LMFDB - Embedded newform 6036.2.a.h.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.55581 q^{5} -4.67011 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.55581 q^{5} -4.67011 q^{7} +1.00000 q^{9} +5.87081 q^{11} -6.77432 q^{13} +2.55581 q^{15} -1.34496 q^{17} +1.52113 q^{19} -4.67011 q^{21} +4.01277 q^{23} +1.53216 q^{25} +1.00000 q^{27} -0.478914 q^{29} +9.88381 q^{31} +5.87081 q^{33} -11.9359 q^{35} +11.0233 q^{37} -6.77432 q^{39} -7.58350 q^{41} -7.58490 q^{43} +2.55581 q^{45} -1.82436 q^{47} +14.8099 q^{49} -1.34496 q^{51} -2.50332 q^{53} +15.0047 q^{55} +1.52113 q^{57} +7.69989 q^{59} +11.4374 q^{61} -4.67011 q^{63} -17.3139 q^{65} -13.9979 q^{67} +4.01277 q^{69} +9.90027 q^{71} +9.02444 q^{73} +1.53216 q^{75} -27.4173 q^{77} -1.26841 q^{79} +1.00000 q^{81} +12.2389 q^{83} -3.43746 q^{85} -0.478914 q^{87} +4.19856 q^{89} +31.6368 q^{91} +9.88381 q^{93} +3.88771 q^{95} +1.74327 q^{97} +5.87081 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * 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
$5$	2.55581	1.14299	0.571496	−	0.820605i	$-0.306363\pi$
$5$	2.55581	1.14299	0.571496	+	0.820605i	$0.306363\pi$
$6$	0	0
$7$	−4.67011	−1.76514	−0.882568	−	0.470185i	$-0.844187\pi$
$7$	−4.67011	−1.76514	−0.882568	+	0.470185i	$0.844187\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.87081	1.77012	0.885058	−	0.465481i	$-0.154119\pi$
$11$	5.87081	1.77012	0.885058	+	0.465481i	$0.154119\pi$
$12$	0	0
$13$	−6.77432	−1.87886	−0.939430	−	0.342742i	$-0.888644\pi$
$13$	−6.77432	−1.87886	−0.939430	+	0.342742i	$0.888644\pi$
$14$	0	0
$15$	2.55581	0.659907
$16$	0	0
$17$	−1.34496	−0.326201	−0.163100	−	0.986609i	$-0.552149\pi$
$17$	−1.34496	−0.326201	−0.163100	+	0.986609i	$0.552149\pi$
$18$	0	0
$19$	1.52113	0.348971	0.174485	−	0.984660i	$-0.444174\pi$
$19$	1.52113	0.348971	0.174485	+	0.984660i	$0.444174\pi$
$20$	0	0
$21$	−4.67011	−1.01910
$22$	0	0
$23$	4.01277	0.836720	0.418360	−	0.908281i	$-0.362605\pi$
$23$	4.01277	0.836720	0.418360	+	0.908281i	$0.362605\pi$
$24$	0	0
$25$	1.53216	0.306433
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.478914	−0.0889321	−0.0444661	−	0.999011i	$-0.514159\pi$
$29$	−0.478914	−0.0889321	−0.0444661	+	0.999011i	$0.514159\pi$
$30$	0	0
$31$	9.88381	1.77519	0.887593	−	0.460629i	$-0.152376\pi$
$31$	9.88381	1.77519	0.887593	+	0.460629i	$0.152376\pi$
$32$	0	0
$33$	5.87081	1.02198
$34$	0	0
$35$	−11.9359	−2.01754
$36$	0	0
$37$	11.0233	1.81221	0.906107	−	0.423048i	$-0.139040\pi$
$37$	11.0233	1.81221	0.906107	+	0.423048i	$0.139040\pi$
$38$	0	0
$39$	−6.77432	−1.08476
$40$	0	0
$41$	−7.58350	−1.18434	−0.592172	−	0.805812i	$-0.701729\pi$
$41$	−7.58350	−1.18434	−0.592172	+	0.805812i	$0.701729\pi$
$42$	0	0
$43$	−7.58490	−1.15669	−0.578343	−	0.815794i	$-0.696301\pi$
$43$	−7.58490	−1.15669	−0.578343	+	0.815794i	$0.696301\pi$
$44$	0	0
$45$	2.55581	0.380998
$46$	0	0
$47$	−1.82436	−0.266111	−0.133055	−	0.991109i	$-0.542479\pi$
$47$	−1.82436	−0.266111	−0.133055	+	0.991109i	$0.542479\pi$
$48$	0	0
$49$	14.8099	2.11571
$50$	0	0
$51$	−1.34496	−0.188332
$52$	0	0
$53$	−2.50332	−0.343857	−0.171929	−	0.985109i	$-0.555000\pi$
$53$	−2.50332	−0.343857	−0.171929	+	0.985109i	$0.555000\pi$
$54$	0	0
$55$	15.0047	2.02323
$56$	0	0
$57$	1.52113	0.201478
$58$	0	0
$59$	7.69989	1.00244	0.501220	−	0.865320i	$-0.332885\pi$
$59$	7.69989	1.00244	0.501220	+	0.865320i	$0.332885\pi$
$60$	0	0
$61$	11.4374	1.46440	0.732202	−	0.681087i	$-0.238493\pi$
$61$	11.4374	1.46440	0.732202	+	0.681087i	$0.238493\pi$
$62$	0	0
$63$	−4.67011	−0.588379
$64$	0	0
$65$	−17.3139	−2.14752
$66$	0	0
$67$	−13.9979	−1.71012	−0.855060	−	0.518529i	$-0.826480\pi$
$67$	−13.9979	−1.71012	−0.855060	+	0.518529i	$0.826480\pi$
$68$	0	0
$69$	4.01277	0.483081
$70$	0	0
$71$	9.90027	1.17495	0.587473	−	0.809244i	$-0.300123\pi$
$71$	9.90027	1.17495	0.587473	+	0.809244i	$0.300123\pi$
$72$	0	0
$73$	9.02444	1.05623	0.528116	−	0.849172i	$-0.322899\pi$
$73$	9.02444	1.05623	0.528116	+	0.849172i	$0.322899\pi$
$74$	0	0
$75$	1.53216	0.176919
$76$	0	0
$77$	−27.4173	−3.12450
$78$	0	0
$79$	−1.26841	−0.142707	−0.0713537	−	0.997451i	$-0.522732\pi$
$79$	−1.26841	−0.142707	−0.0713537	+	0.997451i	$0.522732\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.2389	1.34340	0.671699	−	0.740824i	$-0.265565\pi$
$83$	12.2389	1.34340	0.671699	+	0.740824i	$0.265565\pi$
$84$	0	0
$85$	−3.43746	−0.372845
$86$	0	0
$87$	−0.478914	−0.0513450
