LMFDB - Embedded newform 8004.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
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} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 $ x^9 - 3x^8 - 13x^7 + 32x^6 + 40x^5 - 79x^4 - 39x^3 + 58x^2 + 9x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.975163$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.900888 q^{5} +3.90501 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.900888 q^{5} +3.90501 q^{7} +1.00000 q^{9} -5.39807 q^{11} -2.18514 q^{13} -0.900888 q^{15} -2.64370 q^{17} +3.37158 q^{19} -3.90501 q^{21} -1.00000 q^{23} -4.18840 q^{25} -1.00000 q^{27} +1.00000 q^{29} +5.08447 q^{31} +5.39807 q^{33} +3.51798 q^{35} -3.79035 q^{37} +2.18514 q^{39} +8.82974 q^{41} +0.613535 q^{43} +0.900888 q^{45} +1.02676 q^{47} +8.24908 q^{49} +2.64370 q^{51} +5.94186 q^{53} -4.86306 q^{55} -3.37158 q^{57} -10.0058 q^{59} -3.09826 q^{61} +3.90501 q^{63} -1.96857 q^{65} -12.6836 q^{67} +1.00000 q^{69} -4.69605 q^{71} +9.03175 q^{73} +4.18840 q^{75} -21.0795 q^{77} -3.30647 q^{79} +1.00000 q^{81} -1.01449 q^{83} -2.38168 q^{85} -1.00000 q^{87} -17.4884 q^{89} -8.53300 q^{91} -5.08447 q^{93} +3.03742 q^{95} +4.91409 q^{97} -5.39807 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * 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
$5$	0.900888	0.402889	0.201445	−	0.979500i	$-0.435436\pi$
$5$	0.900888	0.402889	0.201445	+	0.979500i	$0.435436\pi$
$6$	0	0
$7$	3.90501	1.47595	0.737977	−	0.674826i	$-0.235781\pi$
$7$	3.90501	1.47595	0.737977	+	0.674826i	$0.235781\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.39807	−1.62758	−0.813790	−	0.581159i	$-0.802600\pi$
$11$	−5.39807	−1.62758	−0.813790	+	0.581159i	$0.802600\pi$
$12$	0	0
$13$	−2.18514	−0.606050	−0.303025	−	0.952983i	$-0.597997\pi$
$13$	−2.18514	−0.606050	−0.303025	+	0.952983i	$0.597997\pi$
$14$	0	0
$15$	−0.900888	−0.232608
$16$	0	0
$17$	−2.64370	−0.641191	−0.320595	−	0.947216i	$-0.603883\pi$
$17$	−2.64370	−0.641191	−0.320595	+	0.947216i	$0.603883\pi$
$18$	0	0
$19$	3.37158	0.773494	0.386747	−	0.922186i	$-0.373599\pi$
$19$	3.37158	0.773494	0.386747	+	0.922186i	$0.373599\pi$
$20$	0	0
$21$	−3.90501	−0.852142
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.18840	−0.837680
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.08447	0.913197	0.456599	−	0.889673i	$-0.349068\pi$
$31$	5.08447	0.913197	0.456599	+	0.889673i	$0.349068\pi$
$32$	0	0
$33$	5.39807	0.939684
$34$	0	0
$35$	3.51798	0.594646
$36$	0	0
$37$	−3.79035	−0.623130	−0.311565	−	0.950225i	$-0.600853\pi$
$37$	−3.79035	−0.623130	−0.311565	+	0.950225i	$0.600853\pi$
$38$	0	0
$39$	2.18514	0.349903
$40$	0	0
$41$	8.82974	1.37897	0.689487	−	0.724298i	$-0.257836\pi$
$41$	8.82974	1.37897	0.689487	+	0.724298i	$0.257836\pi$
$42$	0	0
$43$	0.613535	0.0935632	0.0467816	−	0.998905i	$-0.485104\pi$
$43$	0.613535	0.0935632	0.0467816	+	0.998905i	$0.485104\pi$
$44$	0	0
$45$	0.900888	0.134296
$46$	0	0
$47$	1.02676	0.149768	0.0748839	−	0.997192i	$-0.476141\pi$
$47$	1.02676	0.149768	0.0748839	+	0.997192i	$0.476141\pi$
$48$	0	0
$49$	8.24908	1.17844
$50$	0	0
$51$	2.64370	0.370192
$52$	0	0
$53$	5.94186	0.816177	0.408088	−	0.912942i	$-0.366196\pi$
$53$	5.94186	0.816177	0.408088	+	0.912942i	$0.366196\pi$
$54$	0	0
$55$	−4.86306	−0.655735
$56$	0	0
$57$	−3.37158	−0.446577
$58$	0	0
$59$	−10.0058	−1.30265	−0.651323	−	0.758800i	$-0.725786\pi$
$59$	−10.0058	−1.30265	−0.651323	+	0.758800i	$0.725786\pi$
$60$	0	0
$61$	−3.09826	−0.396692	−0.198346	−	0.980132i	$-0.563557\pi$
$61$	−3.09826	−0.396692	−0.198346	+	0.980132i	$0.563557\pi$
$62$	0	0
$63$	3.90501	0.491985
$64$	0	0
$65$	−1.96857	−0.244171
$66$	0	0
$67$	−12.6836	−1.54954	−0.774771	−	0.632242i	$-0.782135\pi$
$67$	−12.6836	−1.54954	−0.774771	+	0.632242i	$0.782135\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−4.69605	−0.557318	−0.278659	−	0.960390i	$-0.589890\pi$
$71$	−4.69605	−0.557318	−0.278659	+	0.960390i	$0.589890\pi$
$72$	0	0
$73$	9.03175	1.05709	0.528543	−	0.848906i	$-0.322738\pi$
$73$	9.03175	1.05709	0.528543	+	0.848906i	$0.322738\pi$
$74$	0	0
$75$	4.18840	0.483635
$76$	0	0
$77$	−21.0795	−2.40223
$78$	0	0
$79$	−3.30647	−0.372007	−0.186004	−	0.982549i	$-0.559554\pi$
$79$	−3.30647	−0.372007	−0.186004	+	0.982549i	$0.559554\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.01449	−0.111354	−0.0556772	−	0.998449i	$-0.517732\pi$
$83$	−1.01449	−0.111354	−0.0556772	+	0.998449i	$0.517732\pi$
$84$	0	0
