	−29.3219	−1.23358
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	1.75201	0.0734482	0.0367241	−	0.999325i	$-0.488308\pi$
$569$	1.75201	0.0734482	0.0367241	+	0.999325i	$0.488308\pi$
$570$	0	0
$571$	3.16640	0.132510	0.0662549	−	0.997803i	$-0.478895\pi$
$571$	3.16640	0.132510	0.0662549	+	0.997803i	$0.478895\pi$
$572$	0	0
$573$	14.5989	0.609880
$574$	0	0
$575$	−28.6123	−1.19321
$576$	0	0
$577$	20.4750	0.852383	0.426192	−	0.904633i	$-0.359855\pi$
$577$	20.4750	0.852383	0.426192	+	0.904633i	$0.359855\pi$
$578$	0	0
$579$	13.4592	0.559346
$580$	0	0
$581$	2.84693	0.118111
$582$	0	0
$583$	3.15307	0.130587
$584$	0	0
$585$	15.5079	0.641172
$586$	0	0
$587$	16.6609	0.687671	0.343835	−	0.939030i	$-0.388274\pi$
$587$	16.6609	0.687671	0.343835	+	0.939030i	$0.388274\pi$
$588$	0	0
$589$	5.51736	0.227339
$590$	0	0
$591$	11.4459	0.470820
$592$	0	0
$593$	27.9075	1.14602	0.573012	−	0.819547i	$-0.305775\pi$
$593$	27.9075	1.14602	0.573012	+	0.819547i	$0.305775\pi$
$594$	0	0
$595$	−18.5989	−0.762482
$596$	0	0
$597$	13.0291	0.533245
$598$	0	0
$599$	24.3376	0.994408	0.497204	−	0.867634i	$-0.334360\pi$
$599$	24.3376	0.994408	0.497204	+	0.867634i	$0.334360\pi$
$600$	0	0
$601$	−9.50787	−0.387834	−0.193917	−	0.981018i	$-0.562119\pi$
$601$	−9.50787	−0.387834	−0.193917	+	0.981018i	$0.562119\pi$
$602$	0	0
$603$	−3.21507	−0.130928
$604$	0	0
$605$	−3.93800	−0.160103
$606$	0	0
$607$	−26.8431	−1.08953	−0.544764	−	0.838590i	$-0.683381\pi$
$607$	−26.8431	−1.08953	−0.544764	+	0.838590i	$0.683381\pi$
$608$	0	0
$609$	7.93800	0.321664
$610$	0	0
$611$	−34.5951	−1.39957
$612$	0	0
$613$	−23.0291	−0.930136	−0.465068	−	0.885275i	$-0.653970\pi$
$613$	−23.0291	−0.930136	−0.465068	+	0.885275i	$0.653970\pi$
$614$	0	0
$615$	−1.69386	−0.0683031
$616$	0	0
$617$	31.4459	1.26596	0.632982	−	0.774167i	$-0.281831\pi$
$617$	31.4459	1.26596	0.632982	+	0.774167i	$0.281831\pi$
$618$	0	0
$619$	−2.26132	−0.0908901	−0.0454450	−	0.998967i	$-0.514471\pi$
$619$	−2.26132	−0.0908901	−0.0454450	+	0.998967i	$0.514471\pi$
$620$	0	0
$621$	−2.72294	−0.109268
$622$	0	0
$623$	12.3061	0.493035
$624$	0	0
$625$	32.8760	1.31504
$626$	0	0
$627$	4.78493	0.191092
$628$	0	0
$629$	−26.0133	−1.03722
$630$	0	0
$631$	−10.3061	−0.410281	−0.205140	−	0.978733i	$-0.565765\pi$
$631$	−10.3061	−0.410281	−0.205140	+	0.978733i	$0.565765\pi$
$632$	0	0
$633$	−2.43013	−0.0965891
$634$	0	0
$635$	48.4091	1.92106
$636$	0	0
$637$	−3.93800	−0.156029
$638$	0	0
$639$	−13.4459	−0.531911
$640$	0	0
$641$	4.13733	0.163415	0.0817073	−	0.996656i	$-0.473963\pi$
$641$	4.13733	0.163415	0.0817073	+	0.996656i	$0.473963\pi$
$642$	0	0
$643$	8.44347	0.332978	0.166489	−	0.986043i	$-0.446757\pi$
$643$	8.44347	0.332978	0.166489	+	0.986043i	$0.446757\pi$
