LMFDB - Embedded newform 6012.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.0060616952$
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} - 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:	no (minimal twist has level 2004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.61974$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.619742 q^{5} +1.05844 q^{7} +O(q^{10})$	$q+0.619742 q^{5} +1.05844 q^{7} +2.94492 q^{11} -6.55099 q^{13} -6.19312 q^{17} +7.10438 q^{19} -4.16491 q^{23} -4.61592 q^{25} +0.996993 q^{29} +7.31224 q^{31} +0.655958 q^{35} +5.89613 q^{37} -9.96986 q^{41} +4.82896 q^{43} -6.61501 q^{47} -5.87971 q^{49} -10.9128 q^{53} +1.82509 q^{55} +9.89351 q^{59} +10.2882 q^{61} -4.05992 q^{65} -10.2505 q^{67} -11.7281 q^{71} -12.2938 q^{73} +3.11701 q^{77} -3.71458 q^{79} +5.90125 q^{83} -3.83814 q^{85} -9.88073 q^{89} -6.93380 q^{91} +4.40288 q^{95} -8.96090 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - 9 * q^5 + 2 * q^7	$ 9 q - 9 q^{5} + 2 q^{7} - 7 q^{11} + 6 q^{13} - 7 q^{17} + 2 q^{19} - 19 q^{23} + 22 q^{25} - 13 q^{29} + 12 q^{31} - 4 q^{35} + 15 q^{37} - 18 q^{41} - 6 q^{43} - 25 q^{47} + 19 q^{49} - 17 q^{53} - 3 q^{55} - 3 q^{59} + 14 q^{61} - 14 q^{65} - 4 q^{67} - 17 q^{71} - 20 q^{73} - 14 q^{77} - 8 q^{79} + q^{83} + 5 q^{85} - 36 q^{89} - 41 q^{91} - 5 q^{95} + 31 q^{97}+O(q^{100}) $ 9 * q - 9 * q^5 + 2 * q^7 - 7 * q^11 + 6 * q^13 - 7 * q^17 + 2 * q^19 - 19 * q^23 + 22 * q^25 - 13 * q^29 + 12 * q^31 - 4 * q^35 + 15 * q^37 - 18 * q^41 - 6 * q^43 - 25 * q^47 + 19 * q^49 - 17 * q^53 - 3 * q^55 - 3 * q^59 + 14 * q^61 - 14 * q^65 - 4 * q^67 - 17 * q^71 - 20 * q^73 - 14 * q^77 - 8 * q^79 + q^83 + 5 * q^85 - 36 * q^89 - 41 * q^91 - 5 * q^95 + 31 * 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
$5$	0.619742	0.277157	0.138579	−	0.990351i	$-0.455747\pi$
$5$	0.619742	0.277157	0.138579	+	0.990351i	$0.455747\pi$
$6$	0	0
$7$	1.05844	0.400051	0.200026	−	0.979791i	$-0.435897\pi$
$7$	1.05844	0.400051	0.200026	+	0.979791i	$0.435897\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.94492	0.887927	0.443963	−	0.896045i	$-0.353572\pi$
$11$	2.94492	0.887927	0.443963	+	0.896045i	$0.353572\pi$
$12$	0	0
$13$	−6.55099	−1.81692	−0.908459	−	0.417975i	$-0.862740\pi$
$13$	−6.55099	−1.81692	−0.908459	+	0.417975i	$0.862740\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.19312	−1.50205	−0.751026	−	0.660272i	$-0.770441\pi$
$17$	−6.19312	−1.50205	−0.751026	+	0.660272i	$0.770441\pi$
$18$	0	0
$19$	7.10438	1.62986	0.814928	−	0.579562i	$-0.196776\pi$
$19$	7.10438	1.62986	0.814928	+	0.579562i	$0.196776\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.16491	−0.868444	−0.434222	−	0.900806i	$-0.642977\pi$
$23$	−4.16491	−0.868444	−0.434222	+	0.900806i	$0.642977\pi$
$24$	0	0
$25$	−4.61592	−0.923184
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.996993	0.185137	0.0925685	−	0.995706i	$-0.470492\pi$
$29$	0.996993	0.185137	0.0925685	+	0.995706i	$0.470492\pi$
$30$	0	0
$31$	7.31224	1.31332	0.656659	−	0.754188i	$-0.271969\pi$
$31$	7.31224	1.31332	0.656659	+	0.754188i	$0.271969\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.655958	0.110877
$36$	0	0
$37$	5.89613	0.969317	0.484659	−	0.874703i	$-0.338944\pi$
$37$	5.89613	0.969317	0.484659	+	0.874703i	$0.338944\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.96986	−1.55703	−0.778515	−	0.627626i	$-0.784027\pi$
$41$	−9.96986	−1.55703	−0.778515	+	0.627626i	$0.784027\pi$
$42$	0	0
$43$	4.82896	0.736409	0.368205	−	0.929745i	$-0.379973\pi$
$43$	4.82896	0.736409	0.368205	+	0.929745i	$0.379973\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.61501	−0.964898	−0.482449	−	0.875924i	$-0.660253\pi$
$47$	−6.61501	−0.964898	−0.482449	+	0.875924i	$0.660253\pi$
$48$	0	0
$49$	−5.87971	−0.839959
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.9128	−1.49899	−0.749497	−	0.662007i	$-0.769705\pi$
$53$	−10.9128	−1.49899	−0.749497	+	0.662007i	$0.769705\pi$
$54$	0	0
$55$	1.82509	0.246095
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.89351	1.28803	0.644013	−	0.765015i	$-0.277268\pi$
$59$	9.89351	1.28803	0.644013	+	0.765015i	$0.277268\pi$
$60$	0	0
$61$	10.2882	1.31727	0.658633	−	0.752464i	$-0.271135\pi$
$61$	10.2882	1.31727	0.658633	+	0.752464i	$0.271135\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.05992	−0.503572
$66$	0	0
$67$	−10.2505	−1.25230	−0.626148	−	0.779705i	$-0.715369\pi$
