LMFDB - Embedded newform 4560.2.a.be.1.2

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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} -3.23607 q^{11} +5.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -1.00000 q^{19} +0.763932 q^{21} +6.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.23607 q^{29} -8.94427 q^{31} +3.23607 q^{33} +0.763932 q^{35} -1.23607 q^{37} -5.23607 q^{39} +3.70820 q^{41} -0.763932 q^{43} -1.00000 q^{45} +6.47214 q^{47} -6.41641 q^{49} +4.47214 q^{51} +8.47214 q^{53} +3.23607 q^{55} +1.00000 q^{57} +1.52786 q^{59} -4.47214 q^{61} -0.763932 q^{63} -5.23607 q^{65} -10.4721 q^{67} -6.47214 q^{69} +3.52786 q^{73} -1.00000 q^{75} +2.47214 q^{77} +1.00000 q^{81} -4.00000 q^{83} +4.47214 q^{85} -5.23607 q^{87} +11.7082 q^{89} -4.00000 q^{91} +8.94427 q^{93} +1.00000 q^{95} -9.23607 q^{97} -3.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{13} + 2 q^{15} - 2 q^{19} + 6 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} + 2 q^{33} + 6 q^{35} + 2 q^{37} - 6 q^{39} - 6 q^{41} - 6 q^{43} - 2 q^{45} + 4 q^{47} + 14 q^{49} + 8 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 6 q^{63} - 6 q^{65} - 12 q^{67} - 4 q^{69} + 16 q^{73} - 2 q^{75} - 4 q^{77} + 2 q^{81} - 8 q^{83} - 6 q^{87} + 10 q^{89} - 8 q^{91} + 2 q^{95} - 14 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 + 6 * q^13 + 2 * q^15 - 2 * q^19 + 6 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 6 * q^29 + 2 * q^33 + 6 * q^35 + 2 * q^37 - 6 * q^39 - 6 * q^41 - 6 * q^43 - 2 * q^45 + 4 * q^47 + 14 * q^49 + 8 * q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 - 6 * q^63 - 6 * q^65 - 12 * q^67 - 4 * q^69 + 16 * q^73 - 2 * q^75 - 4 * q^77 + 2 * q^81 - 8 * q^83 - 6 * q^87 + 10 * q^89 - 8 * q^91 + 2 * q^95 - 14 * 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
$7$	−0.763932	−0.288739	−0.144370	−	0.989524i	$-0.546115\pi$
$7$	−0.763932	−0.288739	−0.144370	+	0.989524i	$0.546115\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.763932	0.166704
$22$	0	0
$23$	6.47214	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.23607	0.972313	0.486157	−	0.873872i	$-0.338398\pi$
$29$	5.23607	0.972313	0.486157	+	0.873872i	$0.338398\pi$
$30$	0	0
$31$	−8.94427	−1.60644	−0.803219	−	0.595683i	$-0.796881\pi$
$31$	−8.94427	−1.60644	−0.803219	+	0.595683i	$0.796881\pi$
$32$	0	0
$33$	3.23607	0.563327
$34$	0	0
$35$	0.763932	0.129128
$36$	0	0
$37$	−1.23607	−0.203208	−0.101604	−	0.994825i	$-0.532398\pi$
$37$	−1.23607	−0.203208	−0.101604	+	0.994825i	$0.532398\pi$
$38$	0	0
$39$	−5.23607	−0.838442
$40$	0	0
$41$	3.70820	0.579124	0.289562	−	0.957159i	$-0.406490\pi$
$41$	3.70820	0.579124	0.289562	+	0.957159i	$0.406490\pi$
$42$	0	0
$43$	−0.763932	−0.116499	−0.0582493	−	0.998302i	$-0.518552\pi$
$43$	−0.763932	−0.116499	−0.0582493	+	0.998302i	$0.518552\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.47214	0.944058	0.472029	−	0.881583i	$-0.343522\pi$
$47$	6.47214	0.944058	0.472029	+	0.881583i	$0.343522\pi$
$48$	0	0
$49$	−6.41641	−0.916630
$50$	0	0
$51$	4.47214	0.626224
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	3.23607	0.436351
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	1.52786	0.198911	0.0994555	−	0.995042i	$-0.468290\pi$
$59$	1.52786	0.198911	0.0994555	+	0.995042i	$0.468290\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	0	0
$63$	−0.763932	−0.0962464
$64$	0	0
$65$	−5.23607	−0.649454
$66$	0	0
$67$	−10.4721	−1.27938	−0.639688	−	0.768635i	$-0.720936\pi$
$67$	−10.4721	−1.27938	−0.639688	+	0.768635i	$0.720936\pi$
$68$	0	0
$69$	−6.47214	−0.779154
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	3.52786	0.412905	0.206453	−	0.978457i	$-0.433808\pi$
$73$	3.52786	0.412905	0.206453	+	0.978457i	$0.433808\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.47214	0.281726
$78$	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
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	0	0
$87$	−5.23607	−0.561365
$88$	0	0
$89$	11.7082	1.24107	0.620534	−	0.784180i	$-0.286916\pi$
$89$	11.7082	1.24107	0.620534	+	0.784180i	$0.286916\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	8.94427	0.927478
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−9.23607	−0.937781	−0.468890	−	0.883256i	$-0.655346\pi$
