LMFDB - Embedded newform 6036.2.a.g.1.9

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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.546893$ of defining polynomial
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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.453107 q^{5} +2.17063 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.453107 q^{5} +2.17063 q^{7} +1.00000 q^{9} -5.35442 q^{11} +2.71175 q^{13} -0.453107 q^{15} +6.07839 q^{17} -2.86135 q^{19} +2.17063 q^{21} -5.42982 q^{23} -4.79469 q^{25} +1.00000 q^{27} -7.45567 q^{29} -5.78314 q^{31} -5.35442 q^{33} -0.983530 q^{35} +1.10695 q^{37} +2.71175 q^{39} -6.10245 q^{41} +6.08829 q^{43} -0.453107 q^{45} -3.32792 q^{47} -2.28835 q^{49} +6.07839 q^{51} -5.31676 q^{53} +2.42613 q^{55} -2.86135 q^{57} +1.97520 q^{59} -14.0797 q^{61} +2.17063 q^{63} -1.22872 q^{65} +3.24688 q^{67} -5.42982 q^{69} +3.70398 q^{71} +2.73959 q^{73} -4.79469 q^{75} -11.6225 q^{77} +2.25178 q^{79} +1.00000 q^{81} +0.0721083 q^{83} -2.75416 q^{85} -7.45567 q^{87} -9.49320 q^{89} +5.88622 q^{91} -5.78314 q^{93} +1.29650 q^{95} -2.29553 q^{97} -5.35442 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * 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
$5$	−0.453107	−0.202636	−0.101318	−	0.994854i	$-0.532306\pi$
$5$	−0.453107	−0.202636	−0.101318	+	0.994854i	$0.532306\pi$
$6$	0	0
$7$	2.17063	0.820422	0.410211	−	0.911991i	$-0.365455\pi$
$7$	2.17063	0.820422	0.410211	+	0.911991i	$0.365455\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.35442	−1.61442	−0.807209	−	0.590266i	$-0.799023\pi$
$11$	−5.35442	−1.61442	−0.807209	+	0.590266i	$0.799023\pi$
$12$	0	0
$13$	2.71175	0.752105	0.376052	−	0.926598i	$-0.377281\pi$
$13$	2.71175	0.752105	0.376052	+	0.926598i	$0.377281\pi$
$14$	0	0
$15$	−0.453107	−0.116992
$16$	0	0
$17$	6.07839	1.47423	0.737113	−	0.675770i	$-0.236189\pi$
$17$	6.07839	1.47423	0.737113	+	0.675770i	$0.236189\pi$
$18$	0	0
$19$	−2.86135	−0.656439	−0.328220	−	0.944601i	$-0.606449\pi$
$19$	−2.86135	−0.656439	−0.328220	+	0.944601i	$0.606449\pi$
$20$	0	0
$21$	2.17063	0.473671
$22$	0	0
$23$	−5.42982	−1.13220	−0.566098	−	0.824338i	$-0.691547\pi$
$23$	−5.42982	−1.13220	−0.566098	+	0.824338i	$0.691547\pi$
$24$	0	0
$25$	−4.79469	−0.958939
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.45567	−1.38448	−0.692242	−	0.721666i	$-0.743377\pi$
$29$	−7.45567	−1.38448	−0.692242	+	0.721666i	$0.743377\pi$
$30$	0	0
$31$	−5.78314	−1.03868	−0.519341	−	0.854567i	$-0.673823\pi$
$31$	−5.78314	−1.03868	−0.519341	+	0.854567i	$0.673823\pi$
$32$	0	0
$33$	−5.35442	−0.932084
$34$	0	0
$35$	−0.983530	−0.166247
$36$	0	0
$37$	1.10695	0.181981	0.0909907	−	0.995852i	$-0.470997\pi$
$37$	1.10695	0.181981	0.0909907	+	0.995852i	$0.470997\pi$
$38$	0	0
$39$	2.71175	0.434228
$40$	0	0
$41$	−6.10245	−0.953043	−0.476521	−	0.879163i	$-0.658103\pi$
$41$	−6.10245	−0.953043	−0.476521	+	0.879163i	$0.658103\pi$
$42$	0	0
$43$	6.08829	0.928456	0.464228	−	0.885716i	$-0.346332\pi$
$43$	6.08829	0.928456	0.464228	+	0.885716i	$0.346332\pi$
$44$	0	0
$45$	−0.453107	−0.0675453
$46$	0	0
$47$	−3.32792	−0.485428	−0.242714	−	0.970098i	$-0.578038\pi$
$47$	−3.32792	−0.485428	−0.242714	+	0.970098i	$0.578038\pi$
$48$	0	0
$49$	−2.28835	−0.326907
$50$	0	0
$51$	6.07839	0.851145
$52$	0	0
$53$	−5.31676	−0.730313	−0.365157	−	0.930946i	$-0.618985\pi$
$53$	−5.31676	−0.730313	−0.365157	+	0.930946i	$0.618985\pi$
$54$	0	0
$55$	2.42613	0.327139
$56$	0	0
$57$	−2.86135	−0.378995
$58$	0	0
$59$	1.97520	0.257150	0.128575	−	0.991700i	$-0.458960\pi$
$59$	1.97520	0.257150	0.128575	+	0.991700i	$0.458960\pi$
$60$	0	0
$61$	−14.0797	−1.80272	−0.901360	−	0.433071i	$-0.857430\pi$
$61$	−14.0797	−1.80272	−0.901360	+	0.433071i	$0.857430\pi$
$62$	0	0
$63$	2.17063	0.273474
$64$	0	0
$65$	−1.22872	−0.152403
$66$	0	0
$67$	3.24688	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$67$	3.24688	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$68$	0	0
$69$	−5.42982	−0.653673
$70$	0	0
$71$	3.70398	0.439581	0.219791	−	0.975547i	$-0.429463\pi$
$71$	3.70398	0.439581	0.219791	+	0.975547i	$0.429463\pi$
$72$	0	0
$73$	2.73959	0.320645	0.160322	−	0.987065i	$-0.448747\pi$
$73$	2.73959	0.320645	0.160322	+	0.987065i	$0.448747\pi$
$74$	0	0
$75$	−4.79469	−0.553644
$76$	0	0
$77$	−11.6225	−1.32450
$78$	0	0
$79$	2.25178	0.253345	0.126673	−	0.991945i	$-0.459570\pi$