$88$	0	0
$89$	4.19856	0.445047	0.222523	−	0.974927i	$-0.428571\pi$
$89$	4.19856	0.445047	0.222523	+	0.974927i	$0.428571\pi$
$90$	0	0
$91$	31.6368	3.31644
$92$	0	0
$93$	9.88381	1.02490
$94$	0	0
$95$	3.88771	0.398871
$96$	0	0
$97$	1.74327	0.177002	0.0885009	−	0.996076i	$-0.471792\pi$
$97$	1.74327	0.177002	0.0885009	+	0.996076i	$0.471792\pi$
$98$	0	0
$99$	5.87081	0.590039
$100$	0	0
$101$	13.5650	1.34977	0.674883	−	0.737924i	$-0.264194\pi$
$101$	13.5650	1.34977	0.674883	+	0.737924i	$0.264194\pi$
$102$	0	0
$103$	5.41153	0.533214	0.266607	−	0.963805i	$-0.414097\pi$
$103$	5.41153	0.533214	0.266607	+	0.963805i	$0.414097\pi$
$104$	0	0
$105$	−11.9359	−1.16483
$106$	0	0
$107$	−19.3141	−1.86716	−0.933580	−	0.358368i	$-0.883333\pi$
$107$	−19.3141	−1.86716	−0.933580	+	0.358368i	$0.883333\pi$
$108$	0	0
$109$	7.62677	0.730512	0.365256	−	0.930907i	$-0.380981\pi$
$109$	7.62677	0.730512	0.365256	+	0.930907i	$0.380981\pi$
$110$	0	0
$111$	11.0233	1.04628
$112$	0	0
$113$	−7.44020	−0.699915	−0.349958	−	0.936765i	$-0.613804\pi$
$113$	−7.44020	−0.699915	−0.349958	+	0.936765i	$0.613804\pi$
$114$	0	0
$115$	10.2559	0.956365
$116$	0	0
$117$	−6.77432	−0.626286
$118$	0	0
$119$	6.28111	0.575789
$120$	0	0
$121$	23.4664	2.13331
$122$	0	0
$123$	−7.58350	−0.683781
$124$	0	0
$125$	−8.86313	−0.792743
$126$	0	0
$127$	2.23492	0.198317	0.0991585	−	0.995072i	$-0.468385\pi$
$127$	2.23492	0.198317	0.0991585	+	0.995072i	$0.468385\pi$
$128$	0	0
$129$	−7.58490	−0.667813
$130$	0	0
$131$	−11.3327	−0.990140	−0.495070	−	0.868853i	$-0.664858\pi$
$131$	−11.3327	−0.990140	−0.495070	+	0.868853i	$0.664858\pi$
$132$	0	0
$133$	−7.10384	−0.615981
$134$	0	0
$135$	2.55581	0.219969
$136$	0	0
$137$	6.99895	0.597961	0.298981	−	0.954259i	$-0.403353\pi$
$137$	6.99895	0.597961	0.298981	+	0.954259i	$0.403353\pi$
$138$	0	0
$139$	6.13501	0.520364	0.260182	−	0.965560i	$-0.416217\pi$
$139$	6.13501	0.520364	0.260182	+	0.965560i	$0.416217\pi$
$140$	0	0
$141$	−1.82436	−0.153639
$142$	0	0
$143$	−39.7708	−3.32580
$144$	0	0
$145$	−1.22401	−0.101649
$146$	0	0
$147$	14.8099	1.22150
$148$	0	0
$149$	8.45081	0.692317	0.346159	−	0.938176i	$-0.387486\pi$
$149$	8.45081	0.692317	0.346159	+	0.938176i	$0.387486\pi$
$150$	0	0
$151$	11.6831	0.950761	0.475380	−	0.879780i	$-0.342310\pi$
$151$	11.6831	0.950761	0.475380	+	0.879780i	$0.342310\pi$
$152$	0	0
$153$	−1.34496	−0.108734
$154$	0	0
$155$	25.2611	2.02902
$156$	0	0
$157$	−7.82531	−0.624528	−0.312264	−	0.949995i	$-0.601087\pi$
$157$	−7.82531	−0.624528	−0.312264	+	0.949995i	$0.601087\pi$
$158$	0	0
$159$	−2.50332	−0.198526
$160$	0	0
$161$	−18.7401	−1.47693
$162$	0	0
$163$	4.14814	0.324907	0.162454	−	0.986716i	$-0.448059\pi$
$163$	4.14814	0.324907	0.162454	+	0.986716i	$0.448059\pi$
$164$	0	0
$165$	15.0047	1.16811
$166$	0	0
$167$	18.2175	1.40971	0.704857	−	0.709350i	$-0.251011\pi$
$167$	18.2175	1.40971	0.704857	+	0.709350i	$0.251011\pi$
$168$	0	0
$169$	32.8915	2.53011
$170$	0	0
$171$	1.52113	0.116324
$172$	0	0
$173$	10.1103	0.768674	0.384337	−	0.923193i	$-0.374430\pi$
$173$	10.1103	0.768674	0.384337	+	0.923193i	$0.374430\pi$
$174$	0	0
$175$	−7.15537	−0.540895
$176$	0	0
$177$	7.69989	0.578759
$178$	0	0
$179$	−2.09723	−0.156754	−0.0783772	−	0.996924i	$-0.524974\pi$
$179$	−2.09723	−0.156754	−0.0783772	+	0.996924i	$0.524974\pi$
$180$	0	0
$181$	−21.6529	−1.60944	−0.804722	−	0.593652i	$-0.797686\pi$
$181$	−21.6529	−1.60944	−0.804722	+	0.593652i	$0.797686\pi$
$182$	0	0
$183$	11.4374	0.845474
$184$	0	0
$185$	28.1734	2.07135
$186$	0	0
$187$	−7.89601	−0.577413
$188$	0	0
$189$	−4.67011	−0.339701
$190$	0	0
$191$	20.9922	1.51894	0.759472	−	0.650540i	$-0.225457\pi$
$191$	20.9922	1.51894	0.759472	+	0.650540i	$0.225457\pi$
$192$	0	0
$193$	19.2532	1.38587	0.692936	−	0.720999i	$-0.256317\pi$
$193$	19.2532	1.38587	0.692936	+	0.720999i	$0.256317\pi$
$194$	0	0
$195$	−17.3139	−1.23987
$196$	0	0
$197$	10.3817	0.739663	0.369832	−	0.929099i	$-0.379415\pi$
$197$	10.3817	0.739663	0.369832	+	0.929099i	$0.379415\pi$
$198$	0	0
$199$	−9.12393	−0.646778	−0.323389	−	0.946266i	$-0.604822\pi$
$199$	−9.12393	−0.646778	−0.323389	+	0.946266i	$0.604822\pi$
$200$	0	0
$201$	−13.9979	−0.987338
$202$	0	0
$203$	2.23658	0.156977
$204$	0	0
$205$	−19.3820	−1.35370
$206$	0	0
$207$	4.01277	0.278907
$208$	0	0
$209$	8.93025	0.617719
$210$	0	0
$211$	−20.7159	−1.42614	−0.713070	−	0.701093i	$-0.752696\pi$
$211$	−20.7159	−1.42614	−0.713070	+	0.701093i	$0.752696\pi$
