LMFDB - Embedded newform 2004.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2004,2,Mod(1,2004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2004, 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("2004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2004 = 2^{2} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2004.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:	$16.0020205651$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 $ x^9 - 29x^7 - 7x^6 + 266x^5 + 69x^4 - 901x^3 - 199x^2 + 875*x + 391
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.529979$ of defining polynomial
Character	$\chi$	$=$	2004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.52998 q^{5} -1.05249 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.52998 q^{5} -1.05249 q^{7} +1.00000 q^{9} +0.162063 q^{11} +3.63900 q^{13} +1.52998 q^{15} +4.89726 q^{17} +1.02632 q^{19} -1.05249 q^{21} -7.59021 q^{23} -2.65917 q^{25} +1.00000 q^{27} +9.59787 q^{29} +2.68185 q^{31} +0.162063 q^{33} -1.61029 q^{35} +4.12252 q^{37} +3.63900 q^{39} -1.08012 q^{41} +2.77829 q^{43} +1.52998 q^{45} -9.50274 q^{47} -5.89226 q^{49} +4.89726 q^{51} +4.80545 q^{53} +0.247952 q^{55} +1.02632 q^{57} +8.22708 q^{59} +0.510754 q^{61} -1.05249 q^{63} +5.56759 q^{65} -0.317219 q^{67} -7.59021 q^{69} +5.83623 q^{71} +11.2379 q^{73} -2.65917 q^{75} -0.170570 q^{77} +5.59029 q^{79} +1.00000 q^{81} +4.98299 q^{83} +7.49271 q^{85} +9.59787 q^{87} -0.199333 q^{89} -3.83003 q^{91} +2.68185 q^{93} +1.57025 q^{95} -10.9371 q^{97} +0.162063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * 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
$5$	1.52998	0.684227	0.342114	−	0.939659i	$-0.388857\pi$
$5$	1.52998	0.684227	0.342114	+	0.939659i	$0.388857\pi$
$6$	0	0
$7$	−1.05249	−0.397806	−0.198903	−	0.980019i	$-0.563738\pi$
$7$	−1.05249	−0.397806	−0.198903	+	0.980019i	$0.563738\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.162063	0.0488637	0.0244318	−	0.999701i	$-0.492222\pi$
$11$	0.162063	0.0488637	0.0244318	+	0.999701i	$0.492222\pi$
$12$	0	0
$13$	3.63900	1.00928	0.504638	−	0.863331i	$-0.331626\pi$
$13$	3.63900	1.00928	0.504638	+	0.863331i	$0.331626\pi$
$14$	0	0
$15$	1.52998	0.395039
$16$	0	0
$17$	4.89726	1.18776	0.593880	−	0.804553i	$-0.297595\pi$
$17$	4.89726	1.18776	0.593880	+	0.804553i	$0.297595\pi$
$18$	0	0
$19$	1.02632	0.235455	0.117727	−	0.993046i	$-0.462439\pi$
$19$	1.02632	0.235455	0.117727	+	0.993046i	$0.462439\pi$
$20$	0	0
$21$	−1.05249	−0.229673
$22$	0	0
$23$	−7.59021	−1.58267	−0.791334	−	0.611384i	$-0.790613\pi$
$23$	−7.59021	−1.58267	−0.791334	+	0.611384i	$0.790613\pi$
$24$	0	0
$25$	−2.65917	−0.531833
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.59787	1.78228	0.891140	−	0.453729i	$-0.149906\pi$
$29$	9.59787	1.78228	0.891140	+	0.453729i	$0.149906\pi$
$30$	0	0
$31$	2.68185	0.481675	0.240837	−	0.970565i	$-0.422578\pi$
$31$	2.68185	0.481675	0.240837	+	0.970565i	$0.422578\pi$
$32$	0	0
$33$	0.162063	0.0282115
$34$	0	0
$35$	−1.61029	−0.272189
$36$	0	0
$37$	4.12252	0.677738	0.338869	−	0.940834i	$-0.389956\pi$
$37$	4.12252	0.677738	0.338869	+	0.940834i	$0.389956\pi$
$38$	0	0
$39$	3.63900	0.582706
$40$	0	0
$41$	−1.08012	−0.168687	−0.0843433	−	0.996437i	$-0.526879\pi$
$41$	−1.08012	−0.168687	−0.0843433	+	0.996437i	$0.526879\pi$
$42$	0	0
$43$	2.77829	0.423686	0.211843	−	0.977304i	$-0.432054\pi$
$43$	2.77829	0.423686	0.211843	+	0.977304i	$0.432054\pi$
$44$	0	0
$45$	1.52998	0.228076
$46$	0	0
$47$	−9.50274	−1.38612	−0.693059	−	0.720881i	$-0.743737\pi$
$47$	−9.50274	−1.38612	−0.693059	+	0.720881i	$0.743737\pi$
$48$	0	0
$49$	−5.89226	−0.841751
$50$	0	0
$51$	4.89726	0.685754
$52$	0	0
$53$	4.80545	0.660080	0.330040	−	0.943967i	$-0.392938\pi$
$53$	4.80545	0.660080	0.330040	+	0.943967i	$0.392938\pi$
$54$	0	0
$55$	0.247952	0.0334339
$56$	0	0
$57$	1.02632	0.135940
$58$	0	0
$59$	8.22708	1.07107	0.535537	−	0.844512i	$-0.320109\pi$
$59$	8.22708	1.07107	0.535537	+	0.844512i	$0.320109\pi$
$60$	0	0
$61$	0.510754	0.0653953	0.0326976	−	0.999465i	$-0.489590\pi$
$61$	0.510754	0.0653953	0.0326976	+	0.999465i	$0.489590\pi$
$62$	0	0
$63$	−1.05249	−0.132602
$64$	0	0
$65$	5.56759	0.690575
$66$	0	0
$67$	−0.317219	−0.0387544	−0.0193772	−	0.999812i	$-0.506168\pi$
$67$	−0.317219	−0.0387544	−0.0193772	+	0.999812i	$0.506168\pi$
$68$	0	0
$69$	−7.59021	−0.913754
$70$	0	0
$71$	5.83623	0.692633	0.346316	−	0.938118i	$-0.387432\pi$
$71$	5.83623	0.692633	0.346316	+	0.938118i	$0.387432\pi$
$72$	0	0
$73$	11.2379	1.31529	0.657647	−	0.753326i	$-0.271552\pi$
