LMFDB - Embedded newform 9072.2.a.cd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.18194 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.18194 q^{5} +1.00000 q^{7} -3.70370 q^{11} +1.00000 q^{13} +6.94282 q^{17} -1.94282 q^{19} -5.60301 q^{23} -3.60301 q^{25} -0.239123 q^{29} -1.66019 q^{31} -1.18194 q^{35} -9.54583 q^{37} +10.1819 q^{41} -2.22545 q^{43} +5.82846 q^{47} +1.00000 q^{49} +11.6030 q^{53} +4.37756 q^{55} +2.60301 q^{59} -7.60301 q^{61} -1.18194 q^{65} -3.50808 q^{67} +8.60301 q^{71} +15.1488 q^{73} -3.70370 q^{77} -7.37756 q^{79} -6.94282 q^{83} -8.20602 q^{85} -2.74720 q^{89} +1.00000 q^{91} +2.29630 q^{95} +7.16827 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{5} + 3 q^{7}+O(q^{10}) $ 3 * q + 5 * q^5 + 3 * q^7	$ 3 q + 5 q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 12 q^{17} + 3 q^{19} + 6 q^{25} - q^{29} + 3 q^{31} + 5 q^{35} - 3 q^{37} + 22 q^{41} + 3 q^{43} - 9 q^{47} + 3 q^{49} + 18 q^{53} + 6 q^{55} - 9 q^{59} - 6 q^{61} + 5 q^{65} + 9 q^{71} + 3 q^{73} - 2 q^{77} - 15 q^{79} - 12 q^{83} + 9 q^{85} + 2 q^{89} + 3 q^{91} + 16 q^{95} + 3 q^{97}+O(q^{100}) $ 3 * q + 5 * q^5 + 3 * q^7 - 2 * q^11 + 3 * q^13 + 12 * q^17 + 3 * q^19 + 6 * q^25 - q^29 + 3 * q^31 + 5 * q^35 - 3 * q^37 + 22 * q^41 + 3 * q^43 - 9 * q^47 + 3 * q^49 + 18 * q^53 + 6 * q^55 - 9 * q^59 - 6 * q^61 + 5 * q^65 + 9 * q^71 + 3 * q^73 - 2 * q^77 - 15 * q^79 - 12 * q^83 + 9 * q^85 + 2 * q^89 + 3 * q^91 + 16 * q^95 + 3 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.18194	−0.528581	−0.264291	−	0.964443i	$-0.585138\pi$
$5$	−1.18194	−0.528581	−0.264291	+	0.964443i	$0.585138\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.70370	−1.11671	−0.558353	−	0.829603i	$-0.688567\pi$
$11$	−3.70370	−1.11671	−0.558353	+	0.829603i	$0.688567\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.94282	1.68388	0.841941	−	0.539570i	$-0.181413\pi$
$17$	6.94282	1.68388	0.841941	+	0.539570i	$0.181413\pi$
$18$	0	0
$19$	−1.94282	−0.445713	−0.222857	−	0.974851i	$-0.571538\pi$
$19$	−1.94282	−0.445713	−0.222857	+	0.974851i	$0.571538\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.60301	−1.16831	−0.584154	−	0.811643i	$-0.698574\pi$
$23$	−5.60301	−1.16831	−0.584154	+	0.811643i	$0.698574\pi$
$24$	0	0
$25$	−3.60301	−0.720602
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.239123	−0.0444041	−0.0222020	−	0.999754i	$-0.507068\pi$
$29$	−0.239123	−0.0444041	−0.0222020	+	0.999754i	$0.507068\pi$
$30$	0	0
$31$	−1.66019	−0.298179	−0.149089	−	0.988824i	$-0.547634\pi$
$31$	−1.66019	−0.298179	−0.149089	+	0.988824i	$0.547634\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.18194	−0.199785
$36$	0	0
$37$	−9.54583	−1.56932	−0.784662	−	0.619923i	$-0.787164\pi$
$37$	−9.54583	−1.56932	−0.784662	+	0.619923i	$0.787164\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.1819	1.59015	0.795076	−	0.606510i	$-0.207431\pi$
$41$	10.1819	1.59015	0.795076	+	0.606510i	$0.207431\pi$
$42$	0	0
$43$	−2.22545	−0.339378	−0.169689	−	0.985498i	$-0.554276\pi$
$43$	−2.22545	−0.339378	−0.169689	+	0.985498i	$0.554276\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.82846	0.850168	0.425084	−	0.905154i	$-0.360245\pi$
$47$	5.82846	0.850168	0.425084	+	0.905154i	$0.360245\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.6030	1.59380	0.796898	−	0.604114i	$-0.206473\pi$
$53$	11.6030	1.59380	0.796898	+	0.604114i	$0.206473\pi$
$54$	0	0
$55$	4.37756	0.590270
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.60301	0.338883	0.169442	−	0.985540i	$-0.445804\pi$
$59$	2.60301	0.338883	0.169442	+	0.985540i	$0.445804\pi$
$60$	0	0
$61$	−7.60301	−0.973466	−0.486733	−	0.873551i	$-0.661811\pi$
$61$	−7.60301	−0.973466	−0.486733	+	0.873551i	$0.661811\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.18194	−0.146602
$66$	0	0
$67$	−3.50808	−0.428580	−0.214290	−	0.976770i	$-0.568744\pi$
$67$	−3.50808	−0.428580	−0.214290	+	0.976770i	$0.568744\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.60301	1.02099	0.510495	−	0.859881i	$-0.329462\pi$
$71$	8.60301	1.02099	0.510495	+	0.859881i	$0.329462\pi$
$72$	0	0
$73$	15.1488	1.77304	0.886519	−	0.462693i	$-0.153117\pi$
$73$	15.1488	1.77304	0.886519	+	0.462693i	$0.153117\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.70370	−0.422075
$78$	0	0
$79$	−7.37756	−0.830040	−0.415020	−	0.909812i	$-0.636225\pi$
$79$	−7.37756	−0.830040	−0.415020	+	0.909812i	$0.636225\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.94282	−0.762074	−0.381037	−	0.924560i	$-0.624433\pi$
$83$	−6.94282	−0.762074	−0.381037	+	0.924560i	$0.624433\pi$
$84$	0	0
$85$	−8.20602	−0.890068
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.74720	−0.291203	−0.145602	−	0.989343i	$-0.546512\pi$