$212$	0	0
$213$	9.90027	0.678355
$214$	0	0
$215$	−19.3856	−1.32208
$216$	0	0
$217$	−46.1585	−3.13344
$218$	0	0
$219$	9.02444	0.609815
$220$	0	0
$221$	9.11119	0.612885
$222$	0	0
$223$	7.32647	0.490617	0.245308	−	0.969445i	$-0.421111\pi$
$223$	7.32647	0.490617	0.245308	+	0.969445i	$0.421111\pi$
$224$	0	0
$225$	1.53216	0.102144
$226$	0	0
$227$	−11.0203	−0.731444	−0.365722	−	0.930724i	$-0.619178\pi$
$227$	−11.0203	−0.731444	−0.365722	+	0.930724i	$0.619178\pi$
$228$	0	0
$229$	3.68774	0.243693	0.121847	−	0.992549i	$-0.461118\pi$
$229$	3.68774	0.243693	0.121847	+	0.992549i	$0.461118\pi$
$230$	0	0
$231$	−27.4173	−1.80393
$232$	0	0
$233$	0.667641	0.0437386	0.0218693	−	0.999761i	$-0.493038\pi$
$233$	0.667641	0.0437386	0.0218693	+	0.999761i	$0.493038\pi$
$234$	0	0
$235$	−4.66273	−0.304163
$236$	0	0
$237$	−1.26841	−0.0823921
$238$	0	0
$239$	1.20310	0.0778222	0.0389111	−	0.999243i	$-0.487611\pi$
$239$	1.20310	0.0778222	0.0389111	+	0.999243i	$0.487611\pi$
$240$	0	0
$241$	9.81010	0.631924	0.315962	−	0.948772i	$-0.397673\pi$
$241$	9.81010	0.631924	0.315962	+	0.948772i	$0.397673\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	37.8514	2.41824
$246$	0	0
$247$	−10.3046	−0.655667
$248$	0	0
$249$	12.2389	0.775611
$250$	0	0
$251$	−27.8900	−1.76040	−0.880200	−	0.474604i	$-0.842591\pi$
$251$	−27.8900	−1.76040	−0.880200	+	0.474604i	$0.842591\pi$
$252$	0	0
$253$	23.5582	1.48109
$254$	0	0
$255$	−3.43746	−0.215262
$256$	0	0
$257$	7.71216	0.481071	0.240536	−	0.970640i	$-0.422677\pi$
$257$	7.71216	0.481071	0.240536	+	0.970640i	$0.422677\pi$
$258$	0	0
$259$	−51.4799	−3.19881
$260$	0	0
$261$	−0.478914	−0.0296440
$262$	0	0
$263$	−21.6089	−1.33246	−0.666232	−	0.745744i	$-0.732094\pi$
$263$	−21.6089	−1.33246	−0.666232	+	0.745744i	$0.732094\pi$
$264$	0	0
$265$	−6.39801	−0.393027
$266$	0	0
$267$	4.19856	0.256948
$268$	0	0
$269$	17.7714	1.08354	0.541772	−	0.840526i	$-0.317754\pi$
$269$	17.7714	1.08354	0.541772	+	0.840526i	$0.317754\pi$
$270$	0	0
$271$	1.92995	0.117236	0.0586181	−	0.998280i	$-0.481331\pi$
$271$	1.92995	0.117236	0.0586181	+	0.998280i	$0.481331\pi$
$272$	0	0
$273$	31.6368	1.91475
$274$	0	0
$275$	8.99504	0.542421
$276$	0	0
$277$	9.14189	0.549283	0.274641	−	0.961547i	$-0.411441\pi$
$277$	9.14189	0.549283	0.274641	+	0.961547i	$0.411441\pi$
$278$	0	0
$279$	9.88381	0.591728
$280$	0	0
$281$	−10.6096	−0.632913	−0.316456	−	0.948607i	$-0.602493\pi$
$281$	−10.6096	−0.632913	−0.316456	+	0.948607i	$0.602493\pi$
$282$	0	0
$283$	−29.5140	−1.75442	−0.877212	−	0.480103i	$-0.840599\pi$
$283$	−29.5140	−1.75442	−0.877212	+	0.480103i	$0.840599\pi$
$284$	0	0
$285$	3.88771	0.230288
$286$	0	0
$287$	35.4158	2.09053
$288$	0	0
$289$	−15.1911	−0.893593
$290$	0	0
$291$	1.74327	0.102192
$292$	0	0
$293$	8.25344	0.482171	0.241086	−	0.970504i	$-0.422497\pi$
$293$	8.25344	0.482171	0.241086	+	0.970504i	$0.422497\pi$
$294$	0	0
$295$	19.6794	1.14578
$296$	0	0
$297$	5.87081	0.340659
$298$	0	0
$299$	−27.1838	−1.57208
$300$	0	0
$301$	35.4223	2.04171
$302$	0	0
$303$	13.5650	0.779288
$304$	0	0
$305$	29.2317	1.67380
$306$	0	0
$307$	5.05056	0.288251	0.144125	−	0.989559i	$-0.453963\pi$
$307$	5.05056	0.288251	0.144125	+	0.989559i	$0.453963\pi$
$308$	0	0
$309$	5.41153	0.307851
$310$	0	0
$311$	4.22978	0.239849	0.119924	−	0.992783i	$-0.461735\pi$
$311$	4.22978	0.239849	0.119924	+	0.992783i	$0.461735\pi$
$312$	0	0
$313$	1.33044	0.0752009	0.0376005	−	0.999293i	$-0.488029\pi$
$313$	1.33044	0.0752009	0.0376005	+	0.999293i	$0.488029\pi$
$314$	0	0
$315$	−11.9359	−0.672513
$316$	0	0
$317$	23.2285	1.30464	0.652321	−	0.757943i	$-0.273796\pi$
$317$	23.2285	1.30464	0.652321	+	0.757943i	$0.273796\pi$
$318$	0	0
$319$	−2.81161	−0.157420
$320$	0	0
$321$	−19.3141	−1.07801
$322$	0	0
$323$	−2.04586	−0.113835
$324$	0	0
$325$	−10.3794	−0.575744
$326$	0	0
$327$	7.62677	0.421761
$328$	0	0
$329$	8.51999	0.469722
$330$	0	0
$331$	12.9244	0.710389	0.355195	−	0.934792i	$-0.384415\pi$
$331$	12.9244	0.710389	0.355195	+	0.934792i	$0.384415\pi$
$332$	0	0
$333$	11.0233	0.604072
$334$	0	0
$335$	−35.7761	−1.95465
$336$	0	0
$337$	14.8928	0.811261	0.405631	−	0.914037i	$-0.367052\pi$
$337$	14.8928	0.811261	0.405631	+	0.914037i	$0.367052\pi$
$338$	0	0
$339$	−7.44020	−0.404096
$340$	0	0
$341$	58.0260	3.14228
$342$	0	0
$343$	−36.4733	−1.96937
$344$	0	0
$345$	10.2559	0.552158
$346$	0	0
$347$	−2.72028	−0.146032	−0.0730162	−	0.997331i	$-0.523262\pi$