$644$	0	0
$645$	26.4750	1.04245
$646$	0	0
$647$	25.2732	0.993593	0.496796	−	0.867867i	$-0.334510\pi$
$647$	25.2732	0.993593	0.496796	+	0.867867i	$0.334510\pi$
$648$	0	0
$649$	−15.0911	−0.592376
$650$	0	0
$651$	−1.15307	−0.0451924
$652$	0	0
$653$	47.9075	1.87477	0.937383	−	0.348302i	$-0.113241\pi$
$653$	47.9075	1.87477	0.937383	+	0.348302i	$0.113241\pi$
$654$	0	0
$655$	28.1163	1.09859
$656$	0	0
$657$	11.9380	0.465746
$658$	0	0
$659$	28.2890	1.10198	0.550991	−	0.834511i	$-0.314250\pi$
$659$	28.2890	1.10198	0.550991	+	0.834511i	$0.314250\pi$
$660$	0	0
$661$	19.7653	0.768783	0.384391	−	0.923170i	$-0.374411\pi$
$661$	19.7653	0.768783	0.384391	+	0.923170i	$0.374411\pi$
$662$	0	0
$663$	−18.5989	−0.722323
$664$	0	0
$665$	18.8431	0.730704
$666$	0	0
$667$	−21.6147	−0.836924
$668$	0	0
$669$	−22.7678	−0.880252
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	−51.5174	−1.98585	−0.992924	−	0.118750i	$-0.962111\pi$
$673$	−51.5174	−1.98585	−0.992924	+	0.118750i	$0.962111\pi$
$674$	0	0
$675$	10.5079	0.404448
$676$	0	0
$677$	46.3376	1.78090	0.890450	−	0.455081i	$-0.150390\pi$
$677$	46.3376	1.78090	0.890450	+	0.455081i	$0.150390\pi$
$678$	0	0
$679$	11.1531	0.428016
$680$	0	0
$681$	15.8760	0.608370
$682$	0	0
$683$	39.2112	1.50038	0.750188	−	0.661225i	$-0.229963\pi$
$683$	39.2112	1.50038	0.750188	+	0.661225i	$0.229963\pi$
$684$	0	0
$685$	−6.23465	−0.238214
$686$	0	0
$687$	−3.15307	−0.120297
$688$	0	0
$689$	12.4168	0.473042
$690$	0	0
$691$	−7.62802	−0.290184	−0.145092	−	0.989418i	$-0.546348\pi$
$691$	−7.62802	−0.290184	−0.145092	+	0.989418i	$0.546348\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−67.7253	−2.56897
$696$	0	0
$697$	2.03149	0.0769480
$698$	0	0
$699$	13.1397	0.496990
$700$	0	0
$701$	2.61228	0.0986644	0.0493322	−	0.998782i	$-0.484291\pi$
$701$	2.61228	0.0986644	0.0493322	+	0.998782i	$0.484291\pi$
$702$	0	0
$703$	26.3548	0.993990
$704$	0	0
$705$	−34.5951	−1.30293
$706$	0	0
$707$	16.4750	0.619604
$708$	0	0
$709$	36.1516	1.35770	0.678852	−	0.734276i	$-0.262478\pi$
$709$	36.1516	1.35770	0.678852	+	0.734276i	$0.262478\pi$
$710$	0	0
$711$	5.44588	0.204236
$712$	0	0
$713$	3.13974	0.117584
$714$	0	0
$715$	−15.5079	−0.579962
$716$	0	0
$717$	10.1068	0.377446
$718$	0	0
$719$	−19.6767	−0.733816	−0.366908	−	0.930257i	$-0.619584\pi$
$719$	−19.6767	−0.733816	−0.366908	+	0.930257i	$0.619584\pi$
$720$	0	0
$721$	1.56987	0.0584649
$722$	0	0
$723$	2.36814	0.0880719
$724$	0	0
$725$	83.4115	3.09783
$726$	0	0
$727$	−2.47495	−0.0917909	−0.0458954	−	0.998946i	$-0.514614\pi$
$727$	−2.47495	−0.0917909	−0.0458954	+	0.998946i	$0.514614\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−31.7520	−1.17439
$732$	0	0
$733$	−40.0315	−1.47860	−0.739298	−	0.673378i	$-0.764843\pi$