$97$	−9.23607	−0.937781	−0.468890	+	0.883256i	$0.655346\pi$
$98$	0	0
$99$	−3.23607	−0.325237
$100$	0	0
$101$	13.4164	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$101$	13.4164	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$102$	0	0
$103$	−18.4721	−1.82011	−0.910057	−	0.414484i	$-0.863962\pi$
$103$	−18.4721	−1.82011	−0.910057	+	0.414484i	$0.863962\pi$
$104$	0	0
$105$	−0.763932	−0.0745521
$106$	0	0
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0	0
$109$	4.47214	0.428353	0.214176	−	0.976795i	$-0.431293\pi$
$109$	4.47214	0.428353	0.214176	+	0.976795i	$0.431293\pi$
$110$	0	0
$111$	1.23607	0.117322
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−6.47214	−0.603530
$116$	0	0
$117$	5.23607	0.484075
$118$	0	0
$119$	3.41641	0.313182
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−3.70820	−0.334357
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	0.763932	0.0672605
$130$	0	0
$131$	8.18034	0.714720	0.357360	−	0.933967i	$-0.383677\pi$
$131$	8.18034	0.714720	0.357360	+	0.933967i	$0.383677\pi$
$132$	0	0
$133$	0.763932	0.0662413
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−19.8885	−1.69919	−0.849596	−	0.527433i	$-0.823154\pi$
$137$	−19.8885	−1.69919	−0.849596	+	0.527433i	$0.823154\pi$
$138$	0	0
$139$	5.52786	0.468867	0.234434	−	0.972132i	$-0.424676\pi$
$139$	5.52786	0.468867	0.234434	+	0.972132i	$0.424676\pi$
$140$	0	0
$141$	−6.47214	−0.545052
$142$	0	0
$143$	−16.9443	−1.41695
$144$	0	0
$145$	−5.23607	−0.434832
$146$	0	0
$147$	6.41641	0.529216
$148$	0	0
$149$	−9.41641	−0.771422	−0.385711	−	0.922620i	$-0.626044\pi$
$149$	−9.41641	−0.771422	−0.385711	+	0.922620i	$0.626044\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	8.94427	0.718421
$156$	0	0
$157$	5.41641	0.432276	0.216138	−	0.976363i	$-0.430654\pi$
$157$	5.41641	0.432276	0.216138	+	0.976363i	$0.430654\pi$
$158$	0	0
$159$	−8.47214	−0.671884
$160$	0	0
$161$	−4.94427	−0.389663
$162$	0	0
$163$	10.6525	0.834366	0.417183	−	0.908822i	$-0.363017\pi$
$163$	10.6525	0.834366	0.417183	+	0.908822i	$0.363017\pi$
$164$	0	0
$165$	−3.23607	−0.251928
$166$	0	0
$167$	21.8885	1.69379	0.846893	−	0.531763i	$-0.178470\pi$
$167$	21.8885	1.69379	0.846893	+	0.531763i	$0.178470\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	−0.763932	−0.0577478
$176$	0	0
$177$	−1.52786	−0.114841
$178$	0	0
$179$	−11.4164	−0.853302	−0.426651	−	0.904416i	$-0.640307\pi$
$179$	−11.4164	−0.853302	−0.426651	+	0.904416i	$0.640307\pi$
$180$	0	0
$181$	−0.472136	−0.0350936	−0.0175468	−	0.999846i	$-0.505586\pi$
$181$	−0.472136	−0.0350936	−0.0175468	+	0.999846i	$0.505586\pi$
$182$	0	0
$183$	4.47214	0.330590
$184$	0	0
$185$	1.23607	0.0908775
$186$	0	0
$187$	14.4721	1.05831
$188$	0	0
$189$	0.763932	0.0555679
$190$	0	0
$191$	−16.1803	−1.17077	−0.585384	−	0.810756i	$-0.699056\pi$
$191$	−16.1803	−1.17077	−0.585384	+	0.810756i	$0.699056\pi$
$192$	0	0
$193$	−7.70820	−0.554849	−0.277424	−	0.960747i	$-0.589481\pi$
$193$	−7.70820	−0.554849	−0.277424	+	0.960747i	$0.589481\pi$
$194$	0	0
$195$	5.23607	0.374963
$196$	0	0
$197$	−3.52786	−0.251350	−0.125675	−	0.992071i	$-0.540110\pi$
$197$	−3.52786	−0.251350	−0.125675	+	0.992071i	$0.540110\pi$
$198$	0	0
$199$	−9.52786	−0.675412	−0.337706	−	0.941252i	$-0.609651\pi$
$199$	−9.52786	−0.675412	−0.337706	+	0.941252i	$0.609651\pi$
$200$	0	0
$201$	10.4721	0.738648
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−3.70820	−0.258992
$206$	0	0
$207$	6.47214	0.449845
$208$	0	0
$209$	3.23607	0.223844
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.763932	0.0520997
$216$	0	0
$217$	6.83282	0.463842
$218$	0	0
$219$	−3.52786	−0.238391
$220$	0	0
$221$	−23.4164	−1.57516
$222$	0	0
$223$	−13.5279	−0.905893	−0.452946	−	0.891538i	$-0.649627\pi$
$223$	−13.5279	−0.905893	−0.452946	+	0.891538i	$0.649627\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	5.41641	0.357926	0.178963	−	0.983856i	$-0.442726\pi$
$229$	5.41641	0.357926	0.178963	+	0.983856i	$0.442726\pi$
$230$	0	0
$231$	−2.47214	−0.162655
$232$	0	0
$233$	−11.8885	−0.778844	−0.389422	−	0.921059i	$-0.627325\pi$
$233$	−11.8885	−0.778844	−0.389422	+	0.921059i	$0.627325\pi$
$234$	0	0
$235$	−6.47214	−0.422196
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.70820	−0.110495	−0.0552473	−	0.998473i	$-0.517595\pi$
