LMFDB - Embedded newform 4560.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -0.732051 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -0.732051 q^{7} +1.00000 q^{9} -1.26795 q^{11} -2.73205 q^{13} -1.00000 q^{15} -1.00000 q^{19} +0.732051 q^{21} +3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.19615 q^{29} +4.92820 q^{31} +1.26795 q^{33} -0.732051 q^{35} +4.19615 q^{37} +2.73205 q^{39} +4.73205 q^{41} +6.19615 q^{43} +1.00000 q^{45} -3.46410 q^{47} -6.46410 q^{49} -2.53590 q^{53} -1.26795 q^{55} +1.00000 q^{57} -9.46410 q^{59} -13.4641 q^{61} -0.732051 q^{63} -2.73205 q^{65} -8.00000 q^{67} -3.46410 q^{69} -16.3923 q^{71} -3.07180 q^{73} -1.00000 q^{75} +0.928203 q^{77} -2.92820 q^{79} +1.00000 q^{81} -0.928203 q^{83} +2.19615 q^{87} +7.26795 q^{89} +2.00000 q^{91} -4.92820 q^{93} -1.00000 q^{95} +4.19615 q^{97} -1.26795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 20 q^{61} + 2 q^{63} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 20 q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 6 q^{87} + 18 q^{89} + 4 q^{91} + 4 q^{93} - 2 q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 4 * q^31 + 6 * q^33 + 2 * q^35 - 2 * q^37 + 2 * q^39 + 6 * q^41 + 2 * q^43 + 2 * q^45 - 6 * q^49 - 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 20 * q^61 + 2 * q^63 - 2 * q^65 - 16 * q^67 - 12 * q^71 - 20 * q^73 - 2 * q^75 - 12 * q^77 + 8 * q^79 + 2 * q^81 + 12 * q^83 - 6 * q^87 + 18 * q^89 + 4 * q^91 + 4 * q^93 - 2 * q^95 - 2 * q^97 - 6 * 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
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	−2.73205	−0.757735	−0.378867	−	0.925451i	$-0.623686\pi$
$13$	−2.73205	−0.757735	−0.378867	+	0.925451i	$0.623686\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.732051	0.159747
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.19615	−0.407815	−0.203908	−	0.978990i	$-0.565364\pi$
$29$	−2.19615	−0.407815	−0.203908	+	0.978990i	$0.565364\pi$
$30$	0	0
$31$	4.92820	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$31$	4.92820	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$32$	0	0
$33$	1.26795	0.220722
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	4.19615	0.689843	0.344922	−	0.938631i	$-0.387905\pi$
$37$	4.19615	0.689843	0.344922	+	0.938631i	$0.387905\pi$
$38$	0	0
$39$	2.73205	0.437478
$40$	0	0
$41$	4.73205	0.739022	0.369511	−	0.929226i	$-0.379525\pi$
$41$	4.73205	0.739022	0.369511	+	0.929226i	$0.379525\pi$
$42$	0	0
$43$	6.19615	0.944904	0.472452	−	0.881356i	$-0.343369\pi$
$43$	6.19615	0.944904	0.472452	+	0.881356i	$0.343369\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.53590	−0.348332	−0.174166	−	0.984716i	$-0.555723\pi$
$53$	−2.53590	−0.348332	−0.174166	+	0.984716i	$0.555723\pi$
$54$	0	0
$55$	−1.26795	−0.170970
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	0	0
$61$	−13.4641	−1.72390	−0.861951	−	0.506992i	$-0.830757\pi$
$61$	−13.4641	−1.72390	−0.861951	+	0.506992i	$0.830757\pi$
$62$	0	0
$63$	−0.732051	−0.0922297
$64$	0	0
$65$	−2.73205	−0.338869
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−3.46410	−0.417029
$70$	0	0
$71$	−16.3923	−1.94541	−0.972704	−	0.232048i	$-0.925457\pi$
$71$	−16.3923	−1.94541	−0.972704	+	0.232048i	$0.925457\pi$
$72$	0	0
$73$	−3.07180	−0.359527	−0.179763	−	0.983710i	$-0.557533\pi$
$73$	−3.07180	−0.359527	−0.179763	+	0.983710i	$0.557533\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0.928203	0.105779
$78$	0	0
$79$	−2.92820	−0.329449	−0.164724	−	0.986340i	$-0.552673\pi$
$79$	−2.92820	−0.329449	−0.164724	+	0.986340i	$0.552673\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.928203	−0.101884	−0.0509418	−	0.998702i	$-0.516222\pi$
$83$	−0.928203	−0.101884	−0.0509418	+	0.998702i	$0.516222\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.19615	0.235452
$88$	0	0
$89$	7.26795	0.770401	0.385201	−	0.922833i	$-0.374132\pi$
$89$	7.26795	0.770401	0.385201	+	0.922833i	$0.374132\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−4.92820	−0.511031
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	4.19615	0.426055	0.213027	−	0.977046i	$-0.431668\pi$
$97$	4.19615	0.426055	0.213027	+	0.977046i	$0.431668\pi$
