LMFDB - Embedded newform 4620.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.17741$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.43630 q^{13} -1.00000 q^{15} +8.09593 q^{17} +4.09593 q^{19} +1.00000 q^{21} -2.09593 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.65964 q^{29} +0.563703 q^{31} -1.00000 q^{33} -1.00000 q^{35} -9.62816 q^{37} +3.43630 q^{39} +8.75557 q^{41} +5.53223 q^{43} -1.00000 q^{45} +7.43630 q^{47} +1.00000 q^{49} +8.09593 q^{51} -10.9685 q^{53} +1.00000 q^{55} +4.09593 q^{57} -9.53223 q^{59} +1.22334 q^{61} +1.00000 q^{63} -3.43630 q^{65} -8.19186 q^{67} -2.09593 q^{69} -8.75557 q^{71} +10.8726 q^{73} +1.00000 q^{75} -1.00000 q^{77} +9.43630 q^{79} +1.00000 q^{81} +8.09593 q^{83} -8.09593 q^{85} -6.65964 q^{87} +0.659636 q^{89} +3.43630 q^{91} +0.563703 q^{93} -4.09593 q^{95} +7.34036 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} + 4 q^{13} - 3 q^{15} + 8 q^{17} - 4 q^{19} + 3 q^{21} + 10 q^{23} + 3 q^{25} + 3 q^{27} - 10 q^{29} + 8 q^{31} - 3 q^{33} - 3 q^{35} + 10 q^{37} + 4 q^{39} - 6 q^{43} - 3 q^{45} + 16 q^{47} + 3 q^{49} + 8 q^{51} - 4 q^{53} + 3 q^{55} - 4 q^{57} - 6 q^{59} + 3 q^{63} - 4 q^{65} + 8 q^{67} + 10 q^{69} + 20 q^{73} + 3 q^{75} - 3 q^{77} + 22 q^{79} + 3 q^{81} + 8 q^{83} - 8 q^{85} - 10 q^{87} - 8 q^{89} + 4 q^{91} + 8 q^{93} + 4 q^{95} + 32 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 - 3 * q^11 + 4 * q^13 - 3 * q^15 + 8 * q^17 - 4 * q^19 + 3 * q^21 + 10 * q^23 + 3 * q^25 + 3 * q^27 - 10 * q^29 + 8 * q^31 - 3 * q^33 - 3 * q^35 + 10 * q^37 + 4 * q^39 - 6 * q^43 - 3 * q^45 + 16 * q^47 + 3 * q^49 + 8 * q^51 - 4 * q^53 + 3 * q^55 - 4 * q^57 - 6 * q^59 + 3 * q^63 - 4 * q^65 + 8 * q^67 + 10 * q^69 + 20 * q^73 + 3 * q^75 - 3 * q^77 + 22 * q^79 + 3 * q^81 + 8 * q^83 - 8 * q^85 - 10 * q^87 - 8 * q^89 + 4 * q^91 + 8 * q^93 + 4 * q^95 + 32 * q^97 - 3 * 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
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.43630	0.953057	0.476529	−	0.879159i	$-0.341895\pi$
$13$	3.43630	0.953057	0.476529	+	0.879159i	$0.341895\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	8.09593	1.96355	0.981776	−	0.190042i	$-0.0608624\pi$
$17$	8.09593	1.96355	0.981776	+	0.190042i	$0.0608624\pi$
$18$	0	0
$19$	4.09593	0.939671	0.469836	−	0.882754i	$-0.344313\pi$
$19$	4.09593	0.939671	0.469836	+	0.882754i	$0.344313\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.09593	−0.437032	−0.218516	−	0.975833i	$-0.570122\pi$
$23$	−2.09593	−0.437032	−0.218516	+	0.975833i	$0.570122\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.65964	−1.23666	−0.618332	−	0.785917i	$-0.712191\pi$
$29$	−6.65964	−1.23666	−0.618332	+	0.785917i	$0.712191\pi$
$30$	0	0
$31$	0.563703	0.101244	0.0506220	−	0.998718i	$-0.483880\pi$
$31$	0.563703	0.101244	0.0506220	+	0.998718i	$0.483880\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.62816	−1.58286	−0.791430	−	0.611260i	$-0.790663\pi$
$37$	−9.62816	−1.58286	−0.791430	+	0.611260i	$0.790663\pi$
$38$	0	0
$39$	3.43630	0.550248
$40$	0	0
$41$	8.75557	1.36739	0.683695	−	0.729768i	$-0.260372\pi$
$41$	8.75557	1.36739	0.683695	+	0.729768i	$0.260372\pi$
$42$	0	0
$43$	5.53223	0.843657	0.421829	−	0.906676i	$-0.361388\pi$
$43$	5.53223	0.843657	0.421829	+	0.906676i	$0.361388\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.43630	1.08470	0.542348	−	0.840154i	$-0.317536\pi$
$47$	7.43630	1.08470	0.542348	+	0.840154i	$0.317536\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.09593	1.13366
$52$	0	0
$53$	−10.9685	−1.50664	−0.753321	−	0.657652i	$-0.771550\pi$
$53$	−10.9685	−1.50664	−0.753321	+	0.657652i	$0.771550\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	4.09593	0.542519
$58$	0	0
$59$	−9.53223	−1.24099	−0.620495	−	0.784210i	$-0.713068\pi$
$59$	−9.53223	−1.24099	−0.620495	+	0.784210i	$0.713068\pi$
$60$	0	0
$61$	1.22334	0.156632	0.0783162	−	0.996929i	$-0.475046\pi$
$61$	1.22334	0.156632	0.0783162	+	0.996929i	$0.475046\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.43630	−0.426220
$66$	0	0
$67$	−8.19186	−1.00080	−0.500398	−	0.865796i	$-0.666813\pi$
$67$	−8.19186	−1.00080	−0.500398	+	0.865796i	$0.666813\pi$
$68$	0	0
$69$	−2.09593	−0.252321
$70$	0	0
$71$	−8.75557	−1.03909	−0.519547	−	0.854442i	$-0.673899\pi$
$71$	−8.75557	−1.03909	−0.519547	+	0.854442i	$0.673899\pi$
$72$	0	0
$73$	10.8726	1.27254	0.636270	−	0.771466i	$-0.280476\pi$
$73$	10.8726	1.27254	0.636270	+	0.771466i	$0.280476\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.43630	1.06167	0.530833	−	0.847476i	$-0.321879\pi$