$239$	−1.70820	−0.110495	−0.0552473	+	0.998473i	$0.517595\pi$
$240$	0	0
$241$	−10.9443	−0.704983	−0.352491	−	0.935815i	$-0.614665\pi$
$241$	−10.9443	−0.704983	−0.352491	+	0.935815i	$0.614665\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.41641	0.409929
$246$	0	0
$247$	−5.23607	−0.333163
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	9.70820	0.612776	0.306388	−	0.951907i	$-0.400879\pi$
$251$	9.70820	0.612776	0.306388	+	0.951907i	$0.400879\pi$
$252$	0	0
$253$	−20.9443	−1.31676
$254$	0	0
$255$	−4.47214	−0.280056
$256$	0	0
$257$	−5.05573	−0.315368	−0.157684	−	0.987490i	$-0.550403\pi$
$257$	−5.05573	−0.315368	−0.157684	+	0.987490i	$0.550403\pi$
$258$	0	0
$259$	0.944272	0.0586742
$260$	0	0
$261$	5.23607	0.324104
$262$	0	0
$263$	−20.9443	−1.29148	−0.645740	−	0.763558i	$-0.723451\pi$
$263$	−20.9443	−1.29148	−0.645740	+	0.763558i	$0.723451\pi$
$264$	0	0
$265$	−8.47214	−0.520439
$266$	0	0
$267$	−11.7082	−0.716530
$268$	0	0
$269$	−7.70820	−0.469977	−0.234989	−	0.971998i	$-0.575505\pi$
$269$	−7.70820	−0.469977	−0.234989	+	0.971998i	$0.575505\pi$
$270$	0	0
$271$	−6.47214	−0.393154	−0.196577	−	0.980488i	$-0.562983\pi$
$271$	−6.47214	−0.393154	−0.196577	+	0.980488i	$0.562983\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	−3.23607	−0.195142
$276$	0	0
$277$	−7.52786	−0.452306	−0.226153	−	0.974092i	$-0.572615\pi$
$277$	−7.52786	−0.452306	−0.226153	+	0.974092i	$0.572615\pi$
$278$	0	0
$279$	−8.94427	−0.535480
$280$	0	0
$281$	−10.7639	−0.642122	−0.321061	−	0.947058i	$-0.604040\pi$
$281$	−10.7639	−0.642122	−0.321061	+	0.947058i	$0.604040\pi$
$282$	0	0
$283$	−30.0689	−1.78741	−0.893705	−	0.448655i	$-0.851903\pi$
$283$	−30.0689	−1.78741	−0.893705	+	0.448655i	$0.851903\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	−2.83282	−0.167216
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	9.23607	0.541428
$292$	0	0
$293$	−31.8885	−1.86295	−0.931474	−	0.363807i	$-0.881477\pi$
$293$	−31.8885	−1.86295	−0.931474	+	0.363807i	$0.881477\pi$
$294$	0	0
$295$	−1.52786	−0.0889557
$296$	0	0
$297$	3.23607	0.187776
$298$	0	0
$299$	33.8885	1.95983
$300$	0	0
$301$	0.583592	0.0336377
$302$	0	0
$303$	−13.4164	−0.770752
$304$	0	0
$305$	4.47214	0.256074
$306$	0	0
$307$	26.4721	1.51084	0.755422	−	0.655238i	$-0.227432\pi$
$307$	26.4721	1.51084	0.755422	+	0.655238i	$0.227432\pi$
$308$	0	0
$309$	18.4721	1.05084
$310$	0	0
$311$	−6.29180	−0.356775	−0.178388	−	0.983960i	$-0.557088\pi$
$311$	−6.29180	−0.356775	−0.178388	+	0.983960i	$0.557088\pi$
$312$	0	0
$313$	11.8885	0.671980	0.335990	−	0.941866i	$-0.390929\pi$
$313$	11.8885	0.671980	0.335990	+	0.941866i	$0.390929\pi$
$314$	0	0
$315$	0.763932	0.0430427
$316$	0	0
$317$	3.52786	0.198145	0.0990723	−	0.995080i	$-0.468412\pi$
$317$	3.52786	0.198145	0.0990723	+	0.995080i	$0.468412\pi$
$318$	0	0
$319$	−16.9443	−0.948697
$320$	0	0
$321$	8.94427	0.499221
$322$	0	0
$323$	4.47214	0.248836
$324$	0	0
$325$	5.23607	0.290445
$326$	0	0
$327$	−4.47214	−0.247310
$328$	0	0
$329$	−4.94427	−0.272587
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−1.23607	−0.0677361
$334$	0	0
$335$	10.4721	0.572154
$336$	0	0
$337$	−33.2361	−1.81048	−0.905242	−	0.424896i	$-0.860311\pi$
$337$	−33.2361	−1.81048	−0.905242	+	0.424896i	$0.860311\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	28.9443	1.56742
$342$	0	0
$343$	10.2492	0.553406
$344$	0	0
$345$	6.47214	0.348448
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−6.94427	−0.371718	−0.185859	−	0.982576i	$-0.559507\pi$
$349$	−6.94427	−0.371718	−0.185859	+	0.982576i	$0.559507\pi$
$350$	0	0
$351$	−5.23607	−0.279481
$352$	0	0
$353$	2.94427	0.156708	0.0783539	−	0.996926i	$-0.475034\pi$
$353$	2.94427	0.156708	0.0783539	+	0.996926i	$0.475034\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.41641	−0.180815
$358$	0	0
$359$	−0.180340	−0.00951798	−0.00475899	−	0.999989i	$-0.501515\pi$
$359$	−0.180340	−0.00951798	−0.00475899	+	0.999989i	$0.501515\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.527864	0.0277057
$364$	0	0
$365$	−3.52786	−0.184657
$366$	0	0
$367$	24.7639	1.29267	0.646333	−	0.763055i	$-0.276302\pi$
$367$	24.7639	1.29267	0.646333	+	0.763055i	$0.276302\pi$
$368$	0	0
$369$	3.70820	0.193041
$370$	0	0