$89$	−2.74720	−0.291203	−0.145602	+	0.989343i	$0.546512\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.29630	0.235596
$96$	0	0
$97$	7.16827	0.727828	0.363914	−	0.931433i	$-0.381440\pi$
$97$	7.16827	0.727828	0.363914	+	0.931433i	$0.381440\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.7850	1.27215	0.636075	−	0.771627i	$-0.280557\pi$
$101$	12.7850	1.27215	0.636075	+	0.771627i	$0.280557\pi$
$102$	0	0
$103$	4.39699	0.433248	0.216624	−	0.976255i	$-0.430495\pi$
$103$	4.39699	0.433248	0.216624	+	0.976255i	$0.430495\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.7278	1.32711	0.663557	−	0.748126i	$-0.269046\pi$
$107$	13.7278	1.32711	0.663557	+	0.748126i	$0.269046\pi$
$108$	0	0
$109$	1.26320	0.120993	0.0604963	−	0.998168i	$-0.480732\pi$
$109$	1.26320	0.120993	0.0604963	+	0.998168i	$0.480732\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.1625	−1.14415	−0.572076	−	0.820200i	$-0.693862\pi$
$113$	−12.1625	−1.14415	−0.572076	+	0.820200i	$0.693862\pi$
$114$	0	0
$115$	6.62244	0.617546
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.94282	0.636447
$120$	0	0
$121$	2.71737	0.247034
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.1683	0.909478
$126$	0	0
$127$	−1.33981	−0.118889	−0.0594445	−	0.998232i	$-0.518933\pi$
$127$	−1.33981	−0.118889	−0.0594445	+	0.998232i	$0.518933\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.96690	−0.433960	−0.216980	−	0.976176i	$-0.569621\pi$
$131$	−4.96690	−0.433960	−0.216980	+	0.976176i	$0.569621\pi$
$132$	0	0
$133$	−1.94282	−0.168464
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.33981	0.370775	0.185387	−	0.982665i	$-0.440646\pi$
$137$	4.33981	0.370775	0.185387	+	0.982665i	$0.440646\pi$
$138$	0	0
$139$	−3.94282	−0.334426	−0.167213	−	0.985921i	$-0.553477\pi$
$139$	−3.94282	−0.334426	−0.167213	+	0.985921i	$0.553477\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.70370	−0.309719
$144$	0	0
$145$	0.282630	0.0234712
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.1111	−0.910256	−0.455128	−	0.890426i	$-0.650406\pi$
$149$	−11.1111	−0.910256	−0.455128	+	0.890426i	$0.650406\pi$
$150$	0	0
$151$	−13.9234	−1.13307	−0.566535	−	0.824038i	$-0.691716\pi$
$151$	−13.9234	−1.13307	−0.566535	+	0.824038i	$0.691716\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.96225	0.157612
$156$	0	0
$157$	0.0571799	0.00456346	0.00228173	−	0.999997i	$-0.499274\pi$
$157$	0.0571799	0.00456346	0.00228173	+	0.999997i	$0.499274\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.60301	−0.441579
$162$	0	0
$163$	1.50808	0.118122	0.0590610	−	0.998254i	$-0.481189\pi$
$163$	1.50808	0.118122	0.0590610	+	0.998254i	$0.481189\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.6843	−1.13630	−0.568151	−	0.822924i	$-0.692341\pi$
$167$	−14.6843	−1.13630	−0.568151	+	0.822924i	$0.692341\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.252796	−0.0192197	−0.00960987	−	0.999954i	$-0.503059\pi$
$173$	−0.252796	−0.0192197	−0.00960987	+	0.999954i	$0.503059\pi$
$174$	0	0
$175$	−3.60301	−0.272362
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.1923	1.06079	0.530393	−	0.847752i	$-0.322044\pi$
$179$	14.1923	1.06079	0.530393	+	0.847752i	$0.322044\pi$
$180$	0	0
$181$	−1.43147	−0.106400	−0.0532002	−	0.998584i	$-0.516942\pi$
$181$	−1.43147	−0.106400	−0.0532002	+	0.998584i	$0.516942\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.2826	0.829515
$186$	0	0
$187$	−25.7141	−1.88040
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.0676	1.09025	0.545126	−	0.838354i	$-0.316482\pi$
$191$	15.0676	1.09025	0.545126	+	0.838354i	$0.316482\pi$
$192$	0	0
$193$	−7.84789	−0.564904	−0.282452	−	0.959282i	$-0.591148\pi$
$193$	−7.84789	−0.564904	−0.282452	+	0.959282i	$0.591148\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.69002	−0.476644	−0.238322	−	0.971186i	$-0.576597\pi$
$197$	−6.69002	−0.476644	−0.238322	+	0.971186i	$0.576597\pi$
$198$	0	0
$199$	19.9396	1.41348	0.706739	−	0.707475i	$-0.250166\pi$
$199$	19.9396	1.41348	0.706739	+	0.707475i	$0.250166\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.239123	−0.0167832
$204$	0	0
$205$	−12.0345	−0.840525
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.19562	0.497731
$210$	0	0
$211$	18.0917	1.24548	0.622741	−	0.782428i	$-0.286019\pi$
$211$	18.0917	1.24548	0.622741	+	0.782428i	$0.286019\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.63036	0.179389
$216$	0	0
$217$	−1.66019	−0.112701
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.94282	0.467025
$222$	0	0
$223$	22.6569	1.51722	0.758610	−	0.651545i	$-0.225879\pi$
$223$	22.6569	1.51722	0.758610	+	0.651545i	$0.225879\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.28263	−0.350620	−0.175310	−	0.984513i	$-0.556093\pi$
