LMFDB - Embedded newform 1536.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.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:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.27133$ of defining polynomial
Character	$\chi$	$=$	1536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.95687 q^{5} -1.63899 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.95687 q^{5} -1.63899 q^{7} +1.00000 q^{9} +4.82843 q^{11} +5.59587 q^{13} +3.95687 q^{15} -0.828427 q^{17} -2.82843 q^{19} -1.63899 q^{21} -7.91375 q^{23} +10.6569 q^{25} +1.00000 q^{27} -7.23486 q^{29} +1.63899 q^{31} +4.82843 q^{33} -6.48528 q^{35} -2.31788 q^{37} +5.59587 q^{39} +3.17157 q^{41} +4.48528 q^{43} +3.95687 q^{45} -7.91375 q^{47} -4.31371 q^{49} -0.828427 q^{51} -0.678892 q^{53} +19.1055 q^{55} -2.82843 q^{57} +9.65685 q^{59} -2.31788 q^{61} -1.63899 q^{63} +22.1421 q^{65} -13.6569 q^{67} -7.91375 q^{69} +3.27798 q^{71} +4.00000 q^{73} +10.6569 q^{75} -7.91375 q^{77} -1.63899 q^{79} +1.00000 q^{81} +8.82843 q^{83} -3.27798 q^{85} -7.23486 q^{87} +10.0000 q^{89} -9.17157 q^{91} +1.63899 q^{93} -11.1917 q^{95} +11.6569 q^{97} +4.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 8 q^{11} + 8 q^{17} + 20 q^{25} + 4 q^{27} + 8 q^{33} + 8 q^{35} + 24 q^{41} - 16 q^{43} + 28 q^{49} + 8 q^{51} + 16 q^{59} + 32 q^{65} - 32 q^{67} + 16 q^{73} + 20 q^{75} + 4 q^{81} + 24 q^{83} + 40 q^{89} - 48 q^{91} + 24 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^11 + 8 * q^17 + 20 * q^25 + 4 * q^27 + 8 * q^33 + 8 * q^35 + 24 * q^41 - 16 * q^43 + 28 * q^49 + 8 * q^51 + 16 * q^59 + 32 * q^65 - 32 * q^67 + 16 * q^73 + 20 * q^75 + 4 * q^81 + 24 * q^83 + 40 * q^89 - 48 * q^91 + 24 * q^97 + 8 * 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
$5$	3.95687	1.76957	0.884784	−	0.466001i	$-0.154306\pi$
$5$	3.95687	1.76957	0.884784	+	0.466001i	$0.154306\pi$
$6$	0	0
$7$	−1.63899	−0.619480	−0.309740	−	0.950821i	$-0.600242\pi$
$7$	−1.63899	−0.619480	−0.309740	+	0.950821i	$0.600242\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	5.59587	1.55201	0.776007	−	0.630724i	$-0.217242\pi$
$13$	5.59587	1.55201	0.776007	+	0.630724i	$0.217242\pi$
$14$	0	0
$15$	3.95687	1.02166
$16$	0	0
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−1.63899	−0.357657
$22$	0	0
$23$	−7.91375	−1.65013	−0.825065	−	0.565037i	$-0.808862\pi$
$23$	−7.91375	−1.65013	−0.825065	+	0.565037i	$0.808862\pi$
$24$	0	0
$25$	10.6569	2.13137
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.23486	−1.34348	−0.671740	−	0.740787i	$-0.734453\pi$
$29$	−7.23486	−1.34348	−0.671740	+	0.740787i	$0.734453\pi$
$30$	0	0
$31$	1.63899	0.294371	0.147186	−	0.989109i	$-0.452978\pi$
$31$	1.63899	0.294371	0.147186	+	0.989109i	$0.452978\pi$
$32$	0	0
$33$	4.82843	0.840521
$34$	0	0
$35$	−6.48528	−1.09621
$36$	0	0
$37$	−2.31788	−0.381058	−0.190529	−	0.981682i	$-0.561020\pi$
$37$	−2.31788	−0.381058	−0.190529	+	0.981682i	$0.561020\pi$
$38$	0	0
$39$	5.59587	0.896056
$40$	0	0
$41$	3.17157	0.495316	0.247658	−	0.968847i	$-0.420339\pi$
$41$	3.17157	0.495316	0.247658	+	0.968847i	$0.420339\pi$
$42$	0	0
$43$	4.48528	0.683999	0.341999	−	0.939700i	$-0.388896\pi$
$43$	4.48528	0.683999	0.341999	+	0.939700i	$0.388896\pi$
$44$	0	0
$45$	3.95687	0.589856
$46$	0	0
$47$	−7.91375	−1.15434	−0.577169	−	0.816624i	$-0.695843\pi$
$47$	−7.91375	−1.15434	−0.577169	+	0.816624i	$0.695843\pi$
$48$	0	0
$49$	−4.31371	−0.616244
$50$	0	0
$51$	−0.828427	−0.116003
$52$	0	0
$53$	−0.678892	−0.0932530	−0.0466265	−	0.998912i	$-0.514847\pi$
$53$	−0.678892	−0.0932530	−0.0466265	+	0.998912i	$0.514847\pi$
$54$	0	0
$55$	19.1055	2.57618
$56$	0	0
$57$	−2.82843	−0.374634
$58$	0	0
$59$	9.65685	1.25722	0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685	1.25722	0.628608	+	0.777723i	$0.283625\pi$
$60$	0	0
$61$	−2.31788	−0.296775	−0.148387	−	0.988929i	$-0.547408\pi$
$61$	−2.31788	−0.296775	−0.148387	+	0.988929i	$0.547408\pi$
$62$	0	0
$63$	−1.63899	−0.206493
$64$	0	0
$65$	22.1421	2.74639
$66$	0	0
$67$	−13.6569	−1.66845	−0.834225	−	0.551424i	$-0.814085\pi$
$67$	−13.6569	−1.66845	−0.834225	+	0.551424i	$0.814085\pi$
$68$	0	0
$69$	−7.91375	−0.952703
$70$	0	0
$71$	3.27798	0.389025	0.194512	−	0.980900i	$-0.437688\pi$
$71$	3.27798	0.389025	0.194512	+	0.980900i	$0.437688\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	10.6569	1.23055
$76$	0	0
$77$	−7.91375	−0.901855
$78$	0	0
$79$	−1.63899	−0.184401	−0.0922004	−	0.995740i	$-0.529390\pi$