$371$	−6.47214	−0.336017
$372$	0	0
$373$	−2.76393	−0.143111	−0.0715555	−	0.997437i	$-0.522796\pi$
$373$	−2.76393	−0.143111	−0.0715555	+	0.997437i	$0.522796\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	27.4164	1.41202
$378$	0	0
$379$	33.8885	1.74074	0.870369	−	0.492400i	$-0.163880\pi$
$379$	33.8885	1.74074	0.870369	+	0.492400i	$0.163880\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	−25.8885	−1.32284	−0.661421	−	0.750014i	$-0.730046\pi$
$383$	−25.8885	−1.32284	−0.661421	+	0.750014i	$0.730046\pi$
$384$	0	0
$385$	−2.47214	−0.125992
$386$	0	0
$387$	−0.763932	−0.0388328
$388$	0	0
$389$	−20.8328	−1.05627	−0.528133	−	0.849162i	$-0.677108\pi$
$389$	−20.8328	−1.05627	−0.528133	+	0.849162i	$0.677108\pi$
$390$	0	0
$391$	−28.9443	−1.46377
$392$	0	0
$393$	−8.18034	−0.412644
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−33.4164	−1.67712	−0.838561	−	0.544808i	$-0.816602\pi$
$397$	−33.4164	−1.67712	−0.838561	+	0.544808i	$0.816602\pi$
$398$	0	0
$399$	−0.763932	−0.0382444
$400$	0	0
$401$	−18.7639	−0.937026	−0.468513	−	0.883457i	$-0.655210\pi$
$401$	−18.7639	−0.937026	−0.468513	+	0.883457i	$0.655210\pi$
$402$	0	0
$403$	−46.8328	−2.33291
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−4.47214	−0.221133	−0.110566	−	0.993869i	$-0.535266\pi$
$409$	−4.47214	−0.221133	−0.110566	+	0.993869i	$0.535266\pi$
$410$	0	0
$411$	19.8885	0.981030
$412$	0	0
$413$	−1.16718	−0.0574334
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−5.52786	−0.270701
$418$	0	0
$419$	−0.180340	−0.00881018	−0.00440509	−	0.999990i	$-0.501402\pi$
$419$	−0.180340	−0.00881018	−0.00440509	+	0.999990i	$0.501402\pi$
$420$	0	0
$421$	28.8328	1.40523	0.702613	−	0.711572i	$-0.252017\pi$
$421$	28.8328	1.40523	0.702613	+	0.711572i	$0.252017\pi$
$422$	0	0
$423$	6.47214	0.314686
$424$	0	0
$425$	−4.47214	−0.216930
$426$	0	0
$427$	3.41641	0.165332
$428$	0	0
$429$	16.9443	0.818077
$430$	0	0
$431$	37.3050	1.79692	0.898458	−	0.439059i	$-0.144688\pi$
$431$	37.3050	1.79692	0.898458	+	0.439059i	$0.144688\pi$
$432$	0	0
$433$	5.23607	0.251629	0.125815	−	0.992054i	$-0.459846\pi$
$433$	5.23607	0.251629	0.125815	+	0.992054i	$0.459846\pi$
$434$	0	0
$435$	5.23607	0.251050
$436$	0	0
$437$	−6.47214	−0.309604
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−6.41641	−0.305543
$442$	0	0
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	0	0
$445$	−11.7082	−0.555022
$446$	0	0
$447$	9.41641	0.445381
$448$	0	0
$449$	3.70820	0.175001	0.0875005	−	0.996164i	$-0.472112\pi$
$449$	3.70820	0.175001	0.0875005	+	0.996164i	$0.472112\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−36.8328	−1.72297	−0.861483	−	0.507786i	$-0.830464\pi$
$457$	−36.8328	−1.72297	−0.861483	+	0.507786i	$0.830464\pi$
$458$	0	0
$459$	4.47214	0.208741
$460$	0	0
$461$	0.472136	0.0219896	0.0109948	−	0.999940i	$-0.496500\pi$
$461$	0.472136	0.0219896	0.0109948	+	0.999940i	$0.496500\pi$
$462$	0	0
$463$	23.5967	1.09663	0.548317	−	0.836271i	$-0.315269\pi$
$463$	23.5967	1.09663	0.548317	+	0.836271i	$0.315269\pi$
$464$	0	0
$465$	−8.94427	−0.414781
$466$	0	0
$467$	31.4164	1.45378	0.726889	−	0.686755i	$-0.240965\pi$
$467$	31.4164	1.45378	0.726889	+	0.686755i	$0.240965\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−5.41641	−0.249575
$472$	0	0
$473$	2.47214	0.113669
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	8.47214	0.387912
$478$	0	0
$479$	−8.18034	−0.373769	−0.186885	−	0.982382i	$-0.559839\pi$
$479$	−8.18034	−0.373769	−0.186885	+	0.982382i	$0.559839\pi$
$480$	0	0
$481$	−6.47214	−0.295104
$482$	0	0
$483$	4.94427	0.224972
$484$	0	0
$485$	9.23607	0.419388
$486$	0	0
$487$	−31.4164	−1.42361	−0.711807	−	0.702375i	$-0.752123\pi$
$487$	−31.4164	−1.42361	−0.711807	+	0.702375i	$0.752123\pi$
$488$	0	0
$489$	−10.6525	−0.481722
$490$	0	0
$491$	1.34752	0.0608129	0.0304065	−	0.999538i	$-0.490320\pi$
$491$	1.34752	0.0608129	0.0304065	+	0.999538i	$0.490320\pi$
$492$	0	0
$493$	−23.4164	−1.05462
$494$	0	0
$495$	3.23607	0.145450
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15.4164	−0.690133	−0.345067	−	0.938578i	$-0.612144\pi$
$499$	−15.4164	−0.690133	−0.345067	+	0.938578i	$0.612144\pi$
$500$	0	0
$501$	−21.8885	−0.977908
$502$	0	0
$503$	−4.94427	−0.220454	−0.110227	−	0.993906i	$-0.535158\pi$