$98$	0	0
$99$	−1.26795	−0.127434
$100$	0	0
$101$	10.3923	1.03407	0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923	1.03407	0.517036	+	0.855963i	$0.327035\pi$
$102$	0	0
$103$	17.8564	1.75944	0.879722	−	0.475488i	$-0.157729\pi$
$103$	17.8564	1.75944	0.879722	+	0.475488i	$0.157729\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	6.39230	0.612272	0.306136	−	0.951988i	$-0.400964\pi$
$109$	6.39230	0.612272	0.306136	+	0.951988i	$0.400964\pi$
$110$	0	0
$111$	−4.19615	−0.398281
$112$	0	0
$113$	5.07180	0.477115	0.238557	−	0.971128i	$-0.423326\pi$
$113$	5.07180	0.477115	0.238557	+	0.971128i	$0.423326\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	0	0
$117$	−2.73205	−0.252578
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	−4.73205	−0.426675
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	−6.19615	−0.545541
$130$	0	0
$131$	−15.1244	−1.32142	−0.660711	−	0.750641i	$-0.729745\pi$
$131$	−15.1244	−1.32142	−0.660711	+	0.750641i	$0.729745\pi$
$132$	0	0
$133$	0.732051	0.0634769
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	0	0
$139$	−12.3923	−1.05110	−0.525551	−	0.850762i	$-0.676141\pi$
$139$	−12.3923	−1.05110	−0.525551	+	0.850762i	$0.676141\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	0	0
$143$	3.46410	0.289683
$144$	0	0
$145$	−2.19615	−0.182381
$146$	0	0
$147$	6.46410	0.533150
$148$	0	0
$149$	7.85641	0.643622	0.321811	−	0.946804i	$-0.395708\pi$
$149$	7.85641	0.643622	0.321811	+	0.946804i	$0.395708\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.92820	0.395843
$156$	0	0
$157$	−14.3923	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$157$	−14.3923	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$158$	0	0
$159$	2.53590	0.201110
$160$	0	0
$161$	−2.53590	−0.199857
$162$	0	0
$163$	−12.7321	−0.997251	−0.498626	−	0.866817i	$-0.666162\pi$
$163$	−12.7321	−0.997251	−0.498626	+	0.866817i	$0.666162\pi$
$164$	0	0
$165$	1.26795	0.0987097
$166$	0	0
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−6.92820	−0.526742	−0.263371	−	0.964695i	$-0.584834\pi$
$173$	−6.92820	−0.526742	−0.263371	+	0.964695i	$0.584834\pi$
$174$	0	0
$175$	−0.732051	−0.0553378
$176$	0	0
$177$	9.46410	0.711365
$178$	0	0
$179$	11.3205	0.846135	0.423067	−	0.906098i	$-0.360953\pi$
$179$	11.3205	0.846135	0.423067	+	0.906098i	$0.360953\pi$
$180$	0	0
$181$	18.3923	1.36709	0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923	1.36709	0.683545	+	0.729909i	$0.260437\pi$
$182$	0	0
$183$	13.4641	0.995295
$184$	0	0
$185$	4.19615	0.308507
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0.732051	0.0532489
$190$	0	0
$191$	−17.6603	−1.27785	−0.638926	−	0.769269i	$-0.720621\pi$
$191$	−17.6603	−1.27785	−0.638926	+	0.769269i	$0.720621\pi$
$192$	0	0
$193$	−7.12436	−0.512822	−0.256411	−	0.966568i	$-0.582540\pi$
$193$	−7.12436	−0.512822	−0.256411	+	0.966568i	$0.582540\pi$
$194$	0	0
$195$	2.73205	0.195646
$196$	0	0
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	−19.3205	−1.36959	−0.684797	−	0.728734i	$-0.740109\pi$
$199$	−19.3205	−1.36959	−0.684797	+	0.728734i	$0.740109\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	1.60770	0.112838
$204$	0	0
$205$	4.73205	0.330501
$206$	0	0
$207$	3.46410	0.240772
$208$	0	0
$209$	1.26795	0.0877059
$210$	0	0
$211$	−14.9282	−1.02770	−0.513850	−	0.857880i	$-0.671781\pi$
$211$	−14.9282	−1.02770	−0.513850	+	0.857880i	$0.671781\pi$
$212$	0	0
$213$	16.3923	1.12318
$214$	0	0
$215$	6.19615	0.422574
$216$	0	0
$217$	−3.60770	−0.244906
$218$	0	0
$219$	3.07180	0.207573
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.85641	−0.660034	−0.330017	−	0.943975i	$-0.607054\pi$
$223$	−9.85641	−0.660034	−0.330017	+	0.943975i	$0.607054\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	0	0
$229$	−25.4641	−1.68272	−0.841358	−	0.540479i	$-0.818243\pi$
$229$	−25.4641	−1.68272	−0.841358	+	0.540479i	$0.818243\pi$
$230$	0	0
$231$	−0.928203	−0.0610713
$232$	0	0
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	−3.46410	−0.225973
$236$	0	0
$237$	2.92820	0.190207
$238$	0	0
$239$	20.1962	1.30638	0.653190	−	0.757194i	$-0.273430\pi$
$239$	20.1962	1.30638	0.653190	+	0.757194i	$0.273430\pi$
$240$	0	0