$79$	−1.63899	−0.184401	−0.0922004	+	0.995740i	$0.529390\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.82843	0.969046	0.484523	−	0.874779i	$-0.338993\pi$
$83$	8.82843	0.969046	0.484523	+	0.874779i	$0.338993\pi$
$84$	0	0
$85$	−3.27798	−0.355547
$86$	0	0
$87$	−7.23486	−0.775658
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−9.17157	−0.961442
$92$	0	0
$93$	1.63899	0.169955
$94$	0	0
$95$	−11.1917	−1.14825
$96$	0	0
$97$	11.6569	1.18357	0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569	1.18357	0.591787	+	0.806094i	$0.298423\pi$
$98$	0	0
$99$	4.82843	0.485275
$100$	0	0
$101$	0.678892	0.0675523	0.0337762	−	0.999429i	$-0.489247\pi$
$101$	0.678892	0.0675523	0.0337762	+	0.999429i	$0.489247\pi$
$102$	0	0
$103$	−17.4665	−1.72102	−0.860512	−	0.509430i	$-0.829856\pi$
$103$	−17.4665	−1.72102	−0.860512	+	0.509430i	$0.829856\pi$
$104$	0	0
$105$	−6.48528	−0.632899
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−16.7876	−1.60796	−0.803980	−	0.594656i	$-0.797288\pi$
$109$	−16.7876	−1.60796	−0.803980	+	0.594656i	$0.797288\pi$
$110$	0	0
$111$	−2.31788	−0.220004
$112$	0	0
$113$	0.343146	0.0322804	0.0161402	−	0.999870i	$-0.494862\pi$
$113$	0.343146	0.0322804	0.0161402	+	0.999870i	$0.494862\pi$
$114$	0	0
$115$	−31.3137	−2.92002
$116$	0	0
$117$	5.59587	0.517338
$118$	0	0
$119$	1.35778	0.124468
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	3.17157	0.285971
$124$	0	0
$125$	22.3835	2.00204
$126$	0	0
$127$	14.1885	1.25903	0.629513	−	0.776990i	$-0.283254\pi$
$127$	14.1885	1.25903	0.629513	+	0.776990i	$0.283254\pi$
$128$	0	0
$129$	4.48528	0.394907
$130$	0	0
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	0	0
$133$	4.63577	0.401972
$134$	0	0
$135$	3.95687	0.340554
$136$	0	0
$137$	−16.1421	−1.37912	−0.689558	−	0.724231i	$-0.742195\pi$
$137$	−16.1421	−1.37912	−0.689558	+	0.724231i	$0.742195\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−7.91375	−0.666458
$142$	0	0
$143$	27.0192	2.25946
$144$	0	0
$145$	−28.6274	−2.37738
$146$	0	0
$147$	−4.31371	−0.355789
$148$	0	0
$149$	3.95687	0.324160	0.162080	−	0.986778i	$-0.448180\pi$
$149$	3.95687	0.324160	0.162080	+	0.986778i	$0.448180\pi$
$150$	0	0
$151$	−14.1885	−1.15464	−0.577322	−	0.816516i	$-0.695902\pi$
$151$	−14.1885	−1.15464	−0.577322	+	0.816516i	$0.695902\pi$
$152$	0	0
$153$	−0.828427	−0.0669744
$154$	0	0
$155$	6.48528	0.520910
$156$	0	0
$157$	18.1454	1.44816	0.724080	−	0.689717i	$-0.242265\pi$
$157$	18.1454	1.44816	0.724080	+	0.689717i	$0.242265\pi$
$158$	0	0
$159$	−0.678892	−0.0538397
$160$	0	0
$161$	12.9706	1.02222
$162$	0	0
$163$	10.1421	0.794393	0.397197	−	0.917734i	$-0.369983\pi$
$163$	10.1421	0.794393	0.397197	+	0.917734i	$0.369983\pi$
$164$	0	0
$165$	19.1055	1.48736
$166$	0	0
$167$	4.63577	0.358726	0.179363	−	0.983783i	$-0.442596\pi$
$167$	4.63577	0.358726	0.179363	+	0.983783i	$0.442596\pi$
$168$	0	0
$169$	18.3137	1.40875
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	11.8706	0.902507	0.451253	−	0.892396i	$-0.350977\pi$
$173$	11.8706	0.902507	0.451253	+	0.892396i	$0.350977\pi$
$174$	0	0
$175$	−17.4665	−1.32034
$176$	0	0
$177$	9.65685	0.725854
$178$	0	0
$179$	12.9706	0.969465	0.484733	−	0.874662i	$-0.338917\pi$
$179$	12.9706	0.969465	0.484733	+	0.874662i	$0.338917\pi$
$180$	0	0
$181$	16.7876	1.24781	0.623906	−	0.781499i	$-0.285545\pi$
$181$	16.7876	1.24781	0.623906	+	0.781499i	$0.285545\pi$
$182$	0	0
$183$	−2.31788	−0.171343
$184$	0	0
$185$	−9.17157	−0.674307
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	−1.63899	−0.119219
$190$	0	0
$191$	17.7477	1.28418	0.642089	−	0.766630i	$-0.278068\pi$
$191$	17.7477	1.28418	0.642089	+	0.766630i	$0.278068\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	22.1421	1.58563
$196$	0	0
$197$	−16.5064	−1.17603	−0.588016	−	0.808849i	$-0.700091\pi$
$197$	−16.5064	−1.17603	−0.588016	+	0.808849i	$0.700091\pi$
$198$	0	0
$199$	−1.63899	−0.116185	−0.0580925	−	0.998311i	$-0.518502\pi$
$199$	−1.63899	−0.116185	−0.0580925	+	0.998311i	$0.518502\pi$
$200$	0	0
$201$	−13.6569	−0.963280
$202$	0	0
$203$	11.8579	0.832259
$204$	0	0
$205$	12.5495	0.876496
$206$	0	0
$207$	−7.91375	−0.550044
$208$	0	0
$209$	−13.6569	−0.944664
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	3.27798	0.224604
$214$	0	0
$215$	17.7477	1.21038
$216$	0	0
$217$	−2.68629	−0.182357
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−4.63577	−0.311835
$222$	0	0
$223$	−14.1885	−0.950133	−0.475066	−	0.879950i	$-0.657576\pi$
$223$	−14.1885	−0.950133	−0.475066	+	0.879950i	$0.657576\pi$
$224$	0	0
$225$	10.6569	0.710457
$226$	0	0