$227$	−5.28263	−0.350620	−0.175310	+	0.984513i	$0.556093\pi$
$228$	0	0
$229$	19.3365	1.27779	0.638897	−	0.769292i	$-0.279391\pi$
$229$	19.3365	1.27779	0.638897	+	0.769292i	$0.279391\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.9806	1.11243	0.556217	−	0.831037i	$-0.312252\pi$
$233$	16.9806	1.11243	0.556217	+	0.831037i	$0.312252\pi$
$234$	0	0
$235$	−6.88891	−0.449383
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.8856	1.09224	0.546121	−	0.837707i	$-0.316104\pi$
$239$	16.8856	1.09224	0.546121	+	0.837707i	$0.316104\pi$
$240$	0	0
$241$	27.1456	1.74860	0.874300	−	0.485386i	$-0.161321\pi$
$241$	27.1456	1.74860	0.874300	+	0.485386i	$0.161321\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.18194	−0.0755116
$246$	0	0
$247$	−1.94282	−0.123619
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.0780	−1.20419	−0.602096	−	0.798424i	$-0.705668\pi$
$251$	−19.0780	−1.20419	−0.602096	+	0.798424i	$0.705668\pi$
$252$	0	0
$253$	20.7518	1.30466
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.8421	0.925827	0.462913	−	0.886404i	$-0.346804\pi$
$257$	14.8421	0.925827	0.462913	+	0.886404i	$0.346804\pi$
$258$	0	0
$259$	−9.54583	−0.593149
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.74145	−0.477358	−0.238679	−	0.971099i	$-0.576714\pi$
$263$	−7.74145	−0.477358	−0.238679	+	0.971099i	$0.576714\pi$
$264$	0	0
$265$	−13.7141	−0.842450
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.51135	0.0921486	0.0460743	−	0.998938i	$-0.485329\pi$
$269$	1.51135	0.0921486	0.0460743	+	0.998938i	$0.485329\pi$
$270$	0	0
$271$	21.9806	1.33522	0.667612	−	0.744509i	$-0.267316\pi$
$271$	21.9806	1.33522	0.667612	+	0.744509i	$0.267316\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.3445	0.804701
$276$	0	0
$277$	−10.8285	−0.650619	−0.325310	−	0.945608i	$-0.605469\pi$
$277$	−10.8285	−0.650619	−0.325310	+	0.945608i	$0.605469\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.8766	1.00677	0.503387	−	0.864061i	$-0.332087\pi$
$281$	16.8766	1.00677	0.503387	+	0.864061i	$0.332087\pi$
$282$	0	0
$283$	15.3171	0.910508	0.455254	−	0.890362i	$-0.349549\pi$
$283$	15.3171	0.910508	0.455254	+	0.890362i	$0.349549\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.1819	0.601021
$288$	0	0
$289$	31.2028	1.83546
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.36964	0.547380	0.273690	−	0.961818i	$-0.411756\pi$
$293$	9.36964	0.547380	0.273690	+	0.961818i	$0.411756\pi$
$294$	0	0
$295$	−3.07661	−0.179127
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.60301	−0.324030
$300$	0	0
$301$	−2.22545	−0.128273
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.98633	0.514556
$306$	0	0
$307$	−2.71410	−0.154902	−0.0774509	−	0.996996i	$-0.524678\pi$
$307$	−2.71410	−0.154902	−0.0774509	+	0.996996i	$0.524678\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.9806	−0.792765	−0.396383	−	0.918085i	$-0.629735\pi$
$311$	−13.9806	−0.792765	−0.396383	+	0.918085i	$0.629735\pi$
$312$	0	0
$313$	−19.0539	−1.07699	−0.538495	−	0.842628i	$-0.681007\pi$
$313$	−19.0539	−1.07699	−0.538495	+	0.842628i	$0.681007\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.01943	0.225754	0.112877	−	0.993609i	$-0.463993\pi$
$317$	4.01943	0.225754	0.112877	+	0.993609i	$0.463993\pi$
$318$	0	0
$319$	0.885640	0.0495863
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.4887	−0.750529
$324$	0	0
$325$	−3.60301	−0.199859
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.82846	0.321333
$330$	0	0
$331$	12.3776	0.680332	0.340166	−	0.940365i	$-0.389517\pi$
$331$	12.3776	0.680332	0.340166	+	0.940365i	$0.389517\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.14635	0.226539
$336$	0	0
$337$	12.2599	0.667841	0.333920	−	0.942601i	$-0.391628\pi$
$337$	12.2599	0.667841	0.333920	+	0.942601i	$0.391628\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.14884	0.332978
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.64979	0.356979	0.178490	−	0.983942i	$-0.442879\pi$
$347$	6.64979	0.356979	0.178490	+	0.983942i	$0.442879\pi$
$348$	0	0
$349$	11.4347	0.612088	0.306044	−	0.952017i	$-0.400995\pi$
$349$	11.4347	0.612088	0.306044	+	0.952017i	$0.400995\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.1956	1.18135	0.590677	−	0.806908i	$-0.298861\pi$
$353$	22.1956	1.18135	0.590677	+	0.806908i	$0.298861\pi$
$354$	0	0
$355$	−10.1683	−0.539676
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.55623	−0.398803	−0.199401	−	0.979918i	$-0.563900\pi$
$359$	−7.55623	−0.398803	−0.199401	+	0.979918i	$0.563900\pi$
$360$	0	0
$361$	−15.2255	−0.801339
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.9051	−0.937194
$366$	0	0
$367$	18.5231	0.966900	0.483450	−	0.875372i	$-0.339384\pi$
$367$	18.5231	0.966900	0.483450	+	0.875372i	$0.339384\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.6030	0.602398
