LMFDB - Embedded newform 4020.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 28x^{4} - 12x^{3} + 209x^{2} + 360x + 144 $ x^6 - x^5 - 28x^4 - 12x^3 + 209x^2 + 360x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.21014$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -2.21014 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -2.21014 q^{7} +1.00000 q^{9} +1.26326 q^{11} +0.106048 q^{13} -1.00000 q^{15} +5.15721 q^{17} -4.68575 q^{19} +2.21014 q^{21} +8.51730 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.17059 q^{29} -5.36736 q^{31} -1.26326 q^{33} -2.21014 q^{35} -6.51730 q^{37} -0.106048 q^{39} +1.30502 q^{41} -3.73887 q^{43} +1.00000 q^{45} -4.77843 q^{47} -2.11527 q^{49} -5.15721 q^{51} +1.73453 q^{53} +1.26326 q^{55} +4.68575 q^{57} +1.89395 q^{59} +6.68575 q^{61} -2.21014 q^{63} +0.106048 q^{65} +1.00000 q^{67} -8.51730 q^{69} -0.801026 q^{71} +7.85634 q^{73} -1.00000 q^{75} -2.79199 q^{77} +15.3625 q^{79} +1.00000 q^{81} -9.14091 q^{83} +5.15721 q^{85} -3.17059 q^{87} +14.8449 q^{89} -0.234381 q^{91} +5.36736 q^{93} -4.68575 q^{95} +10.5549 q^{97} +1.26326 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * q^97 - 7 * 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
$7$	−2.21014	−0.835356	−0.417678	−	0.908595i	$-0.637156\pi$
$7$	−2.21014	−0.835356	−0.417678	+	0.908595i	$0.637156\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.26326	0.380888	0.190444	−	0.981698i	$-0.439007\pi$
$11$	1.26326	0.380888	0.190444	+	0.981698i	$0.439007\pi$
$12$	0	0
$13$	0.106048	0.0294123	0.0147062	−	0.999892i	$-0.495319\pi$
$13$	0.106048	0.0294123	0.0147062	+	0.999892i	$0.495319\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.15721	1.25081	0.625404	−	0.780301i	$-0.284934\pi$
$17$	5.15721	1.25081	0.625404	+	0.780301i	$0.284934\pi$
$18$	0	0
$19$	−4.68575	−1.07499	−0.537493	−	0.843268i	$-0.680628\pi$
$19$	−4.68575	−1.07499	−0.537493	+	0.843268i	$0.680628\pi$
$20$	0	0
$21$	2.21014	0.482293
$22$	0	0
$23$	8.51730	1.77598	0.887990	−	0.459863i	$-0.152101\pi$
$23$	8.51730	1.77598	0.887990	+	0.459863i	$0.152101\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.17059	0.588763	0.294382	−	0.955688i	$-0.404886\pi$
$29$	3.17059	0.588763	0.294382	+	0.955688i	$0.404886\pi$
$30$	0	0
$31$	−5.36736	−0.964006	−0.482003	−	0.876170i	$-0.660091\pi$
$31$	−5.36736	−0.964006	−0.482003	+	0.876170i	$0.660091\pi$
$32$	0	0
$33$	−1.26326	−0.219906
$34$	0	0
$35$	−2.21014	−0.373582
$36$	0	0
$37$	−6.51730	−1.07144	−0.535719	−	0.844396i	$-0.679959\pi$
$37$	−6.51730	−1.07144	−0.535719	+	0.844396i	$0.679959\pi$
$38$	0	0
$39$	−0.106048	−0.0169812
$40$	0	0
$41$	1.30502	0.203810	0.101905	−	0.994794i	$-0.467506\pi$
$41$	1.30502	0.203810	0.101905	+	0.994794i	$0.467506\pi$
$42$	0	0
$43$	−3.73887	−0.570173	−0.285086	−	0.958502i	$-0.592022\pi$
$43$	−3.73887	−0.570173	−0.285086	+	0.958502i	$0.592022\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.77843	−0.697005	−0.348503	−	0.937308i	$-0.613310\pi$
$47$	−4.77843	−0.697005	−0.348503	+	0.937308i	$0.613310\pi$
$48$	0	0
$49$	−2.11527	−0.302181
$50$	0	0
$51$	−5.15721	−0.722154
$52$	0	0
$53$	1.73453	0.238257	0.119128	−	0.992879i	$-0.461990\pi$
$53$	1.73453	0.238257	0.119128	+	0.992879i	$0.461990\pi$
$54$	0	0
$55$	1.26326	0.170338
$56$	0	0
$57$	4.68575	0.620643
$58$	0	0
$59$	1.89395	0.246572	0.123286	−	0.992371i	$-0.460657\pi$
$59$	1.89395	0.246572	0.123286	+	0.992371i	$0.460657\pi$
$60$	0	0
$61$	6.68575	0.856023	0.428012	−	0.903773i	$-0.359214\pi$
$61$	6.68575	0.856023	0.428012	+	0.903773i	$0.359214\pi$
$62$	0	0
$63$	−2.21014	−0.278452
$64$	0	0
$65$	0.106048	0.0131536
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−8.51730	−1.02536
$70$	0	0
$71$	−0.801026	−0.0950642	−0.0475321	−	0.998870i	$-0.515136\pi$
$71$	−0.801026	−0.0950642	−0.0475321	+	0.998870i	$0.515136\pi$
$72$	0	0
$73$	7.85634	0.919515	0.459758	−	0.888044i	$-0.347936\pi$
$73$	7.85634	0.919515	0.459758	+	0.888044i	$0.347936\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−2.79199	−0.318177
$78$	0	0
$79$	15.3625	1.72841	0.864207	−	0.503137i	$-0.167821\pi$
$79$	15.3625	1.72841	0.864207	+	0.503137i	$0.167821\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.14091	−1.00334	−0.501672	−	0.865058i	$-0.667282\pi$
$83$	−9.14091	−1.00334	−0.501672	+	0.865058i	$0.667282\pi$
$84$	0	0
$85$	5.15721	0.559378
$86$	0	0
$87$	−3.17059	−0.339923
$88$	0	0
$89$	14.8449	1.57356	0.786779	−	0.617235i	$-0.211747\pi$
$89$	14.8449	1.57356	0.786779	+	0.617235i	$0.211747\pi$
$90$	0	0
$91$	−0.234381	−0.0245698
$92$	0	0
$93$	5.36736	0.556569