$227$	−12.1421	−0.805902	−0.402951	−	0.915222i	$-0.632015\pi$
$227$	−12.1421	−0.805902	−0.402951	+	0.915222i	$0.632015\pi$
$228$	0	0
$229$	−21.4234	−1.41570	−0.707848	−	0.706365i	$-0.750334\pi$
$229$	−21.4234	−1.41570	−0.707848	+	0.706365i	$0.750334\pi$
$230$	0	0
$231$	−7.91375	−0.520686
$232$	0	0
$233$	−21.3137	−1.39631	−0.698154	−	0.715948i	$-0.745995\pi$
$233$	−21.3137	−1.39631	−0.698154	+	0.715948i	$0.745995\pi$
$234$	0	0
$235$	−31.3137	−2.04268
$236$	0	0
$237$	−1.63899	−0.106464
$238$	0	0
$239$	4.63577	0.299863	0.149931	−	0.988696i	$-0.452095\pi$
$239$	4.63577	0.299863	0.149931	+	0.988696i	$0.452095\pi$
$240$	0	0
$241$	−16.9706	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−17.0688	−1.09049
$246$	0	0
$247$	−15.8275	−1.00708
$248$	0	0
$249$	8.82843	0.559479
$250$	0	0
$251$	20.8284	1.31468	0.657339	−	0.753595i	$-0.271682\pi$
$251$	20.8284	1.31468	0.657339	+	0.753595i	$0.271682\pi$
$252$	0	0
$253$	−38.2110	−2.40230
$254$	0	0
$255$	−3.27798	−0.205275
$256$	0	0
$257$	−19.6569	−1.22616	−0.613080	−	0.790020i	$-0.710070\pi$
$257$	−19.6569	−1.22616	−0.613080	+	0.790020i	$0.710070\pi$
$258$	0	0
$259$	3.79899	0.236058
$260$	0	0
$261$	−7.23486	−0.447826
$262$	0	0
$263$	15.8275	0.975965	0.487983	−	0.872853i	$-0.337733\pi$
$263$	15.8275	0.975965	0.487983	+	0.872853i	$0.337733\pi$
$264$	0	0
$265$	−2.68629	−0.165018
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	7.23486	0.441117	0.220558	−	0.975374i	$-0.429212\pi$
$269$	7.23486	0.441117	0.220558	+	0.975374i	$0.429212\pi$
$270$	0	0
$271$	17.4665	1.06101	0.530507	−	0.847681i	$-0.322002\pi$
$271$	17.4665	1.06101	0.530507	+	0.847681i	$0.322002\pi$
$272$	0	0
$273$	−9.17157	−0.555089
$274$	0	0
$275$	51.4558	3.10290
$276$	0	0
$277$	−14.8674	−0.893295	−0.446648	−	0.894710i	$-0.647382\pi$
$277$	−14.8674	−0.893295	−0.446648	+	0.894710i	$0.647382\pi$
$278$	0	0
$279$	1.63899	0.0981238
$280$	0	0
$281$	0.343146	0.0204704	0.0102352	−	0.999948i	$-0.496742\pi$
$281$	0.343146	0.0204704	0.0102352	+	0.999948i	$0.496742\pi$
$282$	0	0
$283$	−28.9706	−1.72212	−0.861061	−	0.508502i	$-0.830199\pi$
$283$	−28.9706	−1.72212	−0.861061	+	0.508502i	$0.830199\pi$
$284$	0	0
$285$	−11.1917	−0.662941
$286$	0	0
$287$	−5.19818	−0.306839
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	11.6569	0.683337
$292$	0	0
$293$	−0.678892	−0.0396613	−0.0198307	−	0.999803i	$-0.506313\pi$
$293$	−0.678892	−0.0396613	−0.0198307	+	0.999803i	$0.506313\pi$
$294$	0	0
$295$	38.2110	2.22473
$296$	0	0
$297$	4.82843	0.280174
$298$	0	0
$299$	−44.2843	−2.56103
$300$	0	0
$301$	−7.35134	−0.423724
$302$	0	0
$303$	0.678892	0.0390013
$304$	0	0
$305$	−9.17157	−0.525163
$306$	0	0
$307$	−21.6569	−1.23602	−0.618011	−	0.786169i	$-0.712061\pi$
$307$	−21.6569	−1.23602	−0.618011	+	0.786169i	$0.712061\pi$
$308$	0	0
$309$	−17.4665	−0.993634
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	7.31371	0.413395	0.206698	−	0.978405i	$-0.433728\pi$
$313$	7.31371	0.413395	0.206698	+	0.978405i	$0.433728\pi$
$314$	0	0
$315$	−6.48528	−0.365404
$316$	0	0
$317$	−21.7046	−1.21905	−0.609525	−	0.792767i	$-0.708640\pi$
$317$	−21.7046	−1.21905	−0.609525	+	0.792767i	$0.708640\pi$
$318$	0	0
$319$	−34.9330	−1.95587
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	2.34315	0.130376
$324$	0	0
$325$	59.6343	3.30792
$326$	0	0
$327$	−16.7876	−0.928356
$328$	0	0
$329$	12.9706	0.715090
$330$	0	0
$331$	10.3431	0.568511	0.284255	−	0.958749i	$-0.408254\pi$
$331$	10.3431	0.568511	0.284255	+	0.958749i	$0.408254\pi$
$332$	0	0
$333$	−2.31788	−0.127019
$334$	0	0
$335$	−54.0385	−2.95244
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	0.343146	0.0186371
$340$	0	0
$341$	7.91375	0.428554
$342$	0	0
$343$	18.5431	1.00123
$344$	0	0
$345$	−31.3137	−1.68587
$346$	0	0
$347$	0.828427	0.0444723	0.0222361	−	0.999753i	$-0.492921\pi$
$347$	0.828427	0.0444723	0.0222361	+	0.999753i	$0.492921\pi$
$348$	0	0
$349$	4.23808	0.226859	0.113430	−	0.993546i	$-0.463816\pi$
$349$	4.23808	0.226859	0.113430	+	0.993546i	$0.463816\pi$
$350$	0	0
$351$	5.59587	0.298685
$352$	0	0
$353$	1.31371	0.0699216	0.0349608	−	0.999389i	$-0.488869\pi$
$353$	1.31371	0.0699216	0.0349608	+	0.999389i	$0.488869\pi$
$354$	0	0
$355$	12.9706	0.688406
$356$	0	0
$357$	1.35778	0.0718616
$358$	0	0
$359$	−34.9330	−1.84369	−0.921846	−	0.387556i	$-0.873319\pi$
$359$	−34.9330	−1.84369	−0.921846	+	0.387556i	$0.873319\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	12.3137	0.646302
$364$	0	0
$365$	15.8275	0.828449
$366$	0	0
$367$	24.0225	1.25396	0.626981	−	0.779035i	$-0.284290\pi$