$67$	−10.2505	−1.25230	−0.626148	+	0.779705i	$0.715369\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.7281	−1.39187	−0.695936	−	0.718103i	$-0.745010\pi$
$71$	−11.7281	−1.39187	−0.695936	+	0.718103i	$0.745010\pi$
$72$	0	0
$73$	−12.2938	−1.43888	−0.719442	−	0.694552i	$-0.755602\pi$
$73$	−12.2938	−1.43888	−0.719442	+	0.694552i	$0.755602\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.11701	0.355216
$78$	0	0
$79$	−3.71458	−0.417923	−0.208962	−	0.977924i	$-0.567008\pi$
$79$	−3.71458	−0.417923	−0.208962	+	0.977924i	$0.567008\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.90125	0.647747	0.323873	−	0.946100i	$-0.395015\pi$
$83$	5.90125	0.647747	0.323873	+	0.946100i	$0.395015\pi$
$84$	0	0
$85$	−3.83814	−0.416305
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.88073	−1.04735	−0.523677	−	0.851917i	$-0.675440\pi$
$89$	−9.88073	−1.04735	−0.523677	+	0.851917i	$0.675440\pi$
$90$	0	0
$91$	−6.93380	−0.726860
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.40288	0.451726
$96$	0	0
$97$	−8.96090	−0.909841	−0.454921	−	0.890532i	$-0.650332\pi$
$97$	−8.96090	−0.909841	−0.454921	+	0.890532i	$0.650332\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.91541	0.489101	0.244551	−	0.969637i	$-0.421360\pi$
$101$	4.91541	0.489101	0.244551	+	0.969637i	$0.421360\pi$
$102$	0	0
$103$	4.72990	0.466051	0.233026	−	0.972471i	$-0.425137\pi$
$103$	4.72990	0.466051	0.233026	+	0.972471i	$0.425137\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.10628	−0.203622	−0.101811	−	0.994804i	$-0.532464\pi$
$107$	−2.10628	−0.203622	−0.101811	+	0.994804i	$0.532464\pi$
$108$	0	0
$109$	−6.73056	−0.644671	−0.322336	−	0.946625i	$-0.604468\pi$
$109$	−6.73056	−0.644671	−0.322336	+	0.946625i	$0.604468\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.21493	−0.396507	−0.198254	−	0.980151i	$-0.563527\pi$
$113$	−4.21493	−0.396507	−0.198254	+	0.980151i	$0.563527\pi$
$114$	0	0
$115$	−2.58117	−0.240695
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.55503	−0.600898
$120$	0	0
$121$	−2.32744	−0.211586
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.95939	−0.533024
$126$	0	0
$127$	11.2289	0.996401	0.498201	−	0.867062i	$-0.333994\pi$
$127$	11.2289	0.996401	0.498201	+	0.867062i	$0.333994\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.86472	0.425032	0.212516	−	0.977158i	$-0.431834\pi$
$131$	4.86472	0.425032	0.212516	+	0.977158i	$0.431834\pi$
$132$	0	0
$133$	7.51953	0.652026
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.48097	0.126528	0.0632640	−	0.997997i	$-0.479849\pi$
$137$	1.48097	0.126528	0.0632640	+	0.997997i	$0.479849\pi$
$138$	0	0
$139$	−11.8564	−1.00565	−0.502823	−	0.864389i	$-0.667705\pi$
$139$	−11.8564	−1.00565	−0.502823	+	0.864389i	$0.667705\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−19.2921	−1.61329
$144$	0	0
$145$	0.617879	0.0513120
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.25014	−0.102415	−0.0512075	−	0.998688i	$-0.516307\pi$
$149$	−1.25014	−0.102415	−0.0512075	+	0.998688i	$0.516307\pi$
$150$	0	0
$151$	8.74306	0.711500	0.355750	−	0.934581i	$-0.384225\pi$
$151$	8.74306	0.711500	0.355750	+	0.934581i	$0.384225\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.53170	0.363995
$156$	0	0
$157$	1.28243	0.102349	0.0511747	−	0.998690i	$-0.483703\pi$
$157$	1.28243	0.102349	0.0511747	+	0.998690i	$0.483703\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.40829	−0.347422
$162$	0	0
$163$	10.2657	0.804074	0.402037	−	0.915623i	$-0.368302\pi$
$163$	10.2657	0.804074	0.402037	+	0.915623i	$0.368302\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	29.9154	2.30119
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.26599	0.0962515	0.0481257	−	0.998841i	$-0.484675\pi$
$173$	1.26599	0.0962515	0.0481257	+	0.998841i	$0.484675\pi$
$174$	0	0
$175$	−4.88566	−0.369321
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.48456	0.559422	0.279711	−	0.960084i	$-0.409761\pi$
$179$	7.48456	0.559422	0.279711	+	0.960084i	$0.409761\pi$
$180$	0	0
$181$	−4.86471	−0.361591	−0.180796	−	0.983521i	$-0.557867\pi$
$181$	−4.86471	−0.361591	−0.180796	+	0.983521i	$0.557867\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.65408	0.268653
$186$	0	0
$187$	−18.2383	−1.33371
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.37459	−0.605964	−0.302982	−	0.952996i	$-0.597982\pi$
$191$	−8.37459	−0.605964	−0.302982	+	0.952996i	$0.597982\pi$
$192$	0	0
$193$	−6.53375	−0.470309	−0.235155	−	0.971958i	$-0.575560\pi$