$73$	11.2379	1.31529	0.657647	+	0.753326i	$0.271552\pi$
$74$	0	0
$75$	−2.65917	−0.307054
$76$	0	0
$77$	−0.170570	−0.0194382
$78$	0	0
$79$	5.59029	0.628957	0.314478	−	0.949265i	$-0.398170\pi$
$79$	5.59029	0.628957	0.314478	+	0.949265i	$0.398170\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.98299	0.546954	0.273477	−	0.961879i	$-0.411826\pi$
$83$	4.98299	0.546954	0.273477	+	0.961879i	$0.411826\pi$
$84$	0	0
$85$	7.49271	0.812698
$86$	0	0
$87$	9.59787	1.02900
$88$	0	0
$89$	−0.199333	−0.0211292	−0.0105646	−	0.999944i	$-0.503363\pi$
$89$	−0.199333	−0.0211292	−0.0105646	+	0.999944i	$0.503363\pi$
$90$	0	0
$91$	−3.83003	−0.401496
$92$	0	0
$93$	2.68185	0.278095
$94$	0	0
$95$	1.57025	0.161104
$96$	0	0
$97$	−10.9371	−1.11049	−0.555247	−	0.831686i	$-0.687376\pi$
$97$	−10.9371	−1.11049	−0.555247	+	0.831686i	$0.687376\pi$
$98$	0	0
$99$	0.162063	0.0162879
$100$	0	0
$101$	11.8811	1.18221	0.591107	−	0.806593i	$-0.298691\pi$
$101$	11.8811	1.18221	0.591107	+	0.806593i	$0.298691\pi$
$102$	0	0
$103$	−1.54707	−0.152437	−0.0762187	−	0.997091i	$-0.524285\pi$
$103$	−1.54707	−0.152437	−0.0762187	+	0.997091i	$0.524285\pi$
$104$	0	0
$105$	−1.61029	−0.157149
$106$	0	0
$107$	−2.89728	−0.280091	−0.140045	−	0.990145i	$-0.544725\pi$
$107$	−2.89728	−0.280091	−0.140045	+	0.990145i	$0.544725\pi$
$108$	0	0
$109$	8.16825	0.782377	0.391188	−	0.920311i	$-0.372064\pi$
$109$	8.16825	0.782377	0.391188	+	0.920311i	$0.372064\pi$
$110$	0	0
$111$	4.12252	0.391292
$112$	0	0
$113$	−6.73714	−0.633777	−0.316889	−	0.948463i	$-0.602638\pi$
$113$	−6.73714	−0.633777	−0.316889	+	0.948463i	$0.602638\pi$
$114$	0	0
$115$	−11.6129	−1.08290
$116$	0	0
$117$	3.63900	0.336426
$118$	0	0
$119$	−5.15434	−0.472498
$120$	0	0
$121$	−10.9737	−0.997612
$122$	0	0
$123$	−1.08012	−0.0973912
$124$	0	0
$125$	−11.7184	−1.04812
$126$	0	0
$127$	−15.8812	−1.40923	−0.704616	−	0.709588i	$-0.748881\pi$
$127$	−15.8812	−1.40923	−0.704616	+	0.709588i	$0.748881\pi$
$128$	0	0
$129$	2.77829	0.244615
$130$	0	0
$131$	9.81474	0.857517	0.428759	−	0.903419i	$-0.358951\pi$
$131$	9.81474	0.857517	0.428759	+	0.903419i	$0.358951\pi$
$132$	0	0
$133$	−1.08020	−0.0936651
$134$	0	0
$135$	1.52998	0.131680
$136$	0	0
$137$	−2.22904	−0.190440	−0.0952199	−	0.995456i	$-0.530355\pi$
$137$	−2.22904	−0.190440	−0.0952199	+	0.995456i	$0.530355\pi$
$138$	0	0
$139$	−7.71321	−0.654226	−0.327113	−	0.944985i	$-0.606076\pi$
$139$	−7.71321	−0.654226	−0.327113	+	0.944985i	$0.606076\pi$
$140$	0	0
$141$	−9.50274	−0.800275
$142$	0	0
$143$	0.589745	0.0493170
$144$	0	0
$145$	14.6845	1.21948
$146$	0	0
$147$	−5.89226	−0.485985
$148$	0	0
$149$	−19.1731	−1.57072	−0.785360	−	0.619039i	$-0.787522\pi$
$149$	−19.1731	−1.57072	−0.785360	+	0.619039i	$0.787522\pi$
$150$	0	0
$151$	−8.05210	−0.655271	−0.327635	−	0.944804i	$-0.606252\pi$
$151$	−8.05210	−0.655271	−0.327635	+	0.944804i	$0.606252\pi$
$152$	0	0
$153$	4.89726	0.395920
$154$	0	0
$155$	4.10318	0.329575
$156$	0	0
$157$	10.1858	0.812914	0.406457	−	0.913670i	$-0.366764\pi$
$157$	10.1858	0.812914	0.406457	+	0.913670i	$0.366764\pi$
$158$	0	0
$159$	4.80545	0.381097
$160$	0	0
$161$	7.98866	0.629594
$162$	0	0
$163$	0.432916	0.0339086	0.0169543	−	0.999856i	$-0.494603\pi$
$163$	0.432916	0.0339086	0.0169543	+	0.999856i	$0.494603\pi$
$164$	0	0
$165$	0.247952	0.0193031
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	0.242320	0.0186400
$170$	0	0
$171$	1.02632	0.0784848
$172$	0	0
$173$	17.2653	1.31266	0.656329	−	0.754475i	$-0.272108\pi$
$173$	17.2653	1.31266	0.656329	+	0.754475i	$0.272108\pi$
$174$	0	0
$175$	2.79876	0.211566
$176$	0	0
$177$	8.22708	0.618385
$178$	0	0
$179$	7.05820	0.527554	0.263777	−	0.964584i	$-0.415032\pi$
$179$	7.05820	0.527554	0.263777	+	0.964584i	$0.415032\pi$
$180$	0	0
$181$	−14.7420	−1.09577	−0.547883	−	0.836555i	$-0.684566\pi$
$181$	−14.7420	−1.09577	−0.547883	+	0.836555i	$0.684566\pi$
$182$	0	0
$183$	0.510754	0.0377560
$184$	0	0
$185$	6.30737	0.463727
$186$	0	0
$187$	0.793663	0.0580384
$188$	0	0
$189$	−1.05249	−0.0765577
$190$	0	0
$191$	22.5477	1.63149	0.815747	−	0.578409i	$-0.196326\pi$
$191$	22.5477	1.63149	0.815747	+	0.578409i	$0.196326\pi$
$192$	0	0
$193$	3.60996	0.259850	0.129925	−	0.991524i	$-0.458526\pi$
$193$	3.60996	0.259850	0.129925	+	0.991524i	$0.458526\pi$
$194$	0	0
$195$	5.56759	0.398704
$196$	0	0
$197$	12.4286	0.885502	0.442751	−	0.896645i	$-0.354003\pi$
$197$	12.4286	0.885502	0.442751	+	0.896645i	$0.354003\pi$
$198$	0	0
$199$	6.92918	0.491197	0.245598	−	0.969372i	$-0.421016\pi$