$241$	−16.9282	−1.09044	−0.545221	−	0.838293i	$-0.683554\pi$
$241$	−16.9282	−1.09044	−0.545221	+	0.838293i	$0.683554\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.46410	−0.412976
$246$	0	0
$247$	2.73205	0.173836
$248$	0	0
$249$	0.928203	0.0588225
$250$	0	0
$251$	10.0526	0.634512	0.317256	−	0.948340i	$-0.397239\pi$
$251$	10.0526	0.634512	0.317256	+	0.948340i	$0.397239\pi$
$252$	0	0
$253$	−4.39230	−0.276142
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	−3.07180	−0.190872
$260$	0	0
$261$	−2.19615	−0.135938
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−2.53590	−0.155779
$266$	0	0
$267$	−7.26795	−0.444791
$268$	0	0
$269$	−30.5885	−1.86501	−0.932506	−	0.361156i	$-0.882382\pi$
$269$	−30.5885	−1.86501	−0.932506	+	0.361156i	$0.882382\pi$
$270$	0	0
$271$	20.3923	1.23874	0.619372	−	0.785098i	$-0.287387\pi$
$271$	20.3923	1.23874	0.619372	+	0.785098i	$0.287387\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−1.26795	−0.0764602
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	4.92820	0.295044
$280$	0	0
$281$	4.73205	0.282290	0.141145	−	0.989989i	$-0.454922\pi$
$281$	4.73205	0.282290	0.141145	+	0.989989i	$0.454922\pi$
$282$	0	0
$283$	26.9808	1.60384	0.801920	−	0.597432i	$-0.203812\pi$
$283$	26.9808	1.60384	0.801920	+	0.597432i	$0.203812\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−4.19615	−0.245983
$292$	0	0
$293$	−27.7128	−1.61900	−0.809500	−	0.587120i	$-0.800262\pi$
$293$	−27.7128	−1.61900	−0.809500	+	0.587120i	$0.800262\pi$
$294$	0	0
$295$	−9.46410	−0.551021
$296$	0	0
$297$	1.26795	0.0735739
$298$	0	0
$299$	−9.46410	−0.547323
$300$	0	0
$301$	−4.53590	−0.261445
$302$	0	0
$303$	−10.3923	−0.597022
$304$	0	0
$305$	−13.4641	−0.770952
$306$	0	0
$307$	11.6077	0.662486	0.331243	−	0.943545i	$-0.392532\pi$
$307$	11.6077	0.662486	0.331243	+	0.943545i	$0.392532\pi$
$308$	0	0
$309$	−17.8564	−1.01582
$310$	0	0
$311$	26.4449	1.49955	0.749775	−	0.661692i	$-0.230162\pi$
$311$	26.4449	1.49955	0.749775	+	0.661692i	$0.230162\pi$
$312$	0	0
$313$	−14.3923	−0.813501	−0.406751	−	0.913539i	$-0.633338\pi$
$313$	−14.3923	−0.813501	−0.406751	+	0.913539i	$0.633338\pi$
$314$	0	0
$315$	−0.732051	−0.0412464
$316$	0	0
$317$	23.3205	1.30981	0.654905	−	0.755711i	$-0.272709\pi$
$317$	23.3205	1.30981	0.654905	+	0.755711i	$0.272709\pi$
$318$	0	0
$319$	2.78461	0.155908
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.73205	−0.151547
$326$	0	0
$327$	−6.39230	−0.353495
$328$	0	0
$329$	2.53590	0.139809
$330$	0	0
$331$	−29.7128	−1.63316	−0.816582	−	0.577230i	$-0.804134\pi$
$331$	−29.7128	−1.63316	−0.816582	+	0.577230i	$0.804134\pi$
$332$	0	0
$333$	4.19615	0.229948
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−19.1244	−1.04177	−0.520885	−	0.853627i	$-0.674398\pi$
$337$	−19.1244	−1.04177	−0.520885	+	0.853627i	$0.674398\pi$
$338$	0	0
$339$	−5.07180	−0.275462
$340$	0	0
$341$	−6.24871	−0.338387
$342$	0	0
$343$	9.85641	0.532196
$344$	0	0
$345$	−3.46410	−0.186501
$346$	0	0
$347$	−12.9282	−0.694022	−0.347011	−	0.937861i	$-0.612803\pi$
$347$	−12.9282	−0.694022	−0.347011	+	0.937861i	$0.612803\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	2.73205	0.145826
$352$	0	0
$353$	26.7846	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$353$	26.7846	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$354$	0	0
$355$	−16.3923	−0.870013
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.6603	−0.932073	−0.466036	−	0.884766i	$-0.654318\pi$
$359$	−17.6603	−0.932073	−0.466036	+	0.884766i	$0.654318\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.39230	0.492968
$364$	0	0
$365$	−3.07180	−0.160785
$366$	0	0
$367$	−5.80385	−0.302958	−0.151479	−	0.988460i	$-0.548404\pi$
$367$	−5.80385	−0.302958	−0.151479	+	0.988460i	$0.548404\pi$
$368$	0	0
$369$	4.73205	0.246341
$370$	0	0
$371$	1.85641	0.0963798
$372$	0	0
$373$	4.19615	0.217269	0.108634	−	0.994082i	$-0.465352\pi$
$373$	4.19615	0.217269	0.108634	+	0.994082i	$0.465352\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−7.07180	−0.363254	−0.181627	−	0.983368i	$-0.558136\pi$
$379$	−7.07180	−0.363254	−0.181627	+	0.983368i	$0.558136\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	30.9282	1.58036	0.790179	−	0.612877i	$-0.209988\pi$
