LMFDB - Embedded newform 2760.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(1,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, 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("2760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.31662$ of defining polynomial
Character	$\chi$	$=$	2760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -4.31662 q^{11} -2.31662 q^{13} +1.00000 q^{15} -3.31662 q^{17} -1.68338 q^{19} +3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.683375 q^{29} -9.63325 q^{31} +4.31662 q^{33} +3.00000 q^{35} +11.6332 q^{37} +2.31662 q^{39} +3.31662 q^{41} +2.63325 q^{43} -1.00000 q^{45} +10.9499 q^{47} +2.00000 q^{49} +3.31662 q^{51} -9.94987 q^{53} +4.31662 q^{55} +1.68338 q^{57} -11.9499 q^{59} +3.68338 q^{61} -3.00000 q^{63} +2.31662 q^{65} +2.36675 q^{67} -1.00000 q^{69} -7.94987 q^{71} +1.68338 q^{73} -1.00000 q^{75} +12.9499 q^{77} +1.00000 q^{81} +5.31662 q^{83} +3.31662 q^{85} +0.683375 q^{87} +8.63325 q^{89} +6.94987 q^{91} +9.63325 q^{93} +1.68338 q^{95} +12.0000 q^{97} -4.31662 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 6 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} - 10 q^{19} + 6 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} - 2 q^{39} - 8 q^{43} - 2 q^{45} + 2 q^{47} + 4 q^{49} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 14 q^{61} - 6 q^{63} - 2 q^{65} + 18 q^{67} - 2 q^{69} + 4 q^{71} + 10 q^{73} - 2 q^{75} + 6 q^{77} + 2 q^{81} + 4 q^{83} + 8 q^{87} + 4 q^{89} - 6 q^{91} + 6 q^{93} + 10 q^{95} + 24 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 - 10 * q^19 + 6 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 - 6 * q^31 + 2 * q^33 + 6 * q^35 + 10 * q^37 - 2 * q^39 - 8 * q^43 - 2 * q^45 + 2 * q^47 + 4 * q^49 + 2 * q^55 + 10 * q^57 - 4 * q^59 + 14 * q^61 - 6 * q^63 - 2 * q^65 + 18 * q^67 - 2 * q^69 + 4 * q^71 + 10 * q^73 - 2 * q^75 + 6 * q^77 + 2 * q^81 + 4 * q^83 + 8 * q^87 + 4 * q^89 - 6 * q^91 + 6 * q^93 + 10 * q^95 + 24 * q^97 - 2 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.31662	−1.30151	−0.650756	−	0.759287i	$-0.725548\pi$
$11$	−4.31662	−1.30151	−0.650756	+	0.759287i	$0.725548\pi$
$12$	0	0
$13$	−2.31662	−0.642516	−0.321258	−	0.946992i	$-0.604106\pi$
$13$	−2.31662	−0.642516	−0.321258	+	0.946992i	$0.604106\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.31662	−0.804400	−0.402200	−	0.915552i	$-0.631754\pi$
$17$	−3.31662	−0.804400	−0.402200	+	0.915552i	$0.631754\pi$
$18$	0	0
$19$	−1.68338	−0.386193	−0.193096	−	0.981180i	$-0.561853\pi$
$19$	−1.68338	−0.386193	−0.193096	+	0.981180i	$0.561853\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.683375	−0.126900	−0.0634498	−	0.997985i	$-0.520210\pi$
$29$	−0.683375	−0.126900	−0.0634498	+	0.997985i	$0.520210\pi$
$30$	0	0
$31$	−9.63325	−1.73018	−0.865091	−	0.501614i	$-0.832740\pi$
$31$	−9.63325	−1.73018	−0.865091	+	0.501614i	$0.832740\pi$
$32$	0	0
$33$	4.31662	0.751428
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	11.6332	1.91249	0.956247	−	0.292560i	$-0.0945071\pi$
$37$	11.6332	1.91249	0.956247	+	0.292560i	$0.0945071\pi$
$38$	0	0
$39$	2.31662	0.370957
$40$	0	0
$41$	3.31662	0.517970	0.258985	−	0.965881i	$-0.416612\pi$
$41$	3.31662	0.517970	0.258985	+	0.965881i	$0.416612\pi$
$42$	0	0
$43$	2.63325	0.401567	0.200783	−	0.979636i	$-0.435651\pi$
$43$	2.63325	0.401567	0.200783	+	0.979636i	$0.435651\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	10.9499	1.59720	0.798602	−	0.601860i	$-0.205573\pi$
$47$	10.9499	1.59720	0.798602	+	0.601860i	$0.205573\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	3.31662	0.464420
$52$	0	0
$53$	−9.94987	−1.36672	−0.683360	−	0.730081i	$-0.739482\pi$
$53$	−9.94987	−1.36672	−0.683360	+	0.730081i	$0.739482\pi$
$54$	0	0
$55$	4.31662	0.582054
$56$	0	0
$57$	1.68338	0.222968
$58$	0	0
$59$	−11.9499	−1.55574	−0.777871	−	0.628425i	$-0.783700\pi$
$59$	−11.9499	−1.55574	−0.777871	+	0.628425i	$0.783700\pi$
$60$	0	0
$61$	3.68338	0.471608	0.235804	−	0.971801i	$-0.424228\pi$
$61$	3.68338	0.471608	0.235804	+	0.971801i	$0.424228\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	2.31662	0.287342
$66$	0	0
$67$	2.36675	0.289145	0.144572	−	0.989494i	$-0.453819\pi$
$67$	2.36675	0.289145	0.144572	+	0.989494i	$0.453819\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−7.94987	−0.943477	−0.471738	−	0.881739i	$-0.656373\pi$