$85$	−2.38168	−0.258329
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−17.4884	−1.85376	−0.926882	−	0.375352i	$-0.877522\pi$
$89$	−17.4884	−1.85376	−0.926882	+	0.375352i	$0.877522\pi$
$90$	0	0
$91$	−8.53300	−0.894501
$92$	0	0
$93$	−5.08447	−0.527235
$94$	0	0
$95$	3.03742	0.311632
$96$	0	0
$97$	4.91409	0.498950	0.249475	−	0.968381i	$-0.419742\pi$
$97$	4.91409	0.498950	0.249475	+	0.968381i	$0.419742\pi$
$98$	0	0
$99$	−5.39807	−0.542527
$100$	0	0
$101$	−12.7246	−1.26615	−0.633074	−	0.774091i	$-0.718207\pi$
$101$	−12.7246	−1.26615	−0.633074	+	0.774091i	$0.718207\pi$
$102$	0	0
$103$	−8.55800	−0.843245	−0.421622	−	0.906772i	$-0.638539\pi$
$103$	−8.55800	−0.843245	−0.421622	+	0.906772i	$0.638539\pi$
$104$	0	0
$105$	−3.51798	−0.343319
$106$	0	0
$107$	−14.5746	−1.40898	−0.704489	−	0.709715i	$-0.748824\pi$
$107$	−14.5746	−1.40898	−0.704489	+	0.709715i	$0.748824\pi$
$108$	0	0
$109$	8.01651	0.767842	0.383921	−	0.923366i	$-0.374573\pi$
$109$	8.01651	0.767842	0.383921	+	0.923366i	$0.374573\pi$
$110$	0	0
$111$	3.79035	0.359764
$112$	0	0
$113$	−14.3051	−1.34571	−0.672856	−	0.739774i	$-0.734933\pi$
$113$	−14.3051	−1.34571	−0.672856	+	0.739774i	$0.734933\pi$
$114$	0	0
$115$	−0.900888	−0.0840083
$116$	0	0
$117$	−2.18514	−0.202017
$118$	0	0
$119$	−10.3237	−0.946368
$120$	0	0
$121$	18.1392	1.64902
$122$	0	0
$123$	−8.82974	−0.796151
$124$	0	0
$125$	−8.27772	−0.740382
$126$	0	0
$127$	14.7786	1.31139	0.655696	−	0.755025i	$-0.272375\pi$
$127$	14.7786	1.31139	0.655696	+	0.755025i	$0.272375\pi$
$128$	0	0
$129$	−0.613535	−0.0540187
$130$	0	0
$131$	0.115094	0.0100558	0.00502789	−	0.999987i	$-0.498400\pi$
$131$	0.115094	0.0100558	0.00502789	+	0.999987i	$0.498400\pi$
$132$	0	0
$133$	13.1660	1.14164
$134$	0	0
$135$	−0.900888	−0.0775361
$136$	0	0
$137$	7.19790	0.614958	0.307479	−	0.951555i	$-0.400515\pi$
$137$	7.19790	0.614958	0.307479	+	0.951555i	$0.400515\pi$
$138$	0	0
$139$	−4.33778	−0.367925	−0.183963	−	0.982933i	$-0.558893\pi$
$139$	−4.33778	−0.367925	−0.183963	+	0.982933i	$0.558893\pi$
$140$	0	0
$141$	−1.02676	−0.0864684
$142$	0	0
$143$	11.7956	0.986395
$144$	0	0
$145$	0.900888	0.0748147
$146$	0	0
$147$	−8.24908	−0.680373
$148$	0	0
$149$	−11.5364	−0.945099	−0.472550	−	0.881304i	$-0.656666\pi$
$149$	−11.5364	−0.945099	−0.472550	+	0.881304i	$0.656666\pi$
$150$	0	0
$151$	13.6790	1.11318	0.556592	−	0.830786i	$-0.312109\pi$
$151$	13.6790	1.11318	0.556592	+	0.830786i	$0.312109\pi$
$152$	0	0
$153$	−2.64370	−0.213730
$154$	0	0
$155$	4.58054	0.367918
$156$	0	0
$157$	0.0214029	0.00170814	0.000854068	−	1.00000i	$-0.499728\pi$
$157$	0.0214029	0.00170814	0.000854068		1.00000i	$0.499728\pi$
$158$	0	0
$159$	−5.94186	−0.471220
$160$	0	0
$161$	−3.90501	−0.307758
$162$	0	0
$163$	−0.555776	−0.0435318	−0.0217659	−	0.999763i	$-0.506929\pi$
$163$	−0.555776	−0.0435318	−0.0217659	+	0.999763i	$0.506929\pi$
$164$	0	0
$165$	4.86306	0.378589
$166$	0	0
$167$	6.90544	0.534359	0.267179	−	0.963647i	$-0.413908\pi$
$167$	6.90544	0.534359	0.267179	+	0.963647i	$0.413908\pi$
$168$	0	0
$169$	−8.22515	−0.632704
$170$	0	0
$171$	3.37158	0.257831
$172$	0	0
$173$	5.27785	0.401267	0.200634	−	0.979666i	$-0.435700\pi$
$173$	5.27785	0.401267	0.200634	+	0.979666i	$0.435700\pi$
$174$	0	0
$175$	−16.3557	−1.23638
$176$	0	0
$177$	10.0058	0.752083
$178$	0	0
$179$	−6.77665	−0.506511	−0.253255	−	0.967399i	$-0.581501\pi$
$179$	−6.77665	−0.506511	−0.253255	+	0.967399i	$0.581501\pi$
$180$	0	0
$181$	−22.0103	−1.63601	−0.818005	−	0.575211i	$-0.804920\pi$
$181$	−22.0103	−1.63601	−0.818005	+	0.575211i	$0.804920\pi$
$182$	0	0
$183$	3.09826	0.229030
$184$	0	0
$185$	−3.41469	−0.251053
$186$	0	0
$187$	14.2709	1.04359
$188$	0	0
$189$	−3.90501	−0.284047
$190$	0	0
$191$	11.7210	0.848103	0.424051	−	0.905638i	$-0.360608\pi$
$191$	11.7210	0.848103	0.424051	+	0.905638i	$0.360608\pi$
$192$	0	0
$193$	8.21648	0.591435	0.295717	−	0.955275i	$-0.404441\pi$
$193$	8.21648	0.591435	0.295717	+	0.955275i	$0.404441\pi$
$194$	0	0
$195$	1.96857	0.140972
$196$	0	0
$197$	0.510756	0.0363899	0.0181949	−	0.999834i	$-0.494208\pi$
$197$	0.510756	0.0363899	0.0181949	+	0.999834i	$0.494208\pi$
$198$	0	0
$199$	−9.23340	−0.654538	−0.327269	−	0.944931i	$-0.606128\pi$
$199$	−9.23340	−0.654538	−0.327269	+	0.944931i	$0.606128\pi$
$200$	0	0
$201$	12.6836	0.894629
$202$	0	0
$203$	3.90501	0.274078
$204$	0	0
$205$	7.95461	0.555574
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−18.2000	−1.25892
$210$	0	0