$503$	−4.94427	−0.220454	−0.110227	+	0.993906i	$0.535158\pi$
$504$	0	0
$505$	−13.4164	−0.597022
$506$	0	0
$507$	−14.4164	−0.640255
$508$	0	0
$509$	43.7082	1.93733	0.968666	−	0.248367i	$-0.0798939\pi$
$509$	43.7082	1.93733	0.968666	+	0.248367i	$0.0798939\pi$
$510$	0	0
$511$	−2.69505	−0.119222
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	18.4721	0.813980
$516$	0	0
$517$	−20.9443	−0.921128
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	−39.7082	−1.73965	−0.869824	−	0.493362i	$-0.835768\pi$
$521$	−39.7082	−1.73965	−0.869824	+	0.493362i	$0.835768\pi$
$522$	0	0
$523$	−16.9443	−0.740921	−0.370461	−	0.928848i	$-0.620800\pi$
$523$	−16.9443	−0.740921	−0.370461	+	0.928848i	$0.620800\pi$
$524$	0	0
$525$	0.763932	0.0333407
$526$	0	0
$527$	40.0000	1.74243
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	1.52786	0.0663037
$532$	0	0
$533$	19.4164	0.841018
$534$	0	0
$535$	8.94427	0.386695
$536$	0	0
$537$	11.4164	0.492654
$538$	0	0
$539$	20.7639	0.894366
$540$	0	0
$541$	41.7771	1.79614	0.898069	−	0.439855i	$-0.144970\pi$
$541$	41.7771	1.79614	0.898069	+	0.439855i	$0.144970\pi$
$542$	0	0
$543$	0.472136	0.0202613
$544$	0	0
$545$	−4.47214	−0.191565
$546$	0	0
$547$	−25.3050	−1.08196	−0.540981	−	0.841035i	$-0.681947\pi$
$547$	−25.3050	−1.08196	−0.540981	+	0.841035i	$0.681947\pi$
$548$	0	0
$549$	−4.47214	−0.190866
$550$	0	0
$551$	−5.23607	−0.223064
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.23607	−0.0524682
$556$	0	0
$557$	−20.8328	−0.882715	−0.441357	−	0.897331i	$-0.645503\pi$
$557$	−20.8328	−0.882715	−0.441357	+	0.897331i	$0.645503\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−14.4721	−0.611014
$562$	0	0
$563$	−25.8885	−1.09107	−0.545536	−	0.838087i	$-0.683674\pi$
$563$	−25.8885	−1.09107	−0.545536	+	0.838087i	$0.683674\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	−0.763932	−0.0320821
$568$	0	0
$569$	34.5410	1.44803	0.724017	−	0.689782i	$-0.242293\pi$
$569$	34.5410	1.44803	0.724017	+	0.689782i	$0.242293\pi$
$570$	0	0
$571$	−41.3050	−1.72856	−0.864279	−	0.503012i	$-0.832225\pi$
$571$	−41.3050	−1.72856	−0.864279	+	0.503012i	$0.832225\pi$
$572$	0	0
$573$	16.1803	0.675943
$574$	0	0
$575$	6.47214	0.269907
$576$	0	0
$577$	−26.9443	−1.12170	−0.560852	−	0.827916i	$-0.689526\pi$
$577$	−26.9443	−1.12170	−0.560852	+	0.827916i	$0.689526\pi$
$578$	0	0
$579$	7.70820	0.320342
$580$	0	0
$581$	3.05573	0.126773
$582$	0	0
$583$	−27.4164	−1.13547
$584$	0	0
$585$	−5.23607	−0.216485
$586$	0	0
$587$	−15.4164	−0.636303	−0.318152	−	0.948040i	$-0.603062\pi$
$587$	−15.4164	−0.636303	−0.318152	+	0.948040i	$0.603062\pi$
$588$	0	0
$589$	8.94427	0.368542
$590$	0	0
$591$	3.52786	0.145117
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	−3.41641	−0.140059
$596$	0	0
$597$	9.52786	0.389950
$598$	0	0
$599$	−20.5836	−0.841023	−0.420511	−	0.907287i	$-0.638149\pi$
$599$	−20.5836	−0.841023	−0.420511	+	0.907287i	$0.638149\pi$
$600$	0	0
$601$	34.3607	1.40160	0.700801	−	0.713357i	$-0.252826\pi$
$601$	34.3607	1.40160	0.700801	+	0.713357i	$0.252826\pi$
$602$	0	0
$603$	−10.4721	−0.426458
$604$	0	0
$605$	0.527864	0.0214607
$606$	0	0
$607$	42.4721	1.72389	0.861945	−	0.507001i	$-0.169246\pi$
$607$	42.4721	1.72389	0.861945	+	0.507001i	$0.169246\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	33.8885	1.37098
$612$	0	0
$613$	−44.4721	−1.79621	−0.898106	−	0.439778i	$-0.855057\pi$
$613$	−44.4721	−1.79621	−0.898106	+	0.439778i	$0.855057\pi$
$614$	0	0
$615$	3.70820	0.149529
$616$	0	0
$617$	23.3050	0.938222	0.469111	−	0.883139i	$-0.344574\pi$
$617$	23.3050	0.938222	0.469111	+	0.883139i	$0.344574\pi$
$618$	0	0
$619$	12.3607	0.496818	0.248409	−	0.968655i	$-0.420092\pi$
$619$	12.3607	0.496818	0.248409	+	0.968655i	$0.420092\pi$
$620$	0	0
$621$	−6.47214	−0.259718
$622$	0	0
$623$	−8.94427	−0.358345
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.23607	−0.129236
$628$	0	0
$629$	5.52786	0.220410
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	20.0000	0.793676
$636$	0	0
$637$	−33.5967	−1.33115
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−49.2361	−1.94471	−0.972354	−	0.233512i	$-0.924978\pi$
$641$	−49.2361	−1.94471	−0.972354	+	0.233512i	$0.924978\pi$
$642$	0	0
$643$	31.5967	1.24605	0.623027	−	0.782200i	$-0.285903\pi$