$71$	−7.94987	−0.943477	−0.471738	+	0.881739i	$0.656373\pi$
$72$	0	0
$73$	1.68338	0.197024	0.0985121	−	0.995136i	$-0.468592\pi$
$73$	1.68338	0.197024	0.0985121	+	0.995136i	$0.468592\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	12.9499	1.47578
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.31662	0.583575	0.291788	−	0.956483i	$-0.405750\pi$
$83$	5.31662	0.583575	0.291788	+	0.956483i	$0.405750\pi$
$84$	0	0
$85$	3.31662	0.359738
$86$	0	0
$87$	0.683375	0.0732655
$88$	0	0
$89$	8.63325	0.915123	0.457561	−	0.889178i	$-0.348723\pi$
$89$	8.63325	0.915123	0.457561	+	0.889178i	$0.348723\pi$
$90$	0	0
$91$	6.94987	0.728545
$92$	0	0
$93$	9.63325	0.998921
$94$	0	0
$95$	1.68338	0.172711
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	−4.31662	−0.433837
$100$	0	0
$101$	5.31662	0.529024	0.264512	−	0.964382i	$-0.414789\pi$
$101$	5.31662	0.529024	0.264512	+	0.964382i	$0.414789\pi$
$102$	0	0
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−13.9499	−1.34859	−0.674293	−	0.738464i	$-0.735551\pi$
$107$	−13.9499	−1.34859	−0.674293	+	0.738464i	$0.735551\pi$
$108$	0	0
$109$	−10.9499	−1.04881	−0.524404	−	0.851470i	$-0.675712\pi$
$109$	−10.9499	−1.04881	−0.524404	+	0.851470i	$0.675712\pi$
$110$	0	0
$111$	−11.6332	−1.10418
$112$	0	0
$113$	15.3166	1.44087	0.720433	−	0.693524i	$-0.243943\pi$
$113$	15.3166	1.44087	0.720433	+	0.693524i	$0.243943\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−2.31662	−0.214172
$118$	0	0
$119$	9.94987	0.912103
$120$	0	0
$121$	7.63325	0.693932
$122$	0	0
$123$	−3.31662	−0.299050
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.94987	−0.439230	−0.219615	−	0.975587i	$-0.570480\pi$
$127$	−4.94987	−0.439230	−0.219615	+	0.975587i	$0.570480\pi$
$128$	0	0
$129$	−2.63325	−0.231845
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	5.05013	0.437901
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−13.2665	−1.13343	−0.566717	−	0.823913i	$-0.691787\pi$
$137$	−13.2665	−1.13343	−0.566717	+	0.823913i	$0.691787\pi$
$138$	0	0
$139$	−1.63325	−0.138530	−0.0692652	−	0.997598i	$-0.522065\pi$
$139$	−1.63325	−0.138530	−0.0692652	+	0.997598i	$0.522065\pi$
$140$	0	0
$141$	−10.9499	−0.922146
$142$	0	0
$143$	10.0000	0.836242
$144$	0	0
$145$	0.683375	0.0567512
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	2.31662	0.189785	0.0948926	−	0.995488i	$-0.469749\pi$
$149$	2.31662	0.189785	0.0948926	+	0.995488i	$0.469749\pi$
$150$	0	0
$151$	−1.36675	−0.111225	−0.0556123	−	0.998452i	$-0.517711\pi$
$151$	−1.36675	−0.111225	−0.0556123	+	0.998452i	$0.517711\pi$
$152$	0	0
$153$	−3.31662	−0.268133
$154$	0	0
$155$	9.63325	0.773761
$156$	0	0
$157$	6.36675	0.508122	0.254061	−	0.967188i	$-0.418234\pi$
$157$	6.36675	0.508122	0.254061	+	0.967188i	$0.418234\pi$
$158$	0	0
$159$	9.94987	0.789076
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	3.26650	0.255852	0.127926	−	0.991784i	$-0.459168\pi$
$163$	3.26650	0.255852	0.127926	+	0.991784i	$0.459168\pi$
$164$	0	0
$165$	−4.31662	−0.336049
$166$	0	0
$167$	19.5831	1.51539	0.757694	−	0.652610i	$-0.226326\pi$
$167$	19.5831	1.51539	0.757694	+	0.652610i	$0.226326\pi$
$168$	0	0
$169$	−7.63325	−0.587173
$170$	0	0
$171$	−1.68338	−0.128731
$172$	0	0
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	11.9499	0.898208
$178$	0	0
$179$	−9.36675	−0.700104	−0.350052	−	0.936730i	$-0.613836\pi$
$179$	−9.36675	−0.700104	−0.350052	+	0.936730i	$0.613836\pi$
$180$	0	0
$181$	20.6332	1.53366	0.766829	−	0.641852i	$-0.221834\pi$
$181$	20.6332	1.53366	0.766829	+	0.641852i	$0.221834\pi$
$182$	0	0
$183$	−3.68338	−0.272283
$184$	0	0
$185$	−11.6332	−0.855294
$186$	0	0
$187$	14.3166	1.04694
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	19.5831	1.41699	0.708493	−	0.705718i	$-0.249376\pi$
$191$	19.5831	1.41699	0.708493	+	0.705718i	$0.249376\pi$
$192$	0	0
$193$	12.6332	0.909361	0.454681	−	0.890655i	$-0.349753\pi$
$193$	12.6332	0.909361	0.454681	+	0.890655i	$0.349753\pi$
$194$	0	0
$195$	−2.31662	−0.165897
$196$	0	0
$197$	−27.8997	−1.98777	−0.993887	−	0.110399i	$-0.964787\pi$
$197$	−27.8997	−1.98777	−0.993887	+	0.110399i	$0.964787\pi$
$198$	0	0
$199$	1.36675	0.0968864	0.0484432	−	0.998826i	$-0.484574\pi$
$199$	1.36675	0.0968864	0.0484432	+	0.998826i	$0.484574\pi$
$200$	0	0
$201$	−2.36675	−0.166938
$202$	0	0
$203$	2.05013	0.143891
$204$	0	0
$205$	−3.31662	−0.231643
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	7.26650	0.502634