$79$	2.25178	0.253345	0.126673	+	0.991945i	$0.459570\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.0721083	0.00791492	0.00395746	−	0.999992i	$-0.498740\pi$
$83$	0.0721083	0.00791492	0.00395746	+	0.999992i	$0.498740\pi$
$84$	0	0
$85$	−2.75416	−0.298731
$86$	0	0
$87$	−7.45567	−0.799332
$88$	0	0
$89$	−9.49320	−1.00628	−0.503139	−	0.864206i	$-0.667821\pi$
$89$	−9.49320	−1.00628	−0.503139	+	0.864206i	$0.667821\pi$
$90$	0	0
$91$	5.88622	0.617044
$92$	0	0
$93$	−5.78314	−0.599683
$94$	0	0
$95$	1.29650	0.133018
$96$	0	0
$97$	−2.29553	−0.233076	−0.116538	−	0.993186i	$-0.537180\pi$
$97$	−2.29553	−0.233076	−0.116538	+	0.993186i	$0.537180\pi$
$98$	0	0
$99$	−5.35442	−0.538139
$100$	0	0
$101$	3.49019	0.347286	0.173643	−	0.984809i	$-0.444446\pi$
$101$	3.49019	0.347286	0.173643	+	0.984809i	$0.444446\pi$
$102$	0	0
$103$	14.8256	1.46081	0.730404	−	0.683016i	$-0.239332\pi$
$103$	14.8256	1.46081	0.730404	+	0.683016i	$0.239332\pi$
$104$	0	0
$105$	−0.983530	−0.0959827
$106$	0	0
$107$	14.2199	1.37469	0.687345	−	0.726331i	$-0.258776\pi$
$107$	14.2199	1.37469	0.687345	+	0.726331i	$0.258776\pi$
$108$	0	0
$109$	15.1497	1.45108	0.725541	−	0.688179i	$-0.241590\pi$
$109$	15.1497	1.45108	0.725541	+	0.688179i	$0.241590\pi$
$110$	0	0
$111$	1.10695	0.105067
$112$	0	0
$113$	−14.1496	−1.33108	−0.665540	−	0.746363i	$-0.731799\pi$
$113$	−14.1496	−1.33108	−0.665540	+	0.746363i	$0.731799\pi$
$114$	0	0
$115$	2.46029	0.229423
$116$	0	0
$117$	2.71175	0.250702
$118$	0	0
$119$	13.1940	1.20949
$120$	0	0
$121$	17.6698	1.60634
$122$	0	0
$123$	−6.10245	−0.550240
$124$	0	0
$125$	4.43805	0.396951
$126$	0	0
$127$	−17.1939	−1.52571	−0.762857	−	0.646568i	$-0.776204\pi$
$127$	−17.1939	−1.52571	−0.762857	+	0.646568i	$0.776204\pi$
$128$	0	0
$129$	6.08829	0.536044
$130$	0	0
$131$	9.99301	0.873093	0.436547	−	0.899682i	$-0.356201\pi$
$131$	9.99301	0.873093	0.436547	+	0.899682i	$0.356201\pi$
$132$	0	0
$133$	−6.21095	−0.538557
$134$	0	0
$135$	−0.453107	−0.0389973
$136$	0	0
$137$	−23.0873	−1.97248	−0.986239	−	0.165328i	$-0.947132\pi$
$137$	−23.0873	−1.97248	−0.986239	+	0.165328i	$0.947132\pi$
$138$	0	0
$139$	−6.63055	−0.562396	−0.281198	−	0.959650i	$-0.590732\pi$
$139$	−6.63055	−0.562396	−0.281198	+	0.959650i	$0.590732\pi$
$140$	0	0
$141$	−3.32792	−0.280262
$142$	0	0
$143$	−14.5199	−1.21421
$144$	0	0
$145$	3.37822	0.280546
$146$	0	0
$147$	−2.28835	−0.188740
$148$	0	0
$149$	−13.9556	−1.14328	−0.571642	−	0.820503i	$-0.693693\pi$
$149$	−13.9556	−1.14328	−0.571642	+	0.820503i	$0.693693\pi$
$150$	0	0
$151$	−1.16809	−0.0950575	−0.0475287	−	0.998870i	$-0.515135\pi$
$151$	−1.16809	−0.0950575	−0.0475287	+	0.998870i	$0.515135\pi$
$152$	0	0
$153$	6.07839	0.491409
$154$	0	0
$155$	2.62038	0.210474
$156$	0	0
$157$	−7.29448	−0.582163	−0.291081	−	0.956698i	$-0.594015\pi$
$157$	−7.29448	−0.582163	−0.291081	+	0.956698i	$0.594015\pi$
$158$	0	0
$159$	−5.31676	−0.421647
$160$	0	0
$161$	−11.7861	−0.928878
$162$	0	0
$163$	10.0146	0.784407	0.392203	−	0.919879i	$-0.371713\pi$
$163$	10.0146	0.784407	0.392203	+	0.919879i	$0.371713\pi$
$164$	0	0
$165$	2.42613	0.188874
$166$	0	0
$167$	12.8988	0.998137	0.499069	−	0.866562i	$-0.333676\pi$
$167$	12.8988	0.998137	0.499069	+	0.866562i	$0.333676\pi$
$168$	0	0
$169$	−5.64640	−0.434338
$170$	0	0
$171$	−2.86135	−0.218813
$172$	0	0
$173$	−11.0976	−0.843737	−0.421868	−	0.906657i	$-0.638626\pi$
$173$	−11.0976	−0.843737	−0.421868	+	0.906657i	$0.638626\pi$
$174$	0	0
$175$	−10.4075	−0.786735
$176$	0	0
$177$	1.97520	0.148465
$178$	0	0
$179$	−19.9102	−1.48816	−0.744081	−	0.668090i	$-0.767112\pi$
$179$	−19.9102	−1.48816	−0.744081	+	0.668090i	$0.767112\pi$
$180$	0	0
$181$	13.2772	0.986884	0.493442	−	0.869779i	$-0.335739\pi$
$181$	13.2772	0.986884	0.493442	+	0.869779i	$0.335739\pi$
$182$	0	0
$183$	−14.0797	−1.04080
$184$	0	0
$185$	−0.501567	−0.0368760
$186$	0	0
$187$	−32.5462	−2.38002
$188$	0	0
$189$	2.17063	0.157890
$190$	0	0
$191$	−9.57891	−0.693105	−0.346553	−	0.938031i	$-0.612648\pi$
$191$	−9.57891	−0.693105	−0.346553	+	0.938031i	$0.612648\pi$
$192$	0	0
$193$	−2.29142	−0.164940	−0.0824701	−	0.996594i	$-0.526281\pi$
$193$	−2.29142	−0.164940	−0.0824701	+	0.996594i	$0.526281\pi$
$194$	0	0
$195$	−1.22872	−0.0879901
$196$	0	0
$197$	−8.32164	−0.592893	−0.296446	−	0.955050i	$-0.595802\pi$
$197$	−8.32164	−0.592893	−0.296446	+	0.955050i	$0.595802\pi$
$198$	0	0
$199$	24.0433	1.70439	0.852193	−	0.523227i	$-0.175272\pi$
$199$	24.0433	1.70439	0.852193	+	0.523227i	$0.175272\pi$
$200$	0	0
$201$	3.24688	0.229017
$202$	0	0
$203$	−16.1835	−1.13586