$199$	6.92918	0.491197	0.245598	+	0.969372i	$0.421016\pi$
$200$	0	0
$201$	−0.317219	−0.0223749
$202$	0	0
$203$	−10.1017	−0.709001
$204$	0	0
$205$	−1.65256	−0.115420
$206$	0	0
$207$	−7.59021	−0.527556
$208$	0	0
$209$	0.166328	0.0115052
$210$	0	0
$211$	−20.3772	−1.40283	−0.701414	−	0.712754i	$-0.747447\pi$
$211$	−20.3772	−1.40283	−0.701414	+	0.712754i	$0.747447\pi$
$212$	0	0
$213$	5.83623	0.399892
$214$	0	0
$215$	4.25073	0.289897
$216$	0	0
$217$	−2.82264	−0.191613
$218$	0	0
$219$	11.2379	0.759385
$220$	0	0
$221$	17.8211	1.19878
$222$	0	0
$223$	−20.2826	−1.35823	−0.679113	−	0.734033i	$-0.737636\pi$
$223$	−20.2826	−1.35823	−0.679113	+	0.734033i	$0.737636\pi$
$224$	0	0
$225$	−2.65917	−0.177278
$226$	0	0
$227$	7.39895	0.491085	0.245543	−	0.969386i	$-0.421034\pi$
$227$	7.39895	0.491085	0.245543	+	0.969386i	$0.421034\pi$
$228$	0	0
$229$	−10.7782	−0.712247	−0.356123	−	0.934439i	$-0.615902\pi$
$229$	−10.7782	−0.712247	−0.356123	+	0.934439i	$0.615902\pi$
$230$	0	0
$231$	−0.170570	−0.0112227
$232$	0	0
$233$	−7.03570	−0.460924	−0.230462	−	0.973081i	$-0.574024\pi$
$233$	−7.03570	−0.460924	−0.230462	+	0.973081i	$0.574024\pi$
$234$	0	0
$235$	−14.5390	−0.948420
$236$	0	0
$237$	5.59029	0.363128
$238$	0	0
$239$	7.21369	0.466615	0.233307	−	0.972403i	$-0.425045\pi$
$239$	7.21369	0.466615	0.233307	+	0.972403i	$0.425045\pi$
$240$	0	0
$241$	−6.07039	−0.391028	−0.195514	−	0.980701i	$-0.562638\pi$
$241$	−6.07039	−0.391028	−0.195514	+	0.980701i	$0.562638\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−9.01502	−0.575949
$246$	0	0
$247$	3.73479	0.237639
$248$	0	0
$249$	4.98299	0.315784
$250$	0	0
$251$	−1.20193	−0.0758652	−0.0379326	−	0.999280i	$-0.512077\pi$
$251$	−1.20193	−0.0758652	−0.0379326	+	0.999280i	$0.512077\pi$
$252$	0	0
$253$	−1.23009	−0.0773350
$254$	0	0
$255$	7.49271	0.469212
$256$	0	0
$257$	−14.5799	−0.909471	−0.454735	−	0.890627i	$-0.650266\pi$
$257$	−14.5799	−0.909471	−0.454735	+	0.890627i	$0.650266\pi$
$258$	0	0
$259$	−4.33893	−0.269608
$260$	0	0
$261$	9.59787	0.594093
$262$	0	0
$263$	16.6235	1.02505	0.512524	−	0.858673i	$-0.328711\pi$
$263$	16.6235	1.02505	0.512524	+	0.858673i	$0.328711\pi$
$264$	0	0
$265$	7.35224	0.451644
$266$	0	0
$267$	−0.199333	−0.0121990
$268$	0	0
$269$	−8.32925	−0.507843	−0.253922	−	0.967225i	$-0.581721\pi$
$269$	−8.32925	−0.507843	−0.253922	+	0.967225i	$0.581721\pi$
$270$	0	0
$271$	−26.8577	−1.63149	−0.815744	−	0.578414i	$-0.803672\pi$
$271$	−26.8577	−1.63149	−0.815744	+	0.578414i	$0.803672\pi$
$272$	0	0
$273$	−3.83003	−0.231804
$274$	0	0
$275$	−0.430951	−0.0259873
$276$	0	0
$277$	7.42647	0.446213	0.223107	−	0.974794i	$-0.428380\pi$
$277$	7.42647	0.446213	0.223107	+	0.974794i	$0.428380\pi$
$278$	0	0
$279$	2.68185	0.160558
$280$	0	0
$281$	2.07711	0.123910	0.0619551	−	0.998079i	$-0.480266\pi$
$281$	2.07711	0.123910	0.0619551	+	0.998079i	$0.480266\pi$
$282$	0	0
$283$	−22.8081	−1.35580	−0.677900	−	0.735154i	$-0.737110\pi$
$283$	−22.8081	−1.35580	−0.677900	+	0.735154i	$0.737110\pi$
$284$	0	0
$285$	1.57025	0.0930137
$286$	0	0
$287$	1.13682	0.0671044
$288$	0	0
$289$	6.98318	0.410775
$290$	0	0
$291$	−10.9371	−0.641144
$292$	0	0
$293$	−19.3193	−1.12864	−0.564322	−	0.825555i	$-0.690862\pi$
$293$	−19.3193	−1.12864	−0.564322	+	0.825555i	$0.690862\pi$
$294$	0	0
$295$	12.5873	0.732858
$296$	0	0
$297$	0.162063	0.00940382
$298$	0	0
$299$	−27.6208	−1.59735
$300$	0	0
$301$	−2.92414	−0.168544
$302$	0	0
$303$	11.8811	0.682552
$304$	0	0
$305$	0.781442	0.0447452
$306$	0	0
$307$	−9.57084	−0.546237	−0.273118	−	0.961980i	$-0.588055\pi$
$307$	−9.57084	−0.546237	−0.273118	+	0.961980i	$0.588055\pi$
$308$	0	0
$309$	−1.54707	−0.0880097
$310$	0	0
$311$	−8.69478	−0.493036	−0.246518	−	0.969138i	$-0.579286\pi$
$311$	−8.69478	−0.493036	−0.246518	+	0.969138i	$0.579286\pi$
$312$	0	0
$313$	−24.7334	−1.39802	−0.699009	−	0.715113i	$-0.746375\pi$
$313$	−24.7334	−1.39802	−0.699009	+	0.715113i	$0.746375\pi$
$314$	0	0
$315$	−1.61029	−0.0907298
$316$	0	0
$317$	30.3559	1.70496	0.852479	−	0.522762i	$-0.175098\pi$
$317$	30.3559	1.70496	0.852479	+	0.522762i	$0.175098\pi$
$318$	0	0
$319$	1.55546	0.0870888
$320$	0	0
$321$	−2.89728	−0.161710
$322$	0	0
$323$	5.02617	0.279664
$324$	0	0
$325$	−9.67670	−0.536767
$326$	0	0
$327$	8.16825	0.451705
$328$	0	0
$329$	10.0016	0.551405
$330$	0	0
$331$	30.2944	1.66513	0.832566	−	0.553926i	$-0.186871\pi$
$331$	30.2944	1.66513	0.832566	+	0.553926i	$0.186871\pi$
$332$	0	0
$333$	4.12252	0.225913
$334$	0	0
$335$	−0.485338	−0.0265168
$336$	0	0
$337$	−0.467044	−0.0254415	−0.0127208	−	0.999919i	$-0.504049\pi$