$733$	−40.0315	−1.47860	−0.739298	+	0.673378i	$0.764843\pi$
$734$	0	0
$735$	−3.93800	−0.145255
$736$	0	0
$737$	3.21507	0.118428
$738$	0	0
$739$	−25.2771	−0.929832	−0.464916	−	0.885355i	$-0.653915\pi$
$739$	−25.2771	−0.929832	−0.464916	+	0.885355i	$0.653915\pi$
$740$	0	0
$741$	18.8431	0.692218
$742$	0	0
$743$	−2.10682	−0.0772916	−0.0386458	−	0.999253i	$-0.512304\pi$
$743$	−2.10682	−0.0772916	−0.0386458	+	0.999253i	$0.512304\pi$
$744$	0	0
$745$	−62.7639	−2.29949
$746$	0	0
$747$	2.84693	0.104164
$748$	0	0
$749$	4.78493	0.174838
$750$	0	0
$751$	45.9828	1.67794	0.838969	−	0.544180i	$-0.183159\pi$
$751$	45.9828	1.67794	0.838969	+	0.544180i	$0.183159\pi$
$752$	0	0
$753$	−10.2308	−0.372831
$754$	0	0
$755$	25.9867	0.945752
$756$	0	0
$757$	−29.2599	−1.06347	−0.531734	−	0.846911i	$-0.678460\pi$
$757$	−29.2599	−1.06347	−0.531734	+	0.846911i	$0.678460\pi$
$758$	0	0
$759$	2.72294	0.0988364
$760$	0	0
$761$	21.8760	0.793005	0.396502	−	0.918034i	$-0.370224\pi$
$761$	21.8760	0.793005	0.396502	+	0.918034i	$0.370224\pi$
$762$	0	0
$763$	−15.0291	−0.544089
$764$	0	0
$765$	−18.5989	−0.672446
$766$	0	0
$767$	−59.4287	−2.14585
$768$	0	0
$769$	−16.2756	−0.586914	−0.293457	−	0.955972i	$-0.594806\pi$
$769$	−16.2756	−0.586914	−0.293457	+	0.955972i	$0.594806\pi$
$770$	0	0
$771$	−11.0777	−0.398955
$772$	0	0
$773$	−30.4921	−1.09673	−0.548363	−	0.836241i	$-0.684749\pi$
$773$	−30.4921	−1.09673	−0.548363	+	0.836241i	$0.684749\pi$
$774$	0	0
$775$	−12.1163	−0.435231
$776$	0	0
$777$	−5.50787	−0.197594
$778$	0	0
$779$	−2.05815	−0.0737410
$780$	0	0
$781$	13.4459	0.481131
$782$	0	0
$783$	7.93800	0.283681
$784$	0	0
$785$	−28.8336	−1.02912
$786$	0	0
$787$	−35.2151	−1.25528	−0.627641	−	0.778503i	$-0.715979\pi$
$787$	−35.2151	−1.25528	−0.627641	+	0.778503i	$0.715979\pi$
$788$	0	0
$789$	8.78493	0.312752
$790$	0	0
$791$	7.44588	0.264745
$792$	0	0
$793$	−23.6280	−0.839056
$794$	0	0
$795$	12.4168	0.440378
$796$	0	0
$797$	−14.9804	−0.530633	−0.265317	−	0.964161i	$-0.585477\pi$
$797$	−14.9804	−0.530633	−0.265317	+	0.964161i	$0.585477\pi$
$798$	0	0
$799$	41.4907	1.46784
$800$	0	0
$801$	12.3061	0.434816
$802$	0	0
$803$	−11.9380	−0.421283
$804$	0	0
$805$	10.7229	0.377934
$806$	0	0
$807$	5.75201	0.202480
$808$	0	0
$809$	−25.2599	−0.888090	−0.444045	−	0.896004i	$-0.646457\pi$
$809$	−25.2599	−0.888090	−0.444045	+	0.896004i	$0.646457\pi$
$810$	0	0
$811$	−29.5212	−1.03663	−0.518315	−	0.855190i	$-0.673440\pi$
$811$	−29.5212	−1.03663	−0.518315	+	0.855190i	$0.673440\pi$
$812$	0	0
$813$	−0.784934	−0.0275288
$814$	0	0
$815$	55.5527	1.94593
$816$	0	0
$817$	32.1688	1.12544
$818$	0	0
$819$	−3.93800	−0.137605
$820$	0	0
$821$	23.7873	0.830184	0.415092	−	0.909779i	$-0.363749\pi$