$193$	−6.53375	−0.470309	−0.235155	+	0.971958i	$0.575560\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.01768	−0.143754	−0.0718768	−	0.997414i	$-0.522899\pi$
$197$	−2.01768	−0.143754	−0.0718768	+	0.997414i	$0.522899\pi$
$198$	0	0
$199$	−18.2648	−1.29476	−0.647379	−	0.762168i	$-0.724135\pi$
$199$	−18.2648	−1.29476	−0.647379	+	0.762168i	$0.724135\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.05525	0.0740643
$204$	0	0
$205$	−6.17874	−0.431542
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.9218	1.44719
$210$	0	0
$211$	8.93507	0.615116	0.307558	−	0.951529i	$-0.400488\pi$
$211$	8.93507	0.615116	0.307558	+	0.951529i	$0.400488\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.99271	0.204101
$216$	0	0
$217$	7.73954	0.525394
$218$	0	0
$219$	0	0
$220$	0	0
$221$	40.5711	2.72911
$222$	0	0
$223$	−17.7607	−1.18934	−0.594671	−	0.803969i	$-0.702718\pi$
$223$	−17.7607	−1.18934	−0.594671	+	0.803969i	$0.702718\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.3559	0.753716	0.376858	−	0.926271i	$-0.377004\pi$
$227$	11.3559	0.753716	0.376858	+	0.926271i	$0.377004\pi$
$228$	0	0
$229$	14.3141	0.945900	0.472950	−	0.881089i	$-0.343189\pi$
$229$	14.3141	0.945900	0.472950	+	0.881089i	$0.343189\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.6800	−1.35479	−0.677395	−	0.735620i	$-0.736891\pi$
$233$	−20.6800	−1.35479	−0.677395	+	0.735620i	$0.736891\pi$
$234$	0	0
$235$	−4.09960	−0.267428
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.67387	−0.625750	−0.312875	−	0.949794i	$-0.601292\pi$
$239$	−9.67387	−0.625750	−0.312875	+	0.949794i	$0.601292\pi$
$240$	0	0
$241$	7.89334	0.508455	0.254227	−	0.967144i	$-0.418179\pi$
$241$	7.89334	0.508455	0.254227	+	0.967144i	$0.418179\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.64391	−0.232801
$246$	0	0
$247$	−46.5407	−2.96131
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.8625	1.94802	0.974011	−	0.226500i	$-0.0727282\pi$
$251$	30.8625	1.94802	0.974011	+	0.226500i	$0.0727282\pi$
$252$	0	0
$253$	−12.2653	−0.771115
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.5611	1.03305	0.516525	−	0.856272i	$-0.327225\pi$
$257$	16.5611	1.03305	0.516525	+	0.856272i	$0.327225\pi$
$258$	0	0
$259$	6.24067	0.387777
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.43522	0.150162	0.0750812	−	0.997177i	$-0.476078\pi$
$263$	2.43522	0.150162	0.0750812	+	0.997177i	$0.476078\pi$
$264$	0	0
$265$	−6.76315	−0.415457
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−32.6765	−1.99232	−0.996162	−	0.0875285i	$-0.972103\pi$
$269$	−32.6765	−1.99232	−0.996162	+	0.0875285i	$0.972103\pi$
$270$	0	0
$271$	−0.00431859	−0.000262335 0	−0.000131168	−	1.00000i	$-0.500042\pi$
$271$	−0.00431859	−0.000262335 0	−0.000131168		1.00000i	$0.500042\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.5935	−0.819720
$276$	0	0
$277$	−25.8726	−1.55453	−0.777266	−	0.629172i	$-0.783394\pi$
$277$	−25.8726	−1.55453	−0.777266	+	0.629172i	$0.783394\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−23.4091	−1.39647	−0.698236	−	0.715868i	$-0.746031\pi$
$281$	−23.4091	−1.39647	−0.698236	+	0.715868i	$0.746031\pi$
$282$	0	0
$283$	−26.8435	−1.59568	−0.797842	−	0.602867i	$-0.794025\pi$
$283$	−26.8435	−1.59568	−0.797842	+	0.602867i	$0.794025\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.5525	−0.622892
$288$	0	0
$289$	21.3548	1.25616
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−23.7852	−1.38954	−0.694772	−	0.719230i	$-0.744495\pi$
$293$	−23.7852	−1.38954	−0.694772	+	0.719230i	$0.744495\pi$
$294$	0	0
$295$	6.13143	0.356985
$296$	0	0
$297$	0	0
$298$	0	0
$299$	27.2843	1.57789
$300$	0	0
$301$	5.11115	0.294602
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.37602	0.365090
$306$	0	0
$307$	−10.0558	−0.573914	−0.286957	−	0.957943i	$-0.592644\pi$
$307$	−10.0558	−0.573914	−0.286957	+	0.957943i	$0.592644\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.7831	−1.06509	−0.532547	−	0.846401i	$-0.678765\pi$
$311$	−18.7831	−1.06509	−0.532547	+	0.846401i	$0.678765\pi$
$312$	0	0
$313$	14.5269	0.821110	0.410555	−	0.911836i	$-0.365335\pi$
$313$	14.5269	0.821110	0.410555	+	0.911836i	$0.365335\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.74455	0.491143	0.245571	−	0.969379i	$-0.421024\pi$
$317$	8.74455	0.491143	0.245571	+	0.969379i	$0.421024\pi$
$318$	0	0
$319$	2.93606	0.164388
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−43.9983	−2.44813
$324$	0	0