$337$	−0.467044	−0.0254415	−0.0127208	+	0.999919i	$0.504049\pi$
$338$	0	0
$339$	−6.73714	−0.365911
$340$	0	0
$341$	0.434628	0.0235364
$342$	0	0
$343$	13.5690	0.732659
$344$	0	0
$345$	−11.6129	−0.625215
$346$	0	0
$347$	−2.81839	−0.151299	−0.0756496	−	0.997134i	$-0.524103\pi$
$347$	−2.81839	−0.151299	−0.0756496	+	0.997134i	$0.524103\pi$
$348$	0	0
$349$	37.0275	1.98204	0.991018	−	0.133726i	$-0.0426943\pi$
$349$	37.0275	1.98204	0.991018	+	0.133726i	$0.0426943\pi$
$350$	0	0
$351$	3.63900	0.194235
$352$	0	0
$353$	−4.85955	−0.258648	−0.129324	−	0.991602i	$-0.541281\pi$
$353$	−4.85955	−0.258648	−0.129324	+	0.991602i	$0.541281\pi$
$354$	0	0
$355$	8.92930	0.473918
$356$	0	0
$357$	−5.15434	−0.272797
$358$	0	0
$359$	−0.0868793	−0.00458531	−0.00229266	−	0.999997i	$-0.500730\pi$
$359$	−0.0868793	−0.00458531	−0.00229266	+	0.999997i	$0.500730\pi$
$360$	0	0
$361$	−17.9467	−0.944561
$362$	0	0
$363$	−10.9737	−0.575972
$364$	0	0
$365$	17.1937	0.899960
$366$	0	0
$367$	−15.3217	−0.799787	−0.399894	−	0.916562i	$-0.630953\pi$
$367$	−15.3217	−0.799787	−0.399894	+	0.916562i	$0.630953\pi$
$368$	0	0
$369$	−1.08012	−0.0562288
$370$	0	0
$371$	−5.05771	−0.262583
$372$	0	0
$373$	0.349987	0.0181216	0.00906081	−	0.999959i	$-0.497116\pi$
$373$	0.349987	0.0181216	0.00906081	+	0.999959i	$0.497116\pi$
$374$	0	0
$375$	−11.7184	−0.605133
$376$	0	0
$377$	34.9267	1.79881
$378$	0	0
$379$	−29.6312	−1.52206	−0.761028	−	0.648720i	$-0.775305\pi$
$379$	−29.6312	−1.52206	−0.761028	+	0.648720i	$0.775305\pi$
$380$	0	0
$381$	−15.8812	−0.813621
$382$	0	0
$383$	−28.4092	−1.45164	−0.725821	−	0.687884i	$-0.758540\pi$
$383$	−28.4092	−1.45164	−0.725821	+	0.687884i	$0.758540\pi$
$384$	0	0
$385$	−0.260968	−0.0133002
$386$	0	0
$387$	2.77829	0.141229
$388$	0	0
$389$	18.4318	0.934530	0.467265	−	0.884117i	$-0.345239\pi$
$389$	18.4318	0.934530	0.467265	+	0.884117i	$0.345239\pi$
$390$	0	0
$391$	−37.1713	−1.87983
$392$	0	0
$393$	9.81474	0.495088
$394$	0	0
$395$	8.55302	0.430349
$396$	0	0
$397$	0.947842	0.0475708	0.0237854	−	0.999717i	$-0.492428\pi$
$397$	0.947842	0.0475708	0.0237854	+	0.999717i	$0.492428\pi$
$398$	0	0
$399$	−1.08020	−0.0540776
$400$	0	0
$401$	1.29454	0.0646464	0.0323232	−	0.999477i	$-0.489709\pi$
$401$	1.29454	0.0646464	0.0323232	+	0.999477i	$0.489709\pi$
$402$	0	0
$403$	9.75926	0.486143
$404$	0	0
$405$	1.52998	0.0760253
$406$	0	0
$407$	0.668106	0.0331168
$408$	0	0
$409$	−29.1328	−1.44053	−0.720263	−	0.693701i	$-0.755979\pi$
$409$	−29.1328	−1.44053	−0.720263	+	0.693701i	$0.755979\pi$
$410$	0	0
$411$	−2.22904	−0.109951
$412$	0	0
$413$	−8.65896	−0.426079
$414$	0	0
$415$	7.62386	0.374241
$416$	0	0
$417$	−7.71321	−0.377717
$418$	0	0
$419$	15.3595	0.750362	0.375181	−	0.926952i	$-0.377581\pi$
$419$	15.3595	0.750362	0.375181	+	0.926952i	$0.377581\pi$
$420$	0	0
$421$	−22.9588	−1.11894	−0.559472	−	0.828849i	$-0.688996\pi$
$421$	−22.9588	−1.11894	−0.559472	+	0.828849i	$0.688996\pi$
$422$	0	0
$423$	−9.50274	−0.462039
$424$	0	0
$425$	−13.0226	−0.631690
$426$	0	0
$427$	−0.537565	−0.0260146
$428$	0	0
$429$	0.589745	0.0284732
$430$	0	0
$431$	−23.4208	−1.12814	−0.564070	−	0.825727i	$-0.690765\pi$
$431$	−23.4208	−1.12814	−0.564070	+	0.825727i	$0.690765\pi$
$432$	0	0
$433$	−30.2786	−1.45510	−0.727549	−	0.686056i	$-0.759340\pi$
$433$	−30.2786	−1.45510	−0.727549	+	0.686056i	$0.759340\pi$
$434$	0	0
$435$	14.6845	0.704070
$436$	0	0
$437$	−7.79000	−0.372646
$438$	0	0
$439$	−20.4125	−0.974236	−0.487118	−	0.873336i	$-0.661952\pi$
$439$	−20.4125	−0.974236	−0.487118	+	0.873336i	$0.661952\pi$
$440$	0	0
$441$	−5.89226	−0.280584
$442$	0	0
$443$	−30.1725	−1.43354	−0.716770	−	0.697310i	$-0.754380\pi$
$443$	−30.1725	−1.43354	−0.716770	+	0.697310i	$0.754380\pi$
$444$	0	0
$445$	−0.304975	−0.0144572
$446$	0	0
$447$	−19.1731	−0.906856
$448$	0	0
$449$	−14.7721	−0.697137	−0.348568	−	0.937283i	$-0.613332\pi$
$449$	−14.7721	−0.697137	−0.348568	+	0.937283i	$0.613332\pi$
$450$	0	0
$451$	−0.175047	−0.00824265
$452$	0	0
$453$	−8.05210	−0.378321
$454$	0	0
$455$	−5.85986	−0.274715
$456$	0	0
$457$	33.0643	1.54668	0.773342	−	0.633989i	$-0.218583\pi$
$457$	33.0643	1.54668	0.773342	+	0.633989i	$0.218583\pi$
$458$	0	0
$459$	4.89726	0.228585
$460$	0	0
$461$	−20.9924	−0.977714	−0.488857	−	0.872364i	$-0.662586\pi$
$461$	−20.9924	−0.977714	−0.488857	+	0.872364i	$0.662586\pi$
$462$	0	0
$463$	−38.8607	−1.80601	−0.903005	−	0.429631i	$-0.858644\pi$
$463$	−38.8607	−1.80601	−0.903005	+	0.429631i	$0.858644\pi$
$464$	0	0
$465$	4.10318	0.190280
$466$	0	0
$467$	30.7010	1.42067	0.710335	−	0.703863i	$-0.248543\pi$