$372$	0	0
$373$	15.6602	0.810854	0.405427	−	0.914127i	$-0.367123\pi$
$373$	15.6602	0.810854	0.405427	+	0.914127i	$0.367123\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.239123	−0.0123155
$378$	0	0
$379$	−4.03775	−0.207405	−0.103703	−	0.994608i	$-0.533069\pi$
$379$	−4.03775	−0.207405	−0.103703	+	0.994608i	$0.533069\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.225450	0.0115200	0.00575998	−	0.999983i	$-0.498167\pi$
$383$	0.225450	0.0115200	0.00575998	+	0.999983i	$0.498167\pi$
$384$	0	0
$385$	4.37756	0.223101
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.2632	−1.28090	−0.640448	−	0.768002i	$-0.721251\pi$
$389$	−25.2632	−1.28090	−0.640448	+	0.768002i	$0.721251\pi$
$390$	0	0
$391$	−38.9007	−1.96729
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.71986	0.438744
$396$	0	0
$397$	−20.3009	−1.01888	−0.509438	−	0.860508i	$-0.670147\pi$
$397$	−20.3009	−1.01888	−0.509438	+	0.860508i	$0.670147\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.2255	0.760323	0.380161	−	0.924920i	$-0.375868\pi$
$401$	15.2255	0.760323	0.380161	+	0.924920i	$0.375868\pi$
$402$	0	0
$403$	−1.66019	−0.0826999
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.3549	1.75248
$408$	0	0
$409$	1.65692	0.0819294	0.0409647	−	0.999161i	$-0.486957\pi$
$409$	1.65692	0.0819294	0.0409647	+	0.999161i	$0.486957\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.60301	0.128086
$414$	0	0
$415$	8.20602	0.402818
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.3743	1.63044	0.815220	−	0.579151i	$-0.196616\pi$
$419$	33.3743	1.63044	0.815220	+	0.579151i	$0.196616\pi$
$420$	0	0
$421$	18.2405	0.888988	0.444494	−	0.895782i	$-0.353384\pi$
$421$	18.2405	0.888988	0.444494	+	0.895782i	$0.353384\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−25.0150	−1.21341
$426$	0	0
$427$	−7.60301	−0.367935
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.2826	1.41049	0.705247	−	0.708961i	$-0.250836\pi$
$431$	29.2826	1.41049	0.705247	+	0.708961i	$0.250836\pi$
$432$	0	0
$433$	−12.2449	−0.588451	−0.294226	−	0.955736i	$-0.595062\pi$
$433$	−12.2449	−0.588451	−0.294226	+	0.955736i	$0.595062\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.8856	0.520731
$438$	0	0
$439$	4.83173	0.230606	0.115303	−	0.993330i	$-0.463216\pi$
$439$	4.83173	0.230606	0.115303	+	0.993330i	$0.463216\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.24488	0.0591461	0.0295730	−	0.999563i	$-0.490585\pi$
$443$	1.24488	0.0591461	0.0295730	+	0.999563i	$0.490585\pi$
$444$	0	0
$445$	3.24704	0.153924
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.82846	−0.416641	−0.208320	−	0.978061i	$-0.566800\pi$
$449$	−8.82846	−0.416641	−0.208320	+	0.978061i	$0.566800\pi$
$450$	0	0
$451$	−37.7108	−1.77573
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.18194	−0.0554104
$456$	0	0
$457$	−10.5081	−0.491547	−0.245774	−	0.969327i	$-0.579042\pi$
$457$	−10.5081	−0.491547	−0.245774	+	0.969327i	$0.579042\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.5516	−1.05033	−0.525166	−	0.851000i	$-0.675997\pi$
$461$	−22.5516	−1.05033	−0.525166	+	0.851000i	$0.675997\pi$
$462$	0	0
$463$	−10.3970	−0.483189	−0.241595	−	0.970377i	$-0.577670\pi$
$463$	−10.3970	−0.483189	−0.241595	+	0.970377i	$0.577670\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.3171	0.616242	0.308121	−	0.951347i	$-0.400300\pi$
$467$	13.3171	0.616242	0.308121	+	0.951347i	$0.400300\pi$
$468$	0	0
$469$	−3.50808	−0.161988
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.24239	0.378986
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.5354	0.664141	0.332070	−	0.943255i	$-0.392253\pi$
$479$	14.5354	0.664141	0.332070	+	0.943255i	$0.392253\pi$
$480$	0	0
$481$	−9.54583	−0.435252
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.47249	−0.384716
$486$	0	0
$487$	−13.0539	−0.591529	−0.295765	−	0.955261i	$-0.595574\pi$
$487$	−13.0539	−0.591529	−0.295765	+	0.955261i	$0.595574\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.3445	0.873003	0.436502	−	0.899704i	$-0.356217\pi$
$491$	19.3445	0.873003	0.436502	+	0.899704i	$0.356217\pi$
$492$	0	0
$493$	−1.66019	−0.0747712
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.60301	0.385898
$498$	0	0
$499$	36.2222	1.62153	0.810764	−	0.585374i	$-0.199052\pi$
$499$	36.2222	1.62153	0.810764	+	0.585374i	$0.199052\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.6764	0.698974	0.349487	−	0.936941i	$-0.386356\pi$
$503$	15.6764	0.698974	0.349487	+	0.936941i	$0.386356\pi$
$504$	0	0
$505$	−15.1111	−0.672435
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.3034	−1.52047	−0.760237	−	0.649646i	$-0.774917\pi$
$509$	−34.3034	−1.52047	−0.760237	+	0.649646i	$0.774917\pi$