$383$	30.9282	1.58036	0.790179	+	0.612877i	$0.209988\pi$
$384$	0	0
$385$	0.928203	0.0473056
$386$	0	0
$387$	6.19615	0.314968
$388$	0	0
$389$	−19.8564	−1.00676	−0.503380	−	0.864065i	$-0.667910\pi$
$389$	−19.8564	−1.00676	−0.503380	+	0.864065i	$0.667910\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	15.1244	0.762923
$394$	0	0
$395$	−2.92820	−0.147334
$396$	0	0
$397$	−4.92820	−0.247339	−0.123670	−	0.992323i	$-0.539466\pi$
$397$	−4.92820	−0.247339	−0.123670	+	0.992323i	$0.539466\pi$
$398$	0	0
$399$	−0.732051	−0.0366484
$400$	0	0
$401$	4.05256	0.202375	0.101188	−	0.994867i	$-0.467736\pi$
$401$	4.05256	0.202375	0.101188	+	0.994867i	$0.467736\pi$
$402$	0	0
$403$	−13.4641	−0.670695
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−5.32051	−0.263728
$408$	0	0
$409$	−5.60770	−0.277283	−0.138641	−	0.990343i	$-0.544274\pi$
$409$	−5.60770	−0.277283	−0.138641	+	0.990343i	$0.544274\pi$
$410$	0	0
$411$	7.85641	0.387528
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	−0.928203	−0.0455637
$416$	0	0
$417$	12.3923	0.606854
$418$	0	0
$419$	−10.0526	−0.491100	−0.245550	−	0.969384i	$-0.578968\pi$
$419$	−10.0526	−0.491100	−0.245550	+	0.969384i	$0.578968\pi$
$420$	0	0
$421$	22.7846	1.11045	0.555227	−	0.831699i	$-0.312631\pi$
$421$	22.7846	1.11045	0.555227	+	0.831699i	$0.312631\pi$
$422$	0	0
$423$	−3.46410	−0.168430
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.85641	0.476985
$428$	0	0
$429$	−3.46410	−0.167248
$430$	0	0
$431$	−23.3205	−1.12331	−0.561655	−	0.827372i	$-0.689835\pi$
$431$	−23.3205	−1.12331	−0.561655	+	0.827372i	$0.689835\pi$
$432$	0	0
$433$	20.5885	0.989418	0.494709	−	0.869059i	$-0.335275\pi$
$433$	20.5885	0.989418	0.494709	+	0.869059i	$0.335275\pi$
$434$	0	0
$435$	2.19615	0.105297
$436$	0	0
$437$	−3.46410	−0.165710
$438$	0	0
$439$	−13.0718	−0.623883	−0.311941	−	0.950101i	$-0.600979\pi$
$439$	−13.0718	−0.623883	−0.311941	+	0.950101i	$0.600979\pi$
$440$	0	0
$441$	−6.46410	−0.307814
$442$	0	0
$443$	−29.3205	−1.39306	−0.696530	−	0.717528i	$-0.745274\pi$
$443$	−29.3205	−1.39306	−0.696530	+	0.717528i	$0.745274\pi$
$444$	0	0
$445$	7.26795	0.344534
$446$	0	0
$447$	−7.85641	−0.371595
$448$	0	0
$449$	11.6603	0.550281	0.275141	−	0.961404i	$-0.411276\pi$
$449$	11.6603	0.550281	0.275141	+	0.961404i	$0.411276\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	14.0000	0.657777
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	11.4641	0.536268	0.268134	−	0.963382i	$-0.413593\pi$
$457$	11.4641	0.536268	0.268134	+	0.963382i	$0.413593\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−9.51666	−0.442277	−0.221138	−	0.975242i	$-0.570977\pi$
$463$	−9.51666	−0.442277	−0.221138	+	0.975242i	$0.570977\pi$
$464$	0	0
$465$	−4.92820	−0.228540
$466$	0	0
$467$	−27.4641	−1.27089	−0.635444	−	0.772147i	$-0.719183\pi$
$467$	−27.4641	−1.27089	−0.635444	+	0.772147i	$0.719183\pi$
$468$	0	0
$469$	5.85641	0.270424
$470$	0	0
$471$	14.3923	0.663162
$472$	0	0
$473$	−7.85641	−0.361238
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−2.53590	−0.116111
$478$	0	0
$479$	19.5167	0.891739	0.445869	−	0.895098i	$-0.352895\pi$
$479$	19.5167	0.891739	0.445869	+	0.895098i	$0.352895\pi$
$480$	0	0
$481$	−11.4641	−0.522718
$482$	0	0
$483$	2.53590	0.115387
$484$	0	0
$485$	4.19615	0.190537
$486$	0	0
$487$	11.6077	0.525995	0.262997	−	0.964797i	$-0.415289\pi$
$487$	11.6077	0.525995	0.262997	+	0.964797i	$0.415289\pi$
$488$	0	0
$489$	12.7321	0.575763
$490$	0	0
$491$	22.0526	0.995218	0.497609	−	0.867401i	$-0.334211\pi$
$491$	22.0526	0.995218	0.497609	+	0.867401i	$0.334211\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−1.26795	−0.0569901
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−17.4641	−0.781801	−0.390900	−	0.920433i	$-0.627836\pi$
$499$	−17.4641	−0.781801	−0.390900	+	0.920433i	$0.627836\pi$
$500$	0	0
$501$	−3.46410	−0.154765
$502$	0	0
$503$	36.9282	1.64655	0.823274	−	0.567645i	$-0.192145\pi$
$503$	36.9282	1.64655	0.823274	+	0.567645i	$0.192145\pi$
$504$	0	0
$505$	10.3923	0.462451
$506$	0	0
$507$	5.53590	0.245858
$508$	0	0
$509$	28.0526	1.24341	0.621704	−	0.783252i	$-0.286441\pi$
$509$	28.0526	1.24341	0.621704	+	0.783252i	$0.286441\pi$