$347$	−2.72028	−0.146032	−0.0730162	+	0.997331i	$0.523262\pi$
$348$	0	0
$349$	−1.53029	−0.0819148	−0.0409574	−	0.999161i	$-0.513041\pi$
$349$	−1.53029	−0.0819148	−0.0409574	+	0.999161i	$0.513041\pi$
$350$	0	0
$351$	−6.77432	−0.361587
$352$	0	0
$353$	26.4548	1.40805	0.704025	−	0.710175i	$-0.251384\pi$
$353$	26.4548	1.40805	0.704025	+	0.710175i	$0.251384\pi$
$354$	0	0
$355$	25.3032	1.34295
$356$	0	0
$357$	6.28111	0.332432
$358$	0	0
$359$	10.9899	0.580025	0.290012	−	0.957023i	$-0.406341\pi$
$359$	10.9899	0.580025	0.290012	+	0.957023i	$0.406341\pi$
$360$	0	0
$361$	−16.6862	−0.878219
$362$	0	0
$363$	23.4664	1.23167
$364$	0	0
$365$	23.0648	1.20726
$366$	0	0
$367$	−24.1738	−1.26186	−0.630930	−	0.775840i	$-0.717327\pi$
$367$	−24.1738	−1.26186	−0.630930	+	0.775840i	$0.717327\pi$
$368$	0	0
$369$	−7.58350	−0.394781
$370$	0	0
$371$	11.6908	0.606955
$372$	0	0
$373$	33.1323	1.71552	0.857762	−	0.514047i	$-0.171854\pi$
$373$	33.1323	1.71552	0.857762	+	0.514047i	$0.171854\pi$
$374$	0	0
$375$	−8.86313	−0.457690
$376$	0	0
$377$	3.24432	0.167091
$378$	0	0
$379$	17.8549	0.917146	0.458573	−	0.888657i	$-0.348361\pi$
$379$	17.8549	0.917146	0.458573	+	0.888657i	$0.348361\pi$
$380$	0	0
$381$	2.23492	0.114498
$382$	0	0
$383$	−32.0277	−1.63654	−0.818270	−	0.574833i	$-0.805067\pi$
$383$	−32.0277	−1.63654	−0.818270	+	0.574833i	$0.805067\pi$
$384$	0	0
$385$	−70.0735	−3.57128
$386$	0	0
$387$	−7.58490	−0.385562
$388$	0	0
$389$	−33.0964	−1.67805	−0.839027	−	0.544090i	$-0.816875\pi$
$389$	−33.0964	−1.67805	−0.839027	+	0.544090i	$0.816875\pi$
$390$	0	0
$391$	−5.39702	−0.272939
$392$	0	0
$393$	−11.3327	−0.571658
$394$	0	0
$395$	−3.24182	−0.163113
$396$	0	0
$397$	−0.685814	−0.0344200	−0.0172100	−	0.999852i	$-0.505478\pi$
$397$	−0.685814	−0.0344200	−0.0172100	+	0.999852i	$0.505478\pi$
$398$	0	0
$399$	−7.10384	−0.355637
$400$	0	0
$401$	−24.3447	−1.21571	−0.607857	−	0.794046i	$-0.707971\pi$
$401$	−24.3447	−1.21571	−0.607857	+	0.794046i	$0.707971\pi$
$402$	0	0
$403$	−66.9561	−3.33532
$404$	0	0
$405$	2.55581	0.126999
$406$	0	0
$407$	64.7155	3.20783
$408$	0	0
$409$	0.855618	0.0423076	0.0211538	−	0.999776i	$-0.493266\pi$
$409$	0.855618	0.0423076	0.0211538	+	0.999776i	$0.493266\pi$
$410$	0	0
$411$	6.99895	0.345233
$412$	0	0
$413$	−35.9593	−1.76944
$414$	0	0
$415$	31.2804	1.53549
$416$	0	0
$417$	6.13501	0.300433
$418$	0	0
$419$	0.797171	0.0389443	0.0194722	−	0.999810i	$-0.493801\pi$
$419$	0.797171	0.0389443	0.0194722	+	0.999810i	$0.493801\pi$
$420$	0	0
$421$	−13.9880	−0.681731	−0.340866	−	0.940112i	$-0.610720\pi$
$421$	−13.9880	−0.681731	−0.340866	+	0.940112i	$0.610720\pi$
$422$	0	0
$423$	−1.82436	−0.0887036
$424$	0	0
$425$	−2.06070	−0.0999586
$426$	0	0
$427$	−53.4138	−2.58487
$428$	0	0
$429$	−39.7708	−1.92015
$430$	0	0
$431$	−33.3137	−1.60466	−0.802332	−	0.596878i	$-0.796408\pi$
$431$	−33.3137	−1.60466	−0.802332	+	0.596878i	$0.796408\pi$
$432$	0	0
$433$	0.787293	0.0378349	0.0189174	−	0.999821i	$-0.493978\pi$
$433$	0.787293	0.0378349	0.0189174	+	0.999821i	$0.493978\pi$
$434$	0	0
$435$	−1.22401	−0.0586870
$436$	0	0
$437$	6.10394	0.291991
$438$	0	0
$439$	−17.3844	−0.829715	−0.414857	−	0.909887i	$-0.636168\pi$
$439$	−17.3844	−0.829715	−0.414857	+	0.909887i	$0.636168\pi$
$440$	0	0
$441$	14.8099	0.705235
$442$	0	0
$443$	−18.8819	−0.897109	−0.448554	−	0.893756i	$-0.648061\pi$
$443$	−18.8819	−0.897109	−0.448554	+	0.893756i	$0.648061\pi$
$444$	0	0
$445$	10.7307	0.508685
$446$	0	0
$447$	8.45081	0.399709
$448$	0	0
$449$	−38.2908	−1.80706	−0.903528	−	0.428529i	$-0.859032\pi$
$449$	−38.2908	−1.80706	−0.903528	+	0.428529i	$0.859032\pi$
$450$	0	0
$451$	−44.5213	−2.09643
$452$	0	0
$453$	11.6831	0.548922
$454$	0	0
$455$	80.8577	3.79067
$456$	0	0
$457$	−17.1985	−0.804510	−0.402255	−	0.915528i	$-0.631773\pi$
$457$	−17.1985	−0.804510	−0.402255	+	0.915528i	$0.631773\pi$
$458$	0	0
$459$	−1.34496	−0.0627774
$460$	0	0
$461$	40.1652	1.87068	0.935340	−	0.353751i	$-0.115094\pi$
$461$	40.1652	1.87068	0.935340	+	0.353751i	$0.115094\pi$
$462$	0	0
$463$	4.00180	0.185979	0.0929896	−	0.995667i	$-0.470358\pi$
$463$	4.00180	0.185979	0.0929896	+	0.995667i	$0.470358\pi$
$464$	0	0
$465$	25.2611	1.17146
$466$	0	0
$467$	−5.36203	−0.248125	−0.124063	−	0.992274i	$-0.539592\pi$
$467$	−5.36203	−0.248125	−0.124063	+	0.992274i	$0.539592\pi$
$468$	0	0
$469$	65.3719	3.01859
$470$	0	0
$471$	−7.82531	−0.360571
$472$	0	0