$79$	9.43630	1.06167	0.530833	+	0.847476i	$0.321879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.09593	0.888644	0.444322	−	0.895867i	$-0.353445\pi$
$83$	8.09593	0.888644	0.444322	+	0.895867i	$0.353445\pi$
$84$	0	0
$85$	−8.09593	−0.878127
$86$	0	0
$87$	−6.65964	−0.713988
$88$	0	0
$89$	0.659636	0.0699212	0.0349606	−	0.999389i	$-0.488869\pi$
$89$	0.659636	0.0699212	0.0349606	+	0.999389i	$0.488869\pi$
$90$	0	0
$91$	3.43630	0.360222
$92$	0	0
$93$	0.563703	0.0584533
$94$	0	0
$95$	−4.09593	−0.420234
$96$	0	0
$97$	7.34036	0.745301	0.372651	−	0.927972i	$-0.378449\pi$
$97$	7.34036	0.745301	0.372651	+	0.927972i	$0.378449\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−16.1919	−1.61115	−0.805575	−	0.592493i	$-0.798144\pi$
$101$	−16.1919	−1.61115	−0.805575	+	0.592493i	$0.798144\pi$
$102$	0	0
$103$	18.8515	1.85749	0.928747	−	0.370715i	$-0.120887\pi$
$103$	18.8515	1.85749	0.928747	+	0.370715i	$0.120887\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	14.7556	1.42647	0.713237	−	0.700923i	$-0.247228\pi$
$107$	14.7556	1.42647	0.713237	+	0.700923i	$0.247228\pi$
$108$	0	0
$109$	−9.62816	−0.922211	−0.461105	−	0.887345i	$-0.652547\pi$
$109$	−9.62816	−0.922211	−0.461105	+	0.887345i	$0.652547\pi$
$110$	0	0
$111$	−9.62816	−0.913865
$112$	0	0
$113$	5.22334	0.491370	0.245685	−	0.969350i	$-0.420987\pi$
$113$	5.22334	0.491370	0.245685	+	0.969350i	$0.420987\pi$
$114$	0	0
$115$	2.09593	0.195447
$116$	0	0
$117$	3.43630	0.317686
$118$	0	0
$119$	8.09593	0.742153
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.75557	0.789463
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.53223	−0.135963	−0.0679817	−	0.997687i	$-0.521656\pi$
$127$	−1.53223	−0.135963	−0.0679817	+	0.997687i	$0.521656\pi$
$128$	0	0
$129$	5.53223	0.487086
$130$	0	0
$131$	−4.56370	−0.398733	−0.199366	−	0.979925i	$-0.563888\pi$
$131$	−4.56370	−0.398733	−0.199366	+	0.979925i	$0.563888\pi$
$132$	0	0
$133$	4.09593	0.355162
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	10.1919	0.870750	0.435375	−	0.900249i	$-0.356616\pi$
$137$	10.1919	0.870750	0.435375	+	0.900249i	$0.356616\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	7.43630	0.626249
$142$	0	0
$143$	−3.43630	−0.287358
$144$	0	0
$145$	6.65964	0.553053
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	19.0645	1.56182	0.780911	−	0.624643i	$-0.214755\pi$
$149$	19.0645	1.56182	0.780911	+	0.624643i	$0.214755\pi$
$150$	0	0
$151$	22.3837	1.82156	0.910781	−	0.412890i	$-0.135481\pi$
$151$	22.3837	1.82156	0.910781	+	0.412890i	$0.135481\pi$
$152$	0	0
$153$	8.09593	0.654517
$154$	0	0
$155$	−0.563703	−0.0452777
$156$	0	0
$157$	14.4048	1.14963	0.574815	−	0.818283i	$-0.305074\pi$
$157$	14.4048	1.14963	0.574815	+	0.818283i	$0.305074\pi$
$158$	0	0
$159$	−10.9685	−0.869861
$160$	0	0
$161$	−2.09593	−0.165183
$162$	0	0
$163$	6.30889	0.494150	0.247075	−	0.968996i	$-0.420531\pi$
$163$	6.30889	0.494150	0.247075	+	0.968996i	$0.420531\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	11.7452	0.908870	0.454435	−	0.890780i	$-0.349841\pi$
$167$	11.7452	0.908870	0.454435	+	0.890780i	$0.349841\pi$
$168$	0	0
$169$	−1.19186	−0.0916819
$170$	0	0
$171$	4.09593	0.313224
$172$	0	0
$173$	−4.44668	−0.338075	−0.169037	−	0.985610i	$-0.554066\pi$
$173$	−4.44668	−0.338075	−0.169037	+	0.985610i	$0.554066\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−9.53223	−0.716486
$178$	0	0
$179$	5.88297	0.439714	0.219857	−	0.975532i	$-0.429441\pi$
$179$	5.88297	0.439714	0.219857	+	0.975532i	$0.429441\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	1.22334	0.0904318
$184$	0	0
$185$	9.62816	0.707876
$186$	0	0
$187$	−8.09593	−0.592033
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−11.0645	−0.796437	−0.398219	−	0.917291i	$-0.630371\pi$
$193$	−11.0645	−0.796437	−0.398219	+	0.917291i	$0.630371\pi$
$194$	0	0
$195$	−3.43630	−0.246078
$196$	0	0
$197$	13.0645	0.930804	0.465402	−	0.885099i	$-0.345910\pi$
$197$	13.0645	0.930804	0.465402	+	0.885099i	$0.345910\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−8.19186	−0.577810
$202$	0	0
$203$	−6.65964	−0.467415
$204$	0	0
$205$	−8.75557	−0.611515
$206$	0	0
$207$	−2.09593	−0.145677
$208$	0	0
$209$	−4.09593	−0.283322
$210$	0	0
$211$	−21.2563	−1.46335	−0.731673	−	0.681656i	$-0.761260\pi$
$211$	−21.2563	−1.46335	−0.731673	+	0.681656i	$0.761260\pi$
$212$	0	0
$213$	−8.75557	−0.599922
$214$	0	0
$215$	−5.53223	−0.377295
$216$	0	0
$217$	0.563703	0.0382667
$218$	0	0
$219$	10.8726	0.734702
$220$	0	0
$221$	27.8200	1.87138
$222$	0	0