$325$	30.2388	1.67735
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.00156	−0.386009
$330$	0	0
$331$	−4.49622	−0.247135	−0.123567	−	0.992336i	$-0.539433\pi$
$331$	−4.49622	−0.247135	−0.123567	+	0.992336i	$0.539433\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.35265	−0.347082
$336$	0	0
$337$	11.1496	0.607359	0.303680	−	0.952774i	$-0.401785\pi$
$337$	11.1496	0.607359	0.303680	+	0.952774i	$0.401785\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.5340	1.16613
$342$	0	0
$343$	−13.6324	−0.736078
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.7784	−1.70595	−0.852977	−	0.521949i	$-0.825205\pi$
$347$	−31.7784	−1.70595	−0.852977	+	0.521949i	$0.825205\pi$
$348$	0	0
$349$	−11.4428	−0.612521	−0.306261	−	0.951948i	$-0.599078\pi$
$349$	−11.4428	−0.612521	−0.306261	+	0.951948i	$0.599078\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.68738	−0.302709	−0.151354	−	0.988480i	$-0.548363\pi$
$353$	−5.68738	−0.302709	−0.151354	+	0.988480i	$0.548363\pi$
$354$	0	0
$355$	−7.26842	−0.385767
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.8239	1.94349	0.971745	−	0.236031i	$-0.0758468\pi$
$359$	36.8239	1.94349	0.971745	+	0.236031i	$0.0758468\pi$
$360$	0	0
$361$	31.4722	1.65643
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.61901	−0.398797
$366$	0	0
$367$	−22.4128	−1.16994	−0.584968	−	0.811056i	$-0.698893\pi$
$367$	−22.4128	−1.16994	−0.584968	+	0.811056i	$0.698893\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.5506	−0.599675
$372$	0	0
$373$	−20.0874	−1.04009	−0.520044	−	0.854140i	$-0.674084\pi$
$373$	−20.0874	−1.04009	−0.520044	+	0.854140i	$0.674084\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.53129	−0.336378
$378$	0	0
$379$	−13.4368	−0.690203	−0.345102	−	0.938565i	$-0.612156\pi$
$379$	−13.4368	−0.690203	−0.345102	+	0.938565i	$0.612156\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.1688	1.08168	0.540838	−	0.841127i	$-0.318107\pi$
$383$	21.1688	1.08168	0.540838	+	0.841127i	$0.318107\pi$
$384$	0	0
$385$	1.93174	0.0984508
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.8032	−0.699849	−0.349924	−	0.936778i	$-0.613793\pi$
$389$	−13.8032	−0.699849	−0.349924	+	0.936778i	$0.613793\pi$
$390$	0	0
$391$	25.7938	1.30445
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.30208	−0.115830
$396$	0	0
$397$	23.8793	1.19847	0.599234	−	0.800574i	$-0.295472\pi$
$397$	23.8793	1.19847	0.599234	+	0.800574i	$0.295472\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.6924	1.58265	0.791323	−	0.611399i	$-0.209393\pi$
$401$	31.6924	1.58265	0.791323	+	0.611399i	$0.209393\pi$
$402$	0	0
$403$	−47.9024	−2.38619
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.3636	0.860683
$408$	0	0
$409$	−10.5695	−0.522626	−0.261313	−	0.965254i	$-0.584156\pi$
$409$	−10.5695	−0.522626	−0.261313	+	0.965254i	$0.584156\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.4717	0.515276
$414$	0	0
$415$	3.65726	0.179528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.09660	0.297839	0.148919	−	0.988849i	$-0.452421\pi$
$419$	6.09660	0.297839	0.148919	+	0.988849i	$0.452421\pi$
$420$	0	0
$421$	−8.05451	−0.392553	−0.196276	−	0.980549i	$-0.562885\pi$
$421$	−8.05451	−0.392553	−0.196276	+	0.980549i	$0.562885\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	28.5870	1.38667
$426$	0	0
$427$	10.8894	0.526974
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.89755	−0.380412	−0.190206	−	0.981744i	$-0.560916\pi$
$431$	−7.89755	−0.380412	−0.190206	+	0.981744i	$0.560916\pi$
$432$	0	0
$433$	−20.1848	−0.970021	−0.485010	−	0.874508i	$-0.661184\pi$
$433$	−20.1848	−0.970021	−0.485010	+	0.874508i	$0.661184\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−29.5891	−1.41544
$438$	0	0
$439$	−30.9441	−1.47688	−0.738442	−	0.674317i	$-0.764438\pi$
$439$	−30.9441	−1.47688	−0.738442	+	0.674317i	$0.764438\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.5088	1.59205	0.796025	−	0.605264i	$-0.206933\pi$
$443$	33.5088	1.59205	0.796025	+	0.605264i	$0.206933\pi$
$444$	0	0
$445$	−6.12350	−0.290282
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.8767	1.03243	0.516213	−	0.856460i	$-0.327341\pi$
$449$	21.8767	1.03243	0.516213	+	0.856460i	$0.327341\pi$
$450$	0	0
$451$	−29.3604	−1.38253
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.29717	−0.201454
$456$	0	0
$457$	21.2520	0.994128	0.497064	−	0.867714i	$-0.334411\pi$