$473$	−44.5295	−2.04747
$474$	0	0
$475$	2.33062	0.106936
$476$	0	0
$477$	−2.50332	−0.114619
$478$	0	0
$479$	−24.1261	−1.10235	−0.551175	−	0.834390i	$-0.685820\pi$
$479$	−24.1261	−1.10235	−0.551175	+	0.834390i	$0.685820\pi$
$480$	0	0
$481$	−74.6752	−3.40490
$482$	0	0
$483$	−18.7401	−0.852703
$484$	0	0
$485$	4.45546	0.202312
$486$	0	0
$487$	14.0085	0.634785	0.317392	−	0.948294i	$-0.397193\pi$
$487$	14.0085	0.634785	0.317392	+	0.948294i	$0.397193\pi$
$488$	0	0
$489$	4.14814	0.187585
$490$	0	0
$491$	13.3693	0.603350	0.301675	−	0.953411i	$-0.402454\pi$
$491$	13.3693	0.603350	0.301675	+	0.953411i	$0.402454\pi$
$492$	0	0
$493$	0.644120	0.0290097
$494$	0	0
$495$	15.0047	0.674410
$496$	0	0
$497$	−46.2354	−2.07394
$498$	0	0
$499$	37.2455	1.66734	0.833668	−	0.552266i	$-0.186237\pi$
$499$	37.2455	1.66734	0.833668	+	0.552266i	$0.186237\pi$
$500$	0	0
$501$	18.2175	0.813899
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	34.6695	1.54277
$506$	0	0
$507$	32.8915	1.46076
$508$	0	0
$509$	−35.1896	−1.55975	−0.779877	−	0.625933i	$-0.784718\pi$
$509$	−35.1896	−1.55975	−0.779877	+	0.625933i	$0.784718\pi$
$510$	0	0
$511$	−42.1452	−1.86439
$512$	0	0
$513$	1.52113	0.0671594
$514$	0	0
$515$	13.8308	0.609459
$516$	0	0
$517$	−10.7105	−0.471047
$518$	0	0
$519$	10.1103	0.443794
$520$	0	0
$521$	15.5455	0.681062	0.340531	−	0.940233i	$-0.389393\pi$
$521$	15.5455	0.681062	0.340531	+	0.940233i	$0.389393\pi$
$522$	0	0
$523$	−11.2993	−0.494083	−0.247041	−	0.969005i	$-0.579458\pi$
$523$	−11.2993	−0.494083	−0.247041	+	0.969005i	$0.579458\pi$
$524$	0	0
$525$	−7.15537	−0.312286
$526$	0	0
$527$	−13.2933	−0.579067
$528$	0	0
$529$	−6.89767	−0.299899
$530$	0	0
$531$	7.69989	0.334147
$532$	0	0
$533$	51.3731	2.22521
$534$	0	0
$535$	−49.3631	−2.13415
$536$	0	0
$537$	−2.09723	−0.0905022
$538$	0	0
$539$	86.9464	3.74504
$540$	0	0
$541$	20.4368	0.878647	0.439323	−	0.898329i	$-0.355218\pi$
$541$	20.4368	0.878647	0.439323	+	0.898329i	$0.355218\pi$
$542$	0	0
$543$	−21.6529	−0.929213
$544$	0	0
$545$	19.4926	0.834970
$546$	0	0
$547$	−31.3363	−1.33984	−0.669922	−	0.742432i	$-0.733672\pi$
$547$	−31.3363	−1.33984	−0.669922	+	0.742432i	$0.733672\pi$
$548$	0	0
$549$	11.4374	0.488135
$550$	0	0
$551$	−0.728490	−0.0310347
$552$	0	0
$553$	5.92362	0.251898
$554$	0	0
$555$	28.1734	1.19589
$556$	0	0
$557$	9.63959	0.408443	0.204221	−	0.978925i	$-0.434534\pi$
$557$	9.63959	0.408443	0.204221	+	0.978925i	$0.434534\pi$
$558$	0	0
$559$	51.3826	2.17325
$560$	0	0
$561$	−7.89601	−0.333370
$562$	0	0
$563$	18.3955	0.775278	0.387639	−	0.921811i	$-0.373291\pi$
$563$	18.3955	0.775278	0.387639	+	0.921811i	$0.373291\pi$
$564$	0	0
$565$	−19.0157	−0.799998
$566$	0	0
$567$	−4.67011	−0.196126
$568$	0	0
$569$	−38.0882	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$569$	−38.0882	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$570$	0	0
$571$	6.64219	0.277967	0.138983	−	0.990295i	$-0.455617\pi$
$571$	6.64219	0.277967	0.138983	+	0.990295i	$0.455617\pi$
$572$	0	0
$573$	20.9922	0.876963
$574$	0	0
$575$	6.14822	0.256398
$576$	0	0
$577$	−15.6371	−0.650982	−0.325491	−	0.945545i	$-0.605530\pi$
$577$	−15.6371	−0.650982	−0.325491	+	0.945545i	$0.605530\pi$
$578$	0	0
$579$	19.2532	0.800134
$580$	0	0
$581$	−57.1572	−2.37128
$582$	0	0
$583$	−14.6965	−0.608667
$584$	0	0
$585$	−17.3139	−0.715841
$586$	0	0
$587$	−21.9472	−0.905860	−0.452930	−	0.891546i	$-0.649621\pi$
$587$	−21.9472	−0.905860	−0.452930	+	0.891546i	$0.649621\pi$
$588$	0	0
$589$	15.0345	0.619488
$590$	0	0
$591$	10.3817	0.427045
$592$	0	0
$593$	−33.9007	−1.39214	−0.696068	−	0.717976i	$-0.745069\pi$
$593$	−33.9007	−1.39214	−0.696068	+	0.717976i	$0.745069\pi$
$594$	0	0
$595$	16.0533	0.658122
$596$	0	0
$597$	−9.12393	−0.373418
$598$	0	0
$599$	18.4364	0.753291	0.376645	−	0.926357i	$-0.377077\pi$
$599$	18.4364	0.753291	0.376645	+	0.926357i	$0.377077\pi$
$600$	0	0
$601$	43.8168	1.78732	0.893661	−	0.448742i	$-0.148128\pi$
$601$	43.8168	1.78732	0.893661	+	0.448742i	$0.148128\pi$
$602$	0	0
$603$	−13.9979	−0.570040
$604$	0	0
$605$	59.9757	2.43836
$606$	0	0
$607$	−15.1831	−0.616265	−0.308132	−	0.951343i	$-0.599704\pi$
$607$	−15.1831	−0.616265	−0.308132	+	0.951343i	$0.599704\pi$
$608$	0	0
$609$	2.23658	0.0906309
$610$	0	0
$611$	12.3588	0.499985
$612$	0	0
$613$	−22.3557	−0.902938	−0.451469	−	0.892287i	$-0.649100\pi$
$613$	−22.3557	−0.902938	−0.451469	+	0.892287i	$0.649100\pi$
$614$	0	0