$643$	31.5967	1.24605	0.623027	+	0.782200i	$0.285903\pi$
$644$	0	0
$645$	−0.763932	−0.0300798
$646$	0	0
$647$	38.8328	1.52668	0.763338	−	0.646000i	$-0.223559\pi$
$647$	38.8328	1.52668	0.763338	+	0.646000i	$0.223559\pi$
$648$	0	0
$649$	−4.94427	−0.194080
$650$	0	0
$651$	−6.83282	−0.267799
$652$	0	0
$653$	−40.4721	−1.58380	−0.791899	−	0.610653i	$-0.790907\pi$
$653$	−40.4721	−1.58380	−0.791899	+	0.610653i	$0.790907\pi$
$654$	0	0
$655$	−8.18034	−0.319632
$656$	0	0
$657$	3.52786	0.137635
$658$	0	0
$659$	22.8328	0.889440	0.444720	−	0.895670i	$-0.353303\pi$
$659$	22.8328	0.889440	0.444720	+	0.895670i	$0.353303\pi$
$660$	0	0
$661$	−39.3050	−1.52879	−0.764393	−	0.644751i	$-0.776961\pi$
$661$	−39.3050	−1.52879	−0.764393	+	0.644751i	$0.776961\pi$
$662$	0	0
$663$	23.4164	0.909418
$664$	0	0
$665$	−0.763932	−0.0296240
$666$	0	0
$667$	33.8885	1.31217
$668$	0	0
$669$	13.5279	0.523017
$670$	0	0
$671$	14.4721	0.558691
$672$	0	0
$673$	21.5967	0.832493	0.416247	−	0.909252i	$-0.363345\pi$
$673$	21.5967	0.832493	0.416247	+	0.909252i	$0.363345\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	24.4721	0.940541	0.470270	−	0.882522i	$-0.344156\pi$
$677$	24.4721	0.940541	0.470270	+	0.882522i	$0.344156\pi$
$678$	0	0
$679$	7.05573	0.270774
$680$	0	0
$681$	0	0
$682$	0	0
$683$	39.7771	1.52203	0.761014	−	0.648735i	$-0.224702\pi$
$683$	39.7771	1.52203	0.761014	+	0.648735i	$0.224702\pi$
$684$	0	0
$685$	19.8885	0.759902
$686$	0	0
$687$	−5.41641	−0.206649
$688$	0	0
$689$	44.3607	1.69001
$690$	0	0
$691$	34.4721	1.31138	0.655691	−	0.755029i	$-0.272377\pi$
$691$	34.4721	1.31138	0.655691	+	0.755029i	$0.272377\pi$
$692$	0	0
$693$	2.47214	0.0939087
$694$	0	0
$695$	−5.52786	−0.209684
$696$	0	0
$697$	−16.5836	−0.628148
$698$	0	0
$699$	11.8885	0.449666
$700$	0	0
$701$	−41.4164	−1.56428	−0.782138	−	0.623105i	$-0.785871\pi$
$701$	−41.4164	−1.56428	−0.782138	+	0.623105i	$0.785871\pi$
$702$	0	0
$703$	1.23607	0.0466192
$704$	0	0
$705$	6.47214	0.243755
$706$	0	0
$707$	−10.2492	−0.385462
$708$	0	0
$709$	31.3050	1.17568	0.587841	−	0.808976i	$-0.299978\pi$
$709$	31.3050	1.17568	0.587841	+	0.808976i	$0.299978\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−57.8885	−2.16794
$714$	0	0
$715$	16.9443	0.633680
$716$	0	0
$717$	1.70820	0.0637940
$718$	0	0
$719$	3.23607	0.120685	0.0603425	−	0.998178i	$-0.480781\pi$
$719$	3.23607	0.120685	0.0603425	+	0.998178i	$0.480781\pi$
$720$	0	0
$721$	14.1115	0.525538
$722$	0	0
$723$	10.9443	0.407022
$724$	0	0
$725$	5.23607	0.194463
$726$	0	0
$727$	10.6525	0.395078	0.197539	−	0.980295i	$-0.436705\pi$
$727$	10.6525	0.395078	0.197539	+	0.980295i	$0.436705\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.41641	0.126360
$732$	0	0
$733$	16.8328	0.621734	0.310867	−	0.950453i	$-0.399381\pi$
$733$	16.8328	0.621734	0.310867	+	0.950453i	$0.399381\pi$
$734$	0	0
$735$	−6.41641	−0.236673
$736$	0	0
$737$	33.8885	1.24830
$738$	0	0
$739$	−8.94427	−0.329020	−0.164510	−	0.986375i	$-0.552604\pi$
$739$	−8.94427	−0.329020	−0.164510	+	0.986375i	$0.552604\pi$
$740$	0	0
$741$	5.23607	0.192352
$742$	0	0
$743$	27.0557	0.992578	0.496289	−	0.868157i	$-0.334696\pi$
$743$	27.0557	0.992578	0.496289	+	0.868157i	$0.334696\pi$
$744$	0	0
$745$	9.41641	0.344990
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	6.83282	0.249666
$750$	0	0
$751$	8.94427	0.326381	0.163191	−	0.986595i	$-0.447821\pi$
$751$	8.94427	0.326381	0.163191	+	0.986595i	$0.447821\pi$
$752$	0	0
$753$	−9.70820	−0.353787
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	14.9443	0.543159	0.271579	−	0.962416i	$-0.412454\pi$
$757$	14.9443	0.543159	0.271579	+	0.962416i	$0.412454\pi$
$758$	0	0
$759$	20.9443	0.760229
$760$	0	0
$761$	−17.4164	−0.631344	−0.315672	−	0.948868i	$-0.602230\pi$
$761$	−17.4164	−0.631344	−0.315672	+	0.948868i	$0.602230\pi$
$762$	0	0
$763$	−3.41641	−0.123682
$764$	0	0
$765$	4.47214	0.161690
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	43.8885	1.58266	0.791331	−	0.611388i	$-0.209389\pi$
$769$	43.8885	1.58266	0.791331	+	0.611388i	$0.209389\pi$
$770$	0	0
$771$	5.05573	0.182078
$772$	0	0
$773$	48.4721	1.74342	0.871711	−	0.490021i	$-0.163011\pi$
$773$	48.4721	1.74342	0.871711	+	0.490021i	$0.163011\pi$
$774$	0	0
$775$	−8.94427	−0.321288
$776$	0	0