$467$	30.7010	1.42067	0.710335	+	0.703863i	$0.248543\pi$
$468$	0	0
$469$	0.333871	0.0154167
$470$	0	0
$471$	10.1858	0.469336
$472$	0	0
$473$	0.450257	0.0207028
$474$	0	0
$475$	−2.72916	−0.125222
$476$	0	0
$477$	4.80545	0.220027
$478$	0	0
$479$	−4.10481	−0.187554	−0.0937768	−	0.995593i	$-0.529894\pi$
$479$	−4.10481	−0.187554	−0.0937768	+	0.995593i	$0.529894\pi$
$480$	0	0
$481$	15.0019	0.684026
$482$	0	0
$483$	7.98866	0.363496
$484$	0	0
$485$	−16.7335	−0.759830
$486$	0	0
$487$	10.0473	0.455289	0.227644	−	0.973744i	$-0.426898\pi$
$487$	10.0473	0.455289	0.227644	+	0.973744i	$0.426898\pi$
$488$	0	0
$489$	0.432916	0.0195771
$490$	0	0
$491$	−16.4195	−0.741001	−0.370500	−	0.928832i	$-0.620814\pi$
$491$	−16.4195	−0.741001	−0.370500	+	0.928832i	$0.620814\pi$
$492$	0	0
$493$	47.0033	2.11692
$494$	0	0
$495$	0.247952	0.0111446
$496$	0	0
$497$	−6.14260	−0.275533
$498$	0	0
$499$	11.1636	0.499753	0.249877	−	0.968278i	$-0.419610\pi$
$499$	11.1636	0.499753	0.249877	+	0.968278i	$0.419610\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−22.6818	−1.01133	−0.505666	−	0.862729i	$-0.668753\pi$
$503$	−22.6818	−1.01133	−0.505666	+	0.862729i	$0.668753\pi$
$504$	0	0
$505$	18.1778	0.808903
$506$	0	0
$507$	0.242320	0.0107618
$508$	0	0
$509$	−2.92089	−0.129466	−0.0647331	−	0.997903i	$-0.520620\pi$
$509$	−2.92089	−0.129466	−0.0647331	+	0.997903i	$0.520620\pi$
$510$	0	0
$511$	−11.8278	−0.523231
$512$	0	0
$513$	1.02632	0.0453132
$514$	0	0
$515$	−2.36698	−0.104302
$516$	0	0
$517$	−1.54004	−0.0677308
$518$	0	0
$519$	17.2653	0.757864
$520$	0	0
$521$	28.6454	1.25498	0.627490	−	0.778625i	$-0.284082\pi$
$521$	28.6454	1.25498	0.627490	+	0.778625i	$0.284082\pi$
$522$	0	0
$523$	−5.60009	−0.244875	−0.122437	−	0.992476i	$-0.539071\pi$
$523$	−5.60009	−0.244875	−0.122437	+	0.992476i	$0.539071\pi$
$524$	0	0
$525$	2.79876	0.122148
$526$	0	0
$527$	13.1337	0.572114
$528$	0	0
$529$	34.6113	1.50484
$530$	0	0
$531$	8.22708	0.357025
$532$	0	0
$533$	−3.93056	−0.170251
$534$	0	0
$535$	−4.43278	−0.191646
$536$	0	0
$537$	7.05820	0.304584
$538$	0	0
$539$	−0.954914	−0.0411310
$540$	0	0
$541$	−37.7504	−1.62302	−0.811509	−	0.584341i	$-0.801353\pi$
$541$	−37.7504	−1.62302	−0.811509	+	0.584341i	$0.801353\pi$
$542$	0	0
$543$	−14.7420	−0.632641
$544$	0	0
$545$	12.4973	0.535323
$546$	0	0
$547$	−8.04129	−0.343821	−0.171910	−	0.985113i	$-0.554994\pi$
$547$	−8.04129	−0.343821	−0.171910	+	0.985113i	$0.554994\pi$
$548$	0	0
$549$	0.510754	0.0217984
$550$	0	0
$551$	9.85051	0.419646
$552$	0	0
$553$	−5.88375	−0.250202
$554$	0	0
$555$	6.30737	0.267733
$556$	0	0
$557$	17.6075	0.746052	0.373026	−	0.927821i	$-0.378320\pi$
$557$	17.6075	0.746052	0.373026	+	0.927821i	$0.378320\pi$
$558$	0	0
$559$	10.1102	0.427616
$560$	0	0
$561$	0.793663	0.0335085
$562$	0	0
$563$	−40.8654	−1.72227	−0.861135	−	0.508376i	$-0.830246\pi$
$563$	−40.8654	−1.72227	−0.861135	+	0.508376i	$0.830246\pi$
$564$	0	0
$565$	−10.3077	−0.433648
$566$	0	0
$567$	−1.05249	−0.0442006
$568$	0	0
$569$	−28.5716	−1.19779	−0.598893	−	0.800829i	$-0.704392\pi$
$569$	−28.5716	−1.19779	−0.598893	+	0.800829i	$0.704392\pi$
$570$	0	0
$571$	−15.5883	−0.652352	−0.326176	−	0.945309i	$-0.605760\pi$
$571$	−15.5883	−0.652352	−0.326176	+	0.945309i	$0.605760\pi$
$572$	0	0
$573$	22.5477	0.941943
$574$	0	0
$575$	20.1836	0.841715
$576$	0	0
$577$	−24.1993	−1.00743	−0.503715	−	0.863870i	$-0.668034\pi$
$577$	−24.1993	−1.00743	−0.503715	+	0.863870i	$0.668034\pi$
$578$	0	0
$579$	3.60996	0.150025
$580$	0	0
$581$	−5.24457	−0.217581
$582$	0	0
$583$	0.778784	0.0322539
$584$	0	0
$585$	5.56759	0.230192
$586$	0	0
$587$	−29.3193	−1.21014	−0.605069	−	0.796173i	$-0.706855\pi$
$587$	−29.3193	−1.21014	−0.605069	+	0.796173i	$0.706855\pi$
$588$	0	0
$589$	2.75245	0.113413
$590$	0	0
$591$	12.4286	0.511245
$592$	0	0
$593$	−21.3422	−0.876418	−0.438209	−	0.898873i	$-0.644387\pi$
$593$	−21.3422	−0.876418	−0.438209	+	0.898873i	$0.644387\pi$
$594$	0	0
$595$	−7.88603	−0.323296
$596$	0	0
$597$	6.92918	0.283593
$598$	0	0
$599$	−22.3543	−0.913373	−0.456686	−	0.889628i	$-0.650964\pi$
$599$	−22.3543	−0.913373	−0.456686	+	0.889628i	$0.650964\pi$
$600$	0	0
$601$	41.2799	1.68384	0.841921	−	0.539602i	$-0.181425\pi$
$601$	41.2799	1.68384	0.841921	+	0.539602i	$0.181425\pi$
$602$	0	0
$603$	−0.317219	−0.0129181
$604$	0	0
$605$	−16.7896	−0.682594
$606$	0	0
$607$	6.06522	0.246180	0.123090	−	0.992396i	$-0.460720\pi$
$607$	6.06522	0.246180	0.123090	+	0.992396i	$0.460720\pi$
$608$	0	0
$609$	−10.1017	−0.409342
$610$	0	0
$611$	−34.5805	−1.39898