$210$	0	0
$211$	9.63325	0.663180	0.331590	−	0.943424i	$-0.392415\pi$
$211$	9.63325	0.663180	0.331590	+	0.943424i	$0.392415\pi$
$212$	0	0
$213$	7.94987	0.544716
$214$	0	0
$215$	−2.63325	−0.179586
$216$	0	0
$217$	28.8997	1.96184
$218$	0	0
$219$	−1.68338	−0.113752
$220$	0	0
$221$	7.68338	0.516840
$222$	0	0
$223$	−3.26650	−0.218741	−0.109370	−	0.994001i	$-0.534883\pi$
$223$	−3.26650	−0.218741	−0.109370	+	0.994001i	$0.534883\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	0	0
$229$	16.6332	1.09916	0.549578	−	0.835442i	$-0.314789\pi$
$229$	16.6332	1.09916	0.549578	+	0.835442i	$0.314789\pi$
$230$	0	0
$231$	−12.9499	−0.852039
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	−10.9499	−0.714291
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.94987	0.255496	0.127748	−	0.991807i	$-0.459225\pi$
$239$	3.94987	0.255496	0.127748	+	0.991807i	$0.459225\pi$
$240$	0	0
$241$	20.2164	1.30225	0.651126	−	0.758970i	$-0.274297\pi$
$241$	20.2164	1.30225	0.651126	+	0.758970i	$0.274297\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	3.89975	0.248135
$248$	0	0
$249$	−5.31662	−0.336927
$250$	0	0
$251$	−3.26650	−0.206180	−0.103090	−	0.994672i	$-0.532873\pi$
$251$	−3.26650	−0.206180	−0.103090	+	0.994672i	$0.532873\pi$
$252$	0	0
$253$	−4.31662	−0.271384
$254$	0	0
$255$	−3.31662	−0.207695
$256$	0	0
$257$	−18.9499	−1.18206	−0.591030	−	0.806649i	$-0.701279\pi$
$257$	−18.9499	−1.18206	−0.591030	+	0.806649i	$0.701279\pi$
$258$	0	0
$259$	−34.8997	−2.16856
$260$	0	0
$261$	−0.683375	−0.0422999
$262$	0	0
$263$	−4.58312	−0.282608	−0.141304	−	0.989966i	$-0.545129\pi$
$263$	−4.58312	−0.282608	−0.141304	+	0.989966i	$0.545129\pi$
$264$	0	0
$265$	9.94987	0.611216
$266$	0	0
$267$	−8.63325	−0.528346
$268$	0	0
$269$	10.5831	0.645264	0.322632	−	0.946524i	$-0.395432\pi$
$269$	10.5831	0.645264	0.322632	+	0.946524i	$0.395432\pi$
$270$	0	0
$271$	−2.26650	−0.137680	−0.0688400	−	0.997628i	$-0.521930\pi$
$271$	−2.26650	−0.137680	−0.0688400	+	0.997628i	$0.521930\pi$
$272$	0	0
$273$	−6.94987	−0.420626
$274$	0	0
$275$	−4.31662	−0.260302
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	−9.63325	−0.576728
$280$	0	0
$281$	−16.9499	−1.01114	−0.505572	−	0.862784i	$-0.668719\pi$
$281$	−16.9499	−1.01114	−0.505572	+	0.862784i	$0.668719\pi$
$282$	0	0
$283$	7.63325	0.453750	0.226875	−	0.973924i	$-0.427149\pi$
$283$	7.63325	0.453750	0.226875	+	0.973924i	$0.427149\pi$
$284$	0	0
$285$	−1.68338	−0.0997145
$286$	0	0
$287$	−9.94987	−0.587323
$288$	0	0
$289$	−6.00000	−0.352941
$290$	0	0
$291$	−12.0000	−0.703452
$292$	0	0
$293$	11.9499	0.698119	0.349060	−	0.937101i	$-0.386501\pi$
$293$	11.9499	0.698119	0.349060	+	0.937101i	$0.386501\pi$
$294$	0	0
$295$	11.9499	0.695749
$296$	0	0
$297$	4.31662	0.250476
$298$	0	0
$299$	−2.31662	−0.133974
$300$	0	0
$301$	−7.89975	−0.455334
$302$	0	0
$303$	−5.31662	−0.305432
$304$	0	0
$305$	−3.68338	−0.210909
$306$	0	0
$307$	−5.05013	−0.288226	−0.144113	−	0.989561i	$-0.546033\pi$
$307$	−5.05013	−0.288226	−0.144113	+	0.989561i	$0.546033\pi$
$308$	0	0
$309$	−18.0000	−1.02398
$310$	0	0
$311$	29.2665	1.65955	0.829775	−	0.558097i	$-0.188468\pi$
$311$	29.2665	1.65955	0.829775	+	0.558097i	$0.188468\pi$
$312$	0	0
$313$	20.3668	1.15120	0.575598	−	0.817733i	$-0.304769\pi$
$313$	20.3668	1.15120	0.575598	+	0.817733i	$0.304769\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−9.68338	−0.543873	−0.271936	−	0.962315i	$-0.587664\pi$
$317$	−9.68338	−0.543873	−0.271936	+	0.962315i	$0.587664\pi$
$318$	0	0
$319$	2.94987	0.165161
$320$	0	0
$321$	13.9499	0.778606
$322$	0	0
$323$	5.58312	0.310653
$324$	0	0
$325$	−2.31662	−0.128503
$326$	0	0
$327$	10.9499	0.605529
$328$	0	0
$329$	−32.8496	−1.81106
$330$	0	0
$331$	−31.5330	−1.73321	−0.866605	−	0.498994i	$-0.833703\pi$
$331$	−31.5330	−1.73321	−0.866605	+	0.498994i	$0.833703\pi$
$332$	0	0
$333$	11.6332	0.637498
$334$	0	0
$335$	−2.36675	−0.129309
$336$	0	0
$337$	31.2665	1.70319	0.851597	−	0.524196i	$-0.175634\pi$
$337$	31.2665	1.70319	0.851597	+	0.524196i	$0.175634\pi$
$338$	0	0
$339$	−15.3166	−0.831885
$340$	0	0
$341$	41.5831	2.25185
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−11.3668	−0.610199	−0.305100	−	0.952320i	$-0.598690\pi$
$347$	−11.3668	−0.610199	−0.305100	+	0.952320i	$0.598690\pi$
$348$	0	0
$349$	−21.6332	−1.15800	−0.579001	−	0.815327i	$-0.696557\pi$
$349$	−21.6332	−1.15800	−0.579001	+	0.815327i	$0.696557\pi$