$821$	23.7873	0.830184	0.415092	+	0.909779i	$0.363749\pi$
$822$	0	0
$823$	−17.7692	−0.619395	−0.309698	−	0.950835i	$-0.600228\pi$
$823$	−17.7692	−0.619395	−0.309698	+	0.950835i	$0.600228\pi$
$824$	0	0
$825$	−10.5079	−0.365837
$826$	0	0
$827$	−8.90893	−0.309794	−0.154897	−	0.987931i	$-0.549505\pi$
$827$	−8.90893	−0.309794	−0.154897	+	0.987931i	$0.549505\pi$
$828$	0	0
$829$	−32.7678	−1.13807	−0.569036	−	0.822313i	$-0.692683\pi$
$829$	−32.7678	−1.13807	−0.569036	+	0.822313i	$0.692683\pi$
$830$	0	0
$831$	−11.5699	−0.401354
$832$	0	0
$833$	4.72294	0.163640
$834$	0	0
$835$	−1.64135	−0.0568014
$836$	0	0
$837$	−1.15307	−0.0398559
$838$	0	0
$839$	−6.96708	−0.240530	−0.120265	−	0.992742i	$-0.538374\pi$
$839$	−6.96708	−0.240530	−0.120265	+	0.992742i	$0.538374\pi$
$840$	0	0
$841$	34.0119	1.17282
$842$	0	0
$843$	−25.2599	−0.869997
$844$	0	0
$845$	−9.87601	−0.339745
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	10.9671	0.376389
$850$	0	0
$851$	14.9976	0.514111
$852$	0	0
$853$	11.1979	0.383408	0.191704	−	0.981453i	$-0.438599\pi$
$853$	11.1979	0.383408	0.191704	+	0.981453i	$0.438599\pi$
$854$	0	0
$855$	18.8431	0.644420
$856$	0	0
$857$	−44.0315	−1.50409	−0.752043	−	0.659114i	$-0.770932\pi$
$857$	−44.0315	−1.50409	−0.752043	+	0.659114i	$0.770932\pi$
$858$	0	0
$859$	−7.13974	−0.243605	−0.121802	−	0.992554i	$-0.538867\pi$
$859$	−7.13974	−0.243605	−0.121802	+	0.992554i	$0.538867\pi$
$860$	0	0
$861$	0.430132	0.0146589
$862$	0	0
$863$	5.27706	0.179633	0.0898166	−	0.995958i	$-0.471372\pi$
$863$	5.27706	0.179633	0.0898166	+	0.995958i	$0.471372\pi$
$864$	0	0
$865$	33.1979	1.12876
$866$	0	0
$867$	5.30614	0.180206
$868$	0	0
$869$	−5.44588	−0.184739
$870$	0	0
$871$	12.6609	0.429000
$872$	0	0
$873$	11.1531	0.377474
$874$	0	0
$875$	−21.6900	−0.733256
$876$	0	0
$877$	−32.4301	−1.09509	−0.547544	−	0.836777i	$-0.684437\pi$
$877$	−32.4301	−1.09509	−0.547544	+	0.836777i	$0.684437\pi$
$878$	0	0
$879$	14.7363	0.497042
$880$	0	0
$881$	39.5660	1.33301	0.666507	−	0.745499i	$-0.267789\pi$
$881$	39.5660	1.33301	0.666507	+	0.745499i	$0.267789\pi$
$882$	0	0
$883$	24.4130	0.821561	0.410781	−	0.911734i	$-0.365256\pi$
$883$	24.4130	0.821561	0.410781	+	0.911734i	$0.365256\pi$
$884$	0	0
$885$	−59.4287	−1.99767
$886$	0	0
$887$	−36.2928	−1.21859	−0.609297	−	0.792942i	$-0.708548\pi$
$887$	−36.2928	−1.21859	−0.609297	+	0.792942i	$0.708548\pi$
$888$	0	0
$889$	−12.2928	−0.412287
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−42.0353	−1.40666
$894$	0	0
$895$	−47.2560	−1.57960
$896$	0	0
$897$	10.7229	0.358028
$898$	0	0
$899$	−9.15307	−0.305272
$900$	0	0
$901$	−14.8918	−0.496116
$902$	0	0
$903$	−6.72294	−0.223725
$904$	0	0
$905$	−81.1187	−2.69648
$906$	0	0