$367$	24.0225	1.25396	0.626981	+	0.779035i	$0.284290\pi$
$368$	0	0
$369$	3.17157	0.165105
$370$	0	0
$371$	1.11270	0.0577684
$372$	0	0
$373$	−20.0656	−1.03896	−0.519478	−	0.854484i	$-0.673874\pi$
$373$	−20.0656	−1.03896	−0.519478	+	0.854484i	$0.673874\pi$
$374$	0	0
$375$	22.3835	1.15588
$376$	0	0
$377$	−40.4853	−2.08510
$378$	0	0
$379$	−0.485281	−0.0249272	−0.0124636	−	0.999922i	$-0.503967\pi$
$379$	−0.485281	−0.0249272	−0.0124636	+	0.999922i	$0.503967\pi$
$380$	0	0
$381$	14.1885	0.726899
$382$	0	0
$383$	11.1917	0.571871	0.285935	−	0.958249i	$-0.407696\pi$
$383$	11.1917	0.571871	0.285935	+	0.958249i	$0.407696\pi$
$384$	0	0
$385$	−31.3137	−1.59589
$386$	0	0
$387$	4.48528	0.228000
$388$	0	0
$389$	−3.95687	−0.200621	−0.100311	−	0.994956i	$-0.531984\pi$
$389$	−3.95687	−0.200621	−0.100311	+	0.994956i	$0.531984\pi$
$390$	0	0
$391$	6.55596	0.331549
$392$	0	0
$393$	−7.31371	−0.368928
$394$	0	0
$395$	−6.48528	−0.326310
$396$	0	0
$397$	18.1454	0.910691	0.455345	−	0.890315i	$-0.349516\pi$
$397$	18.1454	0.910691	0.455345	+	0.890315i	$0.349516\pi$
$398$	0	0
$399$	4.63577	0.232079
$400$	0	0
$401$	24.1421	1.20560	0.602800	−	0.797892i	$-0.294052\pi$
$401$	24.1421	1.20560	0.602800	+	0.797892i	$0.294052\pi$
$402$	0	0
$403$	9.17157	0.456869
$404$	0	0
$405$	3.95687	0.196619
$406$	0	0
$407$	−11.1917	−0.554753
$408$	0	0
$409$	−3.65685	−0.180820	−0.0904099	−	0.995905i	$-0.528818\pi$
$409$	−3.65685	−0.180820	−0.0904099	+	0.995905i	$0.528818\pi$
$410$	0	0
$411$	−16.1421	−0.796233
$412$	0	0
$413$	−15.8275	−0.778820
$414$	0	0
$415$	34.9330	1.71479
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	15.1716	0.741180	0.370590	−	0.928797i	$-0.379156\pi$
$419$	15.1716	0.741180	0.370590	+	0.928797i	$0.379156\pi$
$420$	0	0
$421$	−21.4234	−1.04411	−0.522055	−	0.852912i	$-0.674835\pi$
$421$	−21.4234	−1.04411	−0.522055	+	0.852912i	$0.674835\pi$
$422$	0	0
$423$	−7.91375	−0.384780
$424$	0	0
$425$	−8.82843	−0.428242
$426$	0	0
$427$	3.79899	0.183846
$428$	0	0
$429$	27.0192	1.30450
$430$	0	0
$431$	14.4697	0.696982	0.348491	−	0.937312i	$-0.386694\pi$
$431$	14.4697	0.696982	0.348491	+	0.937312i	$0.386694\pi$
$432$	0	0
$433$	24.6274	1.18352	0.591759	−	0.806115i	$-0.298434\pi$
$433$	24.6274	1.18352	0.591759	+	0.806115i	$0.298434\pi$
$434$	0	0
$435$	−28.6274	−1.37258
$436$	0	0
$437$	22.3835	1.07075
$438$	0	0
$439$	−39.8499	−1.90193	−0.950967	−	0.309292i	$-0.899908\pi$
$439$	−39.8499	−1.90193	−0.950967	+	0.309292i	$0.899908\pi$
$440$	0	0
$441$	−4.31371	−0.205415
$442$	0	0
$443$	−18.4853	−0.878262	−0.439131	−	0.898423i	$-0.644714\pi$
$443$	−18.4853	−0.878262	−0.439131	+	0.898423i	$0.644714\pi$
$444$	0	0
$445$	39.5687	1.87574
$446$	0	0
$447$	3.95687	0.187154
$448$	0	0
$449$	17.5147	0.826571	0.413285	−	0.910602i	$-0.364381\pi$
$449$	17.5147	0.826571	0.413285	+	0.910602i	$0.364381\pi$
$450$	0	0
$451$	15.3137	0.721094
$452$	0	0
$453$	−14.1885	−0.666634
$454$	0	0
$455$	−36.2908	−1.70134
$456$	0	0
$457$	−23.6569	−1.10662	−0.553310	−	0.832975i	$-0.686636\pi$
$457$	−23.6569	−1.10662	−0.553310	+	0.832975i	$0.686636\pi$
$458$	0	0
$459$	−0.828427	−0.0386677
$460$	0	0
$461$	−32.8963	−1.53213	−0.766067	−	0.642761i	$-0.777789\pi$
$461$	−32.8963	−1.53213	−0.766067	+	0.642761i	$0.777789\pi$
$462$	0	0
$463$	−10.9105	−0.507055	−0.253528	−	0.967328i	$-0.581591\pi$
$463$	−10.9105	−0.507055	−0.253528	+	0.967328i	$0.581591\pi$
$464$	0	0
$465$	6.48528	0.300748
$466$	0	0
$467$	37.7990	1.74913	0.874564	−	0.484910i	$-0.161147\pi$
$467$	37.7990	1.74913	0.874564	+	0.484910i	$0.161147\pi$
$468$	0	0
$469$	22.3835	1.03357
$470$	0	0
$471$	18.1454	0.836095
$472$	0	0
$473$	21.6569	0.995783
$474$	0	0
$475$	−30.1421	−1.38302
$476$	0	0
$477$	−0.678892	−0.0310843
$478$	0	0
$479$	32.2174	1.47205	0.736025	−	0.676954i	$-0.236700\pi$
$479$	32.2174	1.47205	0.736025	+	0.676954i	$0.236700\pi$
$480$	0	0
$481$	−12.9706	−0.591407
$482$	0	0
$483$	12.9706	0.590181
$484$	0	0
$485$	46.1247	2.09442
$486$	0	0
$487$	20.7445	0.940022	0.470011	−	0.882661i	$-0.344250\pi$
$487$	20.7445	0.940022	0.470011	+	0.882661i	$0.344250\pi$
$488$	0	0
$489$	10.1421	0.458643
$490$	0	0
$491$	1.65685	0.0747728	0.0373864	−	0.999301i	$-0.488097\pi$
$491$	1.65685	0.0747728	0.0373864	+	0.999301i	$0.488097\pi$
$492$	0	0
$493$	5.99355	0.269936
$494$	0	0
$495$	19.1055	0.858727
$496$	0	0
$497$	−5.37258	−0.240993
$498$	0	0
$499$	−30.3431	−1.35835	−0.679173	−	0.733978i	$-0.737661\pi$
$499$	−30.3431	−1.35835	−0.679173	+	0.733978i	$0.737661\pi$
$500$	0	0