$510$	0	0
$511$	2.24871	0.0994771
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	17.8564	0.786847
$516$	0	0
$517$	4.39230	0.193173
$518$	0	0
$519$	6.92820	0.304114
$520$	0	0
$521$	40.7321	1.78450	0.892252	−	0.451538i	$-0.149125\pi$
$521$	40.7321	1.78450	0.892252	+	0.451538i	$0.149125\pi$
$522$	0	0
$523$	−43.3205	−1.89427	−0.947137	−	0.320830i	$-0.896038\pi$
$523$	−43.3205	−1.89427	−0.947137	+	0.320830i	$0.896038\pi$
$524$	0	0
$525$	0.732051	0.0319493
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	−9.46410	−0.410707
$532$	0	0
$533$	−12.9282	−0.559983
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.3205	−0.488516
$538$	0	0
$539$	8.19615	0.353033
$540$	0	0
$541$	−13.7128	−0.589560	−0.294780	−	0.955565i	$-0.595246\pi$
$541$	−13.7128	−0.589560	−0.294780	+	0.955565i	$0.595246\pi$
$542$	0	0
$543$	−18.3923	−0.789289
$544$	0	0
$545$	6.39230	0.273816
$546$	0	0
$547$	−8.67949	−0.371108	−0.185554	−	0.982634i	$-0.559408\pi$
$547$	−8.67949	−0.371108	−0.185554	+	0.982634i	$0.559408\pi$
$548$	0	0
$549$	−13.4641	−0.574634
$550$	0	0
$551$	2.19615	0.0935592
$552$	0	0
$553$	2.14359	0.0911549
$554$	0	0
$555$	−4.19615	−0.178117
$556$	0	0
$557$	−12.9282	−0.547786	−0.273893	−	0.961760i	$-0.588311\pi$
$557$	−12.9282	−0.547786	−0.273893	+	0.961760i	$0.588311\pi$
$558$	0	0
$559$	−16.9282	−0.715987
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.5359	0.865485	0.432742	−	0.901518i	$-0.357546\pi$
$563$	20.5359	0.865485	0.432742	+	0.901518i	$0.357546\pi$
$564$	0	0
$565$	5.07180	0.213372
$566$	0	0
$567$	−0.732051	−0.0307432
$568$	0	0
$569$	−16.0526	−0.672958	−0.336479	−	0.941691i	$-0.609236\pi$
$569$	−16.0526	−0.672958	−0.336479	+	0.941691i	$0.609236\pi$
$570$	0	0
$571$	−14.2487	−0.596290	−0.298145	−	0.954521i	$-0.596368\pi$
$571$	−14.2487	−0.596290	−0.298145	+	0.954521i	$0.596368\pi$
$572$	0	0
$573$	17.6603	0.737768
$574$	0	0
$575$	3.46410	0.144463
$576$	0	0
$577$	−47.1769	−1.96400	−0.982000	−	0.188879i	$-0.939515\pi$
$577$	−47.1769	−1.96400	−0.982000	+	0.188879i	$0.939515\pi$
$578$	0	0
$579$	7.12436	0.296078
$580$	0	0
$581$	0.679492	0.0281901
$582$	0	0
$583$	3.21539	0.133168
$584$	0	0
$585$	−2.73205	−0.112956
$586$	0	0
$587$	−3.46410	−0.142979	−0.0714894	−	0.997441i	$-0.522775\pi$
$587$	−3.46410	−0.142979	−0.0714894	+	0.997441i	$0.522775\pi$
$588$	0	0
$589$	−4.92820	−0.203063
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	2.78461	0.114350	0.0571751	−	0.998364i	$-0.481791\pi$
$593$	2.78461	0.114350	0.0571751	+	0.998364i	$0.481791\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.3205	0.790736
$598$	0	0
$599$	−13.8564	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$599$	−13.8564	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$600$	0	0
$601$	15.1769	0.619079	0.309540	−	0.950887i	$-0.399825\pi$
$601$	15.1769	0.619079	0.309540	+	0.950887i	$0.399825\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	−9.39230	−0.381851
$606$	0	0
$607$	32.3923	1.31476	0.657382	−	0.753558i	$-0.271664\pi$
$607$	32.3923	1.31476	0.657382	+	0.753558i	$0.271664\pi$
$608$	0	0
$609$	−1.60770	−0.0651471
$610$	0	0
$611$	9.46410	0.382877
$612$	0	0
$613$	21.6077	0.872727	0.436363	−	0.899771i	$-0.356266\pi$
$613$	21.6077	0.872727	0.436363	+	0.899771i	$0.356266\pi$
$614$	0	0
$615$	−4.73205	−0.190815
$616$	0	0
$617$	−27.7128	−1.11568	−0.557838	−	0.829950i	$-0.688369\pi$
$617$	−27.7128	−1.11568	−0.557838	+	0.829950i	$0.688369\pi$
$618$	0	0
$619$	−19.3205	−0.776557	−0.388278	−	0.921542i	$-0.626930\pi$
$619$	−19.3205	−0.776557	−0.388278	+	0.921542i	$0.626930\pi$
$620$	0	0
$621$	−3.46410	−0.139010
$622$	0	0
$623$	−5.32051	−0.213162
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.26795	−0.0506370
$628$	0	0
$629$	0	0
$630$	0	0
$631$	21.0718	0.838855	0.419427	−	0.907789i	$-0.362231\pi$
$631$	21.0718	0.838855	0.419427	+	0.907789i	$0.362231\pi$
$632$	0	0
$633$	14.9282	0.593343
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	17.6603	0.699725
$638$	0	0
$639$	−16.3923	−0.648470
$640$	0	0
$641$	17.4115	0.687715	0.343857	−	0.939022i	$-0.388266\pi$
$641$	17.4115	0.687715	0.343857	+	0.939022i	$0.388266\pi$
$642$	0	0