$204$	0	0
$205$	2.76507	0.193121
$206$	0	0
$207$	−5.42982	−0.377398
$208$	0	0
$209$	15.3209	1.05977
$210$	0	0
$211$	19.3329	1.33093	0.665466	−	0.746428i	$-0.268233\pi$
$211$	19.3329	1.33093	0.665466	+	0.746428i	$0.268233\pi$
$212$	0	0
$213$	3.70398	0.253792
$214$	0	0
$215$	−2.75865	−0.188138
$216$	0	0
$217$	−12.5531	−0.852158
$218$	0	0
$219$	2.73959	0.185124
$220$	0	0
$221$	16.4831	1.10877
$222$	0	0
$223$	−7.11843	−0.476686	−0.238343	−	0.971181i	$-0.576604\pi$
$223$	−7.11843	−0.476686	−0.238343	+	0.971181i	$0.576604\pi$
$224$	0	0
$225$	−4.79469	−0.319646
$226$	0	0
$227$	−8.15200	−0.541068	−0.270534	−	0.962710i	$-0.587200\pi$
$227$	−8.15200	−0.541068	−0.270534	+	0.962710i	$0.587200\pi$
$228$	0	0
$229$	−3.40020	−0.224692	−0.112346	−	0.993669i	$-0.535836\pi$
$229$	−3.40020	−0.224692	−0.112346	+	0.993669i	$0.535836\pi$
$230$	0	0
$231$	−11.6225	−0.764703
$232$	0	0
$233$	−1.36410	−0.0893651	−0.0446825	−	0.999001i	$-0.514228\pi$
$233$	−1.36410	−0.0893651	−0.0446825	+	0.999001i	$0.514228\pi$
$234$	0	0
$235$	1.50791	0.0983650
$236$	0	0
$237$	2.25178	0.146269
$238$	0	0
$239$	−10.6613	−0.689621	−0.344811	−	0.938672i	$-0.612057\pi$
$239$	−10.6613	−0.689621	−0.344811	+	0.938672i	$0.612057\pi$
$240$	0	0
$241$	−2.96872	−0.191232	−0.0956160	−	0.995418i	$-0.530482\pi$
$241$	−2.96872	−0.191232	−0.0956160	+	0.995418i	$0.530482\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.03687	0.0662431
$246$	0	0
$247$	−7.75928	−0.493711
$248$	0	0
$249$	0.0721083	0.00456968
$250$	0	0
$251$	−0.703338	−0.0443943	−0.0221971	−	0.999754i	$-0.507066\pi$
$251$	−0.703338	−0.0443943	−0.0221971	+	0.999754i	$0.507066\pi$
$252$	0	0
$253$	29.0735	1.82784
$254$	0	0
$255$	−2.75416	−0.172472
$256$	0	0
$257$	−10.4602	−0.652487	−0.326243	−	0.945286i	$-0.605783\pi$
$257$	−10.4602	−0.652487	−0.326243	+	0.945286i	$0.605783\pi$
$258$	0	0
$259$	2.40278	0.149302
$260$	0	0
$261$	−7.45567	−0.461495
$262$	0	0
$263$	−12.1467	−0.748996	−0.374498	−	0.927228i	$-0.622185\pi$
$263$	−12.1467	−0.748996	−0.374498	+	0.927228i	$0.622185\pi$
$264$	0	0
$265$	2.40906	0.147988
$266$	0	0
$267$	−9.49320	−0.580974
$268$	0	0
$269$	−0.974897	−0.0594405	−0.0297203	−	0.999558i	$-0.509462\pi$
$269$	−0.974897	−0.0594405	−0.0297203	+	0.999558i	$0.509462\pi$
$270$	0	0
$271$	9.06798	0.550841	0.275420	−	0.961324i	$-0.411183\pi$
$271$	9.06798	0.550841	0.275420	+	0.961324i	$0.411183\pi$
$272$	0	0
$273$	5.88622	0.356250
$274$	0	0
$275$	25.6728	1.54813
$276$	0	0
$277$	−14.5217	−0.872526	−0.436263	−	0.899819i	$-0.643698\pi$
$277$	−14.5217	−0.872526	−0.436263	+	0.899819i	$0.643698\pi$
$278$	0	0
$279$	−5.78314	−0.346227
$280$	0	0
$281$	0.623745	0.0372095	0.0186048	−	0.999827i	$-0.494078\pi$
$281$	0.623745	0.0372095	0.0186048	+	0.999827i	$0.494078\pi$
$282$	0	0
$283$	−25.1834	−1.49700	−0.748498	−	0.663137i	$-0.769225\pi$
$283$	−25.1834	−1.49700	−0.748498	+	0.663137i	$0.769225\pi$
$284$	0	0
$285$	1.29650	0.0767980
$286$	0	0
$287$	−13.2462	−0.781897
$288$	0	0
$289$	19.9468	1.17334
$290$	0	0
$291$	−2.29553	−0.134567
$292$	0	0
$293$	4.37810	0.255772	0.127886	−	0.991789i	$-0.459181\pi$
$293$	4.37810	0.255772	0.127886	+	0.991789i	$0.459181\pi$
$294$	0	0
$295$	−0.894980	−0.0521077
$296$	0	0
$297$	−5.35442	−0.310695
$298$	0	0
$299$	−14.7243	−0.851529
$300$	0	0
$301$	13.2155	0.761726
$302$	0	0
$303$	3.49019	0.200506
$304$	0	0
$305$	6.37961	0.365295
$306$	0	0
$307$	3.46257	0.197619	0.0988096	−	0.995106i	$-0.468497\pi$
$307$	3.46257	0.197619	0.0988096	+	0.995106i	$0.468497\pi$
$308$	0	0
$309$	14.8256	0.843398
$310$	0	0
$311$	6.24430	0.354082	0.177041	−	0.984203i	$-0.443347\pi$
$311$	6.24430	0.354082	0.177041	+	0.984203i	$0.443347\pi$
$312$	0	0
$313$	15.9415	0.901066	0.450533	−	0.892760i	$-0.351234\pi$
$313$	15.9415	0.901066	0.450533	+	0.892760i	$0.351234\pi$
$314$	0	0
$315$	−0.983530	−0.0554156
$316$	0	0
$317$	11.5258	0.647353	0.323677	−	0.946168i	$-0.395081\pi$
$317$	11.5258	0.647353	0.323677	+	0.946168i	$0.395081\pi$
$318$	0	0
$319$	39.9208	2.23513
$320$	0	0
$321$	14.2199	0.793678
$322$	0	0
$323$	−17.3924	−0.967739
$324$	0	0
$325$	−13.0020	−0.721222
$326$	0	0
$327$	15.1497	0.837782
$328$	0	0
$329$	−7.22370	−0.398256
$330$	0	0
$331$	20.7940	1.14294	0.571471	−	0.820622i	$-0.306373\pi$
$331$	20.7940	1.14294	0.571471	+	0.820622i	$0.306373\pi$
$332$	0	0
$333$	1.10695	0.0606605
$334$	0	0
$335$	−1.47119	−0.0803795
$336$	0	0
$337$	−8.98189	−0.489275	−0.244637	−	0.969615i	$-0.578669\pi$