$94$	0	0
$95$	−4.68575	−0.480748
$96$	0	0
$97$	10.5549	1.07169	0.535844	−	0.844317i	$-0.319993\pi$
$97$	10.5549	1.07169	0.535844	+	0.844317i	$0.319993\pi$
$98$	0	0
$99$	1.26326	0.126963
$100$	0	0
$101$	3.56608	0.354838	0.177419	−	0.984135i	$-0.443225\pi$
$101$	3.56608	0.354838	0.177419	+	0.984135i	$0.443225\pi$
$102$	0	0
$103$	15.2957	1.50713	0.753566	−	0.657372i	$-0.228332\pi$
$103$	15.2957	1.50713	0.753566	+	0.657372i	$0.228332\pi$
$104$	0	0
$105$	2.21014	0.215688
$106$	0	0
$107$	−2.56828	−0.248285	−0.124143	−	0.992264i	$-0.539618\pi$
$107$	−2.56828	−0.248285	−0.124143	+	0.992264i	$0.539618\pi$
$108$	0	0
$109$	−15.7961	−1.51300	−0.756498	−	0.653996i	$-0.773091\pi$
$109$	−15.7961	−1.51300	−0.756498	+	0.653996i	$0.773091\pi$
$110$	0	0
$111$	6.51730	0.618595
$112$	0	0
$113$	11.9000	1.11946	0.559728	−	0.828676i	$-0.310906\pi$
$113$	11.9000	1.11946	0.559728	+	0.828676i	$0.310906\pi$
$114$	0	0
$115$	8.51730	0.794242
$116$	0	0
$117$	0.106048	0.00980411
$118$	0	0
$119$	−11.3982	−1.04487
$120$	0	0
$121$	−9.40417	−0.854924
$122$	0	0
$123$	−1.30502	−0.117670
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.78570	0.779605	0.389802	−	0.920899i	$-0.372543\pi$
$127$	8.78570	0.779605	0.389802	+	0.920899i	$0.372543\pi$
$128$	0	0
$129$	3.73887	0.329189
$130$	0	0
$131$	−7.15721	−0.625329	−0.312664	−	0.949864i	$-0.601222\pi$
$131$	−7.15721	−0.625329	−0.312664	+	0.949864i	$0.601222\pi$
$132$	0	0
$133$	10.3562	0.897995
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.93961	0.251148	0.125574	−	0.992084i	$-0.459923\pi$
$137$	2.93961	0.251148	0.125574	+	0.992084i	$0.459923\pi$
$138$	0	0
$139$	7.43580	0.630696	0.315348	−	0.948976i	$-0.397879\pi$
$139$	7.43580	0.630696	0.315348	+	0.948976i	$0.397879\pi$
$140$	0	0
$141$	4.77843	0.402416
$142$	0	0
$143$	0.133966	0.0112028
$144$	0	0
$145$	3.17059	0.263303
$146$	0	0
$147$	2.11527	0.174464
$148$	0	0
$149$	7.61413	0.623774	0.311887	−	0.950119i	$-0.399039\pi$
$149$	7.61413	0.623774	0.311887	+	0.950119i	$0.399039\pi$
$150$	0	0
$151$	−24.4197	−1.98724	−0.993622	−	0.112760i	$-0.964031\pi$
$151$	−24.4197	−1.98724	−0.993622	+	0.112760i	$0.964031\pi$
$152$	0	0
$153$	5.15721	0.416936
$154$	0	0
$155$	−5.36736	−0.431117
$156$	0	0
$157$	13.9888	1.11643	0.558215	−	0.829696i	$-0.311486\pi$
$157$	13.9888	1.11643	0.558215	+	0.829696i	$0.311486\pi$
$158$	0	0
$159$	−1.73453	−0.137557
$160$	0	0
$161$	−18.8245	−1.48357
$162$	0	0
$163$	0.0509842	0.00399339	0.00199670	−	0.999998i	$-0.499364\pi$
$163$	0.0509842	0.00399339	0.00199670	+	0.999998i	$0.499364\pi$
$164$	0	0
$165$	−1.26326	−0.0983448
$166$	0	0
$167$	−2.03735	−0.157655	−0.0788275	−	0.996888i	$-0.525118\pi$
$167$	−2.03735	−0.157655	−0.0788275	+	0.996888i	$0.525118\pi$
$168$	0	0
$169$	−12.9888	−0.999135
$170$	0	0
$171$	−4.68575	−0.358329
$172$	0	0
$173$	8.61901	0.655291	0.327646	−	0.944801i	$-0.393745\pi$
$173$	8.61901	0.655291	0.327646	+	0.944801i	$0.393745\pi$
$174$	0	0
$175$	−2.21014	−0.167071
$176$	0	0
$177$	−1.89395	−0.142358
$178$	0	0
$179$	16.9419	1.26630	0.633149	−	0.774030i	$-0.281762\pi$
$179$	16.9419	1.26630	0.633149	+	0.774030i	$0.281762\pi$
$180$	0	0
$181$	8.41339	0.625363	0.312681	−	0.949858i	$-0.398773\pi$
$181$	8.41339	0.625363	0.312681	+	0.949858i	$0.398773\pi$
$182$	0	0
$183$	−6.68575	−0.494225
$184$	0	0
$185$	−6.51730	−0.479161
$186$	0	0
$187$	6.51491	0.476418
$188$	0	0
$189$	2.21014	0.160764
$190$	0	0
$191$	3.24630	0.234894	0.117447	−	0.993079i	$-0.462529\pi$
$191$	3.24630	0.234894	0.117447	+	0.993079i	$0.462529\pi$
$192$	0	0
$193$	12.8338	0.923794	0.461897	−	0.886934i	$-0.347169\pi$
$193$	12.8338	0.923794	0.461897	+	0.886934i	$0.347169\pi$
$194$	0	0
$195$	−0.106048	−0.00759423
$196$	0	0
$197$	13.3511	0.951223	0.475611	−	0.879655i	$-0.342227\pi$
$197$	13.3511	0.951223	0.475611	+	0.879655i	$0.342227\pi$
$198$	0	0
$199$	−5.53647	−0.392470	−0.196235	−	0.980557i	$-0.562872\pi$
$199$	−5.53647	−0.392470	−0.196235	+	0.980557i	$0.562872\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−7.00745	−0.491827
$204$	0	0
$205$	1.30502	0.0911467
$206$	0	0
$207$	8.51730	0.591993
$208$	0	0
$209$	−5.91933	−0.409449
$210$	0	0
$211$	22.9327	1.57875	0.789376	−	0.613910i	$-0.210404\pi$
$211$	22.9327	1.57875	0.789376	+	0.613910i	$0.210404\pi$
$212$	0	0
$213$	0.801026	0.0548854
$214$	0	0
$215$	−3.73887	−0.254989
$216$	0	0
$217$	11.8626	0.805288
$218$	0	0
$219$	−7.85634	−0.530882
$220$	0	0
$221$	0.546911	0.0367892
$222$	0	0
$223$	10.5620	0.707283	0.353642	−	0.935381i	$-0.384943\pi$