$350$	0	0
$351$	2.31662	0.123652
$352$	0	0
$353$	−20.2164	−1.07601	−0.538004	−	0.842942i	$-0.680822\pi$
$353$	−20.2164	−1.07601	−0.538004	+	0.842942i	$0.680822\pi$
$354$	0	0
$355$	7.94987	0.421936
$356$	0	0
$357$	−9.94987	−0.526603
$358$	0	0
$359$	−28.8496	−1.52262	−0.761312	−	0.648385i	$-0.775445\pi$
$359$	−28.8496	−1.52262	−0.761312	+	0.648385i	$0.775445\pi$
$360$	0	0
$361$	−16.1662	−0.850855
$362$	0	0
$363$	−7.63325	−0.400642
$364$	0	0
$365$	−1.68338	−0.0881119
$366$	0	0
$367$	−17.5330	−0.915215	−0.457608	−	0.889154i	$-0.651294\pi$
$367$	−17.5330	−0.915215	−0.457608	+	0.889154i	$0.651294\pi$
$368$	0	0
$369$	3.31662	0.172657
$370$	0	0
$371$	29.8496	1.54972
$372$	0	0
$373$	29.8997	1.54815	0.774075	−	0.633094i	$-0.218215\pi$
$373$	29.8997	1.54815	0.774075	+	0.633094i	$0.218215\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	1.58312	0.0815350
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	4.94987	0.253590
$382$	0	0
$383$	19.3166	0.987033	0.493517	−	0.869736i	$-0.335711\pi$
$383$	19.3166	0.987033	0.493517	+	0.869736i	$0.335711\pi$
$384$	0	0
$385$	−12.9499	−0.659987
$386$	0	0
$387$	2.63325	0.133856
$388$	0	0
$389$	−23.8997	−1.21177	−0.605883	−	0.795554i	$-0.707180\pi$
$389$	−23.8997	−1.21177	−0.605883	+	0.795554i	$0.707180\pi$
$390$	0	0
$391$	−3.31662	−0.167729
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	0	0
$399$	−5.05013	−0.252823
$400$	0	0
$401$	−19.8997	−0.993746	−0.496873	−	0.867823i	$-0.665519\pi$
$401$	−19.8997	−0.993746	−0.496873	+	0.867823i	$0.665519\pi$
$402$	0	0
$403$	22.3166	1.11167
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−50.2164	−2.48913
$408$	0	0
$409$	0.366750	0.0181346	0.00906732	−	0.999959i	$-0.497114\pi$
$409$	0.366750	0.0181346	0.00906732	+	0.999959i	$0.497114\pi$
$410$	0	0
$411$	13.2665	0.654388
$412$	0	0
$413$	35.8496	1.76404
$414$	0	0
$415$	−5.31662	−0.260983
$416$	0	0
$417$	1.63325	0.0799806
$418$	0	0
$419$	−11.6834	−0.570770	−0.285385	−	0.958413i	$-0.592122\pi$
$419$	−11.6834	−0.570770	−0.285385	+	0.958413i	$0.592122\pi$
$420$	0	0
$421$	23.5831	1.14937	0.574686	−	0.818374i	$-0.305124\pi$
$421$	23.5831	1.14937	0.574686	+	0.818374i	$0.305124\pi$
$422$	0	0
$423$	10.9499	0.532401
$424$	0	0
$425$	−3.31662	−0.160880
$426$	0	0
$427$	−11.0501	−0.534753
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−16.6332	−0.801195	−0.400598	−	0.916254i	$-0.631198\pi$
$431$	−16.6332	−0.801195	−0.400598	+	0.916254i	$0.631198\pi$
$432$	0	0
$433$	−10.3668	−0.498194	−0.249097	−	0.968479i	$-0.580134\pi$
$433$	−10.3668	−0.498194	−0.249097	+	0.968479i	$0.580134\pi$
$434$	0	0
$435$	−0.683375	−0.0327653
$436$	0	0
$437$	−1.68338	−0.0805268
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−21.6834	−1.03021	−0.515104	−	0.857128i	$-0.672247\pi$
$443$	−21.6834	−1.03021	−0.515104	+	0.857128i	$0.672247\pi$
$444$	0	0
$445$	−8.63325	−0.409255
$446$	0	0
$447$	−2.31662	−0.109573
$448$	0	0
$449$	−0.683375	−0.0322505	−0.0161252	−	0.999870i	$-0.505133\pi$
$449$	−0.683375	−0.0322505	−0.0161252	+	0.999870i	$0.505133\pi$
$450$	0	0
$451$	−14.3166	−0.674144
$452$	0	0
$453$	1.36675	0.0642155
$454$	0	0
$455$	−6.94987	−0.325815
$456$	0	0
$457$	12.3668	0.578492	0.289246	−	0.957255i	$-0.406595\pi$
$457$	12.3668	0.578492	0.289246	+	0.957255i	$0.406595\pi$
$458$	0	0
$459$	3.31662	0.154807
$460$	0	0
$461$	−10.6332	−0.495240	−0.247620	−	0.968857i	$-0.579648\pi$
$461$	−10.6332	−0.495240	−0.247620	+	0.968857i	$0.579648\pi$
$462$	0	0
$463$	−8.21637	−0.381847	−0.190924	−	0.981605i	$-0.561148\pi$
$463$	−8.21637	−0.381847	−0.190924	+	0.981605i	$0.561148\pi$
$464$	0	0
$465$	−9.63325	−0.446731
$466$	0	0
$467$	−29.9499	−1.38591	−0.692957	−	0.720978i	$-0.743693\pi$
$467$	−29.9499	−1.38591	−0.692957	+	0.720978i	$0.743693\pi$
$468$	0	0
$469$	−7.10025	−0.327859
$470$	0	0
$471$	−6.36675	−0.293364
$472$	0	0
$473$	−11.3668	−0.522644
$474$	0	0
$475$	−1.68338	−0.0772386
$476$	0	0
$477$	−9.94987	−0.455573
$478$	0	0
$479$	18.3166	0.836908	0.418454	−	0.908238i	$-0.362572\pi$
$479$	18.3166	0.836908	0.418454	+	0.908238i	$0.362572\pi$
$480$	0	0
$481$	−26.9499	−1.22881
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	13.5831	0.615510	0.307755	−	0.951466i	$-0.400422\pi$
$487$	13.5831	0.615510	0.307755	+	0.951466i	$0.400422\pi$
$488$	0	0
$489$	−3.26650	−0.147716
$490$	0	0
$491$	9.94987	0.449032	0.224516	−	0.974470i	$-0.427920\pi$