$223$	−17.7241	−1.18689	−0.593447	−	0.804873i	$-0.702233\pi$
$223$	−17.7241	−1.18689	−0.593447	+	0.804873i	$0.702233\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−22.9685	−1.52447	−0.762237	−	0.647298i	$-0.775899\pi$
$227$	−22.9685	−1.52447	−0.762237	+	0.647298i	$0.775899\pi$
$228$	0	0
$229$	10.7556	0.710748	0.355374	−	0.934724i	$-0.384354\pi$
$229$	10.7556	0.710748	0.355374	+	0.934724i	$0.384354\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−21.8200	−1.42948	−0.714739	−	0.699392i	$-0.753454\pi$
$233$	−21.8200	−1.42948	−0.714739	+	0.699392i	$0.753454\pi$
$234$	0	0
$235$	−7.43630	−0.485091
$236$	0	0
$237$	9.43630	0.612953
$238$	0	0
$239$	−25.7241	−1.66395	−0.831977	−	0.554811i	$-0.812791\pi$
$239$	−25.7241	−1.66395	−0.831977	+	0.554811i	$0.812791\pi$
$240$	0	0
$241$	15.3193	0.986801	0.493400	−	0.869802i	$-0.335754\pi$
$241$	15.3193	0.986801	0.493400	+	0.869802i	$0.335754\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	14.0748	0.895561
$248$	0	0
$249$	8.09593	0.513059
$250$	0	0
$251$	17.7452	1.12007	0.560033	−	0.828470i	$-0.310788\pi$
$251$	17.7452	1.12007	0.560033	+	0.828470i	$0.310788\pi$
$252$	0	0
$253$	2.09593	0.131770
$254$	0	0
$255$	−8.09593	−0.506987
$256$	0	0
$257$	−4.30889	−0.268781	−0.134391	−	0.990928i	$-0.542908\pi$
$257$	−4.30889	−0.268781	−0.134391	+	0.990928i	$0.542908\pi$
$258$	0	0
$259$	−9.62816	−0.598265
$260$	0	0
$261$	−6.65964	−0.412221
$262$	0	0
$263$	−19.3193	−1.19128	−0.595639	−	0.803253i	$-0.703101\pi$
$263$	−19.3193	−1.19128	−0.595639	+	0.803253i	$0.703101\pi$
$264$	0	0
$265$	10.9685	0.673791
$266$	0	0
$267$	0.659636	0.0403690
$268$	0	0
$269$	−24.6596	−1.50352	−0.751762	−	0.659434i	$-0.770796\pi$
$269$	−24.6596	−1.50352	−0.751762	+	0.659434i	$0.770796\pi$
$270$	0	0
$271$	−4.28780	−0.260465	−0.130233	−	0.991483i	$-0.541572\pi$
$271$	−4.28780	−0.260465	−0.130233	+	0.991483i	$0.541572\pi$
$272$	0	0
$273$	3.43630	0.207974
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.8726	1.37428	0.687140	−	0.726525i	$-0.258866\pi$
$277$	22.8726	1.37428	0.687140	+	0.726525i	$0.258866\pi$
$278$	0	0
$279$	0.563703	0.0337480
$280$	0	0
$281$	9.12741	0.544495	0.272248	−	0.962227i	$-0.412233\pi$
$281$	9.12741	0.544495	0.272248	+	0.962227i	$0.412233\pi$
$282$	0	0
$283$	−11.4363	−0.679817	−0.339909	−	0.940458i	$-0.610396\pi$
$283$	−11.4363	−0.679817	−0.339909	+	0.940458i	$0.610396\pi$
$284$	0	0
$285$	−4.09593	−0.242622
$286$	0	0
$287$	8.75557	0.516825
$288$	0	0
$289$	48.5441	2.85554
$290$	0	0
$291$	7.34036	0.430300
$292$	0	0
$293$	24.2878	1.41891	0.709454	−	0.704752i	$-0.248942\pi$
$293$	24.2878	1.41891	0.709454	+	0.704752i	$0.248942\pi$
$294$	0	0
$295$	9.53223	0.554988
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−7.20225	−0.416517
$300$	0	0
$301$	5.53223	0.318872
$302$	0	0
$303$	−16.1919	−0.930198
$304$	0	0
$305$	−1.22334	−0.0700482
$306$	0	0
$307$	9.31927	0.531879	0.265939	−	0.963990i	$-0.414318\pi$
$307$	9.31927	0.531879	0.265939	+	0.963990i	$0.414318\pi$
$308$	0	0
$309$	18.8515	1.07242
$310$	0	0
$311$	−17.5111	−0.992965	−0.496483	−	0.868047i	$-0.665375\pi$
$311$	−17.5111	−0.992965	−0.496483	+	0.868047i	$0.665375\pi$
$312$	0	0
$313$	−27.9160	−1.57790	−0.788952	−	0.614455i	$-0.789376\pi$
$313$	−27.9160	−1.57790	−0.788952	+	0.614455i	$0.789376\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−13.0645	−0.733773	−0.366887	−	0.930266i	$-0.619576\pi$
$317$	−13.0645	−0.733773	−0.366887	+	0.930266i	$0.619576\pi$
$318$	0	0
$319$	6.65964	0.372868
$320$	0	0
$321$	14.7556	0.823575
$322$	0	0
$323$	33.1604	1.84509
$324$	0	0
$325$	3.43630	0.190611
$326$	0	0
$327$	−9.62816	−0.532439
$328$	0	0
$329$	7.43630	0.409976
$330$	0	0
$331$	−1.90407	−0.104657	−0.0523285	−	0.998630i	$-0.516664\pi$
$331$	−1.90407	−0.104657	−0.0523285	+	0.998630i	$0.516664\pi$
$332$	0	0
$333$	−9.62816	−0.527620
$334$	0	0
$335$	8.19186	0.447569
$336$	0	0
$337$	15.9789	0.870427	0.435213	−	0.900327i	$-0.356673\pi$
$337$	15.9789	0.870427	0.435213	+	0.900327i	$0.356673\pi$
$338$	0	0
$339$	5.22334	0.283693
$340$	0	0
$341$	−0.563703	−0.0305262
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.09593	0.112841
$346$	0	0
$347$	19.3193	1.03711	0.518556	−	0.855043i	$-0.326470\pi$
$347$	19.3193	1.03711	0.518556	+	0.855043i	$0.326470\pi$
$348$	0	0
$349$	−26.9685	−1.44359	−0.721796	−	0.692106i	$-0.756683\pi$
$349$	−26.9685	−1.44359	−0.721796	+	0.692106i	$0.756683\pi$
$350$	0	0
$351$	3.43630	0.183416
$352$	0	0
$353$	−20.8726	−1.11094	−0.555468	−	0.831538i	$-0.687461\pi$