$501$	4.63577	0.207111
$502$	0	0
$503$	−12.5495	−0.559555	−0.279778	−	0.960065i	$-0.590261\pi$
$503$	−12.5495	−0.559555	−0.279778	+	0.960065i	$0.590261\pi$
$504$	0	0
$505$	2.68629	0.119538
$506$	0	0
$507$	18.3137	0.813340
$508$	0	0
$509$	8.59264	0.380862	0.190431	−	0.981701i	$-0.439011\pi$
$509$	8.59264	0.380862	0.190431	+	0.981701i	$0.439011\pi$
$510$	0	0
$511$	−6.55596	−0.290019
$512$	0	0
$513$	−2.82843	−0.124878
$514$	0	0
$515$	−69.1127	−3.04547
$516$	0	0
$517$	−38.2110	−1.68052
$518$	0	0
$519$	11.8706	0.521063
$520$	0	0
$521$	15.1716	0.664679	0.332339	−	0.943160i	$-0.392162\pi$
$521$	15.1716	0.664679	0.332339	+	0.943160i	$0.392162\pi$
$522$	0	0
$523$	13.1716	0.575953	0.287976	−	0.957638i	$-0.407018\pi$
$523$	13.1716	0.575953	0.287976	+	0.957638i	$0.407018\pi$
$524$	0	0
$525$	−17.4665	−0.762300
$526$	0	0
$527$	−1.35778	−0.0591460
$528$	0	0
$529$	39.6274	1.72293
$530$	0	0
$531$	9.65685	0.419072
$532$	0	0
$533$	17.7477	0.768738
$534$	0	0
$535$	−15.8275	−0.684282
$536$	0	0
$537$	12.9706	0.559721
$538$	0	0
$539$	−20.8284	−0.897144
$540$	0	0
$541$	10.2316	0.439892	0.219946	−	0.975512i	$-0.429412\pi$
$541$	10.2316	0.439892	0.219946	+	0.975512i	$0.429412\pi$
$542$	0	0
$543$	16.7876	0.720425
$544$	0	0
$545$	−66.4264	−2.84539
$546$	0	0
$547$	−7.79899	−0.333461	−0.166730	−	0.986003i	$-0.553321\pi$
$547$	−7.79899	−0.333461	−0.166730	+	0.986003i	$0.553321\pi$
$548$	0	0
$549$	−2.31788	−0.0989248
$550$	0	0
$551$	20.4633	0.871764
$552$	0	0
$553$	2.68629	0.114233
$554$	0	0
$555$	−9.17157	−0.389312
$556$	0	0
$557$	−27.6981	−1.17361	−0.586804	−	0.809729i	$-0.699614\pi$
$557$	−27.6981	−1.17361	−0.586804	+	0.809729i	$0.699614\pi$
$558$	0	0
$559$	25.0990	1.06158
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	4.82843	0.203494	0.101747	−	0.994810i	$-0.467557\pi$
$563$	4.82843	0.203494	0.101747	+	0.994810i	$0.467557\pi$
$564$	0	0
$565$	1.35778	0.0571224
$566$	0	0
$567$	−1.63899	−0.0688312
$568$	0	0
$569$	−27.4558	−1.15101	−0.575504	−	0.817799i	$-0.695194\pi$
$569$	−27.4558	−1.15101	−0.575504	+	0.817799i	$0.695194\pi$
$570$	0	0
$571$	0.686292	0.0287204	0.0143602	−	0.999897i	$-0.495429\pi$
$571$	0.686292	0.0287204	0.0143602	+	0.999897i	$0.495429\pi$
$572$	0	0
$573$	17.7477	0.741421
$574$	0	0
$575$	−84.3357	−3.51704
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−14.4697	−0.600305
$582$	0	0
$583$	−3.27798	−0.135760
$584$	0	0
$585$	22.1421	0.915465
$586$	0	0
$587$	28.9706	1.19574	0.597872	−	0.801592i	$-0.296013\pi$
$587$	28.9706	1.19574	0.597872	+	0.801592i	$0.296013\pi$
$588$	0	0
$589$	−4.63577	−0.191013
$590$	0	0
$591$	−16.5064	−0.678982
$592$	0	0
$593$	30.9706	1.27181	0.635904	−	0.771768i	$-0.280627\pi$
$593$	30.9706	1.27181	0.635904	+	0.771768i	$0.280627\pi$
$594$	0	0
$595$	5.37258	0.220254
$596$	0	0
$597$	−1.63899	−0.0670794
$598$	0	0
$599$	46.1247	1.88460	0.942302	−	0.334763i	$-0.108656\pi$
$599$	46.1247	1.88460	0.942302	+	0.334763i	$0.108656\pi$
$600$	0	0
$601$	−27.3137	−1.11415	−0.557075	−	0.830462i	$-0.688076\pi$
$601$	−27.3137	−1.11415	−0.557075	+	0.830462i	$0.688076\pi$
$602$	0	0
$603$	−13.6569	−0.556150
$604$	0	0
$605$	48.7238	1.98090
$606$	0	0
$607$	−8.19496	−0.332623	−0.166311	−	0.986073i	$-0.553186\pi$
$607$	−8.19496	−0.332623	−0.166311	+	0.986073i	$0.553186\pi$
$608$	0	0
$609$	11.8579	0.480505
$610$	0	0
$611$	−44.2843	−1.79155
$612$	0	0
$613$	−35.8931	−1.44971	−0.724854	−	0.688903i	$-0.758093\pi$
$613$	−35.8931	−1.44971	−0.724854	+	0.688903i	$0.758093\pi$
$614$	0	0
$615$	12.5495	0.506045
$616$	0	0
$617$	23.9411	0.963833	0.481917	−	0.876217i	$-0.339941\pi$
$617$	23.9411	0.963833	0.481917	+	0.876217i	$0.339941\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−7.91375	−0.317568
$622$	0	0
$623$	−16.3899	−0.656648
$624$	0	0
$625$	35.2843	1.41137
$626$	0	0
$627$	−13.6569	−0.545402
$628$	0	0
$629$	1.92020	0.0765633
$630$	0	0
$631$	11.4729	0.456730	0.228365	−	0.973576i	$-0.426662\pi$
$631$	11.4729	0.456730	0.228365	+	0.973576i	$0.426662\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	56.1421	2.22793
$636$	0	0
$637$	−24.1389	−0.956419
$638$	0	0
$639$	3.27798	0.129675
$640$	0	0
$641$	27.1716	1.07321	0.536606	−	0.843833i	$-0.319706\pi$
$641$	27.1716	1.07321	0.536606	+	0.843833i	$0.319706\pi$
$642$	0	0
$643$	20.4853	0.807861	0.403930	−	0.914790i	$-0.367644\pi$
$643$	20.4853	0.807861	0.403930	+	0.914790i	$0.367644\pi$
$644$	0	0
$645$	17.7477	0.698815
$646$	0	0
$647$	−5.19818	−0.204362	−0.102181	−	0.994766i	$-0.532582\pi$