$612$	0	0
$613$	−11.1383	−0.449871	−0.224935	−	0.974374i	$-0.572217\pi$
$613$	−11.1383	−0.449871	−0.224935	+	0.974374i	$0.572217\pi$
$614$	0	0
$615$	−1.65256	−0.0666377
$616$	0	0
$617$	19.1170	0.769621	0.384810	−	0.922996i	$-0.374267\pi$
$617$	19.1170	0.769621	0.384810	+	0.922996i	$0.374267\pi$
$618$	0	0
$619$	30.9654	1.24460	0.622302	−	0.782777i	$-0.286197\pi$
$619$	30.9654	1.24460	0.622302	+	0.782777i	$0.286197\pi$
$620$	0	0
$621$	−7.59021	−0.304585
$622$	0	0
$623$	0.209797	0.00840533
$624$	0	0
$625$	−4.63301	−0.185321
$626$	0	0
$627$	0.166328	0.00664252
$628$	0	0
$629$	20.1891	0.804991
$630$	0	0
$631$	24.1070	0.959686	0.479843	−	0.877354i	$-0.340694\pi$
$631$	24.1070	0.959686	0.479843	+	0.877354i	$0.340694\pi$
$632$	0	0
$633$	−20.3772	−0.809923
$634$	0	0
$635$	−24.2980	−0.964235
$636$	0	0
$637$	−21.4419	−0.849560
$638$	0	0
$639$	5.83623	0.230878
$640$	0	0
$641$	−32.5154	−1.28428	−0.642141	−	0.766586i	$-0.721954\pi$
$641$	−32.5154	−1.28428	−0.642141	+	0.766586i	$0.721954\pi$
$642$	0	0
$643$	−15.8759	−0.626085	−0.313042	−	0.949739i	$-0.601348\pi$
$643$	−15.8759	−0.626085	−0.313042	+	0.949739i	$0.601348\pi$
$644$	0	0
$645$	4.25073	0.167372
$646$	0	0
$647$	25.0625	0.985310	0.492655	−	0.870225i	$-0.336026\pi$
$647$	25.0625	0.985310	0.492655	+	0.870225i	$0.336026\pi$
$648$	0	0
$649$	1.33330	0.0523366
$650$	0	0
$651$	−2.82264	−0.110628
$652$	0	0
$653$	−5.74801	−0.224937	−0.112469	−	0.993655i	$-0.535876\pi$
$653$	−5.74801	−0.224937	−0.112469	+	0.993655i	$0.535876\pi$
$654$	0	0
$655$	15.0163	0.586737
$656$	0	0
$657$	11.2379	0.438431
$658$	0	0
$659$	11.0446	0.430238	0.215119	−	0.976588i	$-0.430986\pi$
$659$	11.0446	0.430238	0.215119	+	0.976588i	$0.430986\pi$
$660$	0	0
$661$	27.4709	1.06849	0.534247	−	0.845329i	$-0.320595\pi$
$661$	27.4709	1.06849	0.534247	+	0.845329i	$0.320595\pi$
$662$	0	0
$663$	17.8211	0.692116
$664$	0	0
$665$	−1.65268	−0.0640882
$666$	0	0
$667$	−72.8499	−2.82076
$668$	0	0
$669$	−20.2826	−0.784172
$670$	0	0
$671$	0.0827740	0.00319545
$672$	0	0
$673$	13.7360	0.529485	0.264743	−	0.964319i	$-0.414713\pi$
$673$	13.7360	0.529485	0.264743	+	0.964319i	$0.414713\pi$
$674$	0	0
$675$	−2.65917	−0.102351
$676$	0	0
$677$	35.5759	1.36729	0.683646	−	0.729814i	$-0.260393\pi$
$677$	35.5759	1.36729	0.683646	+	0.729814i	$0.260393\pi$
$678$	0	0
$679$	11.5112	0.441760
$680$	0	0
$681$	7.39895	0.283528
$682$	0	0
$683$	4.58516	0.175446	0.0877231	−	0.996145i	$-0.472041\pi$
$683$	4.58516	0.175446	0.0877231	+	0.996145i	$0.472041\pi$
$684$	0	0
$685$	−3.41039	−0.130304
$686$	0	0
$687$	−10.7782	−0.411216
$688$	0	0
$689$	17.4870	0.666203
$690$	0	0
$691$	8.54129	0.324926	0.162463	−	0.986715i	$-0.448056\pi$
$691$	8.54129	0.324926	0.162463	+	0.986715i	$0.448056\pi$
$692$	0	0
$693$	−0.170570	−0.00647942
$694$	0	0
$695$	−11.8010	−0.447639
$696$	0	0
$697$	−5.28964	−0.200359
$698$	0	0
$699$	−7.03570	−0.266115
$700$	0	0
$701$	46.3748	1.75155	0.875776	−	0.482718i	$-0.160350\pi$
$701$	46.3748	1.75155	0.875776	+	0.482718i	$0.160350\pi$
$702$	0	0
$703$	4.23104	0.159577
$704$	0	0
$705$	−14.5390	−0.547570
$706$	0	0
$707$	−12.5048	−0.470292
$708$	0	0
$709$	46.4820	1.74567	0.872835	−	0.488016i	$-0.162279\pi$
$709$	46.4820	1.74567	0.872835	+	0.488016i	$0.162279\pi$
$710$	0	0
$711$	5.59029	0.209652
$712$	0	0
$713$	−20.3558	−0.762332
$714$	0	0
$715$	0.902298	0.0337440
$716$	0	0
$717$	7.21369	0.269400
$718$	0	0
$719$	27.2461	1.01611	0.508053	−	0.861326i	$-0.330365\pi$
$719$	27.2461	1.01611	0.508053	+	0.861326i	$0.330365\pi$
$720$	0	0
$721$	1.62828	0.0606404
$722$	0	0
$723$	−6.07039	−0.225760
$724$	0	0
$725$	−25.5223	−0.947875
$726$	0	0
$727$	−21.9595	−0.814432	−0.407216	−	0.913332i	$-0.633500\pi$
$727$	−21.9595	−0.814432	−0.407216	+	0.913332i	$0.633500\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.6060	0.503237
$732$	0	0
$733$	−17.3357	−0.640309	−0.320154	−	0.947365i	$-0.603735\pi$
$733$	−17.3357	−0.640309	−0.320154	+	0.947365i	$0.603735\pi$
$734$	0	0
$735$	−9.01502	−0.332524
$736$	0	0
$737$	−0.0514092	−0.00189368
$738$	0	0
$739$	14.7105	0.541135	0.270568	−	0.962701i	$-0.412789\pi$
$739$	14.7105	0.541135	0.270568	+	0.962701i	$0.412789\pi$
$740$	0	0
$741$	3.73479	0.137201
$742$	0	0
$743$	16.0012	0.587029	0.293514	−	0.955955i	$-0.405175\pi$
$743$	16.0012	0.587029	0.293514	+	0.955955i	$0.405175\pi$
$744$	0	0
$745$	−29.3344	−1.07473
$746$	0	0
$747$	4.98299	0.182318
$748$	0	0
$749$	3.04937	0.111422
$750$	0	0
$751$	43.2600	1.57858	0.789289	−	0.614022i	$-0.210449\pi$
$751$	43.2600	1.57858	0.789289	+	0.614022i	$0.210449\pi$