$337$	−8.98189	−0.489275	−0.244637	+	0.969615i	$0.578669\pi$
$338$	0	0
$339$	−14.1496	−0.768499
$340$	0	0
$341$	30.9653	1.67687
$342$	0	0
$343$	−20.1616	−1.08862
$344$	0	0
$345$	2.46029	0.132458
$346$	0	0
$347$	10.2501	0.550256	0.275128	−	0.961408i	$-0.411280\pi$
$347$	10.2501	0.550256	0.275128	+	0.961408i	$0.411280\pi$
$348$	0	0
$349$	−2.62046	−0.140270	−0.0701351	−	0.997538i	$-0.522343\pi$
$349$	−2.62046	−0.140270	−0.0701351	+	0.997538i	$0.522343\pi$
$350$	0	0
$351$	2.71175	0.144743
$352$	0	0
$353$	30.2356	1.60928	0.804640	−	0.593763i	$-0.202358\pi$
$353$	30.2356	1.60928	0.804640	+	0.593763i	$0.202358\pi$
$354$	0	0
$355$	−1.67830	−0.0890749
$356$	0	0
$357$	13.1940	0.698298
$358$	0	0
$359$	−2.60842	−0.137667	−0.0688336	−	0.997628i	$-0.521928\pi$
$359$	−2.60842	−0.137667	−0.0688336	+	0.997628i	$0.521928\pi$
$360$	0	0
$361$	−10.8127	−0.569088
$362$	0	0
$363$	17.6698	0.927423
$364$	0	0
$365$	−1.24133	−0.0649741
$366$	0	0
$367$	−12.4100	−0.647798	−0.323899	−	0.946092i	$-0.604994\pi$
$367$	−12.4100	−0.647798	−0.323899	+	0.946092i	$0.604994\pi$
$368$	0	0
$369$	−6.10245	−0.317681
$370$	0	0
$371$	−11.5407	−0.599165
$372$	0	0
$373$	−33.2918	−1.72378	−0.861892	−	0.507091i	$-0.830721\pi$
$373$	−33.2918	−1.72378	−0.861892	+	0.507091i	$0.830721\pi$
$374$	0	0
$375$	4.43805	0.229180
$376$	0	0
$377$	−20.2179	−1.04128
$378$	0	0
$379$	−5.23029	−0.268662	−0.134331	−	0.990937i	$-0.542889\pi$
$379$	−5.23029	−0.268662	−0.134331	+	0.990937i	$0.542889\pi$
$380$	0	0
$381$	−17.1939	−0.880871
$382$	0	0
$383$	25.1088	1.28300	0.641501	−	0.767123i	$-0.278312\pi$
$383$	25.1088	1.28300	0.641501	+	0.767123i	$0.278312\pi$
$384$	0	0
$385$	5.26623	0.268392
$386$	0	0
$387$	6.08829	0.309485
$388$	0	0
$389$	4.58702	0.232571	0.116286	−	0.993216i	$-0.462901\pi$
$389$	4.58702	0.232571	0.116286	+	0.993216i	$0.462901\pi$
$390$	0	0
$391$	−33.0045	−1.66911
$392$	0	0
$393$	9.99301	0.504081
$394$	0	0
$395$	−1.02030	−0.0513368
$396$	0	0
$397$	22.0046	1.10438	0.552189	−	0.833719i	$-0.313793\pi$
$397$	22.0046	1.10438	0.552189	+	0.833719i	$0.313793\pi$
$398$	0	0
$399$	−6.21095	−0.310936
$400$	0	0
$401$	−37.9076	−1.89302	−0.946508	−	0.322680i	$-0.895416\pi$
$401$	−37.9076	−1.89302	−0.946508	+	0.322680i	$0.895416\pi$
$402$	0	0
$403$	−15.6824	−0.781198
$404$	0	0
$405$	−0.453107	−0.0225151
$406$	0	0
$407$	−5.92707	−0.293794
$408$	0	0
$409$	−23.8490	−1.17926	−0.589629	−	0.807674i	$-0.700726\pi$
$409$	−23.8490	−1.17926	−0.589629	+	0.807674i	$0.700726\pi$
$410$	0	0
$411$	−23.0873	−1.13881
$412$	0	0
$413$	4.28744	0.210971
$414$	0	0
$415$	−0.0326728	−0.00160384
$416$	0	0
$417$	−6.63055	−0.324699
$418$	0	0
$419$	−4.22924	−0.206612	−0.103306	−	0.994650i	$-0.532942\pi$
$419$	−4.22924	−0.206612	−0.103306	+	0.994650i	$0.532942\pi$
$420$	0	0
$421$	16.8136	0.819447	0.409723	−	0.912210i	$-0.365625\pi$
$421$	16.8136	0.819447	0.409723	+	0.912210i	$0.365625\pi$
$422$	0	0
$423$	−3.32792	−0.161809
$424$	0	0
$425$	−29.1440	−1.41369
$426$	0	0
$427$	−30.5618	−1.47899
$428$	0	0
$429$	−14.5199	−0.701025
$430$	0	0
$431$	20.0376	0.965178	0.482589	−	0.875847i	$-0.339697\pi$
$431$	20.0376	0.965178	0.482589	+	0.875847i	$0.339697\pi$
$432$	0	0
$433$	14.9602	0.718942	0.359471	−	0.933156i	$-0.382957\pi$
$433$	14.9602	0.718942	0.359471	+	0.933156i	$0.382957\pi$
$434$	0	0
$435$	3.37822	0.161973
$436$	0	0
$437$	15.5366	0.743217
$438$	0	0
$439$	−32.5369	−1.55290	−0.776452	−	0.630177i	$-0.782982\pi$
$439$	−32.5369	−1.55290	−0.776452	+	0.630177i	$0.782982\pi$
$440$	0	0
$441$	−2.28835	−0.108969
$442$	0	0
$443$	18.7623	0.891425	0.445713	−	0.895176i	$-0.352950\pi$
$443$	18.7623	0.891425	0.445713	+	0.895176i	$0.352950\pi$
$444$	0	0
$445$	4.30144	0.203908
$446$	0	0
$447$	−13.9556	−0.660075
$448$	0	0
$449$	18.3823	0.867514	0.433757	−	0.901030i	$-0.357188\pi$
$449$	18.3823	0.867514	0.433757	+	0.901030i	$0.357188\pi$
$450$	0	0
$451$	32.6751	1.53861
$452$	0	0
$453$	−1.16809	−0.0548815
$454$	0	0
$455$	−2.66709	−0.125035
$456$	0	0
$457$	−19.0186	−0.889653	−0.444826	−	0.895617i	$-0.646735\pi$
$457$	−19.0186	−0.889653	−0.444826	+	0.895617i	$0.646735\pi$
$458$	0	0
$459$	6.07839	0.283715
$460$	0	0
$461$	29.6257	1.37980	0.689902	−	0.723902i	$-0.257653\pi$
$461$	29.6257	1.37980	0.689902	+	0.723902i	$0.257653\pi$
$462$	0	0
$463$	−0.0705096	−0.00327686	−0.00163843	−	0.999999i	$-0.500522\pi$
$463$	−0.0705096	−0.00327686	−0.00163843	+	0.999999i	$0.500522\pi$
$464$	0	0
$465$	2.62038	0.121517
$466$	0	0