$643$	1.80385	0.0711368	0.0355684	−	0.999367i	$-0.488676\pi$
$643$	1.80385	0.0711368	0.0355684	+	0.999367i	$0.488676\pi$
$644$	0	0
$645$	−6.19615	−0.243973
$646$	0	0
$647$	31.8564	1.25240	0.626202	−	0.779661i	$-0.284608\pi$
$647$	31.8564	1.25240	0.626202	+	0.779661i	$0.284608\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	3.60770	0.141397
$652$	0	0
$653$	30.9282	1.21031	0.605157	−	0.796106i	$-0.293110\pi$
$653$	30.9282	1.21031	0.605157	+	0.796106i	$0.293110\pi$
$654$	0	0
$655$	−15.1244	−0.590957
$656$	0	0
$657$	−3.07180	−0.119842
$658$	0	0
$659$	−18.9282	−0.737338	−0.368669	−	0.929561i	$-0.620186\pi$
$659$	−18.9282	−0.737338	−0.368669	+	0.929561i	$0.620186\pi$
$660$	0	0
$661$	−23.1769	−0.901477	−0.450739	−	0.892656i	$-0.648839\pi$
$661$	−23.1769	−0.901477	−0.450739	+	0.892656i	$0.648839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.732051	0.0283877
$666$	0	0
$667$	−7.60770	−0.294571
$668$	0	0
$669$	9.85641	0.381071
$670$	0	0
$671$	17.0718	0.659049
$672$	0	0
$673$	−7.12436	−0.274624	−0.137312	−	0.990528i	$-0.543846\pi$
$673$	−7.12436	−0.274624	−0.137312	+	0.990528i	$0.543846\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	35.3205	1.35748	0.678739	−	0.734380i	$-0.262527\pi$
$677$	35.3205	1.35748	0.678739	+	0.734380i	$0.262527\pi$
$678$	0	0
$679$	−3.07180	−0.117885
$680$	0	0
$681$	−10.3923	−0.398234
$682$	0	0
$683$	18.9282	0.724268	0.362134	−	0.932126i	$-0.382048\pi$
$683$	18.9282	0.724268	0.362134	+	0.932126i	$0.382048\pi$
$684$	0	0
$685$	−7.85641	−0.300178
$686$	0	0
$687$	25.4641	0.971516
$688$	0	0
$689$	6.92820	0.263944
$690$	0	0
$691$	8.39230	0.319258	0.159629	−	0.987177i	$-0.448970\pi$
$691$	8.39230	0.319258	0.159629	+	0.987177i	$0.448970\pi$
$692$	0	0
$693$	0.928203	0.0352595
$694$	0	0
$695$	−12.3923	−0.470067
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−19.8564	−0.751038
$700$	0	0
$701$	21.7128	0.820082	0.410041	−	0.912067i	$-0.365514\pi$
$701$	21.7128	0.820082	0.410041	+	0.912067i	$0.365514\pi$
$702$	0	0
$703$	−4.19615	−0.158261
$704$	0	0
$705$	3.46410	0.130466
$706$	0	0
$707$	−7.60770	−0.286117
$708$	0	0
$709$	33.1769	1.24599	0.622993	−	0.782228i	$-0.285917\pi$
$709$	33.1769	1.24599	0.622993	+	0.782228i	$0.285917\pi$
$710$	0	0
$711$	−2.92820	−0.109816
$712$	0	0
$713$	17.0718	0.639344
$714$	0	0
$715$	3.46410	0.129550
$716$	0	0
$717$	−20.1962	−0.754239
$718$	0	0
$719$	−5.66025	−0.211092	−0.105546	−	0.994414i	$-0.533659\pi$
$719$	−5.66025	−0.211092	−0.105546	+	0.994414i	$0.533659\pi$
$720$	0	0
$721$	−13.0718	−0.486819
$722$	0	0
$723$	16.9282	0.629567
$724$	0	0
$725$	−2.19615	−0.0815631
$726$	0	0
$727$	−8.33975	−0.309304	−0.154652	−	0.987969i	$-0.549426\pi$
$727$	−8.33975	−0.309304	−0.154652	+	0.987969i	$0.549426\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	22.7846	0.841569	0.420784	−	0.907161i	$-0.361755\pi$
$733$	22.7846	0.841569	0.420784	+	0.907161i	$0.361755\pi$
$734$	0	0
$735$	6.46410	0.238432
$736$	0	0
$737$	10.1436	0.373644
$738$	0	0
$739$	−33.8564	−1.24543	−0.622714	−	0.782450i	$-0.713970\pi$
$739$	−33.8564	−1.24543	−0.622714	+	0.782450i	$0.713970\pi$
$740$	0	0
$741$	−2.73205	−0.100364
$742$	0	0
$743$	44.7846	1.64299	0.821494	−	0.570217i	$-0.193141\pi$
$743$	44.7846	1.64299	0.821494	+	0.570217i	$0.193141\pi$
$744$	0	0
$745$	7.85641	0.287836
$746$	0	0
$747$	−0.928203	−0.0339612
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	−10.0526	−0.366336
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	−16.2487	−0.590569	−0.295285	−	0.955409i	$-0.595415\pi$
$757$	−16.2487	−0.590569	−0.295285	+	0.955409i	$0.595415\pi$
$758$	0	0
$759$	4.39230	0.159431
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−4.67949	−0.169409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.8564	0.933621
$768$	0	0
$769$	48.6410	1.75404	0.877020	−	0.480454i	$-0.159528\pi$
$769$	48.6410	1.75404	0.877020	+	0.480454i	$0.159528\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	37.1769	1.33716	0.668580	−	0.743640i	$-0.266902\pi$
$773$	37.1769	1.33716	0.668580	+	0.743640i	$0.266902\pi$
$774$	0	0
$775$	4.92820	0.177026
$776$	0	0
$777$	3.07180	0.110200
$778$	0	0
$779$	−4.73205	−0.169543
$780$	0	0
$781$	20.7846	0.743732
$782$	0	0