$615$	−19.3820	−0.781557
$616$	0	0
$617$	14.3140	0.576259	0.288129	−	0.957592i	$-0.406967\pi$
$617$	14.3140	0.576259	0.288129	+	0.957592i	$0.406967\pi$
$618$	0	0
$619$	−24.0696	−0.967437	−0.483718	−	0.875224i	$-0.660714\pi$
$619$	−24.0696	−0.967437	−0.483718	+	0.875224i	$0.660714\pi$
$620$	0	0
$621$	4.01277	0.161027
$622$	0	0
$623$	−19.6078	−0.785568
$624$	0	0
$625$	−30.3133	−1.21253
$626$	0	0
$627$	8.93025	0.356640
$628$	0	0
$629$	−14.8259	−0.591146
$630$	0	0
$631$	22.5406	0.897326	0.448663	−	0.893701i	$-0.351900\pi$
$631$	22.5406	0.897326	0.448663	+	0.893701i	$0.351900\pi$
$632$	0	0
$633$	−20.7159	−0.823383
$634$	0	0
$635$	5.71203	0.226675
$636$	0	0
$637$	−100.327	−3.97511
$638$	0	0
$639$	9.90027	0.391649
$640$	0	0
$641$	−33.0620	−1.30587	−0.652935	−	0.757414i	$-0.726462\pi$
$641$	−33.0620	−1.30587	−0.652935	+	0.757414i	$0.726462\pi$
$642$	0	0
$643$	40.3677	1.59195	0.795974	−	0.605331i	$-0.206959\pi$
$643$	40.3677	1.59195	0.795974	+	0.605331i	$0.206959\pi$
$644$	0	0
$645$	−19.3856	−0.763306
$646$	0	0
$647$	40.4653	1.59085	0.795427	−	0.606049i	$-0.207247\pi$
$647$	40.4653	1.59085	0.795427	+	0.606049i	$0.207247\pi$
$648$	0	0
$649$	45.2046	1.77444
$650$	0	0
$651$	−46.1585	−1.80909
$652$	0	0
$653$	42.3262	1.65635	0.828176	−	0.560468i	$-0.189379\pi$
$653$	42.3262	1.65635	0.828176	+	0.560468i	$0.189379\pi$
$654$	0	0
$655$	−28.9641	−1.13172
$656$	0	0
$657$	9.02444	0.352077
$658$	0	0
$659$	20.3155	0.791380	0.395690	−	0.918384i	$-0.370505\pi$
$659$	20.3155	0.791380	0.395690	+	0.918384i	$0.370505\pi$
$660$	0	0
$661$	34.3592	1.33642	0.668210	−	0.743973i	$-0.267061\pi$
$661$	34.3592	1.33642	0.668210	+	0.743973i	$0.267061\pi$
$662$	0	0
$663$	9.11119	0.353849
$664$	0	0
$665$	−18.1561	−0.704062
$666$	0	0
$667$	−1.92177	−0.0744113
$668$	0	0
$669$	7.32647	0.283258
$670$	0	0
$671$	67.1466	2.59217
$672$	0	0
$673$	44.8383	1.72839	0.864195	−	0.503156i	$-0.167828\pi$
$673$	44.8383	1.72839	0.864195	+	0.503156i	$0.167828\pi$
$674$	0	0
$675$	1.53216	0.0589730
$676$	0	0
$677$	−20.1485	−0.774370	−0.387185	−	0.922002i	$-0.626553\pi$
$677$	−20.1485	−0.774370	−0.387185	+	0.922002i	$0.626553\pi$
$678$	0	0
$679$	−8.14125	−0.312432
$680$	0	0
$681$	−11.0203	−0.422299
$682$	0	0
$683$	25.4928	0.975455	0.487728	−	0.872996i	$-0.337826\pi$
$683$	25.4928	0.975455	0.487728	+	0.872996i	$0.337826\pi$
$684$	0	0
$685$	17.8880	0.683465
$686$	0	0
$687$	3.68774	0.140696
$688$	0	0
$689$	16.9583	0.646060
$690$	0	0
$691$	31.6717	1.20485	0.602425	−	0.798175i	$-0.294201\pi$
$691$	31.6717	1.20485	0.602425	+	0.798175i	$0.294201\pi$
$692$	0	0
$693$	−27.4173	−1.04150
$694$	0	0
$695$	15.6799	0.594773
$696$	0	0
$697$	10.1995	0.386334
$698$	0	0
$699$	0.667641	0.0252525
$700$	0	0
$701$	−15.2465	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$701$	−15.2465	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$702$	0	0
$703$	16.7678	0.632410
$704$	0	0
$705$	−4.66273	−0.175609
$706$	0	0
$707$	−63.3500	−2.38252
$708$	0	0
$709$	4.53740	0.170406	0.0852028	−	0.996364i	$-0.472846\pi$
$709$	4.53740	0.170406	0.0852028	+	0.996364i	$0.472846\pi$
$710$	0	0
$711$	−1.26841	−0.0475691
$712$	0	0
$713$	39.6615	1.48533
$714$	0	0
$715$	−101.647	−3.80136
$716$	0	0
$717$	1.20310	0.0449306
$718$	0	0
$719$	−41.5349	−1.54899	−0.774495	−	0.632580i	$-0.781996\pi$
$719$	−41.5349	−1.54899	−0.774495	+	0.632580i	$0.781996\pi$
$720$	0	0
$721$	−25.2724	−0.941195
$722$	0	0
$723$	9.81010	0.364842
$724$	0	0
$725$	−0.733775	−0.0272517
$726$	0	0
$727$	−8.39492	−0.311350	−0.155675	−	0.987808i	$-0.549755\pi$
$727$	−8.39492	−0.311350	−0.155675	+	0.987808i	$0.549755\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.2014	0.377312
$732$	0	0
$733$	21.4259	0.791382	0.395691	−	0.918384i	$-0.370505\pi$
$733$	21.4259	0.791382	0.395691	+	0.918384i	$0.370505\pi$
$734$	0	0
$735$	37.8514	1.39617
$736$	0	0
$737$	−82.1792	−3.02711
$738$	0	0
$739$	−6.73457	−0.247735	−0.123867	−	0.992299i	$-0.539530\pi$
$739$	−6.73457	−0.247735	−0.123867	+	0.992299i	$0.539530\pi$
$740$	0	0
$741$	−10.3046	−0.378549
$742$	0	0
$743$	4.81731	0.176730	0.0883650	−	0.996088i	$-0.471836\pi$
$743$	4.81731	0.176730	0.0883650	+	0.996088i	$0.471836\pi$
$744$	0	0
$745$	21.5987	0.791313
$746$	0	0
$747$	12.2389	0.447799
$748$	0	0
$749$	90.1988	3.29579
$750$	0	0
$751$	−15.8984	−0.580141	−0.290071	−	0.957005i	$-0.593679\pi$
$751$	−15.8984	−0.580141	−0.290071	+	0.957005i	$0.593679\pi$