$223$	10.5620	0.707283	0.353642	+	0.935381i	$0.384943\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	24.9122	1.65348	0.826742	−	0.562581i	$-0.190192\pi$
$227$	24.9122	1.65348	0.826742	+	0.562581i	$0.190192\pi$
$228$	0	0
$229$	2.43580	0.160962	0.0804812	−	0.996756i	$-0.474354\pi$
$229$	2.43580	0.160962	0.0804812	+	0.996756i	$0.474354\pi$
$230$	0	0
$231$	2.79199	0.183699
$232$	0	0
$233$	18.6633	1.22268	0.611338	−	0.791369i	$-0.290632\pi$
$233$	18.6633	1.22268	0.611338	+	0.791369i	$0.290632\pi$
$234$	0	0
$235$	−4.77843	−0.311710
$236$	0	0
$237$	−15.3625	−0.997900
$238$	0	0
$239$	−1.81402	−0.117339	−0.0586697	−	0.998277i	$-0.518686\pi$
$239$	−1.81402	−0.117339	−0.0586697	+	0.998277i	$0.518686\pi$
$240$	0	0
$241$	3.92961	0.253129	0.126564	−	0.991958i	$-0.459605\pi$
$241$	3.92961	0.253129	0.126564	+	0.991958i	$0.459605\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.11527	−0.135139
$246$	0	0
$247$	−0.496913	−0.0316178
$248$	0	0
$249$	9.14091	0.579281
$250$	0	0
$251$	−20.0684	−1.26670	−0.633352	−	0.773864i	$-0.718321\pi$
$251$	−20.0684	−1.26670	−0.633352	+	0.773864i	$0.718321\pi$
$252$	0	0
$253$	10.7596	0.676449
$254$	0	0
$255$	−5.15721	−0.322957
$256$	0	0
$257$	−11.7430	−0.732505	−0.366253	−	0.930515i	$-0.619359\pi$
$257$	−11.7430	−0.732505	−0.366253	+	0.930515i	$0.619359\pi$
$258$	0	0
$259$	14.4042	0.895032
$260$	0	0
$261$	3.17059	0.196254
$262$	0	0
$263$	22.5517	1.39060	0.695299	−	0.718720i	$-0.255272\pi$
$263$	22.5517	1.39060	0.695299	+	0.718720i	$0.255272\pi$
$264$	0	0
$265$	1.73453	0.106552
$266$	0	0
$267$	−14.8449	−0.908494
$268$	0	0
$269$	−15.9422	−0.972012	−0.486006	−	0.873956i	$-0.661547\pi$
$269$	−15.9422	−0.972012	−0.486006	+	0.873956i	$0.661547\pi$
$270$	0	0
$271$	−7.97400	−0.484386	−0.242193	−	0.970228i	$-0.577867\pi$
$271$	−7.97400	−0.484386	−0.242193	+	0.970228i	$0.577867\pi$
$272$	0	0
$273$	0.234381	0.0141854
$274$	0	0
$275$	1.26326	0.0761776
$276$	0	0
$277$	14.1664	0.851178	0.425589	−	0.904916i	$-0.360067\pi$
$277$	14.1664	0.851178	0.425589	+	0.904916i	$0.360067\pi$
$278$	0	0
$279$	−5.36736	−0.321335
$280$	0	0
$281$	4.49062	0.267888	0.133944	−	0.990989i	$-0.457236\pi$
$281$	4.49062	0.267888	0.133944	+	0.990989i	$0.457236\pi$
$282$	0	0
$283$	−5.30300	−0.315231	−0.157615	−	0.987501i	$-0.550381\pi$
$283$	−5.30300	−0.315231	−0.157615	+	0.987501i	$0.550381\pi$
$284$	0	0
$285$	4.68575	0.277560
$286$	0	0
$287$	−2.88429	−0.170254
$288$	0	0
$289$	9.59686	0.564521
$290$	0	0
$291$	−10.5549	−0.618740
$292$	0	0
$293$	19.7963	1.15651	0.578257	−	0.815855i	$-0.303733\pi$
$293$	19.7963	1.15651	0.578257	+	0.815855i	$0.303733\pi$
$294$	0	0
$295$	1.89395	0.110270
$296$	0	0
$297$	−1.26326	−0.0733019
$298$	0	0
$299$	0.903240	0.0522357
$300$	0	0
$301$	8.26344	0.476297
$302$	0	0
$303$	−3.56608	−0.204866
$304$	0	0
$305$	6.68575	0.382825
$306$	0	0
$307$	23.0792	1.31720	0.658600	−	0.752493i	$-0.271149\pi$
$307$	23.0792	1.31720	0.658600	+	0.752493i	$0.271149\pi$
$308$	0	0
$309$	−15.2957	−0.870144
$310$	0	0
$311$	17.5043	0.992578	0.496289	−	0.868157i	$-0.334696\pi$
$311$	17.5043	0.992578	0.496289	+	0.868157i	$0.334696\pi$
$312$	0	0
$313$	10.1196	0.571994	0.285997	−	0.958230i	$-0.407675\pi$
$313$	10.1196	0.571994	0.285997	+	0.958230i	$0.407675\pi$
$314$	0	0
$315$	−2.21014	−0.124527
$316$	0	0
$317$	−3.44549	−0.193518	−0.0967589	−	0.995308i	$-0.530848\pi$
$317$	−3.44549	−0.193518	−0.0967589	+	0.995308i	$0.530848\pi$
$318$	0	0
$319$	4.00528	0.224253
$320$	0	0
$321$	2.56828	0.143348
$322$	0	0
$323$	−24.1654	−1.34460
$324$	0	0
$325$	0.106048	0.00588247
$326$	0	0
$327$	15.7961	0.873528
$328$	0	0
$329$	10.5610	0.582247
$330$	0	0
$331$	−35.5084	−1.95172	−0.975858	−	0.218404i	$-0.929915\pi$
$331$	−35.5084	−1.95172	−0.975858	+	0.218404i	$0.929915\pi$
$332$	0	0
$333$	−6.51730	−0.357146
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	−12.2675	−0.668255	−0.334127	−	0.942528i	$-0.608442\pi$
$337$	−12.2675	−0.668255	−0.334127	+	0.942528i	$0.608442\pi$
$338$	0	0
$339$	−11.9000	−0.646318
$340$	0	0
$341$	−6.78038	−0.367178
$342$	0	0
$343$	20.1460	1.08778
$344$	0	0
$345$	−8.51730	−0.458556
$346$	0	0
$347$	−5.58503	−0.299820	−0.149910	−	0.988700i	$-0.547898\pi$
$347$	−5.58503	−0.299820	−0.149910	+	0.988700i	$0.547898\pi$
$348$	0	0
$349$	−12.8036	−0.685362	−0.342681	−	0.939452i	$-0.611335\pi$
$349$	−12.8036	−0.685362	−0.342681	+	0.939452i	$0.611335\pi$
$350$	0	0
$351$	−0.106048	−0.00566041
$352$	0	0
$353$	15.1444	0.806055	0.403028	−	0.915188i	$-0.367958\pi$
$353$	15.1444	0.806055	0.403028	+	0.915188i	$0.367958\pi$
$354$	0	0
$355$	−0.801026	−0.0425140
$356$	0	0
$357$	11.3982	0.603256
$358$	0	0