$457$	21.2520	0.994128	0.497064	+	0.867714i	$0.334411\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.74130	0.127675	0.0638375	−	0.997960i	$-0.479666\pi$
$461$	2.74130	0.127675	0.0638375	+	0.997960i	$0.479666\pi$
$462$	0	0
$463$	0.939856	0.0436788	0.0218394	−	0.999761i	$-0.493048\pi$
$463$	0.939856	0.0436788	0.0218394	+	0.999761i	$0.493048\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.00247	−0.185212	−0.0926061	−	0.995703i	$-0.529520\pi$
$467$	−4.00247	−0.185212	−0.0926061	+	0.995703i	$0.529520\pi$
$468$	0	0
$469$	−10.8495	−0.500982
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.2209	0.653878
$474$	0	0
$475$	−32.7932	−1.50466
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.13204	0.417254	0.208627	−	0.977995i	$-0.433101\pi$
$479$	9.13204	0.417254	0.208627	+	0.977995i	$0.433101\pi$
$480$	0	0
$481$	−38.6255	−1.76117
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.55345	−0.252169
$486$	0	0
$487$	−35.6996	−1.61770	−0.808852	−	0.588012i	$-0.799911\pi$
$487$	−35.6996	−1.61770	−0.808852	+	0.588012i	$0.799911\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.1762	−1.72287	−0.861434	−	0.507870i	$-0.830433\pi$
$491$	−38.1762	−1.72287	−0.861434	+	0.507870i	$0.830433\pi$
$492$	0	0
$493$	−6.17450	−0.278085
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.4135	−0.556821
$498$	0	0
$499$	−1.03459	−0.0463144	−0.0231572	−	0.999732i	$-0.507372\pi$
$499$	−1.03459	−0.0463144	−0.0231572	+	0.999732i	$0.507372\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.2992	−1.26180	−0.630900	−	0.775864i	$-0.717314\pi$
$503$	−28.2992	−1.26180	−0.630900	+	0.775864i	$0.717314\pi$
$504$	0	0
$505$	3.04629	0.135558
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.65536	−0.427966	−0.213983	−	0.976837i	$-0.568644\pi$
$509$	−9.65536	−0.427966	−0.213983	+	0.976837i	$0.568644\pi$
$510$	0	0
$511$	−13.0122	−0.575627
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.93132	0.129169
$516$	0	0
$517$	−19.4807	−0.856759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.841870	0.0368830	0.0184415	−	0.999830i	$-0.494130\pi$
$521$	0.841870	0.0368830	0.0184415	+	0.999830i	$0.494130\pi$
$522$	0	0
$523$	−12.6622	−0.553679	−0.276840	−	0.960916i	$-0.589287\pi$
$523$	−12.6622	−0.553679	−0.276840	+	0.960916i	$0.589287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−45.2856	−1.97267
$528$	0	0
$529$	−5.65352	−0.245805
$530$	0	0
$531$	0	0
$532$	0	0
$533$	65.3124	2.82900
$534$	0	0
$535$	−1.30535	−0.0564352
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−17.3153	−0.745822
$540$	0	0
$541$	10.6265	0.456869	0.228435	−	0.973559i	$-0.426639\pi$
$541$	10.6265	0.456869	0.228435	+	0.973559i	$0.426639\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.17122	−0.178675
$546$	0	0
$547$	46.1517	1.97330	0.986652	−	0.162841i	$-0.0520657\pi$
$547$	46.1517	1.97330	0.986652	+	0.162841i	$0.0520657\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.08301	0.301747
$552$	0	0
$553$	−3.93165	−0.167191
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.0103	0.805494	0.402747	−	0.915311i	$-0.368056\pi$
$557$	19.0103	0.805494	0.402747	+	0.915311i	$0.368056\pi$
$558$	0	0
$559$	−31.6345	−1.33800
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.1476	1.01770	0.508850	−	0.860855i	$-0.330071\pi$
$563$	24.1476	1.01770	0.508850	+	0.860855i	$0.330071\pi$
$564$	0	0
$565$	−2.61217	−0.109895
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.7175	0.952369	0.476185	−	0.879345i	$-0.342019\pi$
$569$	22.7175	0.952369	0.476185	+	0.879345i	$0.342019\pi$
$570$	0	0
$571$	−12.5374	−0.524673	−0.262337	−	0.964976i	$-0.584493\pi$
$571$	−12.5374	−0.524673	−0.262337	+	0.964976i	$0.584493\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	19.2249	0.801733
$576$	0	0
$577$	41.0884	1.71053	0.855267	−	0.518188i	$-0.173393\pi$
$577$	41.0884	1.71053	0.855267	+	0.518188i	$0.173393\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.24610	0.259132
$582$	0	0
$583$	−32.1375	−1.33100
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.04147	0.0429861	0.0214931	−	0.999769i	$-0.493158\pi$
$587$	1.04147	0.0429861	0.0214931	+	0.999769i	$0.493158\pi$
$588$	0	0
$589$	51.9489	2.14052
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.8470	−1.43100	−0.715498	−	0.698615i	$-0.753800\pi$
$593$	−34.8470	−1.43100	−0.715498	+	0.698615i	$0.753800\pi$
$594$	0	0
$595$	−4.06243	−0.166543
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.0595	1.59593	0.797965	−	0.602704i	$-0.205910\pi$