$491$	9.94987	0.449032	0.224516	+	0.974470i	$0.427920\pi$
$492$	0	0
$493$	2.26650	0.102078
$494$	0	0
$495$	4.31662	0.194018
$496$	0	0
$497$	23.8496	1.06980
$498$	0	0
$499$	18.2665	0.817721	0.408860	−	0.912597i	$-0.365926\pi$
$499$	18.2665	0.817721	0.408860	+	0.912597i	$0.365926\pi$
$500$	0	0
$501$	−19.5831	−0.874909
$502$	0	0
$503$	37.8496	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$503$	37.8496	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$504$	0	0
$505$	−5.31662	−0.236587
$506$	0	0
$507$	7.63325	0.339005
$508$	0	0
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	−5.05013	−0.223404
$512$	0	0
$513$	1.68338	0.0743228
$514$	0	0
$515$	−18.0000	−0.793175
$516$	0	0
$517$	−47.2665	−2.07878
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	−0.316625	−0.0138716	−0.00693579	−	0.999976i	$-0.502208\pi$
$521$	−0.316625	−0.0138716	−0.00693579	+	0.999976i	$0.502208\pi$
$522$	0	0
$523$	−22.5330	−0.985299	−0.492650	−	0.870228i	$-0.663972\pi$
$523$	−22.5330	−0.985299	−0.492650	+	0.870228i	$0.663972\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	31.9499	1.39176
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.9499	−0.518580
$532$	0	0
$533$	−7.68338	−0.332804
$534$	0	0
$535$	13.9499	0.603106
$536$	0	0
$537$	9.36675	0.404205
$538$	0	0
$539$	−8.63325	−0.371860
$540$	0	0
$541$	21.2665	0.914318	0.457159	−	0.889385i	$-0.348867\pi$
$541$	21.2665	0.914318	0.457159	+	0.889385i	$0.348867\pi$
$542$	0	0
$543$	−20.6332	−0.885457
$544$	0	0
$545$	10.9499	0.469041
$546$	0	0
$547$	−25.2665	−1.08032	−0.540159	−	0.841563i	$-0.681636\pi$
$547$	−25.2665	−1.08032	−0.540159	+	0.841563i	$0.681636\pi$
$548$	0	0
$549$	3.68338	0.157203
$550$	0	0
$551$	1.15038	0.0490077
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11.6332	0.493804
$556$	0	0
$557$	−25.9499	−1.09953	−0.549766	−	0.835319i	$-0.685283\pi$
$557$	−25.9499	−1.09953	−0.549766	+	0.835319i	$0.685283\pi$
$558$	0	0
$559$	−6.10025	−0.258013
$560$	0	0
$561$	−14.3166	−0.604448
$562$	0	0
$563$	36.6834	1.54602	0.773010	−	0.634394i	$-0.218750\pi$
$563$	36.6834	1.54602	0.773010	+	0.634394i	$0.218750\pi$
$564$	0	0
$565$	−15.3166	−0.644375
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	3.89975	0.163486	0.0817430	−	0.996653i	$-0.473951\pi$
$569$	3.89975	0.163486	0.0817430	+	0.996653i	$0.473951\pi$
$570$	0	0
$571$	24.9499	1.04412	0.522060	−	0.852909i	$-0.325164\pi$
$571$	24.9499	1.04412	0.522060	+	0.852909i	$0.325164\pi$
$572$	0	0
$573$	−19.5831	−0.818097
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−9.26650	−0.385769	−0.192885	−	0.981221i	$-0.561784\pi$
$577$	−9.26650	−0.385769	−0.192885	+	0.981221i	$0.561784\pi$
$578$	0	0
$579$	−12.6332	−0.525020
$580$	0	0
$581$	−15.9499	−0.661712
$582$	0	0
$583$	42.9499	1.77880
$584$	0	0
$585$	2.31662	0.0957806
$586$	0	0
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	0	0
$589$	16.2164	0.668184
$590$	0	0
$591$	27.8997	1.14764
$592$	0	0
$593$	−22.3166	−0.916434	−0.458217	−	0.888840i	$-0.651512\pi$
$593$	−22.3166	−0.916434	−0.458217	+	0.888840i	$0.651512\pi$
$594$	0	0
$595$	−9.94987	−0.407905
$596$	0	0
$597$	−1.36675	−0.0559374
$598$	0	0
$599$	28.6332	1.16992	0.584961	−	0.811061i	$-0.301110\pi$
$599$	28.6332	1.16992	0.584961	+	0.811061i	$0.301110\pi$
$600$	0	0
$601$	9.63325	0.392948	0.196474	−	0.980509i	$-0.437051\pi$
$601$	9.63325	0.392948	0.196474	+	0.980509i	$0.437051\pi$
$602$	0	0
$603$	2.36675	0.0963815
$604$	0	0
$605$	−7.63325	−0.310336
$606$	0	0
$607$	−29.5831	−1.20074	−0.600371	−	0.799722i	$-0.704980\pi$
$607$	−29.5831	−1.20074	−0.600371	+	0.799722i	$0.704980\pi$
$608$	0	0
$609$	−2.05013	−0.0830753
$610$	0	0
$611$	−25.3668	−1.02623
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	3.31662	0.133739
$616$	0	0
$617$	31.9499	1.28625	0.643127	−	0.765760i	$-0.277637\pi$
$617$	31.9499	1.28625	0.643127	+	0.765760i	$0.277637\pi$
$618$	0	0
$619$	43.7995	1.76045	0.880225	−	0.474556i	$-0.157391\pi$
$619$	43.7995	1.76045	0.880225	+	0.474556i	$0.157391\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−25.8997	−1.03765
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.26650	−0.290196
$628$	0	0
$629$	−38.5831	−1.53841
$630$	0	0
$631$	16.9499	0.674764	0.337382	−	0.941368i	$-0.390459\pi$
$631$	16.9499	0.674764	0.337382	+	0.941368i	$0.390459\pi$
$632$	0	0
$633$	−9.63325	−0.382887
$634$	0	0
$635$	4.94987	0.196430
$636$	0	0
$637$	−4.63325	−0.183576
$638$	0	0