$359$	−24.9062	−1.31450	−0.657249	−	0.753673i	$-0.728280\pi$
$359$	−24.9062	−1.31450	−0.657249	+	0.753673i	$0.728280\pi$
$360$	0	0
$361$	2.95629	0.155594
$362$	0	0
$363$	9.40417	0.493591
$364$	0	0
$365$	7.85634	0.411220
$366$	0	0
$367$	23.1861	1.21030	0.605152	−	0.796110i	$-0.293112\pi$
$367$	23.1861	1.21030	0.605152	+	0.796110i	$0.293112\pi$
$368$	0	0
$369$	1.30502	0.0679367
$370$	0	0
$371$	−3.83357	−0.199029
$372$	0	0
$373$	−20.4267	−1.05765	−0.528827	−	0.848730i	$-0.677368\pi$
$373$	−20.4267	−1.05765	−0.528827	+	0.848730i	$0.677368\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0.336234	0.0173169
$378$	0	0
$379$	−26.8514	−1.37927	−0.689633	−	0.724159i	$-0.742228\pi$
$379$	−26.8514	−1.37927	−0.689633	+	0.724159i	$0.742228\pi$
$380$	0	0
$381$	−8.78570	−0.450105
$382$	0	0
$383$	−18.7870	−0.959973	−0.479986	−	0.877276i	$-0.659358\pi$
$383$	−18.7870	−0.959973	−0.479986	+	0.877276i	$0.659358\pi$
$384$	0	0
$385$	−2.79199	−0.142293
$386$	0	0
$387$	−3.73887	−0.190058
$388$	0	0
$389$	6.73233	0.341343	0.170671	−	0.985328i	$-0.445406\pi$
$389$	6.73233	0.341343	0.170671	+	0.985328i	$0.445406\pi$
$390$	0	0
$391$	43.9255	2.22141
$392$	0	0
$393$	7.15721	0.361034
$394$	0	0
$395$	15.3625	0.772970
$396$	0	0
$397$	16.6711	0.836700	0.418350	−	0.908286i	$-0.362608\pi$
$397$	16.6711	0.836700	0.418350	+	0.908286i	$0.362608\pi$
$398$	0	0
$399$	−10.3562	−0.518458
$400$	0	0
$401$	−24.5527	−1.22610	−0.613052	−	0.790043i	$-0.710058\pi$
$401$	−24.5527	−1.22610	−0.613052	+	0.790043i	$0.710058\pi$
$402$	0	0
$403$	−0.569196	−0.0283537
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.23306	−0.408098
$408$	0	0
$409$	9.78647	0.483910	0.241955	−	0.970287i	$-0.422211\pi$
$409$	9.78647	0.483910	0.241955	+	0.970287i	$0.422211\pi$
$410$	0	0
$411$	−2.93961	−0.145000
$412$	0	0
$413$	−4.18591	−0.205975
$414$	0	0
$415$	−9.14091	−0.448709
$416$	0	0
$417$	−7.43580	−0.364132
$418$	0	0
$419$	40.2559	1.96663	0.983315	−	0.181913i	$-0.0582288\pi$
$419$	40.2559	1.96663	0.983315	+	0.181913i	$0.0582288\pi$
$420$	0	0
$421$	25.3165	1.23385	0.616925	−	0.787022i	$-0.288378\pi$
$421$	25.3165	1.23385	0.616925	+	0.787022i	$0.288378\pi$
$422$	0	0
$423$	−4.77843	−0.232335
$424$	0	0
$425$	5.15721	0.250162
$426$	0	0
$427$	−14.7765	−0.715084
$428$	0	0
$429$	−0.133966	−0.00646794
$430$	0	0
$431$	−25.0995	−1.20900	−0.604500	−	0.796605i	$-0.706627\pi$
$431$	−25.0995	−1.20900	−0.604500	+	0.796605i	$0.706627\pi$
$432$	0	0
$433$	−16.4636	−0.791189	−0.395594	−	0.918425i	$-0.629461\pi$
$433$	−16.4636	−0.791189	−0.395594	+	0.918425i	$0.629461\pi$
$434$	0	0
$435$	−3.17059	−0.152018
$436$	0	0
$437$	−39.9100	−1.90915
$438$	0	0
$439$	29.9044	1.42726	0.713629	−	0.700524i	$-0.247050\pi$
$439$	29.9044	1.42726	0.713629	+	0.700524i	$0.247050\pi$
$440$	0	0
$441$	−2.11527	−0.100727
$442$	0	0
$443$	−25.3626	−1.20501	−0.602507	−	0.798114i	$-0.705831\pi$
$443$	−25.3626	−1.20501	−0.602507	+	0.798114i	$0.705831\pi$
$444$	0	0
$445$	14.8449	0.703716
$446$	0	0
$447$	−7.61413	−0.360136
$448$	0	0
$449$	14.2634	0.673134	0.336567	−	0.941660i	$-0.390734\pi$
$449$	14.2634	0.673134	0.336567	+	0.941660i	$0.390734\pi$
$450$	0	0
$451$	1.64858	0.0776288
$452$	0	0
$453$	24.4197	1.14734
$454$	0	0
$455$	−0.234381	−0.0109879
$456$	0	0
$457$	4.26308	0.199418	0.0997092	−	0.995017i	$-0.468209\pi$
$457$	4.26308	0.199418	0.0997092	+	0.995017i	$0.468209\pi$
$458$	0	0
$459$	−5.15721	−0.240718
$460$	0	0
$461$	6.71388	0.312697	0.156348	−	0.987702i	$-0.450028\pi$
$461$	6.71388	0.312697	0.156348	+	0.987702i	$0.450028\pi$
$462$	0	0
$463$	35.4205	1.64613	0.823064	−	0.567948i	$-0.192263\pi$
$463$	35.4205	1.64613	0.823064	+	0.567948i	$0.192263\pi$
$464$	0	0
$465$	5.36736	0.248905
$466$	0	0
$467$	−27.6049	−1.27740	−0.638701	−	0.769455i	$-0.720528\pi$
$467$	−27.6049	−1.27740	−0.638701	+	0.769455i	$0.720528\pi$
$468$	0	0
$469$	−2.21014	−0.102055
$470$	0	0
$471$	−13.9888	−0.644571
$472$	0	0
$473$	−4.72318	−0.217172
$474$	0	0
$475$	−4.68575	−0.214997
$476$	0	0
$477$	1.73453	0.0794188
$478$	0	0
$479$	10.6471	0.486480	0.243240	−	0.969966i	$-0.421790\pi$
$479$	10.6471	0.486480	0.243240	+	0.969966i	$0.421790\pi$
$480$	0	0
$481$	−0.691145	−0.0315135
$482$	0	0
$483$	18.8245	0.856542
$484$	0	0
$485$	10.5549	0.479274
$486$	0	0
$487$	−5.55198	−0.251584	−0.125792	−	0.992057i	$-0.540147\pi$
$487$	−5.55198	−0.251584	−0.125792	+	0.992057i	$0.540147\pi$
$488$	0	0
$489$	−0.0509842	−0.00230559
$490$	0	0
$491$	−32.1128	−1.44923	−0.724615	−	0.689154i	$-0.757982\pi$
$491$	−32.1128	−1.44923	−0.724615	+	0.689154i	$0.757982\pi$
$492$	0	0
$493$	16.3514	0.736430