$467$	9.88942	0.457628	0.228814	−	0.973470i	$-0.426515\pi$
$467$	9.88942	0.457628	0.228814	+	0.973470i	$0.426515\pi$
$468$	0	0
$469$	7.04779	0.325437
$470$	0	0
$471$	−7.29448	−0.336112
$472$	0	0
$473$	−32.5993	−1.49892
$474$	0	0
$475$	13.7193	0.629485
$476$	0	0
$477$	−5.31676	−0.243438
$478$	0	0
$479$	−29.8615	−1.36441	−0.682203	−	0.731162i	$-0.738978\pi$
$479$	−29.8615	−1.36441	−0.682203	+	0.731162i	$0.738978\pi$
$480$	0	0
$481$	3.00177	0.136869
$482$	0	0
$483$	−11.7861	−0.536288
$484$	0	0
$485$	1.04012	0.0472296
$486$	0	0
$487$	−5.45499	−0.247189	−0.123595	−	0.992333i	$-0.539442\pi$
$487$	−5.45499	−0.247189	−0.123595	+	0.992333i	$0.539442\pi$
$488$	0	0
$489$	10.0146	0.452877
$490$	0	0
$491$	21.0330	0.949207	0.474604	−	0.880200i	$-0.342591\pi$
$491$	21.0330	0.949207	0.474604	+	0.880200i	$0.342591\pi$
$492$	0	0
$493$	−45.3185	−2.04104
$494$	0	0
$495$	2.42613	0.109046
$496$	0	0
$497$	8.03998	0.360642
$498$	0	0
$499$	−12.1234	−0.542720	−0.271360	−	0.962478i	$-0.587473\pi$
$499$	−12.1234	−0.542720	−0.271360	+	0.962478i	$0.587473\pi$
$500$	0	0
$501$	12.8988	0.576275
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−1.58143	−0.0703727
$506$	0	0
$507$	−5.64640	−0.250765
$508$	0	0
$509$	13.8830	0.615355	0.307677	−	0.951491i	$-0.400448\pi$
$509$	13.8830	0.615355	0.307677	+	0.951491i	$0.400448\pi$
$510$	0	0
$511$	5.94664	0.263064
$512$	0	0
$513$	−2.86135	−0.126332
$514$	0	0
$515$	−6.71758	−0.296012
$516$	0	0
$517$	17.8191	0.783683
$518$	0	0
$519$	−11.0976	−0.487132
$520$	0	0
$521$	−26.5791	−1.16445	−0.582226	−	0.813027i	$-0.697818\pi$
$521$	−26.5791	−1.16445	−0.582226	+	0.813027i	$0.697818\pi$
$522$	0	0
$523$	−9.81327	−0.429104	−0.214552	−	0.976713i	$-0.568829\pi$
$523$	−9.81327	−0.429104	−0.214552	+	0.976713i	$0.568829\pi$
$524$	0	0
$525$	−10.4075	−0.454221
$526$	0	0
$527$	−35.1521	−1.53125
$528$	0	0
$529$	6.48292	0.281866
$530$	0	0
$531$	1.97520	0.0857166
$532$	0	0
$533$	−16.5483	−0.716788
$534$	0	0
$535$	−6.44315	−0.278561
$536$	0	0
$537$	−19.9102	−0.859190
$538$	0	0
$539$	12.2528	0.527765
$540$	0	0
$541$	0.490807	0.0211015	0.0105507	−	0.999944i	$-0.496642\pi$
$541$	0.490807	0.0211015	0.0105507	+	0.999944i	$0.496642\pi$
$542$	0	0
$543$	13.2772	0.569778
$544$	0	0
$545$	−6.86446	−0.294041
$546$	0	0
$547$	−42.3002	−1.80863	−0.904314	−	0.426868i	$-0.859617\pi$
$547$	−42.3002	−1.80863	−0.904314	+	0.426868i	$0.859617\pi$
$548$	0	0
$549$	−14.0797	−0.600906
$550$	0	0
$551$	21.3333	0.908829
$552$	0	0
$553$	4.88779	0.207850
$554$	0	0
$555$	−0.501567	−0.0212903
$556$	0	0
$557$	−13.6538	−0.578531	−0.289265	−	0.957249i	$-0.593411\pi$
$557$	−13.6538	−0.578531	−0.289265	+	0.957249i	$0.593411\pi$
$558$	0	0
$559$	16.5099	0.698296
$560$	0	0
$561$	−32.5462	−1.37410
$562$	0	0
$563$	31.3148	1.31976	0.659881	−	0.751370i	$-0.270606\pi$
$563$	31.3148	1.31976	0.659881	+	0.751370i	$0.270606\pi$
$564$	0	0
$565$	6.41127	0.269724
$566$	0	0
$567$	2.17063	0.0911580
$568$	0	0
$569$	22.0677	0.925127	0.462563	−	0.886586i	$-0.346930\pi$
$569$	22.0677	0.925127	0.462563	+	0.886586i	$0.346930\pi$
$570$	0	0
$571$	36.6681	1.53451	0.767257	−	0.641340i	$-0.221621\pi$
$571$	36.6681	1.53451	0.767257	+	0.641340i	$0.221621\pi$
$572$	0	0
$573$	−9.57891	−0.400165
$574$	0	0
$575$	26.0343	1.08571
$576$	0	0
$577$	31.2547	1.30115	0.650575	−	0.759442i	$-0.274528\pi$
$577$	31.2547	1.30115	0.650575	+	0.759442i	$0.274528\pi$
$578$	0	0
$579$	−2.29142	−0.0952282
$580$	0	0
$581$	0.156521	0.00649357
$582$	0	0
$583$	28.4682	1.17903
$584$	0	0
$585$	−1.22872	−0.0508011
$586$	0	0
$587$	18.1818	0.750442	0.375221	−	0.926935i	$-0.377567\pi$
$587$	18.1818	0.750442	0.375221	+	0.926935i	$0.377567\pi$
$588$	0	0
$589$	16.5476	0.681831
$590$	0	0
$591$	−8.32164	−0.342307
$592$	0	0
$593$	−37.4236	−1.53680	−0.768402	−	0.639968i	$-0.778948\pi$
$593$	−37.4236	−1.53680	−0.768402	+	0.639968i	$0.778948\pi$
$594$	0	0
$595$	−5.97828	−0.245085
$596$	0	0
$597$	24.0433	0.984028
$598$	0	0
$599$	−20.5507	−0.839678	−0.419839	−	0.907599i	$-0.637914\pi$
$599$	−20.5507	−0.839678	−0.419839	+	0.907599i	$0.637914\pi$
$600$	0	0
$601$	3.59515	0.146649	0.0733247	−	0.997308i	$-0.476639\pi$
$601$	3.59515	0.146649	0.0733247	+	0.997308i	$0.476639\pi$
$602$	0	0
$603$	3.24688	0.132223
$604$	0	0
$605$	−8.00631	−0.325503
$606$	0	0
$607$	4.24278	0.172209	0.0861046	−	0.996286i	$-0.472558\pi$
$607$	4.24278	0.172209	0.0861046	+	0.996286i	$0.472558\pi$
$608$	0	0
$609$	−16.1835	−0.655790
$610$	0	0
$611$	−9.02451	−0.365092
$612$	0	0