$752$	0	0
$753$	−27.8900	−1.01637
$754$	0	0
$755$	29.8599	1.08671
$756$	0	0
$757$	13.1696	0.478657	0.239329	−	0.970939i	$-0.423073\pi$
$757$	13.1696	0.478657	0.239329	+	0.970939i	$0.423073\pi$
$758$	0	0
$759$	23.5582	0.855109
$760$	0	0
$761$	35.8406	1.29922	0.649610	−	0.760268i	$-0.274932\pi$
$761$	35.8406	1.29922	0.649610	+	0.760268i	$0.274932\pi$
$762$	0	0
$763$	−35.6179	−1.28945
$764$	0	0
$765$	−3.43746	−0.124282
$766$	0	0
$767$	−52.1615	−1.88344
$768$	0	0
$769$	21.6510	0.780754	0.390377	−	0.920655i	$-0.372345\pi$
$769$	21.6510	0.780754	0.390377	+	0.920655i	$0.372345\pi$
$770$	0	0
$771$	7.71216	0.277747
$772$	0	0
$773$	−5.24131	−0.188517	−0.0942585	−	0.995548i	$-0.530048\pi$
$773$	−5.24131	−0.188517	−0.0942585	+	0.995548i	$0.530048\pi$
$774$	0	0
$775$	15.1436	0.543975
$776$	0	0
$777$	−51.4799	−1.84683
$778$	0	0
$779$	−11.5355	−0.413301
$780$	0	0
$781$	58.1226	2.07979
$782$	0	0
$783$	−0.478914	−0.0171150
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	−17.1625	−0.611777	−0.305889	−	0.952067i	$-0.598953\pi$
$787$	−17.1625	−0.611777	−0.305889	+	0.952067i	$0.598953\pi$
$788$	0	0
$789$	−21.6089	−0.769299
$790$	0	0
$791$	34.7466	1.23545
$792$	0	0
$793$	−77.4804	−2.75141
$794$	0	0
$795$	−6.39801	−0.226914
$796$	0	0
$797$	8.86646	0.314066	0.157033	−	0.987593i	$-0.449807\pi$
$797$	8.86646	0.314066	0.157033	+	0.987593i	$0.449807\pi$
$798$	0	0
$799$	2.45370	0.0868056
$800$	0	0
$801$	4.19856	0.148349
$802$	0	0
$803$	52.9808	1.86965
$804$	0	0
$805$	−47.8961	−1.68812
$806$	0	0
$807$	17.7714	0.625584
$808$	0	0
$809$	−38.6406	−1.35853	−0.679266	−	0.733892i	$-0.737702\pi$
$809$	−38.6406	−1.35853	−0.679266	+	0.733892i	$0.737702\pi$
$810$	0	0
$811$	9.61047	0.337469	0.168735	−	0.985662i	$-0.446032\pi$
$811$	9.61047	0.337469	0.168735	+	0.985662i	$0.446032\pi$
$812$	0	0
$813$	1.92995	0.0676864
$814$	0	0
$815$	10.6018	0.371367
$816$	0	0
$817$	−11.5376	−0.403650
$818$	0	0
$819$	31.6368	1.10548
$820$	0	0
$821$	−2.48821	−0.0868390	−0.0434195	−	0.999057i	$-0.513825\pi$
$821$	−2.48821	−0.0868390	−0.0434195	+	0.999057i	$0.513825\pi$
$822$	0	0
$823$	36.8236	1.28359	0.641795	−	0.766877i	$-0.278190\pi$
$823$	36.8236	1.28359	0.641795	+	0.766877i	$0.278190\pi$
$824$	0	0
$825$	8.99504	0.313167
$826$	0	0
$827$	−8.18795	−0.284723	−0.142361	−	0.989815i	$-0.545470\pi$
$827$	−8.18795	−0.284723	−0.142361	+	0.989815i	$0.545470\pi$
$828$	0	0
$829$	−43.7931	−1.52100	−0.760498	−	0.649340i	$-0.775045\pi$
$829$	−43.7931	−1.52100	−0.760498	+	0.649340i	$0.775045\pi$
$830$	0	0
$831$	9.14189	0.317129
$832$	0	0
$833$	−19.9188	−0.690145
$834$	0	0
$835$	46.5605	1.61129
$836$	0	0
$837$	9.88381	0.341635
$838$	0	0
$839$	35.1339	1.21296	0.606478	−	0.795100i	$-0.292582\pi$
$839$	35.1339	1.21296	0.606478	+	0.795100i	$0.292582\pi$
$840$	0	0
$841$	−28.7706	−0.992091
$842$	0	0
$843$	−10.6096	−0.365412
$844$	0	0
$845$	84.0643	2.89190
$846$	0	0
$847$	−109.591	−3.76558
$848$	0	0
$849$	−29.5140	−1.01292
$850$	0	0
$851$	44.2339	1.51632
$852$	0	0
$853$	47.7915	1.63635	0.818175	−	0.574969i	$-0.194986\pi$
$853$	47.7915	1.63635	0.818175	+	0.574969i	$0.194986\pi$
$854$	0	0
$855$	3.88771	0.132957
$856$	0	0
$857$	39.7029	1.35623	0.678113	−	0.734958i	$-0.262798\pi$
$857$	39.7029	1.35623	0.678113	+	0.734958i	$0.262798\pi$
$858$	0	0
$859$	20.8534	0.711509	0.355755	−	0.934579i	$-0.384224\pi$
$859$	20.8534	0.711509	0.355755	+	0.934579i	$0.384224\pi$
$860$	0	0
$861$	35.4158	1.20697
$862$	0	0
$863$	14.2827	0.486188	0.243094	−	0.970003i	$-0.421838\pi$
$863$	14.2827	0.486188	0.243094	+	0.970003i	$0.421838\pi$
$864$	0	0
$865$	25.8401	0.878589
$866$	0	0
$867$	−15.1911	−0.515916
$868$	0	0
$869$	−7.44660	−0.252609
$870$	0	0
$871$	94.8265	3.21307
$872$	0	0
$873$	1.74327	0.0590006
$874$	0	0
$875$	41.3918	1.39930
$876$	0	0
$877$	−10.6732	−0.360407	−0.180204	−	0.983629i	$-0.557676\pi$
$877$	−10.6732	−0.360407	−0.180204	+	0.983629i	$0.557676\pi$
$878$	0	0
$879$	8.25344	0.278382
$880$	0	0
$881$	−34.1833	−1.15166	−0.575832	−	0.817568i	$-0.695322\pi$
$881$	−34.1833	−1.15166	−0.575832	+	0.817568i	$0.695322\pi$
$882$	0	0
$883$	23.2668	0.782989	0.391494	−	0.920181i	$-0.371958\pi$
$883$	23.2668	0.782989	0.391494	+	0.920181i	$0.371958\pi$
$884$	0	0
$885$	19.6794	0.661517
$886$	0	0
$887$	−26.0192	−0.873638	−0.436819	−	0.899549i	$-0.643895\pi$
$887$	−26.0192	−0.873638	−0.436819	+	0.899549i	$0.643895\pi$
$888$	0	0
$889$	−10.4373	−0.350057