$353$	−20.8726	−1.11094	−0.555468	+	0.831538i	$0.687461\pi$
$354$	0	0
$355$	8.75557	0.464697
$356$	0	0
$357$	8.09593	0.428482
$358$	0	0
$359$	7.97891	0.421111	0.210555	−	0.977582i	$-0.432473\pi$
$359$	7.97891	0.421111	0.210555	+	0.977582i	$0.432473\pi$
$360$	0	0
$361$	−2.22334	−0.117018
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−10.8726	−0.569098
$366$	0	0
$367$	5.53223	0.288780	0.144390	−	0.989521i	$-0.453878\pi$
$367$	5.53223	0.288780	0.144390	+	0.989521i	$0.453878\pi$
$368$	0	0
$369$	8.75557	0.455797
$370$	0	0
$371$	−10.9685	−0.569457
$372$	0	0
$373$	−9.34036	−0.483626	−0.241813	−	0.970323i	$-0.577742\pi$
$373$	−9.34036	−0.483626	−0.241813	+	0.970323i	$0.577742\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−22.8845	−1.17861
$378$	0	0
$379$	11.6493	0.598382	0.299191	−	0.954193i	$-0.403283\pi$
$379$	11.6493	0.598382	0.299191	+	0.954193i	$0.403283\pi$
$380$	0	0
$381$	−1.53223	−0.0784985
$382$	0	0
$383$	−3.24443	−0.165783	−0.0828914	−	0.996559i	$-0.526415\pi$
$383$	−3.24443	−0.165783	−0.0828914	+	0.996559i	$0.526415\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	5.53223	0.281219
$388$	0	0
$389$	−14.7556	−0.748137	−0.374068	−	0.927401i	$-0.622038\pi$
$389$	−14.7556	−0.748137	−0.374068	+	0.927401i	$0.622038\pi$
$390$	0	0
$391$	−16.9685	−0.858135
$392$	0	0
$393$	−4.56370	−0.230208
$394$	0	0
$395$	−9.43630	−0.474792
$396$	0	0
$397$	−11.3193	−0.568098	−0.284049	−	0.958810i	$-0.591678\pi$
$397$	−11.3193	−0.568098	−0.284049	+	0.958810i	$0.591678\pi$
$398$	0	0
$399$	4.09593	0.205053
$400$	0	0
$401$	−21.8200	−1.08964	−0.544820	−	0.838553i	$-0.683402\pi$
$401$	−21.8200	−1.08964	−0.544820	+	0.838553i	$0.683402\pi$
$402$	0	0
$403$	1.93705	0.0964914
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	9.62816	0.477250
$408$	0	0
$409$	3.31927	0.164127	0.0820637	−	0.996627i	$-0.473849\pi$
$409$	3.31927	0.164127	0.0820637	+	0.996627i	$0.473849\pi$
$410$	0	0
$411$	10.1919	0.502728
$412$	0	0
$413$	−9.53223	−0.469050
$414$	0	0
$415$	−8.09593	−0.397414
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	31.9789	1.56227	0.781136	−	0.624361i	$-0.214641\pi$
$419$	31.9789	1.56227	0.781136	+	0.624361i	$0.214641\pi$
$420$	0	0
$421$	21.6071	1.05306	0.526532	−	0.850155i	$-0.323492\pi$
$421$	21.6071	1.05306	0.526532	+	0.850155i	$0.323492\pi$
$422$	0	0
$423$	7.43630	0.361565
$424$	0	0
$425$	8.09593	0.392710
$426$	0	0
$427$	1.22334	0.0592015
$428$	0	0
$429$	−3.43630	−0.165906
$430$	0	0
$431$	29.5111	1.42150	0.710751	−	0.703444i	$-0.248355\pi$
$431$	29.5111	1.42150	0.710751	+	0.703444i	$0.248355\pi$
$432$	0	0
$433$	−24.8726	−1.19530	−0.597650	−	0.801757i	$-0.703899\pi$
$433$	−24.8726	−1.19530	−0.597650	+	0.801757i	$0.703899\pi$
$434$	0	0
$435$	6.65964	0.319305
$436$	0	0
$437$	−8.58480	−0.410667
$438$	0	0
$439$	6.96853	0.332590	0.166295	−	0.986076i	$-0.446820\pi$
$439$	6.96853	0.332590	0.166295	+	0.986076i	$0.446820\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−28.1919	−1.33944	−0.669718	−	0.742616i	$-0.733585\pi$
$443$	−28.1919	−1.33944	−0.669718	+	0.742616i	$0.733585\pi$
$444$	0	0
$445$	−0.659636	−0.0312697
$446$	0	0
$447$	19.0645	0.901718
$448$	0	0
$449$	−13.2022	−0.623052	−0.311526	−	0.950238i	$-0.600840\pi$
$449$	−13.2022	−0.623052	−0.311526	+	0.950238i	$0.600840\pi$
$450$	0	0
$451$	−8.75557	−0.412284
$452$	0	0
$453$	22.3837	1.05168
$454$	0	0
$455$	−3.43630	−0.161096
$456$	0	0
$457$	25.3404	1.18537	0.592686	−	0.805434i	$-0.298067\pi$
$457$	25.3404	1.18537	0.592686	+	0.805434i	$0.298067\pi$
$458$	0	0
$459$	8.09593	0.377886
$460$	0	0
$461$	−19.0645	−0.887920	−0.443960	−	0.896047i	$-0.646427\pi$
$461$	−19.0645	−0.887920	−0.443960	+	0.896047i	$0.646427\pi$
$462$	0	0
$463$	−5.31927	−0.247207	−0.123604	−	0.992332i	$-0.539445\pi$
$463$	−5.31927	−0.247207	−0.123604	+	0.992332i	$0.539445\pi$
$464$	0	0
$465$	−0.563703	−0.0261411
$466$	0	0
$467$	−5.51114	−0.255025	−0.127512	−	0.991837i	$-0.540699\pi$
$467$	−5.51114	−0.255025	−0.127512	+	0.991837i	$0.540699\pi$
$468$	0	0
$469$	−8.19186	−0.378265
$470$	0	0
$471$	14.4048	0.663739
$472$	0	0
$473$	−5.53223	−0.254372
$474$	0	0
$475$	4.09593	0.187934
$476$	0	0
$477$	−10.9685	−0.502214
$478$	0	0
$479$	−4.56370	−0.208521	−0.104260	−	0.994550i	$-0.533248\pi$
$479$	−4.56370	−0.208521	−0.104260	+	0.994550i	$0.533248\pi$
$480$	0	0
$481$	−33.0852	−1.50856
$482$	0	0
$483$	−2.09593	−0.0953682
$484$	0	0
$485$	−7.34036	−0.333309
$486$	0	0
$487$	0.425916	0.0193001	0.00965005	−	0.999953i	$-0.496928\pi$
$487$	0.425916	0.0193001	0.00965005	+	0.999953i	$0.496928\pi$
$488$	0	0