$647$	−5.19818	−0.204362	−0.102181	+	0.994766i	$0.532582\pi$
$648$	0	0
$649$	46.6274	1.83029
$650$	0	0
$651$	−2.68629	−0.105284
$652$	0	0
$653$	−13.2284	−0.517668	−0.258834	−	0.965922i	$-0.583338\pi$
$653$	−13.2284	−0.517668	−0.258834	+	0.965922i	$0.583338\pi$
$654$	0	0
$655$	−28.9394	−1.13076
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	1.65685	0.0645419	0.0322709	−	0.999479i	$-0.489726\pi$
$659$	1.65685	0.0645419	0.0322709	+	0.999479i	$0.489726\pi$
$660$	0	0
$661$	−4.23808	−0.164842	−0.0824211	−	0.996598i	$-0.526265\pi$
$661$	−4.23808	−0.164842	−0.0824211	+	0.996598i	$0.526265\pi$
$662$	0	0
$663$	−4.63577	−0.180038
$664$	0	0
$665$	18.3431	0.711317
$666$	0	0
$667$	57.2548	2.21692
$668$	0	0
$669$	−14.1885	−0.548559
$670$	0	0
$671$	−11.1917	−0.432052
$672$	0	0
$673$	14.6863	0.566115	0.283057	−	0.959103i	$-0.408651\pi$
$673$	14.6863	0.566115	0.283057	+	0.959103i	$0.408651\pi$
$674$	0	0
$675$	10.6569	0.410183
$676$	0	0
$677$	−2.59909	−0.0998911	−0.0499456	−	0.998752i	$-0.515905\pi$
$677$	−2.59909	−0.0998911	−0.0499456	+	0.998752i	$0.515905\pi$
$678$	0	0
$679$	−19.1055	−0.733201
$680$	0	0
$681$	−12.1421	−0.465288
$682$	0	0
$683$	−5.51472	−0.211015	−0.105507	−	0.994419i	$-0.533647\pi$
$683$	−5.51472	−0.211015	−0.105507	+	0.994419i	$0.533647\pi$
$684$	0	0
$685$	−63.8724	−2.44044
$686$	0	0
$687$	−21.4234	−0.817352
$688$	0	0
$689$	−3.79899	−0.144730
$690$	0	0
$691$	38.8284	1.47710	0.738551	−	0.674197i	$-0.235510\pi$
$691$	38.8284	1.47710	0.738551	+	0.674197i	$0.235510\pi$
$692$	0	0
$693$	−7.91375	−0.300618
$694$	0	0
$695$	−31.6550	−1.20074
$696$	0	0
$697$	−2.62742	−0.0995205
$698$	0	0
$699$	−21.3137	−0.806158
$700$	0	0
$701$	−30.9761	−1.16995	−0.584976	−	0.811051i	$-0.698896\pi$
$701$	−30.9761	−1.16995	−0.584976	+	0.811051i	$0.698896\pi$
$702$	0	0
$703$	6.55596	0.247263
$704$	0	0
$705$	−31.3137	−1.17934
$706$	0	0
$707$	−1.11270	−0.0418473
$708$	0	0
$709$	26.0591	0.978671	0.489336	−	0.872096i	$-0.337239\pi$
$709$	26.0591	0.978671	0.489336	+	0.872096i	$0.337239\pi$
$710$	0	0
$711$	−1.63899	−0.0614670
$712$	0	0
$713$	−12.9706	−0.485751
$714$	0	0
$715$	106.912	3.99827
$716$	0	0
$717$	4.63577	0.173126
$718$	0	0
$719$	34.9330	1.30278	0.651390	−	0.758743i	$-0.274186\pi$
$719$	34.9330	1.30278	0.651390	+	0.758743i	$0.274186\pi$
$720$	0	0
$721$	28.6274	1.06614
$722$	0	0
$723$	−16.9706	−0.631142
$724$	0	0
$725$	−77.1008	−2.86345
$726$	0	0
$727$	30.5784	1.13409	0.567045	−	0.823687i	$-0.308086\pi$
$727$	30.5784	1.13409	0.567045	+	0.823687i	$0.308086\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.71573	−0.137431
$732$	0	0
$733$	−18.7078	−0.690988	−0.345494	−	0.938421i	$-0.612289\pi$
$733$	−18.7078	−0.690988	−0.345494	+	0.938421i	$0.612289\pi$
$734$	0	0
$735$	−17.0688	−0.629592
$736$	0	0
$737$	−65.9411	−2.42897
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−15.8275	−0.581438
$742$	0	0
$743$	33.5752	1.23175	0.615877	−	0.787842i	$-0.288802\pi$
$743$	33.5752	1.23175	0.615877	+	0.787842i	$0.288802\pi$
$744$	0	0
$745$	15.6569	0.573623
$746$	0	0
$747$	8.82843	0.323015
$748$	0	0
$749$	6.55596	0.239550
$750$	0	0
$751$	−1.63899	−0.0598076	−0.0299038	−	0.999553i	$-0.509520\pi$
$751$	−1.63899	−0.0598076	−0.0299038	+	0.999553i	$0.509520\pi$
$752$	0	0
$753$	20.8284	0.759030
$754$	0	0
$755$	−56.1421	−2.04322
$756$	0	0
$757$	0.960099	0.0348954	0.0174477	−	0.999848i	$-0.494446\pi$
$757$	0.960099	0.0348954	0.0174477	+	0.999848i	$0.494446\pi$
$758$	0	0
$759$	−38.2110	−1.38697
$760$	0	0
$761$	30.7696	1.11540	0.557698	−	0.830044i	$-0.311685\pi$
$761$	30.7696	1.11540	0.557698	+	0.830044i	$0.311685\pi$
$762$	0	0
$763$	27.5147	0.996100
$764$	0	0
$765$	−3.27798	−0.118516
$766$	0	0
$767$	54.0385	1.95122
$768$	0	0
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	−19.6569	−0.707924
$772$	0	0
$773$	−5.87707	−0.211384	−0.105692	−	0.994399i	$-0.533706\pi$
$773$	−5.87707	−0.211384	−0.105692	+	0.994399i	$0.533706\pi$
$774$	0	0
$775$	17.4665	0.627415
$776$	0	0
$777$	3.79899	0.136288
$778$	0	0
$779$	−8.97056	−0.321404
$780$	0	0
$781$	15.8275	0.566352
$782$	0	0
$783$	−7.23486	−0.258553
$784$	0	0
$785$	71.7990	2.56262
$786$	0	0
$787$	−20.7696	−0.740355	−0.370177	−	0.928961i	$-0.620703\pi$
$787$	−20.7696	−0.740355	−0.370177	+	0.928961i	$0.620703\pi$
$788$	0	0
$789$	15.8275	0.563474
$790$	0	0
$791$	−0.562413	−0.0199971
$792$	0	0
$793$	−12.9706	−0.460598
$794$	0	0
$795$	−2.68629	−0.0952729
$796$	0	0
$797$	13.2284	0.468574	0.234287	−	0.972167i	$-0.424724\pi$
$797$	13.2284	0.468574	0.234287	+	0.972167i	$0.424724\pi$