$639$	−7.94987	−0.314492
$640$	0	0
$641$	−9.58312	−0.378511	−0.189255	−	0.981928i	$-0.560607\pi$
$641$	−9.58312	−0.378511	−0.189255	+	0.981928i	$0.560607\pi$
$642$	0	0
$643$	−18.2665	−0.720360	−0.360180	−	0.932883i	$-0.617285\pi$
$643$	−18.2665	−0.720360	−0.360180	+	0.932883i	$0.617285\pi$
$644$	0	0
$645$	2.63325	0.103684
$646$	0	0
$647$	17.5831	0.691264	0.345632	−	0.938370i	$-0.387665\pi$
$647$	17.5831	0.691264	0.345632	+	0.938370i	$0.387665\pi$
$648$	0	0
$649$	51.5831	2.02481
$650$	0	0
$651$	−28.8997	−1.13267
$652$	0	0
$653$	−50.2164	−1.96512	−0.982559	−	0.185950i	$-0.940464\pi$
$653$	−50.2164	−1.96512	−0.982559	+	0.185950i	$0.940464\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.68338	0.0656747
$658$	0	0
$659$	−37.5831	−1.46403	−0.732015	−	0.681288i	$-0.761420\pi$
$659$	−37.5831	−1.46403	−0.732015	+	0.681288i	$0.761420\pi$
$660$	0	0
$661$	−34.6332	−1.34708	−0.673539	−	0.739152i	$-0.735226\pi$
$661$	−34.6332	−1.34708	−0.673539	+	0.739152i	$0.735226\pi$
$662$	0	0
$663$	−7.68338	−0.298398
$664$	0	0
$665$	−5.05013	−0.195835
$666$	0	0
$667$	−0.683375	−0.0264604
$668$	0	0
$669$	3.26650	0.126290
$670$	0	0
$671$	−15.8997	−0.613803
$672$	0	0
$673$	−3.05013	−0.117574	−0.0587869	−	0.998271i	$-0.518723\pi$
$673$	−3.05013	−0.117574	−0.0587869	+	0.998271i	$0.518723\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−26.0501	−1.00119	−0.500594	−	0.865682i	$-0.666885\pi$
$677$	−26.0501	−1.00119	−0.500594	+	0.865682i	$0.666885\pi$
$678$	0	0
$679$	−36.0000	−1.38155
$680$	0	0
$681$	28.0000	1.07296
$682$	0	0
$683$	13.6834	0.523580	0.261790	−	0.965125i	$-0.415687\pi$
$683$	13.6834	0.523580	0.261790	+	0.965125i	$0.415687\pi$
$684$	0	0
$685$	13.2665	0.506887
$686$	0	0
$687$	−16.6332	−0.634598
$688$	0	0
$689$	23.0501	0.878140
$690$	0	0
$691$	10.6332	0.404508	0.202254	−	0.979333i	$-0.435173\pi$
$691$	10.6332	0.404508	0.202254	+	0.979333i	$0.435173\pi$
$692$	0	0
$693$	12.9499	0.491925
$694$	0	0
$695$	1.63325	0.0619527
$696$	0	0
$697$	−11.0000	−0.416655
$698$	0	0
$699$	−8.00000	−0.302588
$700$	0	0
$701$	−10.3166	−0.389654	−0.194827	−	0.980838i	$-0.562414\pi$
$701$	−10.3166	−0.389654	−0.194827	+	0.980838i	$0.562414\pi$
$702$	0	0
$703$	−19.5831	−0.738592
$704$	0	0
$705$	10.9499	0.412396
$706$	0	0
$707$	−15.9499	−0.599857
$708$	0	0
$709$	−46.8496	−1.75947	−0.879737	−	0.475460i	$-0.842282\pi$
$709$	−46.8496	−1.75947	−0.879737	+	0.475460i	$0.842282\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.63325	−0.360768
$714$	0	0
$715$	−10.0000	−0.373979
$716$	0	0
$717$	−3.94987	−0.147511
$718$	0	0
$719$	−37.8496	−1.41155	−0.705776	−	0.708435i	$-0.749402\pi$
$719$	−37.8496	−1.41155	−0.705776	+	0.708435i	$0.749402\pi$
$720$	0	0
$721$	−54.0000	−2.01107
$722$	0	0
$723$	−20.2164	−0.751855
$724$	0	0
$725$	−0.683375	−0.0253799
$726$	0	0
$727$	10.3668	0.384481	0.192241	−	0.981348i	$-0.438425\pi$
$727$	10.3668	0.384481	0.192241	+	0.981348i	$0.438425\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.73350	−0.323020
$732$	0	0
$733$	−42.2665	−1.56115	−0.780574	−	0.625063i	$-0.785073\pi$
$733$	−42.2665	−1.56115	−0.780574	+	0.625063i	$0.785073\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	−10.2164	−0.376325
$738$	0	0
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	0	0
$741$	−3.89975	−0.143261
$742$	0	0
$743$	47.7995	1.75359	0.876797	−	0.480861i	$-0.159676\pi$
$743$	47.7995	1.75359	0.876797	+	0.480861i	$0.159676\pi$
$744$	0	0
$745$	−2.31662	−0.0848746
$746$	0	0
$747$	5.31662	0.194525
$748$	0	0
$749$	41.8496	1.52915
$750$	0	0
$751$	48.8496	1.78255	0.891274	−	0.453465i	$-0.149812\pi$
$751$	48.8496	1.78255	0.891274	+	0.453465i	$0.149812\pi$
$752$	0	0
$753$	3.26650	0.119038
$754$	0	0
$755$	1.36675	0.0497411
$756$	0	0
$757$	29.5330	1.07340	0.536698	−	0.843775i	$-0.319672\pi$
$757$	29.5330	1.07340	0.536698	+	0.843775i	$0.319672\pi$
$758$	0	0
$759$	4.31662	0.156684
$760$	0	0
$761$	38.5831	1.39864	0.699319	−	0.714810i	$-0.253487\pi$
$761$	38.5831	1.39864	0.699319	+	0.714810i	$0.253487\pi$
$762$	0	0
$763$	32.8496	1.18924
$764$	0	0
$765$	3.31662	0.119913
$766$	0	0
$767$	27.6834	0.999589
$768$	0	0
$769$	−21.5831	−0.778307	−0.389154	−	0.921173i	$-0.627232\pi$
$769$	−21.5831	−0.778307	−0.389154	+	0.921173i	$0.627232\pi$
$770$	0	0
$771$	18.9499	0.682463
$772$	0	0
$773$	21.1662	0.761297	0.380649	−	0.924720i	$-0.375701\pi$
$773$	21.1662	0.761297	0.380649	+	0.924720i	$0.375701\pi$