$613$	−26.3003	−1.06226	−0.531130	−	0.847290i	$-0.678232\pi$
$613$	−26.3003	−1.06226	−0.531130	+	0.847290i	$0.678232\pi$
$614$	0	0
$615$	2.76507	0.111498
$616$	0	0
$617$	−18.1905	−0.732322	−0.366161	−	0.930552i	$-0.619328\pi$
$617$	−18.1905	−0.732322	−0.366161	+	0.930552i	$0.619328\pi$
$618$	0	0
$619$	1.89833	0.0763004	0.0381502	−	0.999272i	$-0.487853\pi$
$619$	1.89833	0.0763004	0.0381502	+	0.999272i	$0.487853\pi$
$620$	0	0
$621$	−5.42982	−0.217891
$622$	0	0
$623$	−20.6063	−0.825572
$624$	0	0
$625$	21.9626	0.878502
$626$	0	0
$627$	15.3209	0.611857
$628$	0	0
$629$	6.72847	0.268282
$630$	0	0
$631$	4.86232	0.193566	0.0967829	−	0.995306i	$-0.469145\pi$
$631$	4.86232	0.193566	0.0967829	+	0.995306i	$0.469145\pi$
$632$	0	0
$633$	19.3329	0.768414
$634$	0	0
$635$	7.79069	0.309164
$636$	0	0
$637$	−6.20544	−0.245869
$638$	0	0
$639$	3.70398	0.146527
$640$	0	0
$641$	−32.1797	−1.27102	−0.635512	−	0.772091i	$-0.719211\pi$
$641$	−32.1797	−1.27102	−0.635512	+	0.772091i	$0.719211\pi$
$642$	0	0
$643$	20.0389	0.790255	0.395128	−	0.918626i	$-0.370700\pi$
$643$	20.0389	0.790255	0.395128	+	0.918626i	$0.370700\pi$
$644$	0	0
$645$	−2.75865	−0.108622
$646$	0	0
$647$	−24.7994	−0.974966	−0.487483	−	0.873132i	$-0.662085\pi$
$647$	−24.7994	−0.974966	−0.487483	+	0.873132i	$0.662085\pi$
$648$	0	0
$649$	−10.5761	−0.415147
$650$	0	0
$651$	−12.5531	−0.491993
$652$	0	0
$653$	0.457411	0.0178999	0.00894994	−	0.999960i	$-0.497151\pi$
$653$	0.457411	0.0178999	0.00894994	+	0.999960i	$0.497151\pi$
$654$	0	0
$655$	−4.52791	−0.176920
$656$	0	0
$657$	2.73959	0.106882
$658$	0	0
$659$	−21.3180	−0.830431	−0.415216	−	0.909723i	$-0.636294\pi$
$659$	−21.3180	−0.830431	−0.415216	+	0.909723i	$0.636294\pi$
$660$	0	0
$661$	−12.5620	−0.488606	−0.244303	−	0.969699i	$-0.578559\pi$
$661$	−12.5620	−0.488606	−0.244303	+	0.969699i	$0.578559\pi$
$662$	0	0
$663$	16.4831	0.640150
$664$	0	0
$665$	2.81423	0.109131
$666$	0	0
$667$	40.4829	1.56751
$668$	0	0
$669$	−7.11843	−0.275215
$670$	0	0
$671$	75.3885	2.91034
$672$	0	0
$673$	19.3809	0.747080	0.373540	−	0.927614i	$-0.378144\pi$
$673$	19.3809	0.747080	0.373540	+	0.927614i	$0.378144\pi$
$674$	0	0
$675$	−4.79469	−0.184548
$676$	0	0
$677$	1.26226	0.0485124	0.0242562	−	0.999706i	$-0.492278\pi$
$677$	1.26226	0.0485124	0.0242562	+	0.999706i	$0.492278\pi$
$678$	0	0
$679$	−4.98276	−0.191221
$680$	0	0
$681$	−8.15200	−0.312385
$682$	0	0
$683$	−11.6336	−0.445149	−0.222575	−	0.974916i	$-0.571446\pi$
$683$	−11.6336	−0.445149	−0.222575	+	0.974916i	$0.571446\pi$
$684$	0	0
$685$	10.4610	0.399694
$686$	0	0
$687$	−3.40020	−0.129726
$688$	0	0
$689$	−14.4177	−0.549272
$690$	0	0
$691$	27.7137	1.05428	0.527139	−	0.849779i	$-0.323265\pi$
$691$	27.7137	1.05428	0.527139	+	0.849779i	$0.323265\pi$
$692$	0	0
$693$	−11.6225	−0.441501
$694$	0	0
$695$	3.00435	0.113962
$696$	0	0
$697$	−37.0931	−1.40500
$698$	0	0
$699$	−1.36410	−0.0515949
$700$	0	0
$701$	18.0567	0.681993	0.340997	−	0.940065i	$-0.389236\pi$
$701$	18.0567	0.681993	0.340997	+	0.940065i	$0.389236\pi$
$702$	0	0
$703$	−3.16737	−0.119460
$704$	0	0
$705$	1.50791	0.0567911
$706$	0	0
$707$	7.57591	0.284922
$708$	0	0
$709$	9.56909	0.359375	0.179687	−	0.983724i	$-0.442491\pi$
$709$	9.56909	0.359375	0.179687	+	0.983724i	$0.442491\pi$
$710$	0	0
$711$	2.25178	0.0844484
$712$	0	0
$713$	31.4014	1.17599
$714$	0	0
$715$	6.57905	0.246043
$716$	0	0
$717$	−10.6613	−0.398153
$718$	0	0
$719$	−22.6323	−0.844043	−0.422021	−	0.906586i	$-0.638679\pi$
$719$	−22.6323	−0.844043	−0.422021	+	0.906586i	$0.638679\pi$
$720$	0	0
$721$	32.1809	1.19848
$722$	0	0
$723$	−2.96872	−0.110408
$724$	0	0
$725$	35.7477	1.32764
$726$	0	0
$727$	17.2755	0.640714	0.320357	−	0.947297i	$-0.396197\pi$
$727$	17.2755	0.640714	0.320357	+	0.947297i	$0.396197\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	37.0070	1.36875
$732$	0	0
$733$	1.37565	0.0508107	0.0254053	−	0.999677i	$-0.491912\pi$
$733$	1.37565	0.0508107	0.0254053	+	0.999677i	$0.491912\pi$
$734$	0	0
$735$	1.03687	0.0382455
$736$	0	0
$737$	−17.3852	−0.640391
$738$	0	0
$739$	41.6690	1.53282	0.766409	−	0.642353i	$-0.222042\pi$
$739$	41.6690	1.53282	0.766409	+	0.642353i	$0.222042\pi$
$740$	0	0
$741$	−7.75928	−0.285044
$742$	0	0
$743$	−29.5159	−1.08283	−0.541417	−	0.840754i	$-0.682112\pi$
$743$	−29.5159	−1.08283	−0.541417	+	0.840754i	$0.682112\pi$
$744$	0	0
$745$	6.32337	0.231670
$746$	0	0
$747$	0.0721083	0.00263831
$748$	0	0
$749$	30.8662	1.12783
$750$	0	0