$211$	13.8507	0.953520	0.476760	−	0.879033i	$-0.341811\pi$
$211$	13.8507	0.953520	0.476760	+	0.879033i	$0.341811\pi$
$212$	0	0
$213$	4.69605	0.321768
$214$	0	0
$215$	0.552726	0.0376956
$216$	0	0
$217$	19.8549	1.34784
$218$	0	0
$219$	−9.03175	−0.610309
$220$	0	0
$221$	5.77686	0.388593
$222$	0	0
$223$	0.400408	0.0268133	0.0134066	−	0.999910i	$-0.495732\pi$
$223$	0.400408	0.0268133	0.0134066	+	0.999910i	$0.495732\pi$
$224$	0	0
$225$	−4.18840	−0.279227
$226$	0	0
$227$	6.20230	0.411661	0.205830	−	0.978588i	$-0.434010\pi$
$227$	6.20230	0.411661	0.205830	+	0.978588i	$0.434010\pi$
$228$	0	0
$229$	7.12470	0.470813	0.235407	−	0.971897i	$-0.424358\pi$
$229$	7.12470	0.470813	0.235407	+	0.971897i	$0.424358\pi$
$230$	0	0
$231$	21.0795	1.38693
$232$	0	0
$233$	−8.64191	−0.566150	−0.283075	−	0.959098i	$-0.591355\pi$
$233$	−8.64191	−0.566150	−0.283075	+	0.959098i	$0.591355\pi$
$234$	0	0
$235$	0.924992	0.0603398
$236$	0	0
$237$	3.30647	0.214778
$238$	0	0
$239$	15.2043	0.983486	0.491743	−	0.870741i	$-0.336360\pi$
$239$	15.2043	0.983486	0.491743	+	0.870741i	$0.336360\pi$
$240$	0	0
$241$	−23.0248	−1.48316	−0.741580	−	0.670865i	$-0.765923\pi$
$241$	−23.0248	−1.48316	−0.741580	+	0.670865i	$0.765923\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	7.43150	0.474781
$246$	0	0
$247$	−7.36739	−0.468776
$248$	0	0
$249$	1.01449	0.0642905
$250$	0	0
$251$	18.5440	1.17049	0.585245	−	0.810857i	$-0.300998\pi$
$251$	18.5440	1.17049	0.585245	+	0.810857i	$0.300998\pi$
$252$	0	0
$253$	5.39807	0.339374
$254$	0	0
$255$	2.38168	0.149146
$256$	0	0
$257$	−12.8335	−0.800531	−0.400265	−	0.916399i	$-0.631082\pi$
$257$	−12.8335	−0.800531	−0.400265	+	0.916399i	$0.631082\pi$
$258$	0	0
$259$	−14.8014	−0.919712
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−14.8682	−0.916812	−0.458406	−	0.888743i	$-0.651579\pi$
$263$	−14.8682	−0.916812	−0.458406	+	0.888743i	$0.651579\pi$
$264$	0	0
$265$	5.35295	0.328829
$266$	0	0
$267$	17.4884	1.07027
$268$	0	0
$269$	−15.3258	−0.934430	−0.467215	−	0.884144i	$-0.654743\pi$
$269$	−15.3258	−0.934430	−0.467215	+	0.884144i	$0.654743\pi$
$270$	0	0
$271$	−25.3597	−1.54049	−0.770246	−	0.637747i	$-0.779867\pi$
$271$	−25.3597	−1.54049	−0.770246	+	0.637747i	$0.779867\pi$
$272$	0	0
$273$	8.53300	0.516441
$274$	0	0
$275$	22.6093	1.36339
$276$	0	0
$277$	−14.5368	−0.873430	−0.436715	−	0.899600i	$-0.643858\pi$
$277$	−14.5368	−0.873430	−0.436715	+	0.899600i	$0.643858\pi$
$278$	0	0
$279$	5.08447	0.304399
$280$	0	0
$281$	−27.7945	−1.65808	−0.829040	−	0.559189i	$-0.811113\pi$
$281$	−27.7945	−1.65808	−0.829040	+	0.559189i	$0.811113\pi$
$282$	0	0
$283$	−4.01747	−0.238814	−0.119407	−	0.992845i	$-0.538099\pi$
$283$	−4.01747	−0.238814	−0.119407	+	0.992845i	$0.538099\pi$
$284$	0	0
$285$	−3.03742	−0.179921
$286$	0	0
$287$	34.4802	2.03530
$288$	0	0
$289$	−10.0109	−0.588875
$290$	0	0
$291$	−4.91409	−0.288069
$292$	0	0
$293$	−32.9948	−1.92757	−0.963787	−	0.266673i	$-0.914076\pi$
$293$	−32.9948	−1.92757	−0.963787	+	0.266673i	$0.914076\pi$
$294$	0	0
$295$	−9.01412	−0.524823
$296$	0	0
$297$	5.39807	0.313228
$298$	0	0
$299$	2.18514	0.126370
$300$	0	0
$301$	2.39586	0.138095
$302$	0	0
$303$	12.7246	0.731011
$304$	0	0
$305$	−2.79119	−0.159823
$306$	0	0
$307$	−9.56599	−0.545960	−0.272980	−	0.962020i	$-0.588009\pi$
$307$	−9.56599	−0.545960	−0.272980	+	0.962020i	$0.588009\pi$
$308$	0	0
$309$	8.55800	0.486848
$310$	0	0
$311$	4.28315	0.242875	0.121438	−	0.992599i	$-0.461250\pi$
$311$	4.28315	0.242875	0.121438	+	0.992599i	$0.461250\pi$
$312$	0	0
$313$	0.673397	0.0380626	0.0190313	−	0.999819i	$-0.493942\pi$
$313$	0.673397	0.0380626	0.0190313	+	0.999819i	$0.493942\pi$
$314$	0	0
$315$	3.51798	0.198215
$316$	0	0
$317$	−17.0792	−0.959263	−0.479632	−	0.877470i	$-0.659230\pi$
$317$	−17.0792	−0.959263	−0.479632	+	0.877470i	$0.659230\pi$
$318$	0	0
$319$	−5.39807	−0.302234
$320$	0	0
$321$	14.5746	0.813474
$322$	0	0
$323$	−8.91344	−0.495957
$324$	0	0
$325$	9.15225	0.507676
$326$	0	0
$327$	−8.01651	−0.443314
$328$	0	0
$329$	4.00949	0.221050
$330$	0	0
$331$	7.39283	0.406347	0.203173	−	0.979143i	$-0.434875\pi$
$331$	7.39283	0.406347	0.203173	+	0.979143i	$0.434875\pi$
$332$	0	0
$333$	−3.79035	−0.207710
$334$	0	0
$335$	−11.4265	−0.624294
$336$	0	0
$337$	14.4508	0.787186	0.393593	−	0.919285i	$-0.371232\pi$
$337$	14.4508	0.787186	0.393593	+	0.919285i	$0.371232\pi$
$338$	0	0
$339$	14.3051	0.776947
$340$	0	0
$341$	−27.4463	−1.48630
$342$	0	0
$343$	4.87767	0.263369
$344$	0	0