$774$	0	0
$775$	−9.63325	−0.346037
$776$	0	0
$777$	34.8997	1.25202
$778$	0	0
$779$	−5.58312	−0.200036
$780$	0	0
$781$	34.3166	1.22795
$782$	0	0
$783$	0.683375	0.0244218
$784$	0	0
$785$	−6.36675	−0.227239
$786$	0	0
$787$	53.5330	1.90825	0.954123	−	0.299416i	$-0.0967918\pi$
$787$	53.5330	1.90825	0.954123	+	0.299416i	$0.0967918\pi$
$788$	0	0
$789$	4.58312	0.163164
$790$	0	0
$791$	−45.9499	−1.63379
$792$	0	0
$793$	−8.53300	−0.303016
$794$	0	0
$795$	−9.94987	−0.352886
$796$	0	0
$797$	−41.8496	−1.48239	−0.741195	−	0.671290i	$-0.765741\pi$
$797$	−41.8496	−1.48239	−0.741195	+	0.671290i	$0.765741\pi$
$798$	0	0
$799$	−36.3166	−1.28479
$800$	0	0
$801$	8.63325	0.305041
$802$	0	0
$803$	−7.26650	−0.256429
$804$	0	0
$805$	3.00000	0.105736
$806$	0	0
$807$	−10.5831	−0.372544
$808$	0	0
$809$	−4.78363	−0.168183	−0.0840917	−	0.996458i	$-0.526799\pi$
$809$	−4.78363	−0.168183	−0.0840917	+	0.996458i	$0.526799\pi$
$810$	0	0
$811$	−42.1662	−1.48066	−0.740329	−	0.672245i	$-0.765330\pi$
$811$	−42.1662	−1.48066	−0.740329	+	0.672245i	$0.765330\pi$
$812$	0	0
$813$	2.26650	0.0794896
$814$	0	0
$815$	−3.26650	−0.114420
$816$	0	0
$817$	−4.43275	−0.155082
$818$	0	0
$819$	6.94987	0.242848
$820$	0	0
$821$	29.2665	1.02141	0.510704	−	0.859757i	$-0.329385\pi$
$821$	29.2665	1.02141	0.510704	+	0.859757i	$0.329385\pi$
$822$	0	0
$823$	53.2665	1.85675	0.928377	−	0.371641i	$-0.121205\pi$
$823$	53.2665	1.85675	0.928377	+	0.371641i	$0.121205\pi$
$824$	0	0
$825$	4.31662	0.150286
$826$	0	0
$827$	18.5831	0.646199	0.323099	−	0.946365i	$-0.395275\pi$
$827$	18.5831	0.646199	0.323099	+	0.946365i	$0.395275\pi$
$828$	0	0
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	−6.63325	−0.229828
$834$	0	0
$835$	−19.5831	−0.677702
$836$	0	0
$837$	9.63325	0.332974
$838$	0	0
$839$	7.26650	0.250867	0.125434	−	0.992102i	$-0.459968\pi$
$839$	7.26650	0.250867	0.125434	+	0.992102i	$0.459968\pi$
$840$	0	0
$841$	−28.5330	−0.983896
$842$	0	0
$843$	16.9499	0.583785
$844$	0	0
$845$	7.63325	0.262592
$846$	0	0
$847$	−22.8997	−0.786845
$848$	0	0
$849$	−7.63325	−0.261972
$850$	0	0
$851$	11.6332	0.398783
$852$	0	0
$853$	26.6332	0.911905	0.455953	−	0.890004i	$-0.349299\pi$
$853$	26.6332	0.911905	0.455953	+	0.890004i	$0.349299\pi$
$854$	0	0
$855$	1.68338	0.0575702
$856$	0	0
$857$	24.5330	0.838031	0.419016	−	0.907979i	$-0.362375\pi$
$857$	24.5330	0.838031	0.419016	+	0.907979i	$0.362375\pi$
$858$	0	0
$859$	20.3668	0.694905	0.347452	−	0.937698i	$-0.387047\pi$
$859$	20.3668	0.694905	0.347452	+	0.937698i	$0.387047\pi$
$860$	0	0
$861$	9.94987	0.339091
$862$	0	0
$863$	−1.36675	−0.0465247	−0.0232624	−	0.999729i	$-0.507405\pi$
$863$	−1.36675	−0.0465247	−0.0232624	+	0.999729i	$0.507405\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	6.00000	0.203771
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−5.48287	−0.185780
$872$	0	0
$873$	12.0000	0.406138
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−27.8997	−0.942108	−0.471054	−	0.882104i	$-0.656126\pi$
$877$	−27.8997	−0.942108	−0.471054	+	0.882104i	$0.656126\pi$
$878$	0	0
$879$	−11.9499	−0.403059
$880$	0	0
$881$	−8.41688	−0.283572	−0.141786	−	0.989897i	$-0.545284\pi$
$881$	−8.41688	−0.283572	−0.141786	+	0.989897i	$0.545284\pi$
$882$	0	0
$883$	−11.6834	−0.393177	−0.196588	−	0.980486i	$-0.562986\pi$
$883$	−11.6834	−0.393177	−0.196588	+	0.980486i	$0.562986\pi$
$884$	0	0
$885$	−11.9499	−0.401691
$886$	0	0
$887$	33.3668	1.12035	0.560173	−	0.828376i	$-0.310735\pi$
$887$	33.3668	1.12035	0.560173	+	0.828376i	$0.310735\pi$
$888$	0	0
$889$	14.8496	0.498040
$890$	0	0
$891$	−4.31662	−0.144612
$892$	0	0
$893$	−18.4327	−0.616828
$894$	0	0
$895$	9.36675	0.313096
$896$	0	0
$897$	2.31662	0.0773499
$898$	0	0
$899$	6.58312	0.219559
$900$	0	0
$901$	33.0000	1.09939
$902$	0	0
$903$	7.89975	0.262887
$904$	0	0
$905$	−20.6332	−0.685872
$906$	0	0
$907$	7.63325	0.253458	0.126729	−	0.991937i	$-0.459552\pi$
$907$	7.63325	0.253458	0.126729	+	0.991937i	$0.459552\pi$
$908$	0	0
$909$	5.31662	0.176341
$910$	0	0
$911$	47.7995	1.58367	0.791834	−	0.610736i	$-0.209126\pi$
$911$	47.7995	1.58367	0.791834	+	0.610736i	$0.209126\pi$
$912$	0	0
$913$	−22.9499	−0.759530
$914$	0	0
$915$	3.68338	0.121769
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−17.2665	−0.569569	−0.284785	−	0.958592i	$-0.591922\pi$
$919$	−17.2665	−0.569569	−0.284785	+	0.958592i	$0.591922\pi$
$920$	0	0
$921$	5.05013	0.166407