$599$	39.0595	1.59593	0.797965	+	0.602704i	$0.205910\pi$
$600$	0	0
$601$	9.49751	0.387411	0.193706	−	0.981060i	$-0.437949\pi$
$601$	9.49751	0.387411	0.193706	+	0.981060i	$0.437949\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.44241	−0.0586425
$606$	0	0
$607$	18.2746	0.741742	0.370871	−	0.928684i	$-0.379059\pi$
$607$	18.2746	0.741742	0.370871	+	0.928684i	$0.379059\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	43.3348	1.75314
$612$	0	0
$613$	−33.5367	−1.35454	−0.677268	−	0.735736i	$-0.736836\pi$
$613$	−33.5367	−1.35454	−0.677268	+	0.735736i	$0.736836\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.66780	−0.0671429	−0.0335715	−	0.999436i	$-0.510688\pi$
$617$	−1.66780	−0.0671429	−0.0335715	+	0.999436i	$0.510688\pi$
$618$	0	0
$619$	−22.5977	−0.908278	−0.454139	−	0.890931i	$-0.650053\pi$
$619$	−22.5977	−0.908278	−0.454139	+	0.890931i	$0.650053\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.4581	−0.418996
$624$	0	0
$625$	19.3863	0.775452
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−36.5154	−1.45597
$630$	0	0
$631$	9.50108	0.378232	0.189116	−	0.981955i	$-0.439438\pi$
$631$	9.50108	0.378232	0.189116	+	0.981955i	$0.439438\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.95901	0.276160
$636$	0	0
$637$	38.5179	1.52614
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.9083	−0.865325	−0.432663	−	0.901556i	$-0.642426\pi$
$641$	−21.9083	−0.865325	−0.432663	+	0.901556i	$0.642426\pi$
$642$	0	0
$643$	41.7641	1.64701	0.823507	−	0.567306i	$-0.192014\pi$
$643$	41.7641	1.64701	0.823507	+	0.567306i	$0.192014\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.32773	0.170141	0.0850703	−	0.996375i	$-0.472889\pi$
$647$	4.32773	0.170141	0.0850703	+	0.996375i	$0.472889\pi$
$648$	0	0
$649$	29.1356	1.14367
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.5986	−1.54961	−0.774807	−	0.632198i	$-0.782153\pi$
$653$	−39.5986	−1.54961	−0.774807	+	0.632198i	$0.782153\pi$
$654$	0	0
$655$	3.01487	0.117801
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.0788	1.28857	0.644284	−	0.764786i	$-0.277155\pi$
$659$	33.0788	1.28857	0.644284	+	0.764786i	$0.277155\pi$
$660$	0	0
$661$	−14.2897	−0.555803	−0.277902	−	0.960610i	$-0.589639\pi$
$661$	−14.2897	−0.555803	−0.277902	+	0.960610i	$0.589639\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.66017	0.180714
$666$	0	0
$667$	−4.15239	−0.160781
$668$	0	0
$669$	0	0
$670$	0	0
$671$	30.2979	1.16964
$672$	0	0
$673$	−43.1679	−1.66400	−0.832000	−	0.554776i	$-0.812804\pi$
$673$	−43.1679	−1.66400	−0.832000	+	0.554776i	$0.812804\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.8027	−1.10698	−0.553489	−	0.832857i	$-0.686704\pi$
$677$	−28.8027	−1.10698	−0.553489	+	0.832857i	$0.686704\pi$
$678$	0	0
$679$	−9.48454	−0.363983
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.6388	1.09583	0.547916	−	0.836533i	$-0.315421\pi$
$683$	28.6388	1.09583	0.547916	+	0.836533i	$0.315421\pi$
$684$	0	0
$685$	0.917821	0.0350682
$686$	0	0
$687$	0	0
$688$	0	0
$689$	71.4899	2.72355
$690$	0	0
$691$	38.6489	1.47027	0.735137	−	0.677919i	$-0.237118\pi$
$691$	38.6489	1.47027	0.735137	+	0.677919i	$0.237118\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.34791	−0.278722
$696$	0	0
$697$	61.7446	2.33874
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.43773	0.129841	0.0649206	−	0.997890i	$-0.479321\pi$
$701$	3.43773	0.129841	0.0649206	+	0.997890i	$0.479321\pi$
$702$	0	0
$703$	41.8883	1.57985
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.20265	0.195666
$708$	0	0
$709$	20.2732	0.761375	0.380687	−	0.924704i	$-0.375687\pi$
$709$	20.2732	0.761375	0.380687	+	0.924704i	$0.375687\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−30.4548	−1.14054
$714$	0	0
$715$	−11.9562	−0.447135
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.6813	1.14422	0.572110	−	0.820177i	$-0.306125\pi$
$719$	30.6813	1.14422	0.572110	+	0.820177i	$0.306125\pi$
$720$	0	0
$721$	5.00630	0.186444
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.60204	−0.170915
$726$	0	0
$727$	11.9584	0.443512	0.221756	−	0.975102i	$-0.428821\pi$
$727$	11.9584	0.443512	0.221756	+	0.975102i	$0.428821\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.9063	−1.10613
$732$	0	0
$733$	9.76983	0.360857	0.180429	−	0.983588i	$-0.442252\pi$
$733$	9.76983	0.360857	0.180429	+	0.983588i	$0.442252\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−30.1868	−1.11195
$738$	0	0
$739$	−36.8792	−1.35662	−0.678312	−	0.734774i	$-0.737288\pi$