$777$	−0.944272	−0.0338756
$778$	0	0
$779$	−3.70820	−0.132860
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−5.23607	−0.187122
$784$	0	0
$785$	−5.41641	−0.193320
$786$	0	0
$787$	42.8328	1.52682	0.763412	−	0.645911i	$-0.223522\pi$
$787$	42.8328	1.52682	0.763412	+	0.645911i	$0.223522\pi$
$788$	0	0
$789$	20.9443	0.745636
$790$	0	0
$791$	1.52786	0.0543246
$792$	0	0
$793$	−23.4164	−0.831541
$794$	0	0
$795$	8.47214	0.300476
$796$	0	0
$797$	−33.7771	−1.19645	−0.598223	−	0.801330i	$-0.704126\pi$
$797$	−33.7771	−1.19645	−0.598223	+	0.801330i	$0.704126\pi$
$798$	0	0
$799$	−28.9443	−1.02397
$800$	0	0
$801$	11.7082	0.413689
$802$	0	0
$803$	−11.4164	−0.402876
$804$	0	0
$805$	4.94427	0.174263
$806$	0	0
$807$	7.70820	0.271342
$808$	0	0
$809$	21.0557	0.740280	0.370140	−	0.928976i	$-0.379310\pi$
$809$	21.0557	0.740280	0.370140	+	0.928976i	$0.379310\pi$
$810$	0	0
$811$	16.9443	0.594994	0.297497	−	0.954723i	$-0.403848\pi$
$811$	16.9443	0.594994	0.297497	+	0.954723i	$0.403848\pi$
$812$	0	0
$813$	6.47214	0.226988
$814$	0	0
$815$	−10.6525	−0.373140
$816$	0	0
$817$	0.763932	0.0267266
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	24.4721	0.854083	0.427042	−	0.904232i	$-0.359556\pi$
$821$	24.4721	0.854083	0.427042	+	0.904232i	$0.359556\pi$
$822$	0	0
$823$	−20.1803	−0.703442	−0.351721	−	0.936105i	$-0.614403\pi$
$823$	−20.1803	−0.703442	−0.351721	+	0.936105i	$0.614403\pi$
$824$	0	0
$825$	3.23607	0.112665
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−15.3050	−0.531563	−0.265781	−	0.964033i	$-0.585630\pi$
$829$	−15.3050	−0.531563	−0.265781	+	0.964033i	$0.585630\pi$
$830$	0	0
$831$	7.52786	0.261139
$832$	0	0
$833$	28.6950	0.994224
$834$	0	0
$835$	−21.8885	−0.757484
$836$	0	0
$837$	8.94427	0.309159
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−1.58359	−0.0546066
$842$	0	0
$843$	10.7639	0.370730
$844$	0	0
$845$	−14.4164	−0.495940
$846$	0	0
$847$	0.403252	0.0138559
$848$	0	0
$849$	30.0689	1.03196
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−46.7214	−1.59971	−0.799854	−	0.600194i	$-0.795090\pi$
$853$	−46.7214	−1.59971	−0.799854	+	0.600194i	$0.795090\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	15.5279	0.530422	0.265211	−	0.964190i	$-0.414558\pi$
$857$	15.5279	0.530422	0.265211	+	0.964190i	$0.414558\pi$
$858$	0	0
$859$	42.8328	1.46144	0.730718	−	0.682679i	$-0.239185\pi$
$859$	42.8328	1.46144	0.730718	+	0.682679i	$0.239185\pi$
$860$	0	0
$861$	2.83282	0.0965421
$862$	0	0
$863$	6.83282	0.232592	0.116296	−	0.993215i	$-0.462898\pi$
$863$	6.83282	0.232592	0.116296	+	0.993215i	$0.462898\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−54.8328	−1.85794
$872$	0	0
$873$	−9.23607	−0.312594
$874$	0	0
$875$	0.763932	0.0258256
$876$	0	0
$877$	24.2918	0.820276	0.410138	−	0.912024i	$-0.365481\pi$
$877$	24.2918	0.820276	0.410138	+	0.912024i	$0.365481\pi$
$878$	0	0
$879$	31.8885	1.07557
$880$	0	0
$881$	7.30495	0.246110	0.123055	−	0.992400i	$-0.460731\pi$
$881$	7.30495	0.246110	0.123055	+	0.992400i	$0.460731\pi$
$882$	0	0
$883$	23.5967	0.794094	0.397047	−	0.917798i	$-0.370035\pi$
$883$	23.5967	0.794094	0.397047	+	0.917798i	$0.370035\pi$
$884$	0	0
$885$	1.52786	0.0513586
$886$	0	0
$887$	−9.88854	−0.332025	−0.166012	−	0.986124i	$-0.553089\pi$
$887$	−9.88854	−0.332025	−0.166012	+	0.986124i	$0.553089\pi$
$888$	0	0
$889$	15.2786	0.512429
$890$	0	0
$891$	−3.23607	−0.108412
$892$	0	0
$893$	−6.47214	−0.216582
$894$	0	0
$895$	11.4164	0.381608
$896$	0	0
$897$	−33.8885	−1.13151
$898$	0	0
$899$	−46.8328	−1.56196
$900$	0	0
$901$	−37.8885	−1.26225
$902$	0	0
$903$	−0.583592	−0.0194207
$904$	0	0
$905$	0.472136	0.0156943
$906$	0	0
$907$	−45.8885	−1.52370	−0.761852	−	0.647751i	$-0.775710\pi$
$907$	−45.8885	−1.52370	−0.761852	+	0.647751i	$0.775710\pi$
$908$	0	0
$909$	13.4164	0.444994
$910$	0	0
$911$	−27.7771	−0.920296	−0.460148	−	0.887842i	$-0.652204\pi$
$911$	−27.7771	−0.920296	−0.460148	+	0.887842i	$0.652204\pi$
$912$	0	0
$913$	12.9443	0.428393
$914$	0	0
$915$	−4.47214	−0.147844
$916$	0	0
$917$	−6.24922	−0.206368
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−26.4721	−0.872287
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1.23607	−0.0406417
$926$	0	0
$927$	−18.4721	−0.606705
$928$	0	0