$907$	30.3958	1.00928	0.504638	−	0.863331i	$-0.331626\pi$
$907$	30.3958	1.00928	0.504638	+	0.863331i	$0.331626\pi$
$908$	0	0
$909$	16.4750	0.546440
$910$	0	0
$911$	56.8259	1.88273	0.941363	−	0.337395i	$-0.109546\pi$
$911$	56.8259	1.88273	0.941363	+	0.337395i	$0.109546\pi$
$912$	0	0
$913$	−2.84693	−0.0942196
$914$	0	0
$915$	−23.6280	−0.781118
$916$	0	0
$917$	−7.13974	−0.235775
$918$	0	0
$919$	40.1688	1.32505	0.662523	−	0.749041i	$-0.269486\pi$
$919$	40.1688	1.32505	0.662523	+	0.749041i	$0.269486\pi$
$920$	0	0
$921$	−0.860264	−0.0283467
$922$	0	0
$923$	52.9499	1.74287
$924$	0	0
$925$	−57.8760	−1.90295
$926$	0	0
$927$	1.56987	0.0515612
$928$	0	0
$929$	−27.3180	−0.896276	−0.448138	−	0.893964i	$-0.647913\pi$
$929$	−27.3180	−0.896276	−0.448138	+	0.893964i	$0.647913\pi$
$930$	0	0
$931$	−4.78493	−0.156820
$932$	0	0
$933$	−26.8918	−0.880396
$934$	0	0
$935$	18.5989	0.608251
$936$	0	0
$937$	−5.75201	−0.187910	−0.0939551	−	0.995576i	$-0.529951\pi$
$937$	−5.75201	−0.187910	−0.0939551	+	0.995576i	$0.529951\pi$
$938$	0	0
$939$	10.1688	0.331847
$940$	0	0
$941$	−27.5699	−0.898752	−0.449376	−	0.893343i	$-0.648354\pi$
$941$	−27.5699	−0.898752	−0.449376	+	0.893343i	$0.648354\pi$
$942$	0	0
$943$	−1.17122	−0.0381402
$944$	0	0
$945$	−3.93800	−0.128103
$946$	0	0
$947$	−37.0739	−1.20474	−0.602370	−	0.798217i	$-0.705777\pi$
$947$	−37.0739	−1.20474	−0.602370	+	0.798217i	$0.705777\pi$
$948$	0	0
$949$	−47.0119	−1.52607
$950$	0	0
$951$	−27.4907	−0.891447
$952$	0	0
$953$	−12.0620	−0.390726	−0.195363	−	0.980731i	$-0.562589\pi$
$953$	−12.0620	−0.390726	−0.195363	+	0.980731i	$0.562589\pi$
$954$	0	0
$955$	−57.4907	−1.86036
$956$	0	0
$957$	−7.93800	−0.256599
$958$	0	0
$959$	1.58320	0.0511242
$960$	0	0
$961$	−29.6704	−0.957111
$962$	0	0
$963$	4.78493	0.154192
$964$	0	0
$965$	−53.0024	−1.70621
$966$	0	0
$967$	−6.43013	−0.206779	−0.103390	−	0.994641i	$-0.532969\pi$
$967$	−6.43013	−0.206779	−0.103390	+	0.994641i	$0.532969\pi$
$968$	0	0
$969$	−22.5989	−0.725983
$970$	0	0
$971$	35.3047	1.13298	0.566491	−	0.824068i	$-0.308301\pi$
$971$	35.3047	1.13298	0.566491	+	0.824068i	$0.308301\pi$
$972$	0	0
$973$	17.1979	0.551339
$974$	0	0
$975$	−41.3800	−1.32522
$976$	0	0
$977$	0.261319	0.00836035	0.00418017	−	0.999991i	$-0.498669\pi$
$977$	0.261319	0.00836035	0.00418017	+	0.999991i	$0.498669\pi$
$978$	0	0
$979$	−12.3061	−0.393306
$980$	0	0
$981$	−15.0291	−0.479841
$982$	0	0
$983$	16.3376	0.521089	0.260545	−	0.965462i	$-0.416098\pi$
$983$	16.3376	0.521089	0.260545	+	0.965462i	$0.416098\pi$
$984$	0	0
$985$	−45.0739	−1.43617
$986$	0	0
$987$	8.78493	0.279628
$988$	0	0
$989$	18.3061	0.582101
$990$	0	0
$991$	36.2623	1.15191	0.575955	−	0.817481i	$-0.304630\pi$
$991$	36.2623	1.15191	0.575955	+	0.817481i	$0.304630\pi$