$890$	0	0
$891$	5.87081	0.196680
$892$	0	0
$893$	−2.77509	−0.0928649
$894$	0	0
$895$	−5.36012	−0.179169
$896$	0	0
$897$	−27.1838	−0.907641
$898$	0	0
$899$	−4.73350	−0.157871
$900$	0	0
$901$	3.36687	0.112167
$902$	0	0
$903$	35.4223	1.17878
$904$	0	0
$905$	−55.3406	−1.83958
$906$	0	0
$907$	−14.3682	−0.477089	−0.238544	−	0.971132i	$-0.576670\pi$
$907$	−14.3682	−0.477089	−0.238544	+	0.971132i	$0.576670\pi$
$908$	0	0
$909$	13.5650	0.449922
$910$	0	0
$911$	3.65314	0.121034	0.0605170	−	0.998167i	$-0.480725\pi$
$911$	3.65314	0.121034	0.0605170	+	0.998167i	$0.480725\pi$
$912$	0	0
$913$	71.8525	2.37797
$914$	0	0
$915$	29.2317	0.966371
$916$	0	0
$917$	52.9248	1.74773
$918$	0	0
$919$	−18.4798	−0.609591	−0.304796	−	0.952418i	$-0.598588\pi$
$919$	−18.4798	−0.609591	−0.304796	+	0.952418i	$0.598588\pi$
$920$	0	0
$921$	5.05056	0.166422
$922$	0	0
$923$	−67.0676	−2.20756
$924$	0	0
$925$	16.8895	0.555322
$926$	0	0
$927$	5.41153	0.177738
$928$	0	0
$929$	−1.43953	−0.0472294	−0.0236147	−	0.999721i	$-0.507517\pi$
$929$	−1.43953	−0.0472294	−0.0236147	+	0.999721i	$0.507517\pi$
$930$	0	0
$931$	22.5278	0.738319
$932$	0	0
$933$	4.22978	0.138477
$934$	0	0
$935$	−20.1807	−0.659979
$936$	0	0
$937$	3.41623	0.111603	0.0558016	−	0.998442i	$-0.482229\pi$
$937$	3.41623	0.111603	0.0558016	+	0.998442i	$0.482229\pi$
$938$	0	0
$939$	1.33044	0.0434173
$940$	0	0
$941$	13.4749	0.439269	0.219635	−	0.975582i	$-0.429513\pi$
$941$	13.4749	0.439269	0.219635	+	0.975582i	$0.429513\pi$
$942$	0	0
$943$	−30.4308	−0.990964
$944$	0	0
$945$	−11.9359	−0.388275
$946$	0	0
$947$	−29.7186	−0.965724	−0.482862	−	0.875696i	$-0.660403\pi$
$947$	−29.7186	−0.965724	−0.482862	+	0.875696i	$0.660403\pi$
$948$	0	0
$949$	−61.1345	−1.98451
$950$	0	0
$951$	23.2285	0.753235
$952$	0	0
$953$	15.5651	0.504203	0.252102	−	0.967701i	$-0.418878\pi$
$953$	15.5651	0.504203	0.252102	+	0.967701i	$0.418878\pi$
$954$	0	0
$955$	53.6521	1.73614
$956$	0	0
$957$	−2.81161	−0.0908866
$958$	0	0
$959$	−32.6859	−1.05548
$960$	0	0
$961$	66.6898	2.15128
$962$	0	0
$963$	−19.3141	−0.622387
$964$	0	0
$965$	49.2074	1.58404
$966$	0	0
$967$	−40.6246	−1.30640	−0.653200	−	0.757186i	$-0.726574\pi$
$967$	−40.6246	−1.30640	−0.653200	+	0.757186i	$0.726574\pi$
$968$	0	0
$969$	−2.04586	−0.0657224
$970$	0	0
$971$	−57.7556	−1.85346	−0.926732	−	0.375722i	$-0.877395\pi$
$971$	−57.7556	−1.85346	−0.926732	+	0.375722i	$0.877395\pi$
$972$	0	0
$973$	−28.6512	−0.918514
$974$	0	0
$975$	−10.3794	−0.332406
$976$	0	0
$977$	−30.6355	−0.980117	−0.490059	−	0.871689i	$-0.663025\pi$
$977$	−30.6355	−0.980117	−0.490059	+	0.871689i	$0.663025\pi$
$978$	0	0
$979$	24.6490	0.787784
$980$	0	0
$981$	7.62677	0.243504
$982$	0	0
$983$	−56.0822	−1.78874	−0.894372	−	0.447325i	$-0.852377\pi$
$983$	−56.0822	−1.78874	−0.894372	+	0.447325i	$0.852377\pi$
$984$	0	0
$985$	26.5336	0.845430
$986$	0	0
$987$	8.51999	0.271194
$988$	0	0
$989$	−30.4365	−0.967823
$990$	0	0
$991$	−25.5858	−0.812758	−0.406379	−	0.913705i	$-0.633209\pi$
$991$	−25.5858	−0.812758	−0.406379	+	0.913705i	$0.633209\pi$
$992$	0	0
$993$	12.9244	0.410143
$994$	0	0
$995$	−23.3190	−0.739263
$996$	0	0
$997$	−50.2844	−1.59252	−0.796261	−	0.604953i	$-0.793192\pi$
$997$	−50.2844	−1.59252	−0.796261	+	0.604953i	$0.793192\pi$
$998$	0	0
$999$	11.0233	0.348761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.h.1.18	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.18	✓	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.55581 q^{5} -4.67011 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.55581 q^{5} -4.67011 q^{7} +1.00000 q^{9} +5.87081 q^{11} -6.77432 q^{13} +2.55581 q^{15} -1.34496 q^{17} +1.52113 q^{19} -4.67011 q^{21} +4.01277 q^{23} +1.53216 q^{25} +1.00000 q^{27} -0.478914 q^{29} +9.88381 q^{31} +5.87081 q^{33} -11.9359 q^{35} +11.0233 q^{37} -6.77432 q^{39} -7.58350 q^{41} -7.58490 q^{43} +2.55581 q^{45} -1.82436 q^{47} +14.8099 q^{49} -1.34496 q^{51} -2.50332 q^{53} +15.0047 q^{55} +1.52113 q^{57} +7.69989 q^{59} +11.4374 q^{61} -4.67011 q^{63} -17.3139 q^{65} -13.9979 q^{67} +4.01277 q^{69} +9.90027 q^{71} +9.02444 q^{73} +1.53216 q^{75} -27.4173 q^{77} -1.26841 q^{79} +1.00000 q^{81} +12.2389 q^{83} -3.43746 q^{85} -0.478914 q^{87} +4.19856 q^{89} +31.6368 q^{91} +9.88381 q^{93} +3.88771 q^{95} +1.74327 q^{97} +5.87081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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