$752$	0	0
$753$	−1.20193	−0.0438008
$754$	0	0
$755$	−12.3195	−0.448354
$756$	0	0
$757$	−9.81138	−0.356601	−0.178300	−	0.983976i	$-0.557060\pi$
$757$	−9.81138	−0.356601	−0.178300	+	0.983976i	$0.557060\pi$
$758$	0	0
$759$	−1.23009	−0.0446494
$760$	0	0
$761$	15.1070	0.547627	0.273814	−	0.961783i	$-0.411715\pi$
$761$	15.1070	0.547627	0.273814	+	0.961783i	$0.411715\pi$
$762$	0	0
$763$	−8.59704	−0.311234
$764$	0	0
$765$	7.49271	0.270899
$766$	0	0
$767$	29.9383	1.08101
$768$	0	0
$769$	20.1675	0.727258	0.363629	−	0.931544i	$-0.381538\pi$
$769$	20.1675	0.727258	0.363629	+	0.931544i	$0.381538\pi$
$770$	0	0
$771$	−14.5799	−0.525083
$772$	0	0
$773$	26.4289	0.950582	0.475291	−	0.879829i	$-0.342343\pi$
$773$	26.4289	0.950582	0.475291	+	0.879829i	$0.342343\pi$
$774$	0	0
$775$	−7.13149	−0.256171
$776$	0	0
$777$	−4.33893	−0.155658
$778$	0	0
$779$	−1.10855	−0.0397180
$780$	0	0
$781$	0.945833	0.0338446
$782$	0	0
$783$	9.59787	0.343000
$784$	0	0
$785$	15.5840	0.556218
$786$	0	0
$787$	−23.9094	−0.852280	−0.426140	−	0.904657i	$-0.640127\pi$
$787$	−23.9094	−0.852280	−0.426140	+	0.904657i	$0.640127\pi$
$788$	0	0
$789$	16.6235	0.591811
$790$	0	0
$791$	7.09081	0.252120
$792$	0	0
$793$	1.85863	0.0660020
$794$	0	0
$795$	7.35224	0.260757
$796$	0	0
$797$	26.2713	0.930578	0.465289	−	0.885159i	$-0.345950\pi$
$797$	26.2713	0.930578	0.465289	+	0.885159i	$0.345950\pi$
$798$	0	0
$799$	−46.5374	−1.64638
$800$	0	0
$801$	−0.199333	−0.00704308
$802$	0	0
$803$	1.82124	0.0642701
$804$	0	0
$805$	12.2225	0.430786
$806$	0	0
$807$	−8.32925	−0.293203
$808$	0	0
$809$	−8.70973	−0.306218	−0.153109	−	0.988209i	$-0.548929\pi$
$809$	−8.70973	−0.306218	−0.153109	+	0.988209i	$0.548929\pi$
$810$	0	0
$811$	−26.7512	−0.939363	−0.469682	−	0.882836i	$-0.655631\pi$
$811$	−26.7512	−0.939363	−0.469682	+	0.882836i	$0.655631\pi$
$812$	0	0
$813$	−26.8577	−0.941940
$814$	0	0
$815$	0.662352	0.0232012
$816$	0	0
$817$	2.85142	0.0997587
$818$	0	0
$819$	−3.83003	−0.133832
$820$	0	0
$821$	4.36277	0.152262	0.0761309	−	0.997098i	$-0.475743\pi$
$821$	4.36277	0.152262	0.0761309	+	0.997098i	$0.475743\pi$
$822$	0	0
$823$	−8.76960	−0.305689	−0.152844	−	0.988250i	$-0.548843\pi$
$823$	−8.76960	−0.305689	−0.152844	+	0.988250i	$0.548843\pi$
$824$	0	0
$825$	−0.430951	−0.0150038
$826$	0	0
$827$	−2.60546	−0.0906009	−0.0453004	−	0.998973i	$-0.514425\pi$
$827$	−2.60546	−0.0906009	−0.0453004	+	0.998973i	$0.514425\pi$
$828$	0	0
$829$	−21.2663	−0.738608	−0.369304	−	0.929309i	$-0.620404\pi$
$829$	−21.2663	−0.738608	−0.369304	+	0.929309i	$0.620404\pi$
$830$	0	0
$831$	7.42647	0.257621
$832$	0	0
$833$	−28.8559	−0.999798
$834$	0	0
$835$	1.52998	0.0529471
$836$	0	0
$837$	2.68185	0.0926984
$838$	0	0
$839$	−29.8458	−1.03039	−0.515195	−	0.857073i	$-0.672281\pi$
$839$	−29.8458	−1.03039	−0.515195	+	0.857073i	$0.672281\pi$
$840$	0	0
$841$	63.1191	2.17652
$842$	0	0
$843$	2.07711	0.0715396
$844$	0	0
$845$	0.370744	0.0127540
$846$	0	0
$847$	11.5498	0.396856
$848$	0	0
$849$	−22.8081	−0.782771
$850$	0	0
$851$	−31.2908	−1.07263
$852$	0	0
$853$	7.85903	0.269088	0.134544	−	0.990908i	$-0.457043\pi$
$853$	7.85903	0.269088	0.134544	+	0.990908i	$0.457043\pi$
$854$	0	0
$855$	1.57025	0.0537015
$856$	0	0
$857$	−27.2600	−0.931183	−0.465591	−	0.885000i	$-0.654158\pi$
$857$	−27.2600	−0.931183	−0.465591	+	0.885000i	$0.654158\pi$
$858$	0	0
$859$	1.16871	0.0398758	0.0199379	−	0.999801i	$-0.493653\pi$
$859$	1.16871	0.0398758	0.0199379	+	0.999801i	$0.493653\pi$
$860$	0	0
$861$	1.13682	0.0387428
$862$	0	0
$863$	57.0280	1.94126	0.970628	−	0.240587i	$-0.0773399\pi$
$863$	57.0280	1.94126	0.970628	+	0.240587i	$0.0773399\pi$
$864$	0	0
$865$	26.4156	0.898157
$866$	0	0
$867$	6.98318	0.237161
$868$	0	0
$869$	0.905976	0.0307331
$870$	0	0
$871$	−1.15436	−0.0391139
$872$	0	0
$873$	−10.9371	−0.370164
$874$	0	0
$875$	12.3335	0.416949
$876$	0	0
$877$	25.3813	0.857065	0.428533	−	0.903526i	$-0.359031\pi$
$877$	25.3813	0.857065	0.428533	+	0.903526i	$0.359031\pi$
$878$	0	0
$879$	−19.3193	−0.651623
$880$	0	0
$881$	−22.2777	−0.750554	−0.375277	−	0.926913i	$-0.622452\pi$
$881$	−22.2777	−0.750554	−0.375277	+	0.926913i	$0.622452\pi$
$882$	0	0
$883$	21.8081	0.733900	0.366950	−	0.930241i	$-0.380402\pi$
$883$	21.8081	0.733900	0.366950	+	0.930241i	$0.380402\pi$
$884$	0	0
$885$	12.5873	0.423116
$886$	0	0
$887$	31.0024	1.04096	0.520480	−	0.853874i	$-0.325753\pi$
$887$	31.0024	1.04096	0.520480	+	0.853874i	$0.325753\pi$
$888$	0	0
$889$	16.7149	0.560601
$890$	0	0
$891$	0.162063	0.00542930
$892$	0	0
$893$	−9.75288	−0.326368
$894$	0	0