$783$	2.19615	0.0784841
$784$	0	0
$785$	−14.3923	−0.513683
$786$	0	0
$787$	−43.3205	−1.54421	−0.772105	−	0.635495i	$-0.780796\pi$
$787$	−43.3205	−1.54421	−0.772105	+	0.635495i	$0.780796\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	−3.71281	−0.132012
$792$	0	0
$793$	36.7846	1.30626
$794$	0	0
$795$	2.53590	0.0899390
$796$	0	0
$797$	3.21539	0.113895	0.0569475	−	0.998377i	$-0.481863\pi$
$797$	3.21539	0.113895	0.0569475	+	0.998377i	$0.481863\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	7.26795	0.256800
$802$	0	0
$803$	3.89488	0.137447
$804$	0	0
$805$	−2.53590	−0.0893787
$806$	0	0
$807$	30.5885	1.07676
$808$	0	0
$809$	−26.7846	−0.941697	−0.470848	−	0.882214i	$-0.656052\pi$
$809$	−26.7846	−0.941697	−0.470848	+	0.882214i	$0.656052\pi$
$810$	0	0
$811$	45.5692	1.60015	0.800076	−	0.599899i	$-0.204793\pi$
$811$	45.5692	1.60015	0.800076	+	0.599899i	$0.204793\pi$
$812$	0	0
$813$	−20.3923	−0.715189
$814$	0	0
$815$	−12.7321	−0.445984
$816$	0	0
$817$	−6.19615	−0.216776
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−39.4641	−1.37731	−0.688653	−	0.725091i	$-0.741798\pi$
$821$	−39.4641	−1.37731	−0.688653	+	0.725091i	$0.741798\pi$
$822$	0	0
$823$	38.9808	1.35878	0.679392	−	0.733776i	$-0.262244\pi$
$823$	38.9808	1.35878	0.679392	+	0.733776i	$0.262244\pi$
$824$	0	0
$825$	1.26795	0.0441443
$826$	0	0
$827$	−29.3205	−1.01957	−0.509787	−	0.860301i	$-0.670276\pi$
$827$	−29.3205	−1.01957	−0.509787	+	0.860301i	$0.670276\pi$
$828$	0	0
$829$	−42.1051	−1.46237	−0.731186	−	0.682179i	$-0.761033\pi$
$829$	−42.1051	−1.46237	−0.731186	+	0.682179i	$0.761033\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.46410	0.119880
$836$	0	0
$837$	−4.92820	−0.170344
$838$	0	0
$839$	40.3923	1.39450	0.697249	−	0.716829i	$-0.254407\pi$
$839$	40.3923	1.39450	0.697249	+	0.716829i	$0.254407\pi$
$840$	0	0
$841$	−24.1769	−0.833687
$842$	0	0
$843$	−4.73205	−0.162980
$844$	0	0
$845$	−5.53590	−0.190441
$846$	0	0
$847$	6.87564	0.236250
$848$	0	0
$849$	−26.9808	−0.925977
$850$	0	0
$851$	14.5359	0.498284
$852$	0	0
$853$	−35.1769	−1.20443	−0.602217	−	0.798332i	$-0.705716\pi$
$853$	−35.1769	−1.20443	−0.602217	+	0.798332i	$0.705716\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	6.24871	0.213452	0.106726	−	0.994288i	$-0.465963\pi$
$857$	6.24871	0.213452	0.106726	+	0.994288i	$0.465963\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	3.46410	0.118056
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	−6.92820	−0.235566
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	3.71281	0.125949
$870$	0	0
$871$	21.8564	0.740576
$872$	0	0
$873$	4.19615	0.142018
$874$	0	0
$875$	−0.732051	−0.0247478
$876$	0	0
$877$	28.8756	0.975061	0.487531	−	0.873106i	$-0.337898\pi$
$877$	28.8756	0.975061	0.487531	+	0.873106i	$0.337898\pi$
$878$	0	0
$879$	27.7128	0.934730
$880$	0	0
$881$	15.4641	0.520999	0.260499	−	0.965474i	$-0.416113\pi$
$881$	15.4641	0.520999	0.260499	+	0.965474i	$0.416113\pi$
$882$	0	0
$883$	14.9808	0.504143	0.252071	−	0.967709i	$-0.418888\pi$
$883$	14.9808	0.504143	0.252071	+	0.967709i	$0.418888\pi$
$884$	0	0
$885$	9.46410	0.318132
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−2.92820	−0.0982088
$890$	0	0
$891$	−1.26795	−0.0424779
$892$	0	0
$893$	3.46410	0.115922
$894$	0	0
$895$	11.3205	0.378403
$896$	0	0
$897$	9.46410	0.315997
$898$	0	0
$899$	−10.8231	−0.360970
$900$	0	0
$901$	0	0
$902$	0	0
$903$	4.53590	0.150945
$904$	0	0
$905$	18.3923	0.611381
$906$	0	0
$907$	11.6077	0.385427	0.192714	−	0.981255i	$-0.438271\pi$
$907$	11.6077	0.385427	0.192714	+	0.981255i	$0.438271\pi$
$908$	0	0
$909$	10.3923	0.344691
$910$	0	0
$911$	41.0718	1.36077	0.680385	−	0.732855i	$-0.261813\pi$
$911$	41.0718	1.36077	0.680385	+	0.732855i	$0.261813\pi$
$912$	0	0
$913$	1.17691	0.0389502
$914$	0	0
$915$	13.4641	0.445109
$916$	0	0
$917$	11.0718	0.365623
$918$	0	0
$919$	59.4256	1.96027	0.980135	−	0.198330i	$-0.0635518\pi$
$919$	59.4256	1.96027	0.980135	+	0.198330i	$0.0635518\pi$
$920$	0	0
$921$	−11.6077	−0.382487
$922$	0	0
$923$	44.7846	1.47410
$924$	0	0
$925$	4.19615	0.137969
$926$	0	0
$927$	17.8564	0.586481
$928$	0	0
$929$	−22.3923	−0.734668	−0.367334	−	0.930089i	$-0.619729\pi$