$992$	0	0
$993$	17.1979	0.545759
$994$	0	0
$995$	−51.3085	−1.62659
$996$	0	0
$997$	−3.47254	−0.109977	−0.0549883	−	0.998487i	$-0.517512\pi$
$997$	−3.47254	−0.109977	−0.0549883	+	0.998487i	$0.517512\pi$
$998$	0	0
$999$	−5.50787	−0.174261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bp.1.1		3
4.3	odd	2		231.2.a.d.1.1	✓	3
12.11	even	2		693.2.a.m.1.3		3
20.19	odd	2		5775.2.a.bw.1.3		3
28.27	even	2		1617.2.a.s.1.1		3
44.43	even	2		2541.2.a.bi.1.3		3
84.83	odd	2		4851.2.a.bp.1.3		3
132.131	odd	2		7623.2.a.cb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.1	✓	3	4.3	odd	2
693.2.a.m.1.3		3	12.11	even	2
1617.2.a.s.1.1		3	28.27	even	2
2541.2.a.bi.1.3		3	44.43	even	2
3696.2.a.bp.1.1		3	1.1	even	1	trivial
4851.2.a.bp.1.3		3	84.83	odd	2
5775.2.a.bw.1.3		3	20.19	odd	2
7623.2.a.cb.1.1		3	132.131	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:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.93800 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.93800 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -3.93800 q^{13} -3.93800 q^{15} +4.72294 q^{17} -4.78493 q^{19} +1.00000 q^{21} -2.72294 q^{23} +10.5079 q^{25} +1.00000 q^{27} +7.93800 q^{29} -1.15307 q^{31} -1.00000 q^{33} -3.93800 q^{35} -5.50787 q^{37} -3.93800 q^{39} +0.430132 q^{41} -6.72294 q^{43} -3.93800 q^{45} +8.78493 q^{47} +1.00000 q^{49} +4.72294 q^{51} -3.15307 q^{53} +3.93800 q^{55} -4.78493 q^{57} +15.0911 q^{59} +6.00000 q^{61} +1.00000 q^{63} +15.5079 q^{65} -3.21507 q^{67} -2.72294 q^{69} -13.4459 q^{71} +11.9380 q^{73} +10.5079 q^{75} -1.00000 q^{77} +5.44588 q^{79} +1.00000 q^{81} +2.84693 q^{83} -18.5989 q^{85} +7.93800 q^{87} +12.3061 q^{89} -3.93800 q^{91} -1.15307 q^{93} +18.8431 q^{95} +11.1531 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 3 q^{11} - 12 q^{19} + 3 q^{21} + 6 q^{23} + 15 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + 6 q^{41} - 6 q^{43} + 24 q^{47} + 3 q^{49} - 12 q^{57} + 24 q^{59} + 18 q^{61} + 3 q^{63} + 30 q^{65} - 12 q^{67} + 6 q^{69} - 12 q^{71} + 24 q^{73} + 15 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 18 q^{83} - 18 q^{85} + 12 q^{87} + 18 q^{89} + 6 q^{93} - 12 q^{95} + 24 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 3 * q^11 - 12 * q^19 + 3 * q^21 + 6 * q^23 + 15 * q^25 + 3 * q^27 + 12 * q^29 + 6 * q^31 - 3 * q^33 + 6 * q^41 - 6 * q^43 + 24 * q^47 + 3 * q^49 - 12 * q^57 + 24 * q^59 + 18 * q^61 + 3 * q^63 + 30 * q^65 - 12 * q^67 + 6 * q^69 - 12 * q^71 + 24 * q^73 + 15 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 18 * q^83 - 18 * q^85 + 12 * q^87 + 18 * q^89 + 6 * q^93 - 12 * q^95 + 24 * q^97 - 3 * q^99

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