$489$	6.30889	0.285298
$490$	0	0
$491$	43.2352	1.95118	0.975589	−	0.219603i	$-0.0704762\pi$
$491$	43.2352	1.95118	0.975589	+	0.219603i	$0.0704762\pi$
$492$	0	0
$493$	−53.9160	−2.42825
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−8.75557	−0.392741
$498$	0	0
$499$	−33.5441	−1.50164	−0.750821	−	0.660506i	$-0.770342\pi$
$499$	−33.5441	−1.50164	−0.750821	+	0.660506i	$0.770342\pi$
$500$	0	0
$501$	11.7452	0.524736
$502$	0	0
$503$	−15.9041	−0.709127	−0.354564	−	0.935032i	$-0.615371\pi$
$503$	−15.9041	−0.709127	−0.354564	+	0.935032i	$0.615371\pi$
$504$	0	0
$505$	16.1919	0.720529
$506$	0	0
$507$	−1.19186	−0.0529326
$508$	0	0
$509$	22.5967	1.00158	0.500790	−	0.865569i	$-0.333043\pi$
$509$	22.5967	1.00158	0.500790	+	0.865569i	$0.333043\pi$
$510$	0	0
$511$	10.8726	0.480975
$512$	0	0
$513$	4.09593	0.180840
$514$	0	0
$515$	−18.8515	−0.830696
$516$	0	0
$517$	−7.43630	−0.327048
$518$	0	0
$519$	−4.44668	−0.195187
$520$	0	0
$521$	11.9160	0.522048	0.261024	−	0.965332i	$-0.415940\pi$
$521$	11.9160	0.522048	0.261024	+	0.965332i	$0.415940\pi$
$522$	0	0
$523$	10.8726	0.475425	0.237713	−	0.971336i	$-0.423602\pi$
$523$	10.8726	0.475425	0.237713	+	0.971336i	$0.423602\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	4.56370	0.198798
$528$	0	0
$529$	−18.6071	−0.809003
$530$	0	0
$531$	−9.53223	−0.413664
$532$	0	0
$533$	30.0867	1.30320
$534$	0	0
$535$	−14.7556	−0.637939
$536$	0	0
$537$	5.88297	0.253869
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	36.8845	1.58579	0.792894	−	0.609360i	$-0.208573\pi$
$541$	36.8845	1.58579	0.792894	+	0.609360i	$0.208573\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	9.62816	0.412425
$546$	0	0
$547$	−43.8530	−1.87502	−0.937510	−	0.347959i	$-0.886875\pi$
$547$	−43.8530	−1.87502	−0.937510	+	0.347959i	$0.886875\pi$
$548$	0	0
$549$	1.22334	0.0522108
$550$	0	0
$551$	−27.2774	−1.16206
$552$	0	0
$553$	9.43630	0.401272
$554$	0	0
$555$	9.62816	0.408693
$556$	0	0
$557$	−35.5111	−1.50466	−0.752328	−	0.658789i	$-0.771069\pi$
$557$	−35.5111	−1.50466	−0.752328	+	0.658789i	$0.771069\pi$
$558$	0	0
$559$	19.0104	0.804053
$560$	0	0
$561$	−8.09593	−0.341811
$562$	0	0
$563$	−0.488864	−0.0206032	−0.0103016	−	0.999947i	$-0.503279\pi$
$563$	−0.488864	−0.0206032	−0.0103016	+	0.999947i	$0.503279\pi$
$564$	0	0
$565$	−5.22334	−0.219748
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−40.5967	−1.70190	−0.850951	−	0.525245i	$-0.823974\pi$
$569$	−40.5967	−1.70190	−0.850951	+	0.525245i	$0.823974\pi$
$570$	0	0
$571$	−31.5652	−1.32096	−0.660482	−	0.750842i	$-0.729648\pi$
$571$	−31.5652	−1.32096	−0.660482	+	0.750842i	$0.729648\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.09593	−0.0874064
$576$	0	0
$577$	−15.7452	−0.655481	−0.327740	−	0.944768i	$-0.606287\pi$
$577$	−15.7452	−0.655481	−0.327740	+	0.944768i	$0.606287\pi$
$578$	0	0
$579$	−11.0645	−0.459823
$580$	0	0
$581$	8.09593	0.335876
$582$	0	0
$583$	10.9685	0.454270
$584$	0	0
$585$	−3.43630	−0.142073
$586$	0	0
$587$	41.3733	1.70766	0.853830	−	0.520551i	$-0.174274\pi$
$587$	41.3733	1.70766	0.853830	+	0.520551i	$0.174274\pi$
$588$	0	0
$589$	2.30889	0.0951362
$590$	0	0
$591$	13.0645	0.537400
$592$	0	0
$593$	−10.9355	−0.449069	−0.224534	−	0.974466i	$-0.572086\pi$
$593$	−10.9355	−0.449069	−0.224534	+	0.974466i	$0.572086\pi$
$594$	0	0
$595$	−8.09593	−0.331901
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	13.3193	0.544211	0.272105	−	0.962267i	$-0.412280\pi$
$599$	13.3193	0.544211	0.272105	+	0.962267i	$0.412280\pi$
$600$	0	0
$601$	−20.4797	−0.835383	−0.417691	−	0.908589i	$-0.637161\pi$
$601$	−20.4797	−0.835383	−0.417691	+	0.908589i	$0.637161\pi$
$602$	0	0
$603$	−8.19186	−0.333599
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	35.9578	1.45948	0.729741	−	0.683723i	$-0.239640\pi$
$607$	35.9578	1.45948	0.729741	+	0.683723i	$0.239640\pi$
$608$	0	0
$609$	−6.65964	−0.269862
$610$	0	0
$611$	25.5533	1.03378
$612$	0	0
$613$	−11.8081	−0.476926	−0.238463	−	0.971152i	$-0.576644\pi$
$613$	−11.8081	−0.476926	−0.238463	+	0.971152i	$0.576644\pi$
$614$	0	0
$615$	−8.75557	−0.353059
$616$	0	0
$617$	−21.9578	−0.883988	−0.441994	−	0.897018i	$-0.645729\pi$
$617$	−21.9578	−0.883988	−0.441994	+	0.897018i	$0.645729\pi$
$618$	0	0
$619$	−5.55332	−0.223207	−0.111603	−	0.993753i	$-0.535599\pi$
$619$	−5.55332	−0.223207	−0.111603	+	0.993753i	$0.535599\pi$
$620$	0	0
$621$	−2.09593	−0.0841069
$622$	0	0
$623$	0.659636	0.0264277
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.09593	−0.163576
$628$	0	0
$629$	−77.9489	−3.10803
$630$	0	0