$510$	0	0
$511$	15.1488	0.670145
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.19699	−0.229007
$516$	0	0
$517$	−21.5868	−0.949389
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.2449	0.448836	0.224418	−	0.974493i	$-0.427952\pi$
$521$	10.2449	0.448836	0.224418	+	0.974493i	$0.427952\pi$
$522$	0	0
$523$	−30.6030	−1.33818	−0.669088	−	0.743183i	$-0.733315\pi$
$523$	−30.6030	−1.33818	−0.669088	+	0.743183i	$0.733315\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.5264	−0.502098
$528$	0	0
$529$	8.39372	0.364944
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.1819	0.441029
$534$	0	0
$535$	−16.2255	−0.701487
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.70370	−0.159530
$540$	0	0
$541$	−26.0917	−1.12177	−0.560884	−	0.827894i	$-0.689539\pi$
$541$	−26.0917	−1.12177	−0.560884	+	0.827894i	$0.689539\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.49303	−0.0639544
$546$	0	0
$547$	10.9234	0.467050	0.233525	−	0.972351i	$-0.424974\pi$
$547$	10.9234	0.467050	0.233525	+	0.972351i	$0.424974\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.464574	0.0197915
$552$	0	0
$553$	−7.37756	−0.313726
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.9442	0.590835	0.295417	−	0.955368i	$-0.404541\pi$
$557$	13.9442	0.590835	0.295417	+	0.955368i	$0.404541\pi$
$558$	0	0
$559$	−2.22545	−0.0941265
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.2574	1.27520	0.637600	−	0.770368i	$-0.279927\pi$
$563$	30.2574	1.27520	0.637600	+	0.770368i	$0.279927\pi$
$564$	0	0
$565$	14.3754	0.604778
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.1352	0.886032	0.443016	−	0.896514i	$-0.353908\pi$
$569$	21.1352	0.886032	0.443016	+	0.896514i	$0.353908\pi$
$570$	0	0
$571$	32.7863	1.37207	0.686033	−	0.727571i	$-0.259351\pi$
$571$	32.7863	1.37207	0.686033	+	0.727571i	$0.259351\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	20.1877	0.841885
$576$	0	0
$577$	−17.3743	−0.723301	−0.361651	−	0.932314i	$-0.617787\pi$
$577$	−17.3743	−0.723301	−0.361651	+	0.932314i	$0.617787\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.94282	−0.288037
$582$	0	0
$583$	−42.9740	−1.77980
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.9759	0.700671	0.350336	−	0.936624i	$-0.386068\pi$
$587$	16.9759	0.700671	0.350336	+	0.936624i	$0.386068\pi$
$588$	0	0
$589$	3.22545	0.132902
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.0733	0.536858	0.268429	−	0.963300i	$-0.413496\pi$
$593$	13.0733	0.536858	0.268429	+	0.963300i	$0.413496\pi$
$594$	0	0
$595$	−8.20602	−0.336414
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.2060	1.19333	0.596663	−	0.802492i	$-0.296493\pi$
$599$	29.2060	1.19333	0.596663	+	0.802492i	$0.296493\pi$
$600$	0	0
$601$	7.79071	0.317790	0.158895	−	0.987296i	$-0.449207\pi$
$601$	7.79071	0.317790	0.158895	+	0.987296i	$0.449207\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.21178	−0.130577
$606$	0	0
$607$	19.6408	0.797194	0.398597	−	0.917126i	$-0.369497\pi$
$607$	19.6408	0.797194	0.398597	+	0.917126i	$0.369497\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.82846	0.235794
$612$	0	0
$613$	23.5653	0.951792	0.475896	−	0.879502i	$-0.342124\pi$
$613$	23.5653	0.951792	0.475896	+	0.879502i	$0.342124\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.6602	0.429163	0.214582	−	0.976706i	$-0.431161\pi$
$617$	10.6602	0.429163	0.214582	+	0.976706i	$0.431161\pi$
$618$	0	0
$619$	18.0150	0.724086	0.362043	−	0.932161i	$-0.382079\pi$
$619$	18.0150	0.724086	0.362043	+	0.932161i	$0.382079\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.74720	−0.110064
$624$	0	0
$625$	5.99673	0.239869
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−66.2750	−2.64256
$630$	0	0
$631$	−12.4703	−0.496436	−0.248218	−	0.968704i	$-0.579845\pi$
$631$	−12.4703	−0.496436	−0.248218	+	0.968704i	$0.579845\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.58358	0.0628424
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19.1456	−0.756205	−0.378102	−	0.925764i	$-0.623423\pi$
$641$	−19.1456	−0.756205	−0.378102	+	0.925764i	$0.623423\pi$
$642$	0	0
$643$	−6.48865	−0.255887	−0.127944	−	0.991781i	$-0.540838\pi$
$643$	−6.48865	−0.255887	−0.127944	+	0.991781i	$0.540838\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.0988	−1.89096	−0.945479	−	0.325682i	$-0.894406\pi$
$647$	−48.0988	−1.89096	−0.945479	+	0.325682i	$0.894406\pi$
$648$	0	0
$649$	−9.64076	−0.378433
$650$	0	0
$651$	0	0
$652$	0	0
$653$	43.2405	1.69213	0.846066	−	0.533079i	$-0.178965\pi$
$653$	43.2405	1.69213	0.846066	+	0.533079i	$0.178965\pi$
$654$	0	0
$655$	5.87059	0.229383
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.50808	−0.0977009	−0.0488505	−	0.998806i	$-0.515556\pi$
$659$	−2.50808	−0.0977009	−0.0488505	+	0.998806i	$0.515556\pi$