$345$	0.900888	0.0485022
$346$	0	0
$347$	16.0698	0.862669	0.431335	−	0.902192i	$-0.358043\pi$
$347$	16.0698	0.862669	0.431335	+	0.902192i	$0.358043\pi$
$348$	0	0
$349$	17.8796	0.957075	0.478537	−	0.878067i	$-0.341167\pi$
$349$	17.8796	0.957075	0.478537	+	0.878067i	$0.341167\pi$
$350$	0	0
$351$	2.18514	0.116634
$352$	0	0
$353$	−31.2236	−1.66187	−0.830933	−	0.556373i	$-0.812193\pi$
$353$	−31.2236	−1.66187	−0.830933	+	0.556373i	$0.812193\pi$
$354$	0	0
$355$	−4.23061	−0.224538
$356$	0	0
$357$	10.3237	0.546386
$358$	0	0
$359$	−23.6558	−1.24851	−0.624253	−	0.781222i	$-0.714596\pi$
$359$	−23.6558	−1.24851	−0.624253	+	0.781222i	$0.714596\pi$
$360$	0	0
$361$	−7.63244	−0.401708
$362$	0	0
$363$	−18.1392	−0.952062
$364$	0	0
$365$	8.13660	0.425889
$366$	0	0
$367$	30.1044	1.57144	0.785719	−	0.618583i	$-0.212293\pi$
$367$	30.1044	1.57144	0.785719	+	0.618583i	$0.212293\pi$
$368$	0	0
$369$	8.82974	0.459658
$370$	0	0
$371$	23.2030	1.20464
$372$	0	0
$373$	−25.6816	−1.32974	−0.664871	−	0.746958i	$-0.731513\pi$
$373$	−25.6816	−1.32974	−0.664871	+	0.746958i	$0.731513\pi$
$374$	0	0
$375$	8.27772	0.427460
$376$	0	0
$377$	−2.18514	−0.112541
$378$	0	0
$379$	24.8901	1.27852	0.639260	−	0.768991i	$-0.279241\pi$
$379$	24.8901	1.27852	0.639260	+	0.768991i	$0.279241\pi$
$380$	0	0
$381$	−14.7786	−0.757133
$382$	0	0
$383$	5.32682	0.272188	0.136094	−	0.990696i	$-0.456545\pi$
$383$	5.32682	0.272188	0.136094	+	0.990696i	$0.456545\pi$
$384$	0	0
$385$	−18.9903	−0.967835
$386$	0	0
$387$	0.613535	0.0311877
$388$	0	0
$389$	31.6443	1.60443	0.802216	−	0.597034i	$-0.203654\pi$
$389$	31.6443	1.60443	0.802216	+	0.597034i	$0.203654\pi$
$390$	0	0
$391$	2.64370	0.133697
$392$	0	0
$393$	−0.115094	−0.00580571
$394$	0	0
$395$	−2.97876	−0.149878
$396$	0	0
$397$	−1.70334	−0.0854881	−0.0427441	−	0.999086i	$-0.513610\pi$
$397$	−1.70334	−0.0854881	−0.0427441	+	0.999086i	$0.513610\pi$
$398$	0	0
$399$	−13.1660	−0.659127
$400$	0	0
$401$	24.1375	1.20537	0.602684	−	0.797980i	$-0.294098\pi$
$401$	24.1375	1.20537	0.602684	+	0.797980i	$0.294098\pi$
$402$	0	0
$403$	−11.1103	−0.553443
$404$	0	0
$405$	0.900888	0.0447655
$406$	0	0
$407$	20.4606	1.01419
$408$	0	0
$409$	−24.7975	−1.22616	−0.613078	−	0.790022i	$-0.710069\pi$
$409$	−24.7975	−1.22616	−0.613078	+	0.790022i	$0.710069\pi$
$410$	0	0
$411$	−7.19790	−0.355046
$412$	0	0
$413$	−39.0728	−1.92265
$414$	0	0
$415$	−0.913939	−0.0448635
$416$	0	0
$417$	4.33778	0.212422
$418$	0	0
$419$	23.5959	1.15274	0.576368	−	0.817190i	$-0.304470\pi$
$419$	23.5959	1.15274	0.576368	+	0.817190i	$0.304470\pi$
$420$	0	0
$421$	−34.6655	−1.68949	−0.844746	−	0.535167i	$-0.820249\pi$
$421$	−34.6655	−1.68949	−0.844746	+	0.535167i	$0.820249\pi$
$422$	0	0
$423$	1.02676	0.0499226
$424$	0	0
$425$	11.0729	0.537113
$426$	0	0
$427$	−12.0987	−0.585499
$428$	0	0
$429$	−11.7956	−0.569495
$430$	0	0
$431$	−2.67396	−0.128800	−0.0644001	−	0.997924i	$-0.520513\pi$
$431$	−2.67396	−0.128800	−0.0644001	+	0.997924i	$0.520513\pi$
$432$	0	0
$433$	26.4396	1.27061	0.635304	−	0.772262i	$-0.280875\pi$
$433$	26.4396	1.27061	0.635304	+	0.772262i	$0.280875\pi$
$434$	0	0
$435$	−0.900888	−0.0431943
$436$	0	0
$437$	−3.37158	−0.161285
$438$	0	0
$439$	−6.00149	−0.286436	−0.143218	−	0.989691i	$-0.545745\pi$
$439$	−6.00149	−0.286436	−0.143218	+	0.989691i	$0.545745\pi$
$440$	0	0
$441$	8.24908	0.392813
$442$	0	0
$443$	22.8584	1.08604	0.543019	−	0.839721i	$-0.317281\pi$
$443$	22.8584	1.08604	0.543019	+	0.839721i	$0.317281\pi$
$444$	0	0
$445$	−15.7551	−0.746862
$446$	0	0
$447$	11.5364	0.545653
$448$	0	0
$449$	9.28471	0.438173	0.219086	−	0.975705i	$-0.429692\pi$
$449$	9.28471	0.438173	0.219086	+	0.975705i	$0.429692\pi$
$450$	0	0
$451$	−47.6636	−2.24439
$452$	0	0
$453$	−13.6790	−0.642697
$454$	0	0
$455$	−7.68728	−0.360385
$456$	0	0
$457$	11.6566	0.545275	0.272637	−	0.962117i	$-0.412104\pi$
$457$	11.6566	0.545275	0.272637	+	0.962117i	$0.412104\pi$
$458$	0	0
$459$	2.64370	0.123397
$460$	0	0
$461$	14.0282	0.653359	0.326680	−	0.945135i	$-0.394070\pi$
$461$	14.0282	0.653359	0.326680	+	0.945135i	$0.394070\pi$
$462$	0	0
$463$	37.9064	1.76166	0.880829	−	0.473434i	$-0.156986\pi$
$463$	37.9064	1.76166	0.880829	+	0.473434i	$0.156986\pi$
$464$	0	0
$465$	−4.58054	−0.212417
$466$	0	0
$467$	−25.5002	−1.18001	−0.590004	−	0.807400i	$-0.700874\pi$
$467$	−25.5002	−1.18001	−0.590004	+	0.807400i	$0.700874\pi$
$468$	0	0
$469$	−49.5294	−2.28705
$470$	0	0
$471$	−0.0214029	−0.000986192 0
$472$	0	0