$739$	−36.8792	−1.35662	−0.678312	+	0.734774i	$0.737288\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.8422	1.16818	0.584089	−	0.811690i	$-0.301452\pi$
$743$	31.8422	1.16818	0.584089	+	0.811690i	$0.301452\pi$
$744$	0	0
$745$	−0.774761	−0.0283851
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.22936	−0.0814591
$750$	0	0
$751$	7.33017	0.267482	0.133741	−	0.991016i	$-0.457301\pi$
$751$	7.33017	0.267482	0.133741	+	0.991016i	$0.457301\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.41844	0.197197
$756$	0	0
$757$	−45.1792	−1.64206	−0.821032	−	0.570882i	$-0.806602\pi$
$757$	−45.1792	−1.64206	−0.821032	+	0.570882i	$0.806602\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.8565	1.04605	0.523024	−	0.852318i	$-0.324804\pi$
$761$	28.8565	1.04605	0.523024	+	0.852318i	$0.324804\pi$
$762$	0	0
$763$	−7.12387	−0.257902
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−64.8123	−2.34024
$768$	0	0
$769$	11.3707	0.410039	0.205019	−	0.978758i	$-0.434274\pi$
$769$	11.3707	0.410039	0.205019	+	0.978758i	$0.434274\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.9885	0.754904	0.377452	−	0.926029i	$-0.376800\pi$
$773$	20.9885	0.754904	0.377452	+	0.926029i	$0.376800\pi$
$774$	0	0
$775$	−33.7527	−1.21243
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−70.8297	−2.53774
$780$	0	0
$781$	−34.5384	−1.23588
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.794779	0.0283669
$786$	0	0
$787$	41.3812	1.47508	0.737540	−	0.675304i	$-0.235988\pi$
$787$	41.3812	1.47508	0.737540	+	0.675304i	$0.235988\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.46123	−0.158623
$792$	0	0
$793$	−67.3977	−2.39336
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.3144	0.967525	0.483762	−	0.875199i	$-0.339270\pi$
$797$	27.3144	0.967525	0.483762	+	0.875199i	$0.339270\pi$
$798$	0	0
$799$	40.9676	1.44933
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−36.2044	−1.27762
$804$	0	0
$805$	−2.73200	−0.0962905
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−33.5282	−1.17879	−0.589395	−	0.807845i	$-0.700634\pi$
$809$	−33.5282	−1.17879	−0.589395	+	0.807845i	$0.700634\pi$
$810$	0	0
$811$	33.7840	1.18632	0.593159	−	0.805086i	$-0.297881\pi$
$811$	33.7840	1.18632	0.593159	+	0.805086i	$0.297881\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.36211	0.222855
$816$	0	0
$817$	34.3068	1.20024
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.8743	−0.937919	−0.468959	−	0.883220i	$-0.655371\pi$
$821$	−26.8743	−0.937919	−0.468959	+	0.883220i	$0.655371\pi$
$822$	0	0
$823$	10.8499	0.378205	0.189103	−	0.981957i	$-0.439442\pi$
$823$	10.8499	0.378205	0.189103	+	0.981957i	$0.439442\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.4359	1.57996	0.789981	−	0.613131i	$-0.210090\pi$
$827$	45.4359	1.57996	0.789981	+	0.613131i	$0.210090\pi$
$828$	0	0
$829$	37.4911	1.30212	0.651060	−	0.759026i	$-0.274325\pi$
$829$	37.4911	1.30212	0.651060	+	0.759026i	$0.274325\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	36.4138	1.26166
$834$	0	0
$835$	−0.619742	−0.0214471
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.1675	−0.799831	−0.399916	−	0.916552i	$-0.630961\pi$
$839$	−23.1675	−0.799831	−0.399916	+	0.916552i	$0.630961\pi$
$840$	0	0
$841$	−28.0060	−0.965724
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.5399	0.637791
$846$	0	0
$847$	−2.46345	−0.0846451
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.5568	−0.841798
$852$	0	0
$853$	24.1133	0.825624	0.412812	−	0.910816i	$-0.364547\pi$
$853$	24.1133	0.825624	0.412812	+	0.910816i	$0.364547\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.6072	0.362335	0.181168	−	0.983452i	$-0.442012\pi$
$857$	10.6072	0.362335	0.181168	+	0.983452i	$0.442012\pi$
$858$	0	0
$859$	37.8886	1.29274	0.646372	−	0.763022i	$-0.276285\pi$
$859$	37.8886	1.29274	0.646372	+	0.763022i	$0.276285\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.3480	−0.420332	−0.210166	−	0.977666i	$-0.567400\pi$
$863$	−12.3480	−0.420332	−0.210166	+	0.977666i	$0.567400\pi$
$864$	0	0
$865$	0.784588	0.0266768
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.9392	−0.371085
$870$	0	0
$871$	67.1508	2.27532
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.30764	−0.213237
$876$	0	0
$877$	−49.0918	−1.65771	−0.828856	−	0.559462i	$-0.811008\pi$
$877$	−49.0918	−1.65771	−0.828856	+	0.559462i	$0.811008\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27.4045	−0.923280	−0.461640	−	0.887067i	$-0.652739\pi$