$660$	0	0
$661$	−42.3354	−1.64666	−0.823329	−	0.567565i	$-0.807886\pi$
$661$	−42.3354	−1.64666	−0.823329	+	0.567565i	$0.807886\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.29630	0.0890468
$666$	0	0
$667$	1.33981	0.0518777
$668$	0	0
$669$	0	0
$670$	0	0
$671$	28.1592	1.08708
$672$	0	0
$673$	13.4153	0.517122	0.258561	−	0.965995i	$-0.416752\pi$
$673$	13.4153	0.517122	0.258561	+	0.965995i	$0.416752\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.96225	−0.0754154	−0.0377077	−	0.999289i	$-0.512006\pi$
$677$	−1.96225	−0.0754154	−0.0377077	+	0.999289i	$0.512006\pi$
$678$	0	0
$679$	7.16827	0.275093
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.1672	1.03952	0.519761	−	0.854312i	$-0.326021\pi$
$683$	27.1672	1.03952	0.519761	+	0.854312i	$0.326021\pi$
$684$	0	0
$685$	−5.12941	−0.195985
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.6030	0.442039
$690$	0	0
$691$	50.3171	1.91415	0.957077	−	0.289835i	$-0.0936005\pi$
$691$	50.3171	1.91415	0.957077	+	0.289835i	$0.0936005\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.66019	0.176771
$696$	0	0
$697$	70.6914	2.67763
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−45.1672	−1.70594	−0.852970	−	0.521960i	$-0.825201\pi$
$701$	−45.1672	−1.70594	−0.852970	+	0.521960i	$0.825201\pi$
$702$	0	0
$703$	18.5458	0.699469
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.7850	0.480828
$708$	0	0
$709$	39.6181	1.48789	0.743944	−	0.668242i	$-0.232953\pi$
$709$	39.6181	1.48789	0.743944	+	0.668242i	$0.232953\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.30206	0.348365
$714$	0	0
$715$	4.37756	0.163711
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.0377	−0.821869	−0.410935	−	0.911665i	$-0.634798\pi$
$719$	−22.0377	−0.821869	−0.410935	+	0.911665i	$0.634798\pi$
$720$	0	0
$721$	4.39699	0.163752
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.861564	0.0319977
$726$	0	0
$727$	−28.1111	−1.04258	−0.521291	−	0.853379i	$-0.674550\pi$
$727$	−28.1111	−1.04258	−0.521291	+	0.853379i	$0.674550\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.4509	−0.571472
$732$	0	0
$733$	11.8695	0.438409	0.219205	−	0.975679i	$-0.429654\pi$
$733$	11.8695	0.438409	0.219205	+	0.975679i	$0.429654\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.9929	0.478598
$738$	0	0
$739$	12.1844	0.448212	0.224106	−	0.974565i	$-0.428054\pi$
$739$	12.1844	0.448212	0.224106	+	0.974565i	$0.428054\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.4854	−1.63201	−0.816005	−	0.578045i	$-0.803816\pi$
$743$	−44.4854	−1.63201	−0.816005	+	0.578045i	$0.803816\pi$
$744$	0	0
$745$	13.1327	0.481144
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.7278	0.501602
$750$	0	0
$751$	−42.8058	−1.56200	−0.781002	−	0.624528i	$-0.785291\pi$
$751$	−42.8058	−1.56200	−0.781002	+	0.624528i	$0.785291\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.4567	0.598919
$756$	0	0
$757$	−22.4919	−0.817483	−0.408741	−	0.912650i	$-0.634032\pi$
$757$	−22.4919	−0.817483	−0.408741	+	0.912650i	$0.634032\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14.3365	0.519699	0.259850	−	0.965649i	$-0.416327\pi$
$761$	14.3365	0.519699	0.259850	+	0.965649i	$0.416327\pi$
$762$	0	0
$763$	1.26320	0.0457309
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.60301	0.0939892
$768$	0	0
$769$	−31.2211	−1.12586	−0.562930	−	0.826504i	$-0.690326\pi$
$769$	−31.2211	−1.12586	−0.562930	+	0.826504i	$0.690326\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.38005	−0.157539	−0.0787697	−	0.996893i	$-0.525099\pi$
$773$	−4.38005	−0.157539	−0.0787697	+	0.996893i	$0.525099\pi$
$774$	0	0
$775$	5.98168	0.214868
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19.7817	−0.708752
$780$	0	0
$781$	−31.8629	−1.14015
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−0.0675835	−0.00241216
$786$	0	0
$787$	−27.6213	−0.984594	−0.492297	−	0.870427i	$-0.663843\pi$
$787$	−27.6213	−0.984594	−0.492297	+	0.870427i	$0.663843\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.1625	−0.432449
$792$	0	0
$793$	−7.60301	−0.269991
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.96363	0.104977	0.0524885	−	0.998622i	$-0.483285\pi$
$797$	2.96363	0.104977	0.0524885	+	0.998622i	$0.483285\pi$
$798$	0	0
$799$	40.4660	1.43158
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−56.1067	−1.97996
$804$	0	0
$805$	6.62244	0.233410
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.7896	0.871556	0.435778	−	0.900054i	$-0.356473\pi$
$809$	24.7896	0.871556	0.435778	+	0.900054i	$0.356473\pi$
$810$	0	0
$811$	−8.24377	−0.289478	−0.144739	−	0.989470i	$-0.546234\pi$
$811$	−8.24377	−0.289478	−0.144739	+	0.989470i	$0.546234\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.78247	−0.0624370