$473$	−3.31191	−0.152282
$474$	0	0
$475$	−14.1215	−0.647940
$476$	0	0
$477$	5.94186	0.272059
$478$	0	0
$479$	−36.0071	−1.64521	−0.822603	−	0.568616i	$-0.807479\pi$
$479$	−36.0071	−1.64521	−0.822603	+	0.568616i	$0.807479\pi$
$480$	0	0
$481$	8.28247	0.377648
$482$	0	0
$483$	3.90501	0.177684
$484$	0	0
$485$	4.42704	0.201022
$486$	0	0
$487$	−20.6281	−0.934747	−0.467373	−	0.884060i	$-0.654800\pi$
$487$	−20.6281	−0.934747	−0.467373	+	0.884060i	$0.654800\pi$
$488$	0	0
$489$	0.555776	0.0251331
$490$	0	0
$491$	−16.8265	−0.759370	−0.379685	−	0.925116i	$-0.623968\pi$
$491$	−16.8265	−0.759370	−0.379685	+	0.925116i	$0.623968\pi$
$492$	0	0
$493$	−2.64370	−0.119066
$494$	0	0
$495$	−4.86306	−0.218578
$496$	0	0
$497$	−18.3381	−0.822576
$498$	0	0
$499$	39.1885	1.75432	0.877160	−	0.480198i	$-0.159435\pi$
$499$	39.1885	1.75432	0.877160	+	0.480198i	$0.159435\pi$
$500$	0	0
$501$	−6.90544	−0.308512
$502$	0	0
$503$	−33.7518	−1.50492	−0.752458	−	0.658640i	$-0.771132\pi$
$503$	−33.7518	−1.50492	−0.752458	+	0.658640i	$0.771132\pi$
$504$	0	0
$505$	−11.4635	−0.510118
$506$	0	0
$507$	8.22515	0.365292
$508$	0	0
$509$	17.6294	0.781407	0.390704	−	0.920517i	$-0.372232\pi$
$509$	17.6294	0.781407	0.390704	+	0.920517i	$0.372232\pi$
$510$	0	0
$511$	35.2690	1.56021
$512$	0	0
$513$	−3.37158	−0.148859
$514$	0	0
$515$	−7.70980	−0.339734
$516$	0	0
$517$	−5.54250	−0.243759
$518$	0	0
$519$	−5.27785	−0.231672
$520$	0	0
$521$	−17.1971	−0.753418	−0.376709	−	0.926332i	$-0.622944\pi$
$521$	−17.1971	−0.753418	−0.376709	+	0.926332i	$0.622944\pi$
$522$	0	0
$523$	18.5539	0.811304	0.405652	−	0.914028i	$-0.367044\pi$
$523$	18.5539	0.811304	0.405652	+	0.914028i	$0.367044\pi$
$524$	0	0
$525$	16.3557	0.713823
$526$	0	0
$527$	−13.4418	−0.585534
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.0058	−0.434215
$532$	0	0
$533$	−19.2942	−0.835726
$534$	0	0
$535$	−13.1301	−0.567662
$536$	0	0
$537$	6.77665	0.292434
$538$	0	0
$539$	−44.5292	−1.91801
$540$	0	0
$541$	3.64277	0.156615	0.0783074	−	0.996929i	$-0.475048\pi$
$541$	3.64277	0.156615	0.0783074	+	0.996929i	$0.475048\pi$
$542$	0	0
$543$	22.0103	0.944551
$544$	0	0
$545$	7.22198	0.309355
$546$	0	0
$547$	−0.827186	−0.0353679	−0.0176840	−	0.999844i	$-0.505629\pi$
$547$	−0.827186	−0.0353679	−0.0176840	+	0.999844i	$0.505629\pi$
$548$	0	0
$549$	−3.09826	−0.132231
$550$	0	0
$551$	3.37158	0.143634
$552$	0	0
$553$	−12.9118	−0.549065
$554$	0	0
$555$	3.41469	0.144945
$556$	0	0
$557$	−31.7654	−1.34594	−0.672972	−	0.739668i	$-0.734983\pi$
$557$	−31.7654	−1.34594	−0.672972	+	0.739668i	$0.734983\pi$
$558$	0	0
$559$	−1.34066	−0.0567039
$560$	0	0
$561$	−14.2709	−0.602517
$562$	0	0
$563$	−21.1838	−0.892789	−0.446395	−	0.894836i	$-0.647292\pi$
$563$	−21.1838	−0.892789	−0.446395	+	0.894836i	$0.647292\pi$
$564$	0	0
$565$	−12.8873	−0.542173
$566$	0	0
$567$	3.90501	0.163995
$568$	0	0
$569$	11.4066	0.478188	0.239094	−	0.970996i	$-0.423150\pi$
$569$	11.4066	0.478188	0.239094	+	0.970996i	$0.423150\pi$
$570$	0	0
$571$	−20.0422	−0.838740	−0.419370	−	0.907815i	$-0.637749\pi$
$571$	−20.0422	−0.838740	−0.419370	+	0.907815i	$0.637749\pi$
$572$	0	0
$573$	−11.7210	−0.489652
$574$	0	0
$575$	4.18840	0.174668
$576$	0	0
$577$	35.4444	1.47557	0.737785	−	0.675036i	$-0.235872\pi$
$577$	35.4444	1.47557	0.737785	+	0.675036i	$0.235872\pi$
$578$	0	0
$579$	−8.21648	−0.341465
$580$	0	0
$581$	−3.96158	−0.164354
$582$	0	0
$583$	−32.0746	−1.32839
$584$	0	0
$585$	−1.96857	−0.0813904
$586$	0	0
$587$	−18.2013	−0.751249	−0.375625	−	0.926772i	$-0.622572\pi$
$587$	−18.2013	−0.751249	−0.375625	+	0.926772i	$0.622572\pi$
$588$	0	0
$589$	17.1427	0.706352
$590$	0	0
$591$	−0.510756	−0.0210097
$592$	0	0
$593$	32.2026	1.32240	0.661200	−	0.750209i	$-0.270047\pi$
$593$	32.2026	1.32240	0.661200	+	0.750209i	$0.270047\pi$
$594$	0	0
$595$	−9.30046	−0.381282
$596$	0	0
$597$	9.23340	0.377898
$598$	0	0
$599$	9.58646	0.391692	0.195846	−	0.980635i	$-0.437255\pi$
$599$	9.58646	0.391692	0.195846	+	0.980635i	$0.437255\pi$
$600$	0	0
$601$	−11.3781	−0.464122	−0.232061	−	0.972701i	$-0.574547\pi$
$601$	−11.3781	−0.464122	−0.232061	+	0.972701i	$0.574547\pi$
$602$	0	0
$603$	−12.6836	−0.516514
$604$	0	0
$605$	16.3414	0.664372
$606$	0	0
$607$	−42.0701	−1.70757	−0.853787	−	0.520623i	$-0.825700\pi$
$607$	−42.0701	−1.70757	−0.853787	+	0.520623i	$0.825700\pi$
$608$	0	0
$609$	−3.90501	−0.158239
$610$	0	0
$611$	−2.24361	−0.0907667
$612$	0	0
$613$	−25.5846	−1.03335	−0.516676	−	0.856181i	$-0.672831\pi$