$494$	0	0
$495$	1.26326	0.0567794
$496$	0	0
$497$	1.77038	0.0794125
$498$	0	0
$499$	15.6755	0.701734	0.350867	−	0.936425i	$-0.385887\pi$
$499$	15.6755	0.701734	0.350867	+	0.936425i	$0.385887\pi$
$500$	0	0
$501$	2.03735	0.0910221
$502$	0	0
$503$	7.45785	0.332529	0.166264	−	0.986081i	$-0.446829\pi$
$503$	7.45785	0.332529	0.166264	+	0.986081i	$0.446829\pi$
$504$	0	0
$505$	3.56608	0.158688
$506$	0	0
$507$	12.9888	0.576851
$508$	0	0
$509$	40.3972	1.79057	0.895287	−	0.445490i	$-0.146970\pi$
$509$	40.3972	1.79057	0.895287	+	0.445490i	$0.146970\pi$
$510$	0	0
$511$	−17.3636	−0.768122
$512$	0	0
$513$	4.68575	0.206881
$514$	0	0
$515$	15.2957	0.674010
$516$	0	0
$517$	−6.03641	−0.265481
$518$	0	0
$519$	−8.61901	−0.378333
$520$	0	0
$521$	5.85579	0.256547	0.128274	−	0.991739i	$-0.459056\pi$
$521$	5.85579	0.256547	0.128274	+	0.991739i	$0.459056\pi$
$522$	0	0
$523$	31.3201	1.36953	0.684766	−	0.728763i	$-0.259905\pi$
$523$	31.3201	1.36953	0.684766	+	0.728763i	$0.259905\pi$
$524$	0	0
$525$	2.21014	0.0964586
$526$	0	0
$527$	−27.6806	−1.20579
$528$	0	0
$529$	49.5444	2.15410
$530$	0	0
$531$	1.89395	0.0821905
$532$	0	0
$533$	0.138395	0.00599453
$534$	0	0
$535$	−2.56828	−0.111037
$536$	0	0
$537$	−16.9419	−0.731098
$538$	0	0
$539$	−2.67213	−0.115097
$540$	0	0
$541$	11.5617	0.497075	0.248537	−	0.968622i	$-0.420050\pi$
$541$	11.5617	0.497075	0.248537	+	0.968622i	$0.420050\pi$
$542$	0	0
$543$	−8.41339	−0.361053
$544$	0	0
$545$	−15.7961	−0.676632
$546$	0	0
$547$	−40.0755	−1.71350	−0.856752	−	0.515728i	$-0.827521\pi$
$547$	−40.0755	−1.71350	−0.856752	+	0.515728i	$0.827521\pi$
$548$	0	0
$549$	6.68575	0.285341
$550$	0	0
$551$	−14.8566	−0.632912
$552$	0	0
$553$	−33.9533	−1.44384
$554$	0	0
$555$	6.51730	0.276644
$556$	0	0
$557$	−33.1141	−1.40309	−0.701544	−	0.712626i	$-0.747506\pi$
$557$	−33.1141	−1.40309	−0.701544	+	0.712626i	$0.747506\pi$
$558$	0	0
$559$	−0.396499	−0.0167701
$560$	0	0
$561$	−6.51491	−0.275060
$562$	0	0
$563$	12.5318	0.528151	0.264076	−	0.964502i	$-0.414933\pi$
$563$	12.5318	0.528151	0.264076	+	0.964502i	$0.414933\pi$
$564$	0	0
$565$	11.9000	0.500636
$566$	0	0
$567$	−2.21014	−0.0928173
$568$	0	0
$569$	5.45405	0.228646	0.114323	−	0.993444i	$-0.463530\pi$
$569$	5.45405	0.228646	0.114323	+	0.993444i	$0.463530\pi$
$570$	0	0
$571$	0.527974	0.0220950	0.0110475	−	0.999939i	$-0.496483\pi$
$571$	0.527974	0.0220950	0.0110475	+	0.999939i	$0.496483\pi$
$572$	0	0
$573$	−3.24630	−0.135616
$574$	0	0
$575$	8.51730	0.355196
$576$	0	0
$577$	−9.10872	−0.379201	−0.189600	−	0.981861i	$-0.560719\pi$
$577$	−9.10872	−0.379201	−0.189600	+	0.981861i	$0.560719\pi$
$578$	0	0
$579$	−12.8338	−0.533352
$580$	0	0
$581$	20.2027	0.838150
$582$	0	0
$583$	2.19117	0.0907490
$584$	0	0
$585$	0.106048	0.00438453
$586$	0	0
$587$	−5.25899	−0.217062	−0.108531	−	0.994093i	$-0.534615\pi$
$587$	−5.25899	−0.217062	−0.108531	+	0.994093i	$0.534615\pi$
$588$	0	0
$589$	25.1501	1.03629
$590$	0	0
$591$	−13.3511	−0.549189
$592$	0	0
$593$	7.30936	0.300159	0.150080	−	0.988674i	$-0.452047\pi$
$593$	7.30936	0.300159	0.150080	+	0.988674i	$0.452047\pi$
$594$	0	0
$595$	−11.3982	−0.467280
$596$	0	0
$597$	5.53647	0.226593
$598$	0	0
$599$	−31.5786	−1.29027	−0.645133	−	0.764070i	$-0.723198\pi$
$599$	−31.5786	−1.29027	−0.645133	+	0.764070i	$0.723198\pi$
$600$	0	0
$601$	−5.84285	−0.238335	−0.119167	−	0.992874i	$-0.538023\pi$
$601$	−5.84285	−0.238335	−0.119167	+	0.992874i	$0.538023\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−9.40417	−0.382334
$606$	0	0
$607$	−5.30106	−0.215163	−0.107582	−	0.994196i	$-0.534311\pi$
$607$	−5.30106	−0.215163	−0.107582	+	0.994196i	$0.534311\pi$
$608$	0	0
$609$	7.00745	0.283956
$610$	0	0
$611$	−0.506741	−0.0205006
$612$	0	0
$613$	3.75933	0.151838	0.0759189	−	0.997114i	$-0.475811\pi$
$613$	3.75933	0.151838	0.0759189	+	0.997114i	$0.475811\pi$
$614$	0	0
$615$	−1.30502	−0.0526236
$616$	0	0
$617$	−29.3256	−1.18060	−0.590302	−	0.807183i	$-0.700991\pi$
$617$	−29.3256	−1.18060	−0.590302	+	0.807183i	$0.700991\pi$
$618$	0	0
$619$	−11.8155	−0.474903	−0.237452	−	0.971399i	$-0.576312\pi$
$619$	−11.8155	−0.474903	−0.237452	+	0.971399i	$0.576312\pi$
$620$	0	0
$621$	−8.51730	−0.341787
$622$	0	0
$623$	−32.8094	−1.31448
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.91933	0.236395
$628$	0	0
$629$	−33.6111	−1.34016
$630$	0	0
$631$	−12.6481	−0.503515	−0.251757	−	0.967790i	$-0.581008\pi$
$631$	−12.6481	−0.503515	−0.251757	+	0.967790i	$0.581008\pi$
$632$	0	0
$633$	−22.9327	−0.911493
$634$	0	0
$635$	8.78570	0.348650
$636$	0	0
$637$	−0.224319	−0.00888784
$638$	0	0