$895$	10.7989	0.360967
$896$	0	0
$897$	−27.6208	−0.922231
$898$	0	0
$899$	25.7401	0.858479
$900$	0	0
$901$	23.5336	0.784017
$902$	0	0
$903$	−2.92414	−0.0973092
$904$	0	0
$905$	−22.5550	−0.749753
$906$	0	0
$907$	−49.9506	−1.65858	−0.829291	−	0.558817i	$-0.811255\pi$
$907$	−49.9506	−1.65858	−0.829291	+	0.558817i	$0.811255\pi$
$908$	0	0
$909$	11.8811	0.394072
$910$	0	0
$911$	31.1409	1.03174	0.515872	−	0.856666i	$-0.327468\pi$
$911$	31.1409	1.03174	0.515872	+	0.856666i	$0.327468\pi$
$912$	0	0
$913$	0.807555	0.0267262
$914$	0	0
$915$	0.781442	0.0258337
$916$	0	0
$917$	−10.3300	−0.341125
$918$	0	0
$919$	−50.0648	−1.65149	−0.825743	−	0.564046i	$-0.809244\pi$
$919$	−50.0648	−1.65149	−0.825743	+	0.564046i	$0.809244\pi$
$920$	0	0
$921$	−9.57084	−0.315370
$922$	0	0
$923$	21.2380	0.699058
$924$	0	0
$925$	−10.9625	−0.360444
$926$	0	0
$927$	−1.54707	−0.0508124
$928$	0	0
$929$	28.2739	0.927638	0.463819	−	0.885930i	$-0.346479\pi$
$929$	28.2739	0.927638	0.463819	+	0.885930i	$0.346479\pi$
$930$	0	0
$931$	−6.04735	−0.198194
$932$	0	0
$933$	−8.69478	−0.284654
$934$	0	0
$935$	1.21429	0.0397114
$936$	0	0
$937$	38.8173	1.26811	0.634053	−	0.773290i	$-0.281390\pi$
$937$	38.8173	1.26811	0.634053	+	0.773290i	$0.281390\pi$
$938$	0	0
$939$	−24.7334	−0.807146
$940$	0	0
$941$	−28.4498	−0.927437	−0.463719	−	0.885983i	$-0.653485\pi$
$941$	−28.4498	−0.927437	−0.463719	+	0.885983i	$0.653485\pi$
$942$	0	0
$943$	8.19834	0.266975
$944$	0	0
$945$	−1.61029	−0.0523829
$946$	0	0
$947$	8.80156	0.286012	0.143006	−	0.989722i	$-0.454323\pi$
$947$	8.80156	0.286012	0.143006	+	0.989722i	$0.454323\pi$
$948$	0	0
$949$	40.8946	1.32750
$950$	0	0
$951$	30.3559	0.984358
$952$	0	0
$953$	−11.0564	−0.358151	−0.179075	−	0.983835i	$-0.557311\pi$
$953$	−11.0564	−0.358151	−0.179075	+	0.983835i	$0.557311\pi$
$954$	0	0
$955$	34.4975	1.11631
$956$	0	0
$957$	1.55546	0.0502807
$958$	0	0
$959$	2.34605	0.0757581
$960$	0	0
$961$	−23.8077	−0.767989
$962$	0	0
$963$	−2.89728	−0.0933636
$964$	0	0
$965$	5.52316	0.177797
$966$	0	0
$967$	24.2114	0.778586	0.389293	−	0.921114i	$-0.372719\pi$
$967$	24.2114	0.778586	0.389293	+	0.921114i	$0.372719\pi$
$968$	0	0
$969$	5.02617	0.161464
$970$	0	0
$971$	7.68129	0.246504	0.123252	−	0.992375i	$-0.460668\pi$
$971$	7.68129	0.246504	0.123252	+	0.992375i	$0.460668\pi$
$972$	0	0
$973$	8.11811	0.260255
$974$	0	0
$975$	−9.67670	−0.309902
$976$	0	0
$977$	−36.4901	−1.16742	−0.583711	−	0.811961i	$-0.698400\pi$
$977$	−36.4901	−1.16742	−0.583711	+	0.811961i	$0.698400\pi$
$978$	0	0
$979$	−0.0323044	−0.00103245
$980$	0	0
$981$	8.16825	0.260792
$982$	0	0
$983$	−27.9487	−0.891424	−0.445712	−	0.895176i	$-0.647049\pi$
$983$	−27.9487	−0.891424	−0.445712	+	0.895176i	$0.647049\pi$
$984$	0	0
$985$	19.0155	0.605885
$986$	0	0
$987$	10.0016	0.318354
$988$	0	0
$989$	−21.0878	−0.670554
$990$	0	0
$991$	1.69672	0.0538980	0.0269490	−	0.999637i	$-0.491421\pi$
$991$	1.69672	0.0538980	0.0269490	+	0.999637i	$0.491421\pi$
$992$	0	0
$993$	30.2944	0.961365
$994$	0	0
$995$	10.6015	0.336090
$996$	0	0
$997$	20.9519	0.663553	0.331777	−	0.943358i	$-0.392352\pi$
$997$	20.9519	0.663553	0.331777	+	0.943358i	$0.392352\pi$
$998$	0	0
$999$	4.12252	0.130431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.d.1.5	✓	9
3.2	odd	2		6012.2.a.h.1.5		9
4.3	odd	2		8016.2.a.bb.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.5	✓	9	1.1	even	1	trivial
6012.2.a.h.1.5		9	3.2	odd	2
8016.2.a.bb.1.5		9	4.3	odd	2

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:	\( N \)	\(=\)	\( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.52998 q^{5} -1.05249 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.52998 q^{5} -1.05249 q^{7} +1.00000 q^{9} +0.162063 q^{11} +3.63900 q^{13} +1.52998 q^{15} +4.89726 q^{17} +1.02632 q^{19} -1.05249 q^{21} -7.59021 q^{23} -2.65917 q^{25} +1.00000 q^{27} +9.59787 q^{29} +2.68185 q^{31} +0.162063 q^{33} -1.61029 q^{35} +4.12252 q^{37} +3.63900 q^{39} -1.08012 q^{41} +2.77829 q^{43} +1.52998 q^{45} -9.50274 q^{47} -5.89226 q^{49} +4.89726 q^{51} +4.80545 q^{53} +0.247952 q^{55} +1.02632 q^{57} +8.22708 q^{59} +0.510754 q^{61} -1.05249 q^{63} +5.56759 q^{65} -0.317219 q^{67} -7.59021 q^{69} +5.83623 q^{71} +11.2379 q^{73} -2.65917 q^{75} -0.170570 q^{77} +5.59029 q^{79} +1.00000 q^{81} +4.98299 q^{83} +7.49271 q^{85} +9.59787 q^{87} -0.199333 q^{89} -3.83003 q^{91} +2.68185 q^{93} +1.57025 q^{95} -10.9371 q^{97} +0.162063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * q^97 + 7 * q^99

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