$798$	0	0
$799$	6.55596	0.231933
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	19.3137	0.681566
$804$	0	0
$805$	51.3229	1.80889
$806$	0	0
$807$	7.23486	0.254679
$808$	0	0
$809$	47.4558	1.66846	0.834229	−	0.551418i	$-0.185913\pi$
$809$	47.4558	1.66846	0.834229	+	0.551418i	$0.185913\pi$
$810$	0	0
$811$	−1.45584	−0.0511216	−0.0255608	−	0.999673i	$-0.508137\pi$
$811$	−1.45584	−0.0511216	−0.0255608	+	0.999673i	$0.508137\pi$
$812$	0	0
$813$	17.4665	0.612576
$814$	0	0
$815$	40.1312	1.40573
$816$	0	0
$817$	−12.6863	−0.443837
$818$	0	0
$819$	−9.17157	−0.320481
$820$	0	0
$821$	−2.03668	−0.0710805	−0.0355403	−	0.999368i	$-0.511315\pi$
$821$	−2.03668	−0.0710805	−0.0355403	+	0.999368i	$0.511315\pi$
$822$	0	0
$823$	14.1885	0.494580	0.247290	−	0.968941i	$-0.420460\pi$
$823$	14.1885	0.494580	0.247290	+	0.968941i	$0.420460\pi$
$824$	0	0
$825$	51.4558	1.79146
$826$	0	0
$827$	13.9411	0.484780	0.242390	−	0.970179i	$-0.422069\pi$
$827$	13.9411	0.484780	0.242390	+	0.970179i	$0.422069\pi$
$828$	0	0
$829$	5.59587	0.194352	0.0971762	−	0.995267i	$-0.469019\pi$
$829$	5.59587	0.194352	0.0971762	+	0.995267i	$0.469019\pi$
$830$	0	0
$831$	−14.8674	−0.515744
$832$	0	0
$833$	3.57359	0.123818
$834$	0	0
$835$	18.3431	0.634791
$836$	0	0
$837$	1.63899	0.0566518
$838$	0	0
$839$	32.2174	1.11227	0.556134	−	0.831092i	$-0.312284\pi$
$839$	32.2174	1.11227	0.556134	+	0.831092i	$0.312284\pi$
$840$	0	0
$841$	23.3431	0.804936
$842$	0	0
$843$	0.343146	0.0118186
$844$	0	0
$845$	72.4650	2.49287
$846$	0	0
$847$	−20.1821	−0.693464
$848$	0	0
$849$	−28.9706	−0.994267
$850$	0	0
$851$	18.3431	0.628795
$852$	0	0
$853$	18.1454	0.621286	0.310643	−	0.950527i	$-0.399456\pi$
$853$	18.1454	0.621286	0.310643	+	0.950527i	$0.399456\pi$
$854$	0	0
$855$	−11.1917	−0.382749
$856$	0	0
$857$	13.5147	0.461654	0.230827	−	0.972995i	$-0.425857\pi$
$857$	13.5147	0.461654	0.230827	+	0.972995i	$0.425857\pi$
$858$	0	0
$859$	−29.4558	−1.00502	−0.502510	−	0.864571i	$-0.667590\pi$
$859$	−29.4558	−1.00502	−0.502510	+	0.864571i	$0.667590\pi$
$860$	0	0
$861$	−5.19818	−0.177153
$862$	0	0
$863$	−4.63577	−0.157803	−0.0789017	−	0.996882i	$-0.525141\pi$
$863$	−4.63577	−0.157803	−0.0789017	+	0.996882i	$0.525141\pi$
$864$	0	0
$865$	46.9706	1.59705
$866$	0	0
$867$	−16.3137	−0.554043
$868$	0	0
$869$	−7.91375	−0.268456
$870$	0	0
$871$	−76.4219	−2.58946
$872$	0	0
$873$	11.6569	0.394525
$874$	0	0
$875$	−36.6863	−1.24022
$876$	0	0
$877$	58.2765	1.96786	0.983929	−	0.178558i	$-0.0571432\pi$
$877$	58.2765	1.96786	0.983929	+	0.178558i	$0.0571432\pi$
$878$	0	0
$879$	−0.678892	−0.0228985
$880$	0	0
$881$	−5.02944	−0.169446	−0.0847230	−	0.996405i	$-0.527001\pi$
$881$	−5.02944	−0.169446	−0.0847230	+	0.996405i	$0.527001\pi$
$882$	0	0
$883$	−48.7696	−1.64123	−0.820613	−	0.571484i	$-0.806368\pi$
$883$	−48.7696	−1.64123	−0.820613	+	0.571484i	$0.806368\pi$
$884$	0	0
$885$	38.2110	1.28445
$886$	0	0
$887$	15.8275	0.531435	0.265718	−	0.964051i	$-0.414391\pi$
$887$	15.8275	0.531435	0.265718	+	0.964051i	$0.414391\pi$
$888$	0	0
$889$	−23.2548	−0.779942
$890$	0	0
$891$	4.82843	0.161758
$892$	0	0
$893$	22.3835	0.749034
$894$	0	0
$895$	51.3229	1.71553
$896$	0	0
$897$	−44.2843	−1.47861
$898$	0	0
$899$	−11.8579	−0.395482
$900$	0	0
$901$	0.562413	0.0187367
$902$	0	0
$903$	−7.35134	−0.244637
$904$	0	0
$905$	66.4264	2.20809
$906$	0	0
$907$	15.7990	0.524597	0.262298	−	0.964987i	$-0.415520\pi$
$907$	15.7990	0.524597	0.262298	+	0.964987i	$0.415520\pi$
$908$	0	0
$909$	0.678892	0.0225174
$910$	0	0
$911$	−1.92020	−0.0636190	−0.0318095	−	0.999494i	$-0.510127\pi$
$911$	−1.92020	−0.0636190	−0.0318095	+	0.999494i	$0.510127\pi$
$912$	0	0
$913$	42.6274	1.41076
$914$	0	0
$915$	−9.17157	−0.303203
$916$	0	0
$917$	11.9871	0.395849
$918$	0	0
$919$	4.91697	0.162196	0.0810980	−	0.996706i	$-0.474157\pi$
$919$	4.91697	0.162196	0.0810980	+	0.996706i	$0.474157\pi$
$920$	0	0
$921$	−21.6569	−0.713618
$922$	0	0
$923$	18.3431	0.603772
$924$	0	0
$925$	−24.7013	−0.812175
$926$	0	0
$927$	−17.4665	−0.573675
$928$	0	0
$929$	21.7990	0.715202	0.357601	−	0.933875i	$-0.383595\pi$
$929$	21.7990	0.715202	0.357601	+	0.933875i	$0.383595\pi$
$930$	0	0
$931$	12.2010	0.399872
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15.8275	−0.517615
$936$	0	0
$937$	−40.6274	−1.32724	−0.663620	−	0.748070i	$-0.730981\pi$
$937$	−40.6274	−1.32724	−0.663620	+	0.748070i	$0.730981\pi$
$938$	0	0
$939$	7.31371	0.238674
$940$	0	0
$941$	52.7972	1.72114	0.860569	−	0.509334i	$-0.170108\pi$
$941$	52.7972	1.72114	0.860569	+	0.509334i	$0.170108\pi$