$929$	−22.3923	−0.734668	−0.367334	+	0.930089i	$0.619729\pi$
$930$	0	0
$931$	6.46410	0.211852
$932$	0	0
$933$	−26.4449	−0.865766
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.2487	1.05352	0.526760	−	0.850014i	$-0.323407\pi$
$937$	32.2487	1.05352	0.526760	+	0.850014i	$0.323407\pi$
$938$	0	0
$939$	14.3923	0.469675
$940$	0	0
$941$	−30.5885	−0.997155	−0.498578	−	0.866845i	$-0.666144\pi$
$941$	−30.5885	−0.997155	−0.498578	+	0.866845i	$0.666144\pi$
$942$	0	0
$943$	16.3923	0.533807
$944$	0	0
$945$	0.732051	0.0238136
$946$	0	0
$947$	−55.8564	−1.81509	−0.907545	−	0.419956i	$-0.862046\pi$
$947$	−55.8564	−1.81509	−0.907545	+	0.419956i	$0.862046\pi$
$948$	0	0
$949$	8.39230	0.272426
$950$	0	0
$951$	−23.3205	−0.756219
$952$	0	0
$953$	10.1436	0.328583	0.164292	−	0.986412i	$-0.447466\pi$
$953$	10.1436	0.328583	0.164292	+	0.986412i	$0.447466\pi$
$954$	0	0
$955$	−17.6603	−0.571472
$956$	0	0
$957$	−2.78461	−0.0900136
$958$	0	0
$959$	5.75129	0.185719
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.12436	−0.229341
$966$	0	0
$967$	−29.1244	−0.936576	−0.468288	−	0.883576i	$-0.655129\pi$
$967$	−29.1244	−0.936576	−0.468288	+	0.883576i	$0.655129\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	0	0
$973$	9.07180	0.290828
$974$	0	0
$975$	2.73205	0.0874957
$976$	0	0
$977$	51.0333	1.63270	0.816350	−	0.577557i	$-0.195994\pi$
$977$	51.0333	1.63270	0.816350	+	0.577557i	$0.195994\pi$
$978$	0	0
$979$	−9.21539	−0.294525
$980$	0	0
$981$	6.39230	0.204091
$982$	0	0
$983$	6.67949	0.213043	0.106521	−	0.994310i	$-0.466029\pi$
$983$	6.67949	0.213043	0.106521	+	0.994310i	$0.466029\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	−2.53590	−0.0807185
$988$	0	0
$989$	21.4641	0.682519
$990$	0	0
$991$	−26.9282	−0.855403	−0.427701	−	0.903920i	$-0.640676\pi$
$991$	−26.9282	−0.855403	−0.427701	+	0.903920i	$0.640676\pi$
$992$	0	0
$993$	29.7128	0.942908
$994$	0	0
$995$	−19.3205	−0.612501
$996$	0	0
$997$	−38.3923	−1.21590	−0.607948	−	0.793977i	$-0.708007\pi$
$997$	−38.3923	−1.21590	−0.607948	+	0.793977i	$0.708007\pi$
$998$	0	0
$999$	−4.19615	−0.132760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bh.1.1		2
4.3	odd	2		285.2.a.e.1.2	✓	2
12.11	even	2		855.2.a.f.1.1		2
20.3	even	4		1425.2.c.k.799.2		4
20.7	even	4		1425.2.c.k.799.3		4
20.19	odd	2		1425.2.a.o.1.1		2
60.59	even	2		4275.2.a.t.1.2		2
76.75	even	2		5415.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.e.1.2	✓	2	4.3	odd	2
855.2.a.f.1.1		2	12.11	even	2
1425.2.a.o.1.1		2	20.19	odd	2
1425.2.c.k.799.2		4	20.3	even	4
1425.2.c.k.799.3		4	20.7	even	4
4275.2.a.t.1.2		2	60.59	even	2
4560.2.a.bh.1.1		2	1.1	even	1	trivial
5415.2.a.r.1.1		2	76.75	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.732051 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.732051 q^{7} +1.00000 q^{9} -1.26795 q^{11} -2.73205 q^{13} -1.00000 q^{15} -1.00000 q^{19} +0.732051 q^{21} +3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.19615 q^{29} +4.92820 q^{31} +1.26795 q^{33} -0.732051 q^{35} +4.19615 q^{37} +2.73205 q^{39} +4.73205 q^{41} +6.19615 q^{43} +1.00000 q^{45} -3.46410 q^{47} -6.46410 q^{49} -2.53590 q^{53} -1.26795 q^{55} +1.00000 q^{57} -9.46410 q^{59} -13.4641 q^{61} -0.732051 q^{63} -2.73205 q^{65} -8.00000 q^{67} -3.46410 q^{69} -16.3923 q^{71} -3.07180 q^{73} -1.00000 q^{75} +0.928203 q^{77} -2.92820 q^{79} +1.00000 q^{81} -0.928203 q^{83} +2.19615 q^{87} +7.26795 q^{89} +2.00000 q^{91} -4.92820 q^{93} -1.00000 q^{95} +4.19615 q^{97} -1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 20 q^{61} + 2 q^{63} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 20 q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 6 q^{87} + 18 q^{89} + 4 q^{91} + 4 q^{93} - 2 q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 4 * q^31 + 6 * q^33 + 2 * q^35 - 2 * q^37 + 2 * q^39 + 6 * q^41 + 2 * q^43 + 2 * q^45 - 6 * q^49 - 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 20 * q^61 + 2 * q^63 - 2 * q^65 - 16 * q^67 - 12 * q^71 - 20 * q^73 - 2 * q^75 - 12 * q^77 + 8 * q^79 + 2 * q^81 + 12 * q^83 - 6 * q^87 + 18 * q^89 + 4 * q^91 + 4 * q^93 - 2 * q^95 - 2 * q^97 - 6 * q^99

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