$639$	−0.801026	−0.0316881
$640$	0	0
$641$	−43.2712	−1.70911	−0.854555	−	0.519360i	$-0.826170\pi$
$641$	−43.2712	−1.70911	−0.854555	+	0.519360i	$0.826170\pi$
$642$	0	0
$643$	12.9473	0.510590	0.255295	−	0.966863i	$-0.417827\pi$
$643$	12.9473	0.510590	0.255295	+	0.966863i	$0.417827\pi$
$644$	0	0
$645$	3.73887	0.147218
$646$	0	0
$647$	12.3794	0.486683	0.243341	−	0.969941i	$-0.421756\pi$
$647$	12.3794	0.486683	0.243341	+	0.969941i	$0.421756\pi$
$648$	0	0
$649$	2.39256	0.0939161
$650$	0	0
$651$	−11.8626	−0.464933
$652$	0	0
$653$	−31.6462	−1.23841	−0.619205	−	0.785229i	$-0.712545\pi$
$653$	−31.6462	−1.23841	−0.619205	+	0.785229i	$0.712545\pi$
$654$	0	0
$655$	−7.15721	−0.279656
$656$	0	0
$657$	7.85634	0.306505
$658$	0	0
$659$	24.5471	0.956219	0.478109	−	0.878300i	$-0.341322\pi$
$659$	24.5471	0.956219	0.478109	+	0.878300i	$0.341322\pi$
$660$	0	0
$661$	−17.6243	−0.685504	−0.342752	−	0.939426i	$-0.611359\pi$
$661$	−17.6243	−0.685504	−0.342752	+	0.939426i	$0.611359\pi$
$662$	0	0
$663$	−0.546911	−0.0212403
$664$	0	0
$665$	10.3562	0.401596
$666$	0	0
$667$	27.0048	1.04563
$668$	0	0
$669$	−10.5620	−0.408350
$670$	0	0
$671$	8.44586	0.326049
$672$	0	0
$673$	9.87009	0.380464	0.190232	−	0.981739i	$-0.439076\pi$
$673$	9.87009	0.380464	0.190232	+	0.981739i	$0.439076\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	12.6008	0.484290	0.242145	−	0.970240i	$-0.422149\pi$
$677$	12.6008	0.484290	0.242145	+	0.970240i	$0.422149\pi$
$678$	0	0
$679$	−23.3279	−0.895241
$680$	0	0
$681$	−24.9122	−0.954639
$682$	0	0
$683$	27.7893	1.06333	0.531664	−	0.846955i	$-0.321567\pi$
$683$	27.7893	1.06333	0.531664	+	0.846955i	$0.321567\pi$
$684$	0	0
$685$	2.93961	0.112317
$686$	0	0
$687$	−2.43580	−0.0929316
$688$	0	0
$689$	0.183943	0.00700768
$690$	0	0
$691$	−29.0496	−1.10510	−0.552549	−	0.833480i	$-0.686345\pi$
$691$	−29.0496	−1.10510	−0.552549	+	0.833480i	$0.686345\pi$
$692$	0	0
$693$	−2.79199	−0.106059
$694$	0	0
$695$	7.43580	0.282056
$696$	0	0
$697$	6.73028	0.254927
$698$	0	0
$699$	−18.6633	−0.705913
$700$	0	0
$701$	−26.7638	−1.01085	−0.505427	−	0.862869i	$-0.668665\pi$
$701$	−26.7638	−1.01085	−0.505427	+	0.862869i	$0.668665\pi$
$702$	0	0
$703$	30.5385	1.15178
$704$	0	0
$705$	4.77843	0.179966
$706$	0	0
$707$	−7.88155	−0.296416
$708$	0	0
$709$	26.7444	1.00441	0.502203	−	0.864750i	$-0.332523\pi$
$709$	26.7444	1.00441	0.502203	+	0.864750i	$0.332523\pi$
$710$	0	0
$711$	15.3625	0.576138
$712$	0	0
$713$	−45.7154	−1.71206
$714$	0	0
$715$	0.133966	0.00501005
$716$	0	0
$717$	1.81402	0.0677459
$718$	0	0
$719$	22.4122	0.835832	0.417916	−	0.908486i	$-0.362761\pi$
$719$	22.4122	0.835832	0.417916	+	0.908486i	$0.362761\pi$
$720$	0	0
$721$	−33.8058	−1.25899
$722$	0	0
$723$	−3.92961	−0.146144
$724$	0	0
$725$	3.17059	0.117753
$726$	0	0
$727$	−4.28202	−0.158811	−0.0794056	−	0.996842i	$-0.525302\pi$
$727$	−4.28202	−0.158811	−0.0794056	+	0.996842i	$0.525302\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.2822	−0.713177
$732$	0	0
$733$	−18.4320	−0.680803	−0.340401	−	0.940280i	$-0.610563\pi$
$733$	−18.4320	−0.680803	−0.340401	+	0.940280i	$0.610563\pi$
$734$	0	0
$735$	2.11527	0.0780227
$736$	0	0
$737$	1.26326	0.0465329
$738$	0	0
$739$	22.1607	0.815194	0.407597	−	0.913162i	$-0.366367\pi$
$739$	22.1607	0.815194	0.407597	+	0.913162i	$0.366367\pi$
$740$	0	0
$741$	0.496913	0.0182546
$742$	0	0
$743$	5.24091	0.192270	0.0961352	−	0.995368i	$-0.469352\pi$
$743$	5.24091	0.192270	0.0961352	+	0.995368i	$0.469352\pi$
$744$	0	0
$745$	7.61413	0.278960
$746$	0	0
$747$	−9.14091	−0.334448
$748$	0	0
$749$	5.67628	0.207407
$750$	0	0
$751$	−26.2377	−0.957426	−0.478713	−	0.877972i	$-0.658896\pi$
$751$	−26.2377	−0.957426	−0.478713	+	0.877972i	$0.658896\pi$
$752$	0	0
$753$	20.0684	0.731332
$754$	0	0
$755$	−24.4197	−0.888723
$756$	0	0
$757$	18.8780	0.686133	0.343066	−	0.939311i	$-0.388534\pi$
$757$	18.8780	0.686133	0.343066	+	0.939311i	$0.388534\pi$
$758$	0	0
$759$	−10.7596	−0.390548
$760$	0	0
$761$	−11.8626	−0.430018	−0.215009	−	0.976612i	$-0.568978\pi$
$761$	−11.8626	−0.430018	−0.215009	+	0.976612i	$0.568978\pi$
$762$	0	0
$763$	34.9117	1.26389
$764$	0	0
$765$	5.15721	0.186459
$766$	0	0
$767$	0.200849	0.00725225
$768$	0	0
$769$	−34.9825	−1.26150	−0.630751	−	0.775985i	$-0.717253\pi$
$769$	−34.9825	−1.26150	−0.630751	+	0.775985i	$0.717253\pi$
$770$	0	0
$771$	11.7430	0.422912
$772$	0	0
$773$	−9.01222	−0.324147	−0.162074	−	0.986779i	$-0.551818\pi$
$773$	−9.01222	−0.324147	−0.162074	+	0.986779i	$0.551818\pi$
$774$	0	0
$775$	−5.36736	−0.192801
$776$	0	0
$777$	−14.4042	−0.516747
$778$	0	0