$942$	0	0
$943$	−25.0990	−0.817337
$944$	0	0
$945$	−6.48528	−0.210966
$946$	0	0
$947$	−28.9706	−0.941417	−0.470708	−	0.882289i	$-0.656002\pi$
$947$	−28.9706	−0.941417	−0.470708	+	0.882289i	$0.656002\pi$
$948$	0	0
$949$	22.3835	0.726598
$950$	0	0
$951$	−21.7046	−0.703819
$952$	0	0
$953$	29.7990	0.965284	0.482642	−	0.875818i	$-0.339677\pi$
$953$	29.7990	0.965284	0.482642	+	0.875818i	$0.339677\pi$
$954$	0	0
$955$	70.2254	2.27244
$956$	0	0
$957$	−34.9330	−1.12922
$958$	0	0
$959$	26.4568	0.854335
$960$	0	0
$961$	−28.3137	−0.913345
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	−15.8275	−0.509505
$966$	0	0
$967$	−43.1279	−1.38690	−0.693450	−	0.720504i	$-0.743910\pi$
$967$	−43.1279	−1.38690	−0.693450	+	0.720504i	$0.743910\pi$
$968$	0	0
$969$	2.34315	0.0752727
$970$	0	0
$971$	−15.4558	−0.496002	−0.248001	−	0.968760i	$-0.579774\pi$
$971$	−15.4558	−0.496002	−0.248001	+	0.968760i	$0.579774\pi$
$972$	0	0
$973$	13.1119	0.420349
$974$	0	0
$975$	59.6343	1.90983
$976$	0	0
$977$	−42.7696	−1.36832	−0.684160	−	0.729332i	$-0.739831\pi$
$977$	−42.7696	−1.36832	−0.684160	+	0.729332i	$0.739831\pi$
$978$	0	0
$979$	48.2843	1.54317
$980$	0	0
$981$	−16.7876	−0.535987
$982$	0	0
$983$	−38.2110	−1.21874	−0.609370	−	0.792886i	$-0.708578\pi$
$983$	−38.2110	−1.21874	−0.609370	+	0.792886i	$0.708578\pi$
$984$	0	0
$985$	−65.3137	−2.08107
$986$	0	0
$987$	12.9706	0.412858
$988$	0	0
$989$	−35.4954	−1.12869
$990$	0	0
$991$	−46.4059	−1.47413	−0.737066	−	0.675821i	$-0.763789\pi$
$991$	−46.4059	−1.47413	−0.737066	+	0.675821i	$0.763789\pi$
$992$	0	0
$993$	10.3431	0.328230
$994$	0	0
$995$	−6.48528	−0.205597
$996$	0	0
$997$	29.3371	0.929116	0.464558	−	0.885543i	$-0.346213\pi$
$997$	29.3371	0.929116	0.464558	+	0.885543i	$0.346213\pi$
$998$	0	0
$999$	−2.31788	−0.0733346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.a.n.1.4	yes	4
3.2	odd	2		4608.2.a.t.1.1		4
4.3	odd	2		1536.2.a.m.1.4	yes	4
8.3	odd	2	inner	1536.2.a.n.1.1	yes	4
8.5	even	2		1536.2.a.m.1.1	✓	4
12.11	even	2		4608.2.a.ba.1.1		4
16.3	odd	4		1536.2.d.g.769.1		8
16.5	even	4		1536.2.d.g.769.4		8
16.11	odd	4		1536.2.d.g.769.8		8
16.13	even	4		1536.2.d.g.769.5		8
24.5	odd	2		4608.2.a.ba.1.4		4
24.11	even	2		4608.2.a.t.1.4		4
48.5	odd	4		4608.2.d.p.2305.2		8
48.11	even	4		4608.2.d.p.2305.1		8
48.29	odd	4		4608.2.d.p.2305.8		8
48.35	even	4		4608.2.d.p.2305.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.m.1.1	✓	4	8.5	even	2
1536.2.a.m.1.4	yes	4	4.3	odd	2
1536.2.a.n.1.1	yes	4	8.3	odd	2	inner
1536.2.a.n.1.4	yes	4	1.1	even	1	trivial
1536.2.d.g.769.1		8	16.3	odd	4
1536.2.d.g.769.4		8	16.5	even	4
1536.2.d.g.769.5		8	16.13	even	4
1536.2.d.g.769.8		8	16.11	odd	4
4608.2.a.t.1.1		4	3.2	odd	2
4608.2.a.t.1.4		4	24.11	even	2
4608.2.a.ba.1.1		4	12.11	even	2
4608.2.a.ba.1.4		4	24.5	odd	2
4608.2.d.p.2305.1		8	48.11	even	4
4608.2.d.p.2305.2		8	48.5	odd	4
4608.2.d.p.2305.7		8	48.35	even	4
4608.2.d.p.2305.8		8	48.29	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.95687 q^{5} -1.63899 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.95687 q^{5} -1.63899 q^{7} +1.00000 q^{9} +4.82843 q^{11} +5.59587 q^{13} +3.95687 q^{15} -0.828427 q^{17} -2.82843 q^{19} -1.63899 q^{21} -7.91375 q^{23} +10.6569 q^{25} +1.00000 q^{27} -7.23486 q^{29} +1.63899 q^{31} +4.82843 q^{33} -6.48528 q^{35} -2.31788 q^{37} +5.59587 q^{39} +3.17157 q^{41} +4.48528 q^{43} +3.95687 q^{45} -7.91375 q^{47} -4.31371 q^{49} -0.828427 q^{51} -0.678892 q^{53} +19.1055 q^{55} -2.82843 q^{57} +9.65685 q^{59} -2.31788 q^{61} -1.63899 q^{63} +22.1421 q^{65} -13.6569 q^{67} -7.91375 q^{69} +3.27798 q^{71} +4.00000 q^{73} +10.6569 q^{75} -7.91375 q^{77} -1.63899 q^{79} +1.00000 q^{81} +8.82843 q^{83} -3.27798 q^{85} -7.23486 q^{87} +10.0000 q^{89} -9.17157 q^{91} +1.63899 q^{93} -11.1917 q^{95} +11.6569 q^{97} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 8 q^{11} + 8 q^{17} + 20 q^{25} + 4 q^{27} + 8 q^{33} + 8 q^{35} + 24 q^{41} - 16 q^{43} + 28 q^{49} + 8 q^{51} + 16 q^{59} + 32 q^{65} - 32 q^{67} + 16 q^{73} + 20 q^{75} + 4 q^{81} + 24 q^{83} + 40 q^{89} - 48 q^{91} + 24 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^11 + 8 * q^17 + 20 * q^25 + 4 * q^27 + 8 * q^33 + 8 * q^35 + 24 * q^41 - 16 * q^43 + 28 * q^49 + 8 * q^51 + 16 * q^59 + 32 * q^65 - 32 * q^67 + 16 * q^73 + 20 * q^75 + 4 * q^81 + 24 * q^83 + 40 * q^89 - 48 * q^91 + 24 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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