$751$	−29.0693	−1.06075	−0.530377	−	0.847762i	$-0.677950\pi$
$751$	−29.0693	−1.06075	−0.530377	+	0.847762i	$0.677950\pi$
$752$	0	0
$753$	−0.703338	−0.0256310
$754$	0	0
$755$	0.529268	0.0192621
$756$	0	0
$757$	23.1987	0.843172	0.421586	−	0.906788i	$-0.361474\pi$
$757$	23.1987	0.843172	0.421586	+	0.906788i	$0.361474\pi$
$758$	0	0
$759$	29.0735	1.05530
$760$	0	0
$761$	−50.9946	−1.84855	−0.924276	−	0.381725i	$-0.875330\pi$
$761$	−50.9946	−1.84855	−0.924276	+	0.381725i	$0.875330\pi$
$762$	0	0
$763$	32.8845	1.19050
$764$	0	0
$765$	−2.75416	−0.0995770
$766$	0	0
$767$	5.35627	0.193404
$768$	0	0
$769$	16.3536	0.589726	0.294863	−	0.955540i	$-0.404726\pi$
$769$	16.3536	0.589726	0.294863	+	0.955540i	$0.404726\pi$
$770$	0	0
$771$	−10.4602	−0.376713
$772$	0	0
$773$	26.7218	0.961116	0.480558	−	0.876963i	$-0.340434\pi$
$773$	26.7218	0.961116	0.480558	+	0.876963i	$0.340434\pi$
$774$	0	0
$775$	27.7284	0.996032
$776$	0	0
$777$	2.40278	0.0861993
$778$	0	0
$779$	17.4613	0.625615
$780$	0	0
$781$	−19.8326	−0.709668
$782$	0	0
$783$	−7.45567	−0.266444
$784$	0	0
$785$	3.30518	0.117967
$786$	0	0
$787$	33.6769	1.20045	0.600226	−	0.799831i	$-0.295077\pi$
$787$	33.6769	1.20045	0.600226	+	0.799831i	$0.295077\pi$
$788$	0	0
$789$	−12.1467	−0.432433
$790$	0	0
$791$	−30.7135	−1.09205
$792$	0	0
$793$	−38.1806	−1.35583
$794$	0	0
$795$	2.40906	0.0854407
$796$	0	0
$797$	−18.1019	−0.641202	−0.320601	−	0.947214i	$-0.603885\pi$
$797$	−18.1019	−0.641202	−0.320601	+	0.947214i	$0.603885\pi$
$798$	0	0
$799$	−20.2284	−0.715630
$800$	0	0
$801$	−9.49320	−0.335426
$802$	0	0
$803$	−14.6689	−0.517654
$804$	0	0
$805$	5.34039	0.188224
$806$	0	0
$807$	−0.974897	−0.0343180
$808$	0	0
$809$	−28.4728	−1.00105	−0.500525	−	0.865722i	$-0.666859\pi$
$809$	−28.4728	−1.00105	−0.500525	+	0.865722i	$0.666859\pi$
$810$	0	0
$811$	45.2550	1.58912	0.794559	−	0.607187i	$-0.207702\pi$
$811$	45.2550	1.58912	0.794559	+	0.607187i	$0.207702\pi$
$812$	0	0
$813$	9.06798	0.318028
$814$	0	0
$815$	−4.53770	−0.158949
$816$	0	0
$817$	−17.4208	−0.609475
$818$	0	0
$819$	5.88622	0.205681
$820$	0	0
$821$	40.4969	1.41335	0.706676	−	0.707537i	$-0.250194\pi$
$821$	40.4969	1.41335	0.706676	+	0.707537i	$0.250194\pi$
$822$	0	0
$823$	15.1873	0.529394	0.264697	−	0.964332i	$-0.414728\pi$
$823$	15.1873	0.529394	0.264697	+	0.964332i	$0.414728\pi$
$824$	0	0
$825$	25.6728	0.893812
$826$	0	0
$827$	−27.5058	−0.956471	−0.478236	−	0.878232i	$-0.658724\pi$
$827$	−27.5058	−0.956471	−0.478236	+	0.878232i	$0.658724\pi$
$828$	0	0
$829$	−9.42817	−0.327454	−0.163727	−	0.986506i	$-0.552352\pi$
$829$	−9.42817	−0.327454	−0.163727	+	0.986506i	$0.552352\pi$
$830$	0	0
$831$	−14.5217	−0.503753
$832$	0	0
$833$	−13.9095	−0.481935
$834$	0	0
$835$	−5.84453	−0.202258
$836$	0	0
$837$	−5.78314	−0.199894
$838$	0	0
$839$	44.2263	1.52686	0.763431	−	0.645890i	$-0.223513\pi$
$839$	44.2263	1.52686	0.763431	+	0.645890i	$0.223513\pi$
$840$	0	0
$841$	26.5871	0.916795
$842$	0	0
$843$	0.623745	0.0214829
$844$	0	0
$845$	2.55842	0.0880125
$846$	0	0
$847$	38.3546	1.31788
$848$	0	0
$849$	−25.1834	−0.864291
$850$	0	0
$851$	−6.01054	−0.206039
$852$	0	0
$853$	−0.974354	−0.0333613	−0.0166806	−	0.999861i	$-0.505310\pi$
$853$	−0.974354	−0.0333613	−0.0166806	+	0.999861i	$0.505310\pi$
$854$	0	0
$855$	1.29650	0.0443394
$856$	0	0
$857$	44.8334	1.53148	0.765740	−	0.643150i	$-0.222373\pi$
$857$	44.8334	1.53148	0.765740	+	0.643150i	$0.222373\pi$
$858$	0	0
$859$	50.0429	1.70744	0.853721	−	0.520730i	$-0.174340\pi$
$859$	50.0429	1.70744	0.853721	+	0.520730i	$0.174340\pi$
$860$	0	0
$861$	−13.2462	−0.451429
$862$	0	0
$863$	−10.7593	−0.366251	−0.183126	−	0.983090i	$-0.558622\pi$
$863$	−10.7593	−0.366251	−0.183126	+	0.983090i	$0.558622\pi$
$864$	0	0
$865$	5.02842	0.170971
$866$	0	0
$867$	19.9468	0.677429
$868$	0	0
$869$	−12.0570	−0.409005
$870$	0	0
$871$	8.80474	0.298337
$872$	0	0
$873$	−2.29553	−0.0776921
$874$	0	0
$875$	9.63337	0.325667
$876$	0	0
$877$	−36.9714	−1.24844	−0.624218	−	0.781250i	$-0.714582\pi$
$877$	−36.9714	−1.24844	−0.624218	+	0.781250i	$0.714582\pi$
$878$	0	0
$879$	4.37810	0.147670
$880$	0	0
$881$	−15.3517	−0.517211	−0.258605	−	0.965983i	$-0.583263\pi$
$881$	−15.3517	−0.517211	−0.258605	+	0.965983i	$0.583263\pi$
$882$	0	0
$883$	12.4942	0.420462	0.210231	−	0.977652i	$-0.432578\pi$
$883$	12.4942	0.420462	0.210231	+	0.977652i	$0.432578\pi$
$884$	0	0
$885$	−0.894980	−0.0300844
$886$	0	0
$887$	5.75567	0.193256	0.0966282	−	0.995321i	$-0.469194\pi$