$922$	0	0
$923$	18.4169	0.606199
$924$	0	0
$925$	11.6332	0.382499
$926$	0	0
$927$	18.0000	0.591198
$928$	0	0
$929$	−17.3166	−0.568140	−0.284070	−	0.958804i	$-0.591685\pi$
$929$	−17.3166	−0.568140	−0.284070	+	0.958804i	$0.591685\pi$
$930$	0	0
$931$	−3.36675	−0.110341
$932$	0	0
$933$	−29.2665	−0.958142
$934$	0	0
$935$	−14.3166	−0.468204
$936$	0	0
$937$	40.0000	1.30674	0.653372	−	0.757037i	$-0.273354\pi$
$937$	40.0000	1.30674	0.653372	+	0.757037i	$0.273354\pi$
$938$	0	0
$939$	−20.3668	−0.664644
$940$	0	0
$941$	4.41688	0.143986	0.0719930	−	0.997405i	$-0.477064\pi$
$941$	4.41688	0.143986	0.0719930	+	0.997405i	$0.477064\pi$
$942$	0	0
$943$	3.31662	0.108004
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	0.633250	0.0205778	0.0102889	−	0.999947i	$-0.496725\pi$
$947$	0.633250	0.0205778	0.0102889	+	0.999947i	$0.496725\pi$
$948$	0	0
$949$	−3.89975	−0.126591
$950$	0	0
$951$	9.68338	0.314005
$952$	0	0
$953$	0.532998	0.0172655	0.00863275	−	0.999963i	$-0.497252\pi$
$953$	0.532998	0.0172655	0.00863275	+	0.999963i	$0.497252\pi$
$954$	0	0
$955$	−19.5831	−0.633695
$956$	0	0
$957$	−2.94987	−0.0953559
$958$	0	0
$959$	39.7995	1.28519
$960$	0	0
$961$	61.7995	1.99353
$962$	0	0
$963$	−13.9499	−0.449528
$964$	0	0
$965$	−12.6332	−0.406679
$966$	0	0
$967$	−0.949874	−0.0305459	−0.0152730	−	0.999883i	$-0.504862\pi$
$967$	−0.949874	−0.0305459	−0.0152730	+	0.999883i	$0.504862\pi$
$968$	0	0
$969$	−5.58312	−0.179356
$970$	0	0
$971$	8.63325	0.277054	0.138527	−	0.990359i	$-0.455763\pi$
$971$	8.63325	0.277054	0.138527	+	0.990359i	$0.455763\pi$
$972$	0	0
$973$	4.89975	0.157079
$974$	0	0
$975$	2.31662	0.0741914
$976$	0	0
$977$	13.8496	0.443089	0.221544	−	0.975150i	$-0.428890\pi$
$977$	13.8496	0.443089	0.221544	+	0.975150i	$0.428890\pi$
$978$	0	0
$979$	−37.2665	−1.19104
$980$	0	0
$981$	−10.9499	−0.349603
$982$	0	0
$983$	−18.4829	−0.589512	−0.294756	−	0.955573i	$-0.595238\pi$
$983$	−18.4829	−0.589512	−0.294756	+	0.955573i	$0.595238\pi$
$984$	0	0
$985$	27.8997	0.888960
$986$	0	0
$987$	32.8496	1.04562
$988$	0	0
$989$	2.63325	0.0837325
$990$	0	0
$991$	52.1662	1.65712	0.828558	−	0.559904i	$-0.189162\pi$
$991$	52.1662	1.65712	0.828558	+	0.559904i	$0.189162\pi$
$992$	0	0
$993$	31.5330	1.00067
$994$	0	0
$995$	−1.36675	−0.0433289
$996$	0	0
$997$	24.6332	0.780143	0.390071	−	0.920785i	$-0.372450\pi$
$997$	24.6332	0.780143	0.390071	+	0.920785i	$0.372450\pi$
$998$	0	0
$999$	−11.6332	−0.368060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.a.l.1.1	✓	2
3.2	odd	2		8280.2.a.bc.1.2		2
4.3	odd	2		5520.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.l.1.1	✓	2	1.1	even	1	trivial
5520.2.a.bo.1.2		2	4.3	odd	2
8280.2.a.bc.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} -4.31662 q^{11} -2.31662 q^{13} +1.00000 q^{15} -3.31662 q^{17} -1.68338 q^{19} +3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.683375 q^{29} -9.63325 q^{31} +4.31662 q^{33} +3.00000 q^{35} +11.6332 q^{37} +2.31662 q^{39} +3.31662 q^{41} +2.63325 q^{43} -1.00000 q^{45} +10.9499 q^{47} +2.00000 q^{49} +3.31662 q^{51} -9.94987 q^{53} +4.31662 q^{55} +1.68338 q^{57} -11.9499 q^{59} +3.68338 q^{61} -3.00000 q^{63} +2.31662 q^{65} +2.36675 q^{67} -1.00000 q^{69} -7.94987 q^{71} +1.68338 q^{73} -1.00000 q^{75} +12.9499 q^{77} +1.00000 q^{81} +5.31662 q^{83} +3.31662 q^{85} +0.683375 q^{87} +8.63325 q^{89} +6.94987 q^{91} +9.63325 q^{93} +1.68338 q^{95} +12.0000 q^{97} -4.31662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} - 10 q^{19} + 6 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} - 2 q^{39} - 8 q^{43} - 2 q^{45} + 2 q^{47} + 4 q^{49} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 14 q^{61} - 6 q^{63} - 2 q^{65} + 18 q^{67} - 2 q^{69} + 4 q^{71} + 10 q^{73} - 2 q^{75} + 6 q^{77} + 2 q^{81} + 4 q^{83} + 8 q^{87} + 4 q^{89} - 6 q^{91} + 6 q^{93} + 10 q^{95} + 24 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 - 10 * q^19 + 6 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 - 6 * q^31 + 2 * q^33 + 6 * q^35 + 10 * q^37 - 2 * q^39 - 8 * q^43 - 2 * q^45 + 2 * q^47 + 4 * q^49 + 2 * q^55 + 10 * q^57 - 4 * q^59 + 14 * q^61 - 6 * q^63 - 2 * q^65 + 18 * q^67 - 2 * q^69 + 4 * q^71 + 10 * q^73 - 2 * q^75 + 6 * q^77 + 2 * q^81 + 4 * q^83 + 8 * q^87 + 4 * q^89 - 6 * q^91 + 6 * q^93 + 10 * q^95 + 24 * q^97 - 2 * q^99

Introduction

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