$779$	−6.11501	−0.219093
$780$	0	0
$781$	−1.01191	−0.0362088
$782$	0	0
$783$	−3.17059	−0.113308
$784$	0	0
$785$	13.9888	0.499283
$786$	0	0
$787$	−47.2379	−1.68385	−0.841925	−	0.539595i	$-0.818577\pi$
$787$	−47.2379	−1.68385	−0.841925	+	0.539595i	$0.818577\pi$
$788$	0	0
$789$	−22.5517	−0.802862
$790$	0	0
$791$	−26.3007	−0.935144
$792$	0	0
$793$	0.709009	0.0251776
$794$	0	0
$795$	−1.73453	−0.0615176
$796$	0	0
$797$	20.8198	0.737476	0.368738	−	0.929533i	$-0.379790\pi$
$797$	20.8198	0.737476	0.368738	+	0.929533i	$0.379790\pi$
$798$	0	0
$799$	−24.6434	−0.871820
$800$	0	0
$801$	14.8449	0.524519
$802$	0	0
$803$	9.92462	0.350232
$804$	0	0
$805$	−18.8245	−0.663475
$806$	0	0
$807$	15.9422	0.561191
$808$	0	0
$809$	−12.4120	−0.436383	−0.218192	−	0.975906i	$-0.570016\pi$
$809$	−12.4120	−0.436383	−0.218192	+	0.975906i	$0.570016\pi$
$810$	0	0
$811$	−5.32724	−0.187065	−0.0935324	−	0.995616i	$-0.529816\pi$
$811$	−5.32724	−0.187065	−0.0935324	+	0.995616i	$0.529816\pi$
$812$	0	0
$813$	7.97400	0.279660
$814$	0	0
$815$	0.0509842	0.00178590
$816$	0	0
$817$	17.5194	0.612927
$818$	0	0
$819$	−0.234381	−0.00818992
$820$	0	0
$821$	27.3529	0.954623	0.477312	−	0.878734i	$-0.341611\pi$
$821$	27.3529	0.954623	0.477312	+	0.878734i	$0.341611\pi$
$822$	0	0
$823$	13.7190	0.478216	0.239108	−	0.970993i	$-0.423145\pi$
$823$	13.7190	0.478216	0.239108	+	0.970993i	$0.423145\pi$
$824$	0	0
$825$	−1.26326	−0.0439811
$826$	0	0
$827$	−55.5082	−1.93021	−0.965104	−	0.261865i	$-0.915662\pi$
$827$	−55.5082	−1.93021	−0.965104	+	0.261865i	$0.915662\pi$
$828$	0	0
$829$	−39.1652	−1.36026	−0.680132	−	0.733090i	$-0.738077\pi$
$829$	−39.1652	−1.36026	−0.680132	+	0.733090i	$0.738077\pi$
$830$	0	0
$831$	−14.1664	−0.491428
$832$	0	0
$833$	−10.9089	−0.377970
$834$	0	0
$835$	−2.03735	−0.0705054
$836$	0	0
$837$	5.36736	0.185523
$838$	0	0
$839$	−29.2235	−1.00891	−0.504454	−	0.863439i	$-0.668306\pi$
$839$	−29.2235	−1.00891	−0.504454	+	0.863439i	$0.668306\pi$
$840$	0	0
$841$	−18.9474	−0.653358
$842$	0	0
$843$	−4.49062	−0.154665
$844$	0	0
$845$	−12.9888	−0.446827
$846$	0	0
$847$	20.7846	0.714166
$848$	0	0
$849$	5.30300	0.181998
$850$	0	0
$851$	−55.5098	−1.90285
$852$	0	0
$853$	−29.7267	−1.01782	−0.508912	−	0.860819i	$-0.669952\pi$
$853$	−29.7267	−1.01782	−0.508912	+	0.860819i	$0.669952\pi$
$854$	0	0
$855$	−4.68575	−0.160249
$856$	0	0
$857$	29.0798	0.993347	0.496674	−	0.867937i	$-0.334555\pi$
$857$	29.0798	0.993347	0.496674	+	0.867937i	$0.334555\pi$
$858$	0	0
$859$	14.7526	0.503354	0.251677	−	0.967811i	$-0.419018\pi$
$859$	14.7526	0.503354	0.251677	+	0.967811i	$0.419018\pi$
$860$	0	0
$861$	2.88429	0.0982962
$862$	0	0
$863$	31.6185	1.07631	0.538153	−	0.842847i	$-0.319122\pi$
$863$	31.6185	1.07631	0.538153	+	0.842847i	$0.319122\pi$
$864$	0	0
$865$	8.61901	0.293055
$866$	0	0
$867$	−9.59686	−0.325926
$868$	0	0
$869$	19.4068	0.658332
$870$	0	0
$871$	0.106048	0.00359329
$872$	0	0
$873$	10.5549	0.357230
$874$	0	0
$875$	−2.21014	−0.0747165
$876$	0	0
$877$	−13.9267	−0.470270	−0.235135	−	0.971963i	$-0.575553\pi$
$877$	−13.9267	−0.470270	−0.235135	+	0.971963i	$0.575553\pi$
$878$	0	0
$879$	−19.7963	−0.667713
$880$	0	0
$881$	−24.2254	−0.816174	−0.408087	−	0.912943i	$-0.633804\pi$
$881$	−24.2254	−0.816174	−0.408087	+	0.912943i	$0.633804\pi$
$882$	0	0
$883$	−24.6352	−0.829041	−0.414521	−	0.910040i	$-0.636051\pi$
$883$	−24.6352	−0.829041	−0.414521	+	0.910040i	$0.636051\pi$
$884$	0	0
$885$	−1.89395	−0.0636645
$886$	0	0
$887$	−59.3485	−1.99273	−0.996363	−	0.0852053i	$-0.972845\pi$
$887$	−59.3485	−1.99273	−0.996363	+	0.0852053i	$0.972845\pi$
$888$	0	0
$889$	−19.4177	−0.651247
$890$	0	0
$891$	1.26326	0.0423209
$892$	0	0
$893$	22.3905	0.749271
$894$	0	0
$895$	16.9419	0.566306
$896$	0	0
$897$	−0.903240	−0.0301583
$898$	0	0
$899$	−17.0177	−0.567571
$900$	0	0
$901$	8.94536	0.298013
$902$	0	0
$903$	−8.26344	−0.274990
$904$	0	0
$905$	8.41339	0.279671
$906$	0	0
$907$	−36.3047	−1.20548	−0.602738	−	0.797939i	$-0.705924\pi$
$907$	−36.3047	−1.20548	−0.602738	+	0.797939i	$0.705924\pi$
$908$	0	0
$909$	3.56608	0.118279
$910$	0	0
$911$	44.6717	1.48004	0.740020	−	0.672585i	$-0.234816\pi$
$911$	44.6717	1.48004	0.740020	+	0.672585i	$0.234816\pi$
$912$	0	0
$913$	−11.5474	−0.382162
$914$	0	0
$915$	−6.68575	−0.221024
$916$	0	0
$917$	15.8185	0.522372
$918$	0	0
$919$	9.74069	0.321316	0.160658	−	0.987010i	$-0.448638\pi$
$919$	9.74069	0.321316	0.160658	+	0.987010i	$0.448638\pi$
$920$	0	0
$921$	−23.0792	−0.760486
$922$	0	0
$923$	−0.0849469	−0.00279606
$924$	0	0
$925$	−6.51730	−0.214288
$926$	0	0
$927$	15.2957	0.502378
$928$	0	0