$816$	0	0
$817$	4.32365	0.151265
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.8993	1.00859	0.504296	−	0.863531i	$-0.331752\pi$
$821$	28.8993	1.00859	0.504296	+	0.863531i	$0.331752\pi$
$822$	0	0
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.7108	−1.76339	−0.881694	−	0.471821i	$-0.843597\pi$
$827$	−50.7108	−1.76339	−0.881694	+	0.471821i	$0.843597\pi$
$828$	0	0
$829$	14.8123	0.514452	0.257226	−	0.966351i	$-0.417191\pi$
$829$	14.8123	0.514452	0.257226	+	0.966351i	$0.417191\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.94282	0.240554
$834$	0	0
$835$	17.3560	0.600628
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33.7212	1.16419	0.582093	−	0.813122i	$-0.302234\pi$
$839$	33.7212	1.16419	0.582093	+	0.813122i	$0.302234\pi$
$840$	0	0
$841$	−28.9428	−0.998028
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.1833	0.487921
$846$	0	0
$847$	2.71737	0.0933699
$848$	0	0
$849$	0	0
$850$	0	0
$851$	53.4854	1.83346
$852$	0	0
$853$	11.7896	0.403668	0.201834	−	0.979420i	$-0.435310\pi$
$853$	11.7896	0.403668	0.201834	+	0.979420i	$0.435310\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.3261	1.07008	0.535040	−	0.844827i	$-0.320296\pi$
$857$	31.3261	1.07008	0.535040	+	0.844827i	$0.320296\pi$
$858$	0	0
$859$	50.3893	1.71926	0.859631	−	0.510915i	$-0.170693\pi$
$859$	50.3893	1.71926	0.859631	+	0.510915i	$0.170693\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.13268	0.0385568	0.0192784	−	0.999814i	$-0.493863\pi$
$863$	1.13268	0.0385568	0.0192784	+	0.999814i	$0.493863\pi$
$864$	0	0
$865$	0.298791	0.0101592
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.3242	0.926911
$870$	0	0
$871$	−3.50808	−0.118867
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.1683	0.343750
$876$	0	0
$877$	−27.3937	−0.925020	−0.462510	−	0.886614i	$-0.653051\pi$
$877$	−27.3937	−0.925020	−0.462510	+	0.886614i	$0.653051\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.20929	−0.0407420	−0.0203710	−	0.999792i	$-0.506485\pi$
$881$	−1.20929	−0.0407420	−0.0203710	+	0.999792i	$0.506485\pi$
$882$	0	0
$883$	51.0884	1.71926	0.859631	−	0.510916i	$-0.170694\pi$
$883$	51.0884	1.71926	0.859631	+	0.510916i	$0.170694\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.5757	−1.39597	−0.697987	−	0.716110i	$-0.745921\pi$
$887$	−41.5757	−1.39597	−0.697987	+	0.716110i	$0.745921\pi$
$888$	0	0
$889$	−1.33981	−0.0449358
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.3236	−0.378931
$894$	0	0
$895$	−16.7745	−0.560711
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.396990	0.0132404
$900$	0	0
$901$	80.5576	2.68376
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.69192	0.0562412
$906$	0	0
$907$	−35.4509	−1.17713	−0.588564	−	0.808451i	$-0.700306\pi$
$907$	−35.4509	−1.17713	−0.588564	+	0.808451i	$0.700306\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20.7108	−0.686180	−0.343090	−	0.939302i	$-0.611474\pi$
$911$	−20.7108	−0.686180	−0.343090	+	0.939302i	$0.611474\pi$
$912$	0	0
$913$	25.7141	0.851013
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.96690	−0.164021
$918$	0	0
$919$	−14.3926	−0.474768	−0.237384	−	0.971416i	$-0.576290\pi$
$919$	−14.3926	−0.474768	−0.237384	+	0.971416i	$0.576290\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.60301	0.283172
$924$	0	0
$925$	34.3937	1.13086
$926$	0	0
$927$	0	0
$928$	0	0
$929$	41.7428	1.36954	0.684769	−	0.728760i	$-0.259903\pi$
$929$	41.7428	1.36954	0.684769	+	0.728760i	$0.259903\pi$
$930$	0	0
$931$	−1.94282	−0.0636734
$932$	0	0
$933$	0	0
$934$	0	0
$935$	30.3926	0.993945
$936$	0	0
$937$	3.17154	0.103610	0.0518048	−	0.998657i	$-0.483503\pi$
$937$	3.17154	0.103610	0.0518048	+	0.998657i	$0.483503\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.22080	0.104995	0.0524976	−	0.998621i	$-0.483282\pi$
$941$	3.22080	0.104995	0.0524976	+	0.998621i	$0.483282\pi$
$942$	0	0
$943$	−57.0495	−1.85779
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45.3469	−1.47358	−0.736789	−	0.676123i	$-0.763659\pi$
$947$	−45.3469	−1.47358	−0.736789	+	0.676123i	$0.763659\pi$
$948$	0	0
$949$	15.1488	0.491752
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.2703	1.75799	0.878994	−	0.476832i	$-0.158215\pi$
$953$	54.2703	1.75799	0.878994	+	0.476832i	$0.158215\pi$
$954$	0	0
$955$	−17.8090	−0.576287
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.33981	0.140140
$960$	0	0
$961$	−28.2438	−0.911089
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.27576	0.298597
$966$	0	0
$967$	−25.6591	−0.825140	−0.412570	−	0.910926i	$-0.635369\pi$
$967$	−25.6591	−0.825140	−0.412570	+	0.910926i	$0.635369\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.0183	−0.674510	−0.337255	−	0.941413i	$-0.609498\pi$
$971$	−21.0183	−0.674510	−0.337255	+	0.941413i	$0.609498\pi$