$631$	11.6493	0.463750	0.231875	−	0.972746i	$-0.425514\pi$
$631$	11.6493	0.463750	0.231875	+	0.972746i	$0.425514\pi$
$632$	0	0
$633$	−21.2563	−0.844863
$634$	0	0
$635$	1.53223	0.0608047
$636$	0	0
$637$	3.43630	0.136151
$638$	0	0
$639$	−8.75557	−0.346365
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	26.6596	1.05135	0.525677	−	0.850684i	$-0.323812\pi$
$643$	26.6596	1.05135	0.525677	+	0.850684i	$0.323812\pi$
$644$	0	0
$645$	−5.53223	−0.217831
$646$	0	0
$647$	−2.87259	−0.112933	−0.0564667	−	0.998404i	$-0.517983\pi$
$647$	−2.87259	−0.112933	−0.0564667	+	0.998404i	$0.517983\pi$
$648$	0	0
$649$	9.53223	0.374173
$650$	0	0
$651$	0.563703	0.0220933
$652$	0	0
$653$	19.3523	0.757312	0.378656	−	0.925537i	$-0.376386\pi$
$653$	19.3523	0.757312	0.378656	+	0.925537i	$0.376386\pi$
$654$	0	0
$655$	4.56370	0.178319
$656$	0	0
$657$	10.8726	0.424180
$658$	0	0
$659$	−30.1500	−1.17448	−0.587239	−	0.809414i	$-0.699785\pi$
$659$	−30.1500	−1.17448	−0.587239	+	0.809414i	$0.699785\pi$
$660$	0	0
$661$	31.1393	1.21118	0.605589	−	0.795777i	$-0.292938\pi$
$661$	31.1393	1.21118	0.605589	+	0.795777i	$0.292938\pi$
$662$	0	0
$663$	27.8200	1.08044
$664$	0	0
$665$	−4.09593	−0.158833
$666$	0	0
$667$	13.9581	0.540462
$668$	0	0
$669$	−17.7241	−0.685253
$670$	0	0
$671$	−1.22334	−0.0472265
$672$	0	0
$673$	−13.5322	−0.521629	−0.260815	−	0.965389i	$-0.583991\pi$
$673$	−13.5322	−0.521629	−0.260815	+	0.965389i	$0.583991\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−24.2878	−0.933456	−0.466728	−	0.884401i	$-0.654567\pi$
$677$	−24.2878	−0.933456	−0.466728	+	0.884401i	$0.654567\pi$
$678$	0	0
$679$	7.34036	0.281697
$680$	0	0
$681$	−22.9685	−0.880156
$682$	0	0
$683$	−32.3837	−1.23913	−0.619564	−	0.784946i	$-0.712691\pi$
$683$	−32.3837	−1.23913	−0.619564	+	0.784946i	$0.712691\pi$
$684$	0	0
$685$	−10.1919	−0.389411
$686$	0	0
$687$	10.7556	0.410351
$688$	0	0
$689$	−37.6911	−1.43592
$690$	0	0
$691$	30.4467	1.15825	0.579123	−	0.815240i	$-0.303395\pi$
$691$	30.4467	1.15825	0.579123	+	0.815240i	$0.303395\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−14.0000	−0.531050
$696$	0	0
$697$	70.8845	2.68494
$698$	0	0
$699$	−21.8200	−0.825309
$700$	0	0
$701$	0.404823	0.0152899	0.00764497	−	0.999971i	$-0.497567\pi$
$701$	0.404823	0.0152899	0.00764497	+	0.999971i	$0.497567\pi$
$702$	0	0
$703$	−39.4363	−1.48737
$704$	0	0
$705$	−7.43630	−0.280067
$706$	0	0
$707$	−16.1919	−0.608958
$708$	0	0
$709$	−8.71371	−0.327250	−0.163625	−	0.986523i	$-0.552319\pi$
$709$	−8.71371	−0.327250	−0.163625	+	0.986523i	$0.552319\pi$
$710$	0	0
$711$	9.43630	0.353889
$712$	0	0
$713$	−1.18148	−0.0442469
$714$	0	0
$715$	3.43630	0.128510
$716$	0	0
$717$	−25.7241	−0.960684
$718$	0	0
$719$	43.2352	1.61240	0.806201	−	0.591642i	$-0.201520\pi$
$719$	43.2352	1.61240	0.806201	+	0.591642i	$0.201520\pi$
$720$	0	0
$721$	18.8515	0.702067
$722$	0	0
$723$	15.3193	0.569730
$724$	0	0
$725$	−6.65964	−0.247333
$726$	0	0
$727$	21.9581	0.814383	0.407191	−	0.913343i	$-0.366508\pi$
$727$	21.9581	0.814383	0.407191	+	0.913343i	$0.366508\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.7886	1.65656
$732$	0	0
$733$	9.31927	0.344215	0.172108	−	0.985078i	$-0.444942\pi$
$733$	9.31927	0.344215	0.172108	+	0.985078i	$0.444942\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	8.19186	0.301751
$738$	0	0
$739$	19.5111	0.717729	0.358864	−	0.933390i	$-0.383164\pi$
$739$	19.5111	0.717729	0.358864	+	0.933390i	$0.383164\pi$
$740$	0	0
$741$	14.0748	0.517052
$742$	0	0
$743$	−11.5111	−0.422303	−0.211151	−	0.977453i	$-0.567721\pi$
$743$	−11.5111	−0.422303	−0.211151	+	0.977453i	$0.567721\pi$
$744$	0	0
$745$	−19.0645	−0.698468
$746$	0	0
$747$	8.09593	0.296215
$748$	0	0
$749$	14.7556	0.539157
$750$	0	0
$751$	4.35075	0.158761	0.0793805	−	0.996844i	$-0.474706\pi$
$751$	4.35075	0.158761	0.0793805	+	0.996844i	$0.474706\pi$
$752$	0	0
$753$	17.7452	0.646671
$754$	0	0
$755$	−22.3837	−0.814627
$756$	0	0
$757$	−18.6178	−0.676675	−0.338337	−	0.941025i	$-0.609865\pi$
$757$	−18.6178	−0.676675	−0.338337	+	0.941025i	$0.609865\pi$
$758$	0	0
$759$	2.09593	0.0760775
$760$	0	0
$761$	−19.8081	−0.718044	−0.359022	−	0.933329i	$-0.616890\pi$
$761$	−19.8081	−0.718044	−0.359022	+	0.933329i	$0.616890\pi$
$762$	0	0
$763$	−9.62816	−0.348563
$764$	0	0
$765$	−8.09593	−0.292709
$766$	0	0
$767$	−32.7556	−1.18274
$768$	0	0
$769$	11.1604	0.402454	0.201227	−	0.979545i	$-0.435507\pi$
$769$	11.1604	0.402454	0.201227	+	0.979545i	$0.435507\pi$
$770$	0	0
$771$	−4.30889	−0.155181
$772$	0	0
$773$	−47.5111	−1.70886	−0.854428	−	0.519569i	$-0.826092\pi$