$613$	−25.5846	−1.03335	−0.516676	+	0.856181i	$0.672831\pi$
$614$	0	0
$615$	−7.95461	−0.320761
$616$	0	0
$617$	−21.8215	−0.878502	−0.439251	−	0.898364i	$-0.644756\pi$
$617$	−21.8215	−0.878502	−0.439251	+	0.898364i	$0.644756\pi$
$618$	0	0
$619$	23.7623	0.955087	0.477544	−	0.878608i	$-0.341527\pi$
$619$	23.7623	0.955087	0.477544	+	0.878608i	$0.341527\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−68.2922	−2.73607
$624$	0	0
$625$	13.4847	0.539388
$626$	0	0
$627$	18.2000	0.726840
$628$	0	0
$629$	10.0205	0.399545
$630$	0	0
$631$	19.0191	0.757138	0.378569	−	0.925573i	$-0.376416\pi$
$631$	19.0191	0.757138	0.378569	+	0.925573i	$0.376416\pi$
$632$	0	0
$633$	−13.8507	−0.550515
$634$	0	0
$635$	13.3139	0.528346
$636$	0	0
$637$	−18.0254	−0.714193
$638$	0	0
$639$	−4.69605	−0.185773
$640$	0	0
$641$	−11.2100	−0.442767	−0.221383	−	0.975187i	$-0.571057\pi$
$641$	−11.2100	−0.442767	−0.221383	+	0.975187i	$0.571057\pi$
$642$	0	0
$643$	−22.7058	−0.895430	−0.447715	−	0.894176i	$-0.647762\pi$
$643$	−22.7058	−0.895430	−0.447715	+	0.894176i	$0.647762\pi$
$644$	0	0
$645$	−0.552726	−0.0217636
$646$	0	0
$647$	−13.7388	−0.540127	−0.270064	−	0.962842i	$-0.587045\pi$
$647$	−13.7388	−0.540127	−0.270064	+	0.962842i	$0.587045\pi$
$648$	0	0
$649$	54.0121	2.12016
$650$	0	0
$651$	−19.8549	−0.778174
$652$	0	0
$653$	−3.31245	−0.129626	−0.0648130	−	0.997897i	$-0.520645\pi$
$653$	−3.31245	−0.129626	−0.0648130	+	0.997897i	$0.520645\pi$
$654$	0	0
$655$	0.103687	0.00405137
$656$	0	0
$657$	9.03175	0.352362
$658$	0	0
$659$	4.02358	0.156736	0.0783681	−	0.996924i	$-0.475029\pi$
$659$	4.02358	0.156736	0.0783681	+	0.996924i	$0.475029\pi$
$660$	0	0
$661$	−15.0724	−0.586249	−0.293125	−	0.956074i	$-0.594695\pi$
$661$	−15.0724	−0.586249	−0.293125	+	0.956074i	$0.594695\pi$
$662$	0	0
$663$	−5.77686	−0.224355
$664$	0	0
$665$	11.8611	0.459955
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−0.400408	−0.0154807
$670$	0	0
$671$	16.7246	0.645648
$672$	0	0
$673$	−15.0200	−0.578978	−0.289489	−	0.957181i	$-0.593485\pi$
$673$	−15.0200	−0.578978	−0.289489	+	0.957181i	$0.593485\pi$
$674$	0	0
$675$	4.18840	0.161212
$676$	0	0
$677$	−20.0315	−0.769872	−0.384936	−	0.922943i	$-0.625776\pi$
$677$	−20.0315	−0.769872	−0.384936	+	0.922943i	$0.625776\pi$
$678$	0	0
$679$	19.1895	0.736427
$680$	0	0
$681$	−6.20230	−0.237673
$682$	0	0
$683$	32.3988	1.23971	0.619853	−	0.784718i	$-0.287192\pi$
$683$	32.3988	1.23971	0.619853	+	0.784718i	$0.287192\pi$
$684$	0	0
$685$	6.48450	0.247760
$686$	0	0
$687$	−7.12470	−0.271824
$688$	0	0
$689$	−12.9838	−0.494644
$690$	0	0
$691$	−31.9042	−1.21369	−0.606847	−	0.794818i	$-0.707566\pi$
$691$	−31.9042	−1.21369	−0.606847	+	0.794818i	$0.707566\pi$
$692$	0	0
$693$	−21.0795	−0.800745
$694$	0	0
$695$	−3.90785	−0.148233
$696$	0	0
$697$	−23.3432	−0.884185
$698$	0	0
$699$	8.64191	0.326867
$700$	0	0
$701$	−41.2753	−1.55894	−0.779472	−	0.626437i	$-0.784513\pi$
$701$	−41.2753	−1.55894	−0.779472	+	0.626437i	$0.784513\pi$
$702$	0	0
$703$	−12.7795	−0.481987
$704$	0	0
$705$	−0.924992	−0.0348372
$706$	0	0
$707$	−49.6898	−1.86878
$708$	0	0
$709$	−11.4863	−0.431379	−0.215689	−	0.976462i	$-0.569200\pi$
$709$	−11.4863	−0.431379	−0.215689	+	0.976462i	$0.569200\pi$
$710$	0	0
$711$	−3.30647	−0.124002
$712$	0	0
$713$	−5.08447	−0.190415
$714$	0	0
$715$	10.6265	0.397408
$716$	0	0
$717$	−15.2043	−0.567816
$718$	0	0
$719$	45.5003	1.69687	0.848437	−	0.529297i	$-0.177544\pi$
$719$	45.5003	1.69687	0.848437	+	0.529297i	$0.177544\pi$
$720$	0	0
$721$	−33.4190	−1.24459
$722$	0	0
$723$	23.0248	0.856302
$724$	0	0
$725$	−4.18840	−0.155553
$726$	0	0
$727$	2.52414	0.0936152	0.0468076	−	0.998904i	$-0.485095\pi$
$727$	2.52414	0.0936152	0.0468076	+	0.998904i	$0.485095\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.62200	−0.0599918
$732$	0	0
$733$	−22.9871	−0.849049	−0.424524	−	0.905417i	$-0.639559\pi$
$733$	−22.9871	−0.849049	−0.424524	+	0.905417i	$0.639559\pi$
$734$	0	0
$735$	−7.43150	−0.274115
$736$	0	0
$737$	68.4668	2.52201
$738$	0	0
$739$	28.4478	1.04647	0.523234	−	0.852189i	$-0.324725\pi$
$739$	28.4478	1.04647	0.523234	+	0.852189i	$0.324725\pi$
$740$	0	0
$741$	7.36739	0.270648
$742$	0	0
$743$	9.46727	0.347321	0.173660	−	0.984806i	$-0.444441\pi$
$743$	9.46727	0.347321	0.173660	+	0.984806i	$0.444441\pi$
$744$	0	0
$745$	−10.3930	−0.380771
$746$	0	0
$747$	−1.01449	−0.0371181
$748$	0	0
$749$	−56.9138	−2.07959
$750$	0	0
$751$	−11.2455	−0.410355	−0.205177	−	0.978725i	$-0.565777\pi$