$929$	59.0874	1.93860	0.969298	−	0.245891i	$-0.0790805\pi$
$929$	59.0874	1.93860	0.969298	+	0.245891i	$0.0790805\pi$
$930$	0	0
$931$	9.91161	0.324840
$932$	0	0
$933$	−17.5043	−0.573065
$934$	0	0
$935$	6.51491	0.213060
$936$	0	0
$937$	−57.4884	−1.87806	−0.939032	−	0.343831i	$-0.888275\pi$
$937$	−57.4884	−1.87806	−0.939032	+	0.343831i	$0.888275\pi$
$938$	0	0
$939$	−10.1196	−0.330241
$940$	0	0
$941$	−27.1219	−0.884147	−0.442074	−	0.896979i	$-0.645757\pi$
$941$	−27.1219	−0.884147	−0.442074	+	0.896979i	$0.645757\pi$
$942$	0	0
$943$	11.1153	0.361963
$944$	0	0
$945$	2.21014	0.0718960
$946$	0	0
$947$	25.5476	0.830184	0.415092	−	0.909779i	$-0.363749\pi$
$947$	25.5476	0.830184	0.415092	+	0.909779i	$0.363749\pi$
$948$	0	0
$949$	0.833147	0.0270451
$950$	0	0
$951$	3.44549	0.111728
$952$	0	0
$953$	34.6562	1.12262	0.561312	−	0.827604i	$-0.310297\pi$
$953$	34.6562	1.12262	0.561312	+	0.827604i	$0.310297\pi$
$954$	0	0
$955$	3.24630	0.105048
$956$	0	0
$957$	−4.00528	−0.129472
$958$	0	0
$959$	−6.49696	−0.209798
$960$	0	0
$961$	−2.19147	−0.0706925
$962$	0	0
$963$	−2.56828	−0.0827618
$964$	0	0
$965$	12.8338	0.413133
$966$	0	0
$967$	−13.3646	−0.429778	−0.214889	−	0.976638i	$-0.568939\pi$
$967$	−13.3646	−0.429778	−0.214889	+	0.976638i	$0.568939\pi$
$968$	0	0
$969$	24.1654	0.776306
$970$	0	0
$971$	−18.3244	−0.588059	−0.294029	−	0.955796i	$-0.594996\pi$
$971$	−18.3244	−0.588059	−0.294029	+	0.955796i	$0.594996\pi$
$972$	0	0
$973$	−16.4342	−0.526855
$974$	0	0
$975$	−0.106048	−0.00339624
$976$	0	0
$977$	−7.83133	−0.250546	−0.125273	−	0.992122i	$-0.539981\pi$
$977$	−7.83133	−0.250546	−0.125273	+	0.992122i	$0.539981\pi$
$978$	0	0
$979$	18.7530	0.599349
$980$	0	0
$981$	−15.7961	−0.504332
$982$	0	0
$983$	2.82690	0.0901642	0.0450821	−	0.998983i	$-0.485645\pi$
$983$	2.82690	0.0901642	0.0450821	+	0.998983i	$0.485645\pi$
$984$	0	0
$985$	13.3511	0.425400
$986$	0	0
$987$	−10.5610	−0.336161
$988$	0	0
$989$	−31.8451	−1.01262
$990$	0	0
$991$	11.4824	0.364750	0.182375	−	0.983229i	$-0.441622\pi$
$991$	11.4824	0.364750	0.182375	+	0.983229i	$0.441622\pi$
$992$	0	0
$993$	35.5084	1.12682
$994$	0	0
$995$	−5.53647	−0.175518
$996$	0	0
$997$	−44.3361	−1.40414	−0.702069	−	0.712109i	$-0.747740\pi$
$997$	−44.3361	−1.40414	−0.702069	+	0.712109i	$0.747740\pi$
$998$	0	0
$999$	6.51730	0.206198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.f.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.f.1.2	✓	6	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.21014 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -2.21014 q^{7} +1.00000 q^{9} +1.26326 q^{11} +0.106048 q^{13} -1.00000 q^{15} +5.15721 q^{17} -4.68575 q^{19} +2.21014 q^{21} +8.51730 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.17059 q^{29} -5.36736 q^{31} -1.26326 q^{33} -2.21014 q^{35} -6.51730 q^{37} -0.106048 q^{39} +1.30502 q^{41} -3.73887 q^{43} +1.00000 q^{45} -4.77843 q^{47} -2.11527 q^{49} -5.15721 q^{51} +1.73453 q^{53} +1.26326 q^{55} +4.68575 q^{57} +1.89395 q^{59} +6.68575 q^{61} -2.21014 q^{63} +0.106048 q^{65} +1.00000 q^{67} -8.51730 q^{69} -0.801026 q^{71} +7.85634 q^{73} -1.00000 q^{75} -2.79199 q^{77} +15.3625 q^{79} +1.00000 q^{81} -9.14091 q^{83} +5.15721 q^{85} -3.17059 q^{87} +14.8449 q^{89} -0.234381 q^{91} +5.36736 q^{93} -4.68575 q^{95} +10.5549 q^{97} +1.26326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + q^{7} + 6 q^{9} - 7 q^{11} + 7 q^{13} - 6 q^{15} + 10 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 6 q^{25} - 6 q^{27} + 9 q^{29} + 3 q^{31} + 7 q^{33} + q^{35} + 16 q^{37} - 7 q^{39} + 7 q^{41} + 3 q^{43} + 6 q^{45} + q^{47} + 15 q^{49} - 10 q^{51} + 7 q^{53} - 7 q^{55} + 3 q^{57} + 5 q^{59} + 15 q^{61} + q^{63} + 7 q^{65} + 6 q^{67} + 4 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{75} + 9 q^{77} + 9 q^{79} + 6 q^{81} - 2 q^{83} + 10 q^{85} - 9 q^{87} + 10 q^{89} - 5 q^{91} - 3 q^{93} - 3 q^{95} + 37 q^{97} - 7 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + q^7 + 6 * q^9 - 7 * q^11 + 7 * q^13 - 6 * q^15 + 10 * q^17 - 3 * q^19 - q^21 - 4 * q^23 + 6 * q^25 - 6 * q^27 + 9 * q^29 + 3 * q^31 + 7 * q^33 + q^35 + 16 * q^37 - 7 * q^39 + 7 * q^41 + 3 * q^43 + 6 * q^45 + q^47 + 15 * q^49 - 10 * q^51 + 7 * q^53 - 7 * q^55 + 3 * q^57 + 5 * q^59 + 15 * q^61 + q^63 + 7 * q^65 + 6 * q^67 + 4 * q^69 - 12 * q^71 + 12 * q^73 - 6 * q^75 + 9 * q^77 + 9 * q^79 + 6 * q^81 - 2 * q^83 + 10 * q^85 - 9 * q^87 + 10 * q^89 - 5 * q^91 - 3 * q^93 - 3 * q^95 + 37 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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