$773$	−47.5111	−1.70886	−0.854428	+	0.519569i	$0.826092\pi$
$774$	0	0
$775$	0.563703	0.0202488
$776$	0	0
$777$	−9.62816	−0.345408
$778$	0	0
$779$	35.8622	1.28490
$780$	0	0
$781$	8.75557	0.313299
$782$	0	0
$783$	−6.65964	−0.237996
$784$	0	0
$785$	−14.4048	−0.514130
$786$	0	0
$787$	−7.61627	−0.271491	−0.135745	−	0.990744i	$-0.543343\pi$
$787$	−7.61627	−0.271491	−0.135745	+	0.990744i	$0.543343\pi$
$788$	0	0
$789$	−19.3193	−0.687784
$790$	0	0
$791$	5.22334	0.185721
$792$	0	0
$793$	4.20376	0.149280
$794$	0	0
$795$	10.9685	0.389014
$796$	0	0
$797$	31.8949	1.12977	0.564887	−	0.825168i	$-0.308920\pi$
$797$	31.8949	1.12977	0.564887	+	0.825168i	$0.308920\pi$
$798$	0	0
$799$	60.2038	2.12986
$800$	0	0
$801$	0.659636	0.0233071
$802$	0	0
$803$	−10.8726	−0.383685
$804$	0	0
$805$	2.09593	0.0738719
$806$	0	0
$807$	−24.6596	−0.868060
$808$	0	0
$809$	47.8319	1.68168	0.840840	−	0.541283i	$-0.182061\pi$
$809$	47.8319	1.68168	0.840840	+	0.541283i	$0.182061\pi$
$810$	0	0
$811$	23.9371	0.840544	0.420272	−	0.907398i	$-0.361935\pi$
$811$	23.9371	0.840544	0.420272	+	0.907398i	$0.361935\pi$
$812$	0	0
$813$	−4.28780	−0.150380
$814$	0	0
$815$	−6.30889	−0.220991
$816$	0	0
$817$	22.6596	0.792760
$818$	0	0
$819$	3.43630	0.120074
$820$	0	0
$821$	−21.5322	−0.751480	−0.375740	−	0.926725i	$-0.622611\pi$
$821$	−21.5322	−0.751480	−0.375740	+	0.926725i	$0.622611\pi$
$822$	0	0
$823$	44.2038	1.54085	0.770423	−	0.637533i	$-0.220045\pi$
$823$	44.2038	1.54085	0.770423	+	0.637533i	$0.220045\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−13.0645	−0.454296	−0.227148	−	0.973860i	$-0.572940\pi$
$827$	−13.0645	−0.454296	−0.227148	+	0.973860i	$0.572940\pi$
$828$	0	0
$829$	4.87259	0.169232	0.0846161	−	0.996414i	$-0.473034\pi$
$829$	4.87259	0.169232	0.0846161	+	0.996414i	$0.473034\pi$
$830$	0	0
$831$	22.8726	0.793441
$832$	0	0
$833$	8.09593	0.280507
$834$	0	0
$835$	−11.7452	−0.406459
$836$	0	0
$837$	0.563703	0.0194844
$838$	0	0
$839$	33.2982	1.14958	0.574790	−	0.818301i	$-0.305084\pi$
$839$	33.2982	1.14958	0.574790	+	0.818301i	$0.305084\pi$
$840$	0	0
$841$	15.3507	0.529336
$842$	0	0
$843$	9.12741	0.314365
$844$	0	0
$845$	1.19186	0.0410014
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−11.4363	−0.392493
$850$	0	0
$851$	20.1800	0.691761
$852$	0	0
$853$	16.0119	0.548237	0.274118	−	0.961696i	$-0.411614\pi$
$853$	16.0119	0.548237	0.274118	+	0.961696i	$0.411614\pi$
$854$	0	0
$855$	−4.09593	−0.140078
$856$	0	0
$857$	−14.8934	−0.508747	−0.254374	−	0.967106i	$-0.581869\pi$
$857$	−14.8934	−0.508747	−0.254374	+	0.967106i	$0.581869\pi$
$858$	0	0
$859$	26.9267	0.918726	0.459363	−	0.888249i	$-0.348078\pi$
$859$	26.9267	0.918726	0.459363	+	0.888249i	$0.348078\pi$
$860$	0	0
$861$	8.75557	0.298389
$862$	0	0
$863$	−41.5441	−1.41418	−0.707089	−	0.707124i	$-0.749992\pi$
$863$	−41.5441	−1.41418	−0.707089	+	0.707124i	$0.749992\pi$
$864$	0	0
$865$	4.44668	0.151192
$866$	0	0
$867$	48.5441	1.64864
$868$	0	0
$869$	−9.43630	−0.320104
$870$	0	0
$871$	−28.1497	−0.953815
$872$	0	0
$873$	7.34036	0.248434
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−55.8530	−1.88602	−0.943011	−	0.332761i	$-0.892020\pi$
$877$	−55.8530	−1.88602	−0.943011	+	0.332761i	$0.892020\pi$
$878$	0	0
$879$	24.2878	0.819207
$880$	0	0
$881$	−42.9804	−1.44805	−0.724024	−	0.689775i	$-0.757709\pi$
$881$	−42.9804	−1.44805	−0.724024	+	0.689775i	$0.757709\pi$
$882$	0	0
$883$	−53.8949	−1.81371	−0.906853	−	0.421446i	$-0.861523\pi$
$883$	−53.8949	−1.81371	−0.906853	+	0.421446i	$0.861523\pi$
$884$	0	0
$885$	9.53223	0.320422
$886$	0	0
$887$	−7.35225	−0.246865	−0.123432	−	0.992353i	$-0.539390\pi$
$887$	−7.35225	−0.246865	−0.123432	+	0.992353i	$0.539390\pi$
$888$	0	0
$889$	−1.53223	−0.0513893
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	30.4586	1.01926
$894$	0	0
$895$	−5.88297	−0.196646
$896$	0	0
$897$	−7.20225	−0.240476
$898$	0	0
$899$	−3.75406	−0.125205
$900$	0	0
$901$	−88.8004	−2.95837
$902$	0	0
$903$	5.53223	0.184101
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−23.4363	−0.778189	−0.389095	−	0.921198i	$-0.627212\pi$
$907$	−23.4363	−0.778189	−0.389095	+	0.921198i	$0.627212\pi$
$908$	0	0
$909$	−16.1919	−0.537050
$910$	0	0
$911$	33.1274	1.09756	0.548780	−	0.835967i	$-0.315092\pi$
$911$	33.1274	1.09756	0.548780	+	0.835967i	$0.315092\pi$
$912$	0	0
$913$	−8.09593	−0.267936
$914$	0	0
$915$	−1.22334	−0.0404423
$916$	0	0
$917$	−4.56370	−0.150707
$918$	0	0
$919$	−12.6385	−0.416907	−0.208454	−	0.978032i	$-0.566843\pi$
$919$	−12.6385	−0.416907	−0.208454	+	0.978032i	$0.566843\pi$