$751$	−11.2455	−0.410355	−0.205177	+	0.978725i	$0.565777\pi$
$752$	0	0
$753$	−18.5440	−0.675782
$754$	0	0
$755$	12.3233	0.448490
$756$	0	0
$757$	5.68039	0.206457	0.103229	−	0.994658i	$-0.467083\pi$
$757$	5.68039	0.206457	0.103229	+	0.994658i	$0.467083\pi$
$758$	0	0
$759$	−5.39807	−0.195938
$760$	0	0
$761$	−14.1168	−0.511734	−0.255867	−	0.966712i	$-0.582361\pi$
$761$	−14.1168	−0.511734	−0.255867	+	0.966712i	$0.582361\pi$
$762$	0	0
$763$	31.3045	1.13330
$764$	0	0
$765$	−2.38168	−0.0861097
$766$	0	0
$767$	21.8641	0.789469
$768$	0	0
$769$	−1.36141	−0.0490938	−0.0245469	−	0.999699i	$-0.507814\pi$
$769$	−1.36141	−0.0490938	−0.0245469	+	0.999699i	$0.507814\pi$
$770$	0	0
$771$	12.8335	0.462187
$772$	0	0
$773$	−36.4036	−1.30935	−0.654673	−	0.755912i	$-0.727194\pi$
$773$	−36.4036	−1.30935	−0.654673	+	0.755912i	$0.727194\pi$
$774$	0	0
$775$	−21.2958	−0.764967
$776$	0	0
$777$	14.8014	0.530996
$778$	0	0
$779$	29.7702	1.06663
$780$	0	0
$781$	25.3496	0.907080
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	0.0192816	0.000688190 0
$786$	0	0
$787$	−53.5148	−1.90760	−0.953799	−	0.300447i	$-0.902864\pi$
$787$	−53.5148	−1.90760	−0.953799	+	0.300447i	$0.902864\pi$
$788$	0	0
$789$	14.8682	0.529322
$790$	0	0
$791$	−55.8616	−1.98621
$792$	0	0
$793$	6.77015	0.240415
$794$	0	0
$795$	−5.35295	−0.189850
$796$	0	0
$797$	43.8576	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$797$	43.8576	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$798$	0	0
$799$	−2.71443	−0.0960297
$800$	0	0
$801$	−17.4884	−0.617921
$802$	0	0
$803$	−48.7541	−1.72049
$804$	0	0
$805$	−3.51798	−0.123992
$806$	0	0
$807$	15.3258	0.539493
$808$	0	0
$809$	−20.4766	−0.719920	−0.359960	−	0.932968i	$-0.617210\pi$
$809$	−20.4766	−0.719920	−0.359960	+	0.932968i	$0.617210\pi$
$810$	0	0
$811$	41.0750	1.44234	0.721170	−	0.692758i	$-0.243605\pi$
$811$	41.0750	1.44234	0.721170	+	0.692758i	$0.243605\pi$
$812$	0	0
$813$	25.3597	0.889403
$814$	0	0
$815$	−0.500692	−0.0175385
$816$	0	0
$817$	2.06858	0.0723705
$818$	0	0
$819$	−8.53300	−0.298167
$820$	0	0
$821$	−34.0314	−1.18770	−0.593852	−	0.804574i	$-0.702393\pi$
$821$	−34.0314	−1.18770	−0.593852	+	0.804574i	$0.702393\pi$
$822$	0	0
$823$	−54.7566	−1.90870	−0.954348	−	0.298698i	$-0.903448\pi$
$823$	−54.7566	−1.90870	−0.954348	+	0.298698i	$0.903448\pi$
$824$	0	0
$825$	−22.6093	−0.787155
$826$	0	0
$827$	4.40663	0.153233	0.0766167	−	0.997061i	$-0.475588\pi$
$827$	4.40663	0.153233	0.0766167	+	0.997061i	$0.475588\pi$
$828$	0	0
$829$	−27.7917	−0.965244	−0.482622	−	0.875829i	$-0.660315\pi$
$829$	−27.7917	−0.965244	−0.482622	+	0.875829i	$0.660315\pi$
$830$	0	0
$831$	14.5368	0.504275
$832$	0	0
$833$	−21.8081	−0.755605
$834$	0	0
$835$	6.22103	0.215287
$836$	0	0
$837$	−5.08447	−0.175745
$838$	0	0
$839$	35.7491	1.23420	0.617098	−	0.786886i	$-0.288308\pi$
$839$	35.7491	1.23420	0.617098	+	0.786886i	$0.288308\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	27.7945	0.957293
$844$	0	0
$845$	−7.40994	−0.254910
$846$	0	0
$847$	70.8337	2.43388
$848$	0	0
$849$	4.01747	0.137879
$850$	0	0
$851$	3.79035	0.129932
$852$	0	0
$853$	20.2840	0.694509	0.347255	−	0.937771i	$-0.387114\pi$
$853$	20.2840	0.694509	0.347255	+	0.937771i	$0.387114\pi$
$854$	0	0
$855$	3.03742	0.103877
$856$	0	0
$857$	11.5100	0.393175	0.196588	−	0.980486i	$-0.437014\pi$
$857$	11.5100	0.393175	0.196588	+	0.980486i	$0.437014\pi$
$858$	0	0
$859$	−44.2221	−1.50884	−0.754419	−	0.656393i	$-0.772081\pi$
$859$	−44.2221	−1.50884	−0.754419	+	0.656393i	$0.772081\pi$
$860$	0	0
$861$	−34.4802	−1.17508
$862$	0	0
$863$	−26.5666	−0.904337	−0.452169	−	0.891932i	$-0.649349\pi$
$863$	−26.5666	−0.904337	−0.452169	+	0.891932i	$0.649349\pi$
$864$	0	0
$865$	4.75475	0.161666
$866$	0	0
$867$	10.0109	0.339987
$868$	0	0
$869$	17.8486	0.605471
$870$	0	0
$871$	27.7154	0.939100
$872$	0	0
$873$	4.91409	0.166317
$874$	0	0
$875$	−32.3246	−1.09277
$876$	0	0
$877$	23.1214	0.780754	0.390377	−	0.920655i	$-0.372345\pi$
$877$	23.1214	0.780754	0.390377	+	0.920655i	$0.372345\pi$
$878$	0	0
$879$	32.9948	1.11289
$880$	0	0
$881$	21.4796	0.723666	0.361833	−	0.932243i	$-0.382151\pi$
$881$	21.4796	0.723666	0.361833	+	0.932243i	$0.382151\pi$
$882$	0	0
$883$	−11.9649	−0.402650	−0.201325	−	0.979525i	$-0.564525\pi$
$883$	−11.9649	−0.402650	−0.201325	+	0.979525i	$0.564525\pi$
$884$	0	0
$885$	9.01412	0.303006
$886$	0	0
$887$	43.8213	1.47138	0.735688	−	0.677320i	$-0.236859\pi$
$887$	43.8213	1.47138	0.735688	+	0.677320i	$0.236859\pi$