$972$	0	0
$973$	−3.94282	−0.126401
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.09820	0.0671273	0.0335637	−	0.999437i	$-0.489314\pi$
$977$	2.09820	0.0671273	0.0335637	+	0.999437i	$0.489314\pi$
$978$	0	0
$979$	10.1748	0.325188
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.9923	1.37124	0.685622	−	0.727958i	$-0.259531\pi$
$983$	42.9923	1.37124	0.685622	+	0.727958i	$0.259531\pi$
$984$	0	0
$985$	7.90723	0.251945
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.4692	0.396498
$990$	0	0
$991$	17.2632	0.548384	0.274192	−	0.961675i	$-0.411590\pi$
$991$	17.2632	0.548384	0.274192	+	0.961675i	$0.411590\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−23.5674	−0.747137
$996$	0	0
$997$	−38.9018	−1.23203	−0.616016	−	0.787733i	$-0.711254\pi$
$997$	−38.9018	−1.23203	−0.616016	+	0.787733i	$0.711254\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.cd.1.1		3
3.2	odd	2		9072.2.a.bq.1.3		3
4.3	odd	2		567.2.a.g.1.2		3
9.2	odd	6		1008.2.r.k.337.2		6
9.4	even	3		3024.2.r.g.2017.3		6
9.5	odd	6		1008.2.r.k.673.2		6
9.7	even	3		3024.2.r.g.1009.3		6
12.11	even	2		567.2.a.d.1.2		3
28.27	even	2		3969.2.a.p.1.2		3
36.7	odd	6		189.2.f.a.64.2		6
36.11	even	6		63.2.f.b.22.2	✓	6
36.23	even	6		63.2.f.b.43.2	yes	6
36.31	odd	6		189.2.f.a.127.2		6
84.83	odd	2		3969.2.a.m.1.2		3
252.11	even	6		441.2.h.c.373.2		6
252.23	even	6		441.2.h.c.214.2		6
252.31	even	6		1323.2.g.b.667.2		6
252.47	odd	6		441.2.g.d.67.2		6
252.59	odd	6		441.2.g.d.79.2		6
252.67	odd	6		1323.2.g.c.667.2		6
252.79	odd	6		1323.2.g.c.361.2		6
252.83	odd	6		441.2.f.d.148.2		6
252.95	even	6		441.2.g.e.79.2		6
252.103	even	6		1323.2.h.e.802.2		6
252.115	even	6		1323.2.h.e.226.2		6
252.131	odd	6		441.2.h.b.214.2		6
252.139	even	6		1323.2.f.c.883.2		6
252.151	odd	6		1323.2.h.d.226.2		6
252.167	odd	6		441.2.f.d.295.2		6
252.187	even	6		1323.2.g.b.361.2		6
252.191	even	6		441.2.g.e.67.2		6
252.223	even	6		1323.2.f.c.442.2		6
252.227	odd	6		441.2.h.b.373.2		6
252.247	odd	6		1323.2.h.d.802.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.b.22.2	✓	6	36.11	even	6
63.2.f.b.43.2	yes	6	36.23	even	6
189.2.f.a.64.2		6	36.7	odd	6
189.2.f.a.127.2		6	36.31	odd	6
441.2.f.d.148.2		6	252.83	odd	6
441.2.f.d.295.2		6	252.167	odd	6
441.2.g.d.67.2		6	252.47	odd	6
441.2.g.d.79.2		6	252.59	odd	6
441.2.g.e.67.2		6	252.191	even	6
441.2.g.e.79.2		6	252.95	even	6
441.2.h.b.214.2		6	252.131	odd	6
441.2.h.b.373.2		6	252.227	odd	6
441.2.h.c.214.2		6	252.23	even	6
441.2.h.c.373.2		6	252.11	even	6
567.2.a.d.1.2		3	12.11	even	2
567.2.a.g.1.2		3	4.3	odd	2
1008.2.r.k.337.2		6	9.2	odd	6
1008.2.r.k.673.2		6	9.5	odd	6
1323.2.f.c.442.2		6	252.223	even	6
1323.2.f.c.883.2		6	252.139	even	6
1323.2.g.b.361.2		6	252.187	even	6
1323.2.g.b.667.2		6	252.31	even	6
1323.2.g.c.361.2		6	252.79	odd	6
1323.2.g.c.667.2		6	252.67	odd	6
1323.2.h.d.226.2		6	252.151	odd	6
1323.2.h.d.802.2		6	252.247	odd	6
1323.2.h.e.226.2		6	252.115	even	6
1323.2.h.e.802.2		6	252.103	even	6
3024.2.r.g.1009.3		6	9.7	even	3
3024.2.r.g.2017.3		6	9.4	even	3
3969.2.a.m.1.2		3	84.83	odd	2
3969.2.a.p.1.2		3	28.27	even	2
9072.2.a.bq.1.3		3	3.2	odd	2
9072.2.a.cd.1.1		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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.18194 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.18194 q^{5} +1.00000 q^{7} -3.70370 q^{11} +1.00000 q^{13} +6.94282 q^{17} -1.94282 q^{19} -5.60301 q^{23} -3.60301 q^{25} -0.239123 q^{29} -1.66019 q^{31} -1.18194 q^{35} -9.54583 q^{37} +10.1819 q^{41} -2.22545 q^{43} +5.82846 q^{47} +1.00000 q^{49} +11.6030 q^{53} +4.37756 q^{55} +2.60301 q^{59} -7.60301 q^{61} -1.18194 q^{65} -3.50808 q^{67} +8.60301 q^{71} +15.1488 q^{73} -3.70370 q^{77} -7.37756 q^{79} -6.94282 q^{83} -8.20602 q^{85} -2.74720 q^{89} +1.00000 q^{91} +2.29630 q^{95} +7.16827 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{5} + 3 q^{7}+O(q^{10}) \) 3 * q + 5 * q^5 + 3 * q^7	\( 3 q + 5 q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 12 q^{17} + 3 q^{19} + 6 q^{25} - q^{29} + 3 q^{31} + 5 q^{35} - 3 q^{37} + 22 q^{41} + 3 q^{43} - 9 q^{47} + 3 q^{49} + 18 q^{53} + 6 q^{55} - 9 q^{59} - 6 q^{61} + 5 q^{65} + 9 q^{71} + 3 q^{73} - 2 q^{77} - 15 q^{79} - 12 q^{83} + 9 q^{85} + 2 q^{89} + 3 q^{91} + 16 q^{95} + 3 q^{97}+O(q^{100}) \) 3 * q + 5 * q^5 + 3 * q^7 - 2 * q^11 + 3 * q^13 + 12 * q^17 + 3 * q^19 + 6 * q^25 - q^29 + 3 * q^31 + 5 * q^35 - 3 * q^37 + 22 * q^41 + 3 * q^43 - 9 * q^47 + 3 * q^49 + 18 * q^53 + 6 * q^55 - 9 * q^59 - 6 * q^61 + 5 * q^65 + 9 * q^71 + 3 * q^73 - 2 * q^77 - 15 * q^79 - 12 * q^83 + 9 * q^85 + 2 * q^89 + 3 * q^91 + 16 * q^95 + 3 * q^97

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