$920$	0	0
$921$	9.31927	0.307080
$922$	0	0
$923$	−30.0867	−0.990317
$924$	0	0
$925$	−9.62816	−0.316572
$926$	0	0
$927$	18.8515	0.619164
$928$	0	0
$929$	−13.0645	−0.428631	−0.214315	−	0.976764i	$-0.568752\pi$
$929$	−13.0645	−0.428631	−0.214315	+	0.976764i	$0.568752\pi$
$930$	0	0
$931$	4.09593	0.134239
$932$	0	0
$933$	−17.5111	−0.573289
$934$	0	0
$935$	8.09593	0.264765
$936$	0	0
$937$	40.2459	1.31478	0.657389	−	0.753552i	$-0.271661\pi$
$937$	40.2459	1.31478	0.657389	+	0.753552i	$0.271661\pi$
$938$	0	0
$939$	−27.9160	−0.911003
$940$	0	0
$941$	−53.9489	−1.75869	−0.879343	−	0.476190i	$-0.842018\pi$
$941$	−53.9489	−1.75869	−0.879343	+	0.476190i	$0.842018\pi$
$942$	0	0
$943$	−18.3511	−0.597593
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	29.5441	0.960055	0.480027	−	0.877253i	$-0.340627\pi$
$947$	29.5441	0.960055	0.480027	+	0.877253i	$0.340627\pi$
$948$	0	0
$949$	37.3615	1.21280
$950$	0	0
$951$	−13.0645	−0.423644
$952$	0	0
$953$	−14.3837	−0.465935	−0.232967	−	0.972485i	$-0.574844\pi$
$953$	−14.3837	−0.465935	−0.232967	+	0.972485i	$0.574844\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.65964	0.215275
$958$	0	0
$959$	10.1919	0.329112
$960$	0	0
$961$	−30.6822	−0.989750
$962$	0	0
$963$	14.7556	0.475492
$964$	0	0
$965$	11.0645	0.356178
$966$	0	0
$967$	31.6611	1.01815	0.509077	−	0.860721i	$-0.329987\pi$
$967$	31.6611	1.01815	0.509077	+	0.860721i	$0.329987\pi$
$968$	0	0
$969$	33.1604	1.06527
$970$	0	0
$971$	17.9160	0.574950	0.287475	−	0.957788i	$-0.407184\pi$
$971$	17.9160	0.574950	0.287475	+	0.957788i	$0.407184\pi$
$972$	0	0
$973$	14.0000	0.448819
$974$	0	0
$975$	3.43630	0.110050
$976$	0	0
$977$	−16.7137	−0.534719	−0.267360	−	0.963597i	$-0.586151\pi$
$977$	−16.7137	−0.534719	−0.267360	+	0.963597i	$0.586151\pi$
$978$	0	0
$979$	−0.659636	−0.0210820
$980$	0	0
$981$	−9.62816	−0.307404
$982$	0	0
$983$	−33.7030	−1.07496	−0.537479	−	0.843277i	$-0.680623\pi$
$983$	−33.7030	−1.07496	−0.537479	+	0.843277i	$0.680623\pi$
$984$	0	0
$985$	−13.0645	−0.416268
$986$	0	0
$987$	7.43630	0.236700
$988$	0	0
$989$	−11.5952	−0.368705
$990$	0	0
$991$	−59.0974	−1.87729	−0.938646	−	0.344882i	$-0.887919\pi$
$991$	−59.0974	−1.87729	−0.938646	+	0.344882i	$0.887919\pi$
$992$	0	0
$993$	−1.90407	−0.0604238
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	20.5756	0.651636	0.325818	−	0.945433i	$-0.394360\pi$
$997$	20.5756	0.651636	0.325818	+	0.945433i	$0.394360\pi$
$998$	0	0
$999$	−9.62816	−0.304622

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.v.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.v.1.2	✓	3	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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.43630 q^{13} -1.00000 q^{15} +8.09593 q^{17} +4.09593 q^{19} +1.00000 q^{21} -2.09593 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.65964 q^{29} +0.563703 q^{31} -1.00000 q^{33} -1.00000 q^{35} -9.62816 q^{37} +3.43630 q^{39} +8.75557 q^{41} +5.53223 q^{43} -1.00000 q^{45} +7.43630 q^{47} +1.00000 q^{49} +8.09593 q^{51} -10.9685 q^{53} +1.00000 q^{55} +4.09593 q^{57} -9.53223 q^{59} +1.22334 q^{61} +1.00000 q^{63} -3.43630 q^{65} -8.19186 q^{67} -2.09593 q^{69} -8.75557 q^{71} +10.8726 q^{73} +1.00000 q^{75} -1.00000 q^{77} +9.43630 q^{79} +1.00000 q^{81} +8.09593 q^{83} -8.09593 q^{85} -6.65964 q^{87} +0.659636 q^{89} +3.43630 q^{91} +0.563703 q^{93} -4.09593 q^{95} +7.34036 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} + 4 q^{13} - 3 q^{15} + 8 q^{17} - 4 q^{19} + 3 q^{21} + 10 q^{23} + 3 q^{25} + 3 q^{27} - 10 q^{29} + 8 q^{31} - 3 q^{33} - 3 q^{35} + 10 q^{37} + 4 q^{39} - 6 q^{43} - 3 q^{45} + 16 q^{47} + 3 q^{49} + 8 q^{51} - 4 q^{53} + 3 q^{55} - 4 q^{57} - 6 q^{59} + 3 q^{63} - 4 q^{65} + 8 q^{67} + 10 q^{69} + 20 q^{73} + 3 q^{75} - 3 q^{77} + 22 q^{79} + 3 q^{81} + 8 q^{83} - 8 q^{85} - 10 q^{87} - 8 q^{89} + 4 q^{91} + 8 q^{93} + 4 q^{95} + 32 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 - 3 * q^11 + 4 * q^13 - 3 * q^15 + 8 * q^17 - 4 * q^19 + 3 * q^21 + 10 * q^23 + 3 * q^25 + 3 * q^27 - 10 * q^29 + 8 * q^31 - 3 * q^33 - 3 * q^35 + 10 * q^37 + 4 * q^39 - 6 * q^43 - 3 * q^45 + 16 * q^47 + 3 * q^49 + 8 * q^51 - 4 * q^53 + 3 * q^55 - 4 * q^57 - 6 * q^59 + 3 * q^63 - 4 * q^65 + 8 * q^67 + 10 * q^69 + 20 * q^73 + 3 * q^75 - 3 * q^77 + 22 * q^79 + 3 * q^81 + 8 * q^83 - 8 * q^85 - 10 * q^87 - 8 * q^89 + 4 * q^91 + 8 * q^93 + 4 * q^95 + 32 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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