$887$	5.75567	0.193256	0.0966282	+	0.995321i	$0.469194\pi$
$888$	0	0
$889$	−37.3217	−1.25173
$890$	0	0
$891$	−5.35442	−0.179380
$892$	0	0
$893$	9.52236	0.318654
$894$	0	0
$895$	9.02148	0.301555
$896$	0	0
$897$	−14.7243	−0.491631
$898$	0	0
$899$	43.1172	1.43804
$900$	0	0
$901$	−32.3173	−1.07665
$902$	0	0
$903$	13.2155	0.439783
$904$	0	0
$905$	−6.01598	−0.199978
$906$	0	0
$907$	33.4509	1.11072	0.555360	−	0.831610i	$-0.312581\pi$
$907$	33.4509	1.11072	0.555360	+	0.831610i	$0.312581\pi$
$908$	0	0
$909$	3.49019	0.115762
$910$	0	0
$911$	−58.2620	−1.93031	−0.965153	−	0.261686i	$-0.915721\pi$
$911$	−58.2620	−1.93031	−0.965153	+	0.261686i	$0.915721\pi$
$912$	0	0
$913$	−0.386098	−0.0127780
$914$	0	0
$915$	6.37961	0.210903
$916$	0	0
$917$	21.6912	0.716305
$918$	0	0
$919$	−18.9879	−0.626352	−0.313176	−	0.949695i	$-0.601393\pi$
$919$	−18.9879	−0.626352	−0.313176	+	0.949695i	$0.601393\pi$
$920$	0	0
$921$	3.46257	0.114095
$922$	0	0
$923$	10.0443	0.330611
$924$	0	0
$925$	−5.30749	−0.174509
$926$	0	0
$927$	14.8256	0.486936
$928$	0	0
$929$	−3.14313	−0.103123	−0.0515614	−	0.998670i	$-0.516420\pi$
$929$	−3.14313	−0.103123	−0.0515614	+	0.998670i	$0.516420\pi$
$930$	0	0
$931$	6.54778	0.214595
$932$	0	0
$933$	6.24430	0.204429
$934$	0	0
$935$	14.7469	0.482276
$936$	0	0
$937$	17.4629	0.570489	0.285244	−	0.958455i	$-0.407925\pi$
$937$	17.4629	0.570489	0.285244	+	0.958455i	$0.407925\pi$
$938$	0	0
$939$	15.9415	0.520230
$940$	0	0
$941$	−24.1050	−0.785799	−0.392899	−	0.919581i	$-0.628528\pi$
$941$	−24.1050	−0.785799	−0.392899	+	0.919581i	$0.628528\pi$
$942$	0	0
$943$	33.1352	1.07903
$944$	0	0
$945$	−0.983530	−0.0319942
$946$	0	0
$947$	−3.56033	−0.115695	−0.0578476	−	0.998325i	$-0.518424\pi$
$947$	−3.56033	−0.115695	−0.0578476	+	0.998325i	$0.518424\pi$
$948$	0	0
$949$	7.42909	0.241158
$950$	0	0
$951$	11.5258	0.373749
$952$	0	0
$953$	24.2055	0.784092	0.392046	−	0.919946i	$-0.371767\pi$
$953$	24.2055	0.784092	0.392046	+	0.919946i	$0.371767\pi$
$954$	0	0
$955$	4.34027	0.140448
$956$	0	0
$957$	39.9208	1.29046
$958$	0	0
$959$	−50.1140	−1.61826
$960$	0	0
$961$	2.44466	0.0788600
$962$	0	0
$963$	14.2199	0.458230
$964$	0	0
$965$	1.03826	0.0334228
$966$	0	0
$967$	48.3663	1.55536	0.777678	−	0.628662i	$-0.216397\pi$
$967$	48.3663	1.55536	0.777678	+	0.628662i	$0.216397\pi$
$968$	0	0
$969$	−17.3924	−0.558725
$970$	0	0
$971$	38.2827	1.22855	0.614276	−	0.789092i	$-0.289448\pi$
$971$	38.2827	1.22855	0.614276	+	0.789092i	$0.289448\pi$
$972$	0	0
$973$	−14.3925	−0.461402
$974$	0	0
$975$	−13.0020	−0.416398
$976$	0	0
$977$	−50.3330	−1.61030	−0.805148	−	0.593073i	$-0.797914\pi$
$977$	−50.3330	−1.61030	−0.805148	+	0.593073i	$0.797914\pi$
$978$	0	0
$979$	50.8305	1.62455
$980$	0	0
$981$	15.1497	0.483694
$982$	0	0
$983$	−41.9070	−1.33663	−0.668313	−	0.743880i	$-0.732983\pi$
$983$	−41.9070	−1.33663	−0.668313	+	0.743880i	$0.732983\pi$
$984$	0	0
$985$	3.77060	0.120141
$986$	0	0
$987$	−7.22370	−0.229933
$988$	0	0
$989$	−33.0583	−1.05119
$990$	0	0
$991$	−30.8646	−0.980447	−0.490224	−	0.871597i	$-0.663085\pi$
$991$	−30.8646	−0.980447	−0.490224	+	0.871597i	$0.663085\pi$
$992$	0	0
$993$	20.7940	0.659878
$994$	0	0
$995$	−10.8942	−0.345370
$996$	0	0
$997$	−38.9030	−1.23207	−0.616035	−	0.787719i	$-0.711262\pi$
$997$	−38.9030	−1.23207	−0.616035	+	0.787719i	$0.711262\pi$
$998$	0	0
$999$	1.10695	0.0350224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.9	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.9	✓	15	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.453107 q^{5} +2.17063 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.453107 q^{5} +2.17063 q^{7} +1.00000 q^{9} -5.35442 q^{11} +2.71175 q^{13} -0.453107 q^{15} +6.07839 q^{17} -2.86135 q^{19} +2.17063 q^{21} -5.42982 q^{23} -4.79469 q^{25} +1.00000 q^{27} -7.45567 q^{29} -5.78314 q^{31} -5.35442 q^{33} -0.983530 q^{35} +1.10695 q^{37} +2.71175 q^{39} -6.10245 q^{41} +6.08829 q^{43} -0.453107 q^{45} -3.32792 q^{47} -2.28835 q^{49} +6.07839 q^{51} -5.31676 q^{53} +2.42613 q^{55} -2.86135 q^{57} +1.97520 q^{59} -14.0797 q^{61} +2.17063 q^{63} -1.22872 q^{65} +3.24688 q^{67} -5.42982 q^{69} +3.70398 q^{71} +2.73959 q^{73} -4.79469 q^{75} -11.6225 q^{77} +2.25178 q^{79} +1.00000 q^{81} +0.0721083 q^{83} -2.75416 q^{85} -7.45567 q^{87} -9.49320 q^{89} +5.88622 q^{91} -5.78314 q^{93} +1.29650 q^{95} -2.29553 q^{97} -5.35442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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