$929$	−7.88854	−0.258815	−0.129407	−	0.991592i	$-0.541307\pi$
$929$	−7.88854	−0.258815	−0.129407	+	0.991592i	$0.541307\pi$
$930$	0	0
$931$	6.41641	0.210289
$932$	0	0
$933$	6.29180	0.205984
$934$	0	0
$935$	−14.4721	−0.473289
$936$	0	0
$937$	−10.5836	−0.345751	−0.172875	−	0.984944i	$-0.555306\pi$
$937$	−10.5836	−0.345751	−0.172875	+	0.984944i	$0.555306\pi$
$938$	0	0
$939$	−11.8885	−0.387968
$940$	0	0
$941$	23.1246	0.753841	0.376920	−	0.926246i	$-0.376983\pi$
$941$	23.1246	0.753841	0.376920	+	0.926246i	$0.376983\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	−0.763932	−0.0248507
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	18.4721	0.599631
$950$	0	0
$951$	−3.52786	−0.114399
$952$	0	0
$953$	25.0557	0.811635	0.405817	−	0.913954i	$-0.366987\pi$
$953$	25.0557	0.811635	0.405817	+	0.913954i	$0.366987\pi$
$954$	0	0
$955$	16.1803	0.523584
$956$	0	0
$957$	16.9443	0.547731
$958$	0	0
$959$	15.1935	0.490624
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	−8.94427	−0.288225
$964$	0	0
$965$	7.70820	0.248136
$966$	0	0
$967$	−4.18034	−0.134431	−0.0672153	−	0.997738i	$-0.521411\pi$
$967$	−4.18034	−0.134431	−0.0672153	+	0.997738i	$0.521411\pi$
$968$	0	0
$969$	−4.47214	−0.143666
$970$	0	0
$971$	−14.8328	−0.476008	−0.238004	−	0.971264i	$-0.576493\pi$
$971$	−14.8328	−0.476008	−0.238004	+	0.971264i	$0.576493\pi$
$972$	0	0
$973$	−4.22291	−0.135380
$974$	0	0
$975$	−5.23607	−0.167688
$976$	0	0
$977$	57.4164	1.83691	0.918457	−	0.395521i	$-0.129436\pi$
$977$	57.4164	1.83691	0.918457	+	0.395521i	$0.129436\pi$
$978$	0	0
$979$	−37.8885	−1.21092
$980$	0	0
$981$	4.47214	0.142784
$982$	0	0
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	3.52786	0.112407
$986$	0	0
$987$	4.94427	0.157378
$988$	0	0
$989$	−4.94427	−0.157219
$990$	0	0
$991$	−11.0557	−0.351197	−0.175598	−	0.984462i	$-0.556186\pi$
$991$	−11.0557	−0.351197	−0.175598	+	0.984462i	$0.556186\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.52786	0.302054
$996$	0	0
$997$	−25.4164	−0.804946	−0.402473	−	0.915432i	$-0.631849\pi$
$997$	−25.4164	−0.804946	−0.402473	+	0.915432i	$0.631849\pi$
$998$	0	0
$999$	1.23607	0.0391075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.be.1.2		2
4.3	odd	2		2280.2.a.p.1.1	✓	2
12.11	even	2		6840.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.p.1.1	✓	2	4.3	odd	2
4560.2.a.be.1.2		2	1.1	even	1	trivial
6840.2.a.bd.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -0.763932 q^{7} +1.00000 q^{9} -3.23607 q^{11} +5.23607 q^{13} +1.00000 q^{15} -4.47214 q^{17} -1.00000 q^{19} +0.763932 q^{21} +6.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.23607 q^{29} -8.94427 q^{31} +3.23607 q^{33} +0.763932 q^{35} -1.23607 q^{37} -5.23607 q^{39} +3.70820 q^{41} -0.763932 q^{43} -1.00000 q^{45} +6.47214 q^{47} -6.41641 q^{49} +4.47214 q^{51} +8.47214 q^{53} +3.23607 q^{55} +1.00000 q^{57} +1.52786 q^{59} -4.47214 q^{61} -0.763932 q^{63} -5.23607 q^{65} -10.4721 q^{67} -6.47214 q^{69} +3.52786 q^{73} -1.00000 q^{75} +2.47214 q^{77} +1.00000 q^{81} -4.00000 q^{83} +4.47214 q^{85} -5.23607 q^{87} +11.7082 q^{89} -4.00000 q^{91} +8.94427 q^{93} +1.00000 q^{95} -9.23607 q^{97} -3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 6 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 6 q^{13} + 2 q^{15} - 2 q^{19} + 6 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} + 2 q^{33} + 6 q^{35} + 2 q^{37} - 6 q^{39} - 6 q^{41} - 6 q^{43} - 2 q^{45} + 4 q^{47} + 14 q^{49} + 8 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 6 q^{63} - 6 q^{65} - 12 q^{67} - 4 q^{69} + 16 q^{73} - 2 q^{75} - 4 q^{77} + 2 q^{81} - 8 q^{83} - 6 q^{87} + 10 q^{89} - 8 q^{91} + 2 q^{95} - 14 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 6 * q^7 + 2 * q^9 - 2 * q^11 + 6 * q^13 + 2 * q^15 - 2 * q^19 + 6 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 6 * q^29 + 2 * q^33 + 6 * q^35 + 2 * q^37 - 6 * q^39 - 6 * q^41 - 6 * q^43 - 2 * q^45 + 4 * q^47 + 14 * q^49 + 8 * q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 - 6 * q^63 - 6 * q^65 - 12 * q^67 - 4 * q^69 + 16 * q^73 - 2 * q^75 - 4 * q^77 + 2 * q^81 - 8 * q^83 - 6 * q^87 + 10 * q^89 - 8 * q^91 + 2 * q^95 - 14 * q^97 - 2 * q^99

Introduction

L-functions

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