$888$	0	0
$889$	57.7107	1.93556
$890$	0	0
$891$	−5.39807	−0.180842
$892$	0	0
$893$	3.46179	0.115844
$894$	0	0
$895$	−6.10501	−0.204068
$896$	0	0
$897$	−2.18514	−0.0729598
$898$	0	0
$899$	5.08447	0.169576
$900$	0	0
$901$	−15.7085	−0.523325
$902$	0	0
$903$	−2.39586	−0.0797291
$904$	0	0
$905$	−19.8288	−0.659131
$906$	0	0
$907$	37.5669	1.24739	0.623694	−	0.781668i	$-0.285631\pi$
$907$	37.5669	1.24739	0.623694	+	0.781668i	$0.285631\pi$
$908$	0	0
$909$	−12.7246	−0.422050
$910$	0	0
$911$	−6.78171	−0.224688	−0.112344	−	0.993669i	$-0.535836\pi$
$911$	−6.78171	−0.224688	−0.112344	+	0.993669i	$0.535836\pi$
$912$	0	0
$913$	5.47627	0.181238
$914$	0	0
$915$	2.79119	0.0922738
$916$	0	0
$917$	0.449442	0.0148419
$918$	0	0
$919$	−35.3208	−1.16513	−0.582563	−	0.812786i	$-0.697950\pi$
$919$	−35.3208	−1.16513	−0.582563	+	0.812786i	$0.697950\pi$
$920$	0	0
$921$	9.56599	0.315210
$922$	0	0
$923$	10.2615	0.337762
$924$	0	0
$925$	15.8755	0.521984
$926$	0	0
$927$	−8.55800	−0.281082
$928$	0	0
$929$	−25.8804	−0.849108	−0.424554	−	0.905403i	$-0.639569\pi$
$929$	−25.8804	−0.849108	−0.424554	+	0.905403i	$0.639569\pi$
$930$	0	0
$931$	27.8124	0.911516
$932$	0	0
$933$	−4.28315	−0.140224
$934$	0	0
$935$	12.8565	0.420451
$936$	0	0
$937$	29.1283	0.951582	0.475791	−	0.879558i	$-0.342162\pi$
$937$	29.1283	0.951582	0.475791	+	0.879558i	$0.342162\pi$
$938$	0	0
$939$	−0.673397	−0.0219755
$940$	0	0
$941$	−2.68696	−0.0875925	−0.0437963	−	0.999040i	$-0.513945\pi$
$941$	−2.68696	−0.0875925	−0.0437963	+	0.999040i	$0.513945\pi$
$942$	0	0
$943$	−8.82974	−0.287536
$944$	0	0
$945$	−3.51798	−0.114440
$946$	0	0
$947$	39.4768	1.28282	0.641412	−	0.767197i	$-0.278349\pi$
$947$	39.4768	1.28282	0.641412	+	0.767197i	$0.278349\pi$
$948$	0	0
$949$	−19.7357	−0.640647
$950$	0	0
$951$	17.0792	0.553831
$952$	0	0
$953$	19.7605	0.640106	0.320053	−	0.947400i	$-0.396299\pi$
$953$	19.7605	0.640106	0.320053	+	0.947400i	$0.396299\pi$
$954$	0	0
$955$	10.5593	0.341692
$956$	0	0
$957$	5.39807	0.174495
$958$	0	0
$959$	28.1078	0.907649
$960$	0	0
$961$	−5.14819	−0.166071
$962$	0	0
$963$	−14.5746	−0.469659
$964$	0	0
$965$	7.40213	0.238283
$966$	0	0
$967$	55.6598	1.78990	0.894949	−	0.446169i	$-0.147212\pi$
$967$	55.6598	1.78990	0.894949	+	0.446169i	$0.147212\pi$
$968$	0	0
$969$	8.91344	0.286341
$970$	0	0
$971$	−21.7277	−0.697276	−0.348638	−	0.937257i	$-0.613356\pi$
$971$	−21.7277	−0.697276	−0.348638	+	0.937257i	$0.613356\pi$
$972$	0	0
$973$	−16.9390	−0.543041
$974$	0	0
$975$	−9.15225	−0.293107
$976$	0	0
$977$	8.02371	0.256701	0.128351	−	0.991729i	$-0.459032\pi$
$977$	8.02371	0.256701	0.128351	+	0.991729i	$0.459032\pi$
$978$	0	0
$979$	94.4036	3.01715
$980$	0	0
$981$	8.01651	0.255947
$982$	0	0
$983$	35.6083	1.13573	0.567865	−	0.823122i	$-0.307770\pi$
$983$	35.6083	1.13573	0.567865	+	0.823122i	$0.307770\pi$
$984$	0	0
$985$	0.460134	0.0146611
$986$	0	0
$987$	−4.00949	−0.127623
$988$	0	0
$989$	−0.613535	−0.0195093
$990$	0	0
$991$	27.9781	0.888755	0.444377	−	0.895840i	$-0.353425\pi$
$991$	27.9781	0.888755	0.444377	+	0.895840i	$0.353425\pi$
$992$	0	0
$993$	−7.39283	−0.234604
$994$	0	0
$995$	−8.31826	−0.263707
$996$	0	0
$997$	57.5748	1.82341	0.911707	−	0.410842i	$-0.134765\pi$
$997$	57.5748	1.82341	0.911707	+	0.410842i	$0.134765\pi$
$998$	0	0
$999$	3.79035	0.119921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.e.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.e.1.6	✓	9	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.900888 q^{5} +3.90501 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.900888 q^{5} +3.90501 q^{7} +1.00000 q^{9} -5.39807 q^{11} -2.18514 q^{13} -0.900888 q^{15} -2.64370 q^{17} +3.37158 q^{19} -3.90501 q^{21} -1.00000 q^{23} -4.18840 q^{25} -1.00000 q^{27} +1.00000 q^{29} +5.08447 q^{31} +5.39807 q^{33} +3.51798 q^{35} -3.79035 q^{37} +2.18514 q^{39} +8.82974 q^{41} +0.613535 q^{43} +0.900888 q^{45} +1.02676 q^{47} +8.24908 q^{49} +2.64370 q^{51} +5.94186 q^{53} -4.86306 q^{55} -3.37158 q^{57} -10.0058 q^{59} -3.09826 q^{61} +3.90501 q^{63} -1.96857 q^{65} -12.6836 q^{67} +1.00000 q^{69} -4.69605 q^{71} +9.03175 q^{73} +4.18840 q^{75} -21.0795 q^{77} -3.30647 q^{79} +1.00000 q^{81} -1.01449 q^{83} -2.38168 q^{85} -1.00000 q^{87} -17.4884 q^{89} -8.53300 q^{91} -5.08447 q^{93} +3.03742 q^{95} +4.91409 q^{97} -5.39807 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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