$881$	−27.4045	−0.923280	−0.461640	+	0.887067i	$0.652739\pi$
$882$	0	0
$883$	−34.8248	−1.17195	−0.585974	−	0.810330i	$-0.699288\pi$
$883$	−34.8248	−1.17195	−0.585974	+	0.810330i	$0.699288\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.157909	−0.00530207	−0.00265104	−	0.999996i	$-0.500844\pi$
$887$	−0.157909	−0.00530207	−0.00265104	+	0.999996i	$0.500844\pi$
$888$	0	0
$889$	11.8850	0.398612
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−46.9955	−1.57265
$894$	0	0
$895$	4.63849	0.155048
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.29025	0.243143
$900$	0	0
$901$	67.5846	2.25157
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.01487	−0.100218
$906$	0	0
$907$	−3.38987	−0.112559	−0.0562794	−	0.998415i	$-0.517924\pi$
$907$	−3.38987	−0.112559	−0.0562794	+	0.998415i	$0.517924\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.5128	−1.24285	−0.621427	−	0.783472i	$-0.713447\pi$
$911$	−37.5128	−1.24285	−0.621427	+	0.783472i	$0.713447\pi$
$912$	0	0
$913$	17.3787	0.575152
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.14899	0.170035
$918$	0	0
$919$	3.48351	0.114910	0.0574552	−	0.998348i	$-0.481701\pi$
$919$	3.48351	0.114910	0.0574552	+	0.998348i	$0.481701\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	76.8308	2.52892
$924$	0	0
$925$	−27.2160	−0.894858
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.6068	−0.544851	−0.272425	−	0.962177i	$-0.587826\pi$
$929$	−16.6068	−0.544851	−0.272425	+	0.962177i	$0.587826\pi$
$930$	0	0
$931$	−41.7717	−1.36901
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.3030	−0.369648
$936$	0	0
$937$	−41.7410	−1.36362	−0.681810	−	0.731529i	$-0.738807\pi$
$937$	−41.7410	−1.36362	−0.681810	+	0.731529i	$0.738807\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−60.1088	−1.95949	−0.979746	−	0.200246i	$-0.935826\pi$
$941$	−60.1088	−1.95949	−0.979746	+	0.200246i	$0.935826\pi$
$942$	0	0
$943$	41.5236	1.35219
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.3268	0.465558	0.232779	−	0.972530i	$-0.425218\pi$
$947$	14.3268	0.465558	0.232779	+	0.972530i	$0.425218\pi$
$948$	0	0
$949$	80.5367	2.61433
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53.4836	1.73250	0.866252	−	0.499608i	$-0.166522\pi$
$953$	53.4836	1.73250	0.866252	+	0.499608i	$0.166522\pi$
$954$	0	0
$955$	−5.19009	−0.167947
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.56752	0.0506177
$960$	0	0
$961$	22.4688	0.724802
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.04924	−0.130350
$966$	0	0
$967$	31.0170	0.997440	0.498720	−	0.866763i	$-0.333804\pi$
$967$	31.0170	0.997440	0.498720	+	0.866763i	$0.333804\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.6921	−0.631949	−0.315974	−	0.948768i	$-0.602331\pi$
$971$	−19.6921	−0.631949	−0.315974	+	0.948768i	$0.602331\pi$
$972$	0	0
$973$	−12.5492	−0.402310
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.4720	1.71072	0.855360	−	0.518033i	$-0.173336\pi$
$977$	53.4720	1.71072	0.855360	+	0.518033i	$0.173336\pi$
$978$	0	0
$979$	−29.0980	−0.929975
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.0426	−0.320309	−0.160154	−	0.987092i	$-0.551199\pi$
$983$	−10.0426	−0.320309	−0.160154	+	0.987092i	$0.551199\pi$
$984$	0	0
$985$	−1.25044	−0.0398424
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.1122	−0.639530
$990$	0	0
$991$	−23.1529	−0.735477	−0.367739	−	0.929929i	$-0.619868\pi$
$991$	−23.1529	−0.735477	−0.367739	+	0.929929i	$0.619868\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.3195	−0.358852
$996$	0	0
$997$	−14.0835	−0.446029	−0.223014	−	0.974815i	$-0.571590\pi$
$997$	−14.0835	−0.446029	−0.223014	+	0.974815i	$0.571590\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.h.1.6		9
3.2	odd	2		2004.2.a.d.1.4	✓	9
12.11	even	2		8016.2.a.bb.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.4	✓	9	3.2	odd	2
6012.2.a.h.1.6		9	1.1	even	1	trivial
8016.2.a.bb.1.4		9	12.11	even	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.619742 q^{5} +1.05844 q^{7} +O(q^{10})\)	\(q+0.619742 q^{5} +1.05844 q^{7} +2.94492 q^{11} -6.55099 q^{13} -6.19312 q^{17} +7.10438 q^{19} -4.16491 q^{23} -4.61592 q^{25} +0.996993 q^{29} +7.31224 q^{31} +0.655958 q^{35} +5.89613 q^{37} -9.96986 q^{41} +4.82896 q^{43} -6.61501 q^{47} -5.87971 q^{49} -10.9128 q^{53} +1.82509 q^{55} +9.89351 q^{59} +10.2882 q^{61} -4.05992 q^{65} -10.2505 q^{67} -11.7281 q^{71} -12.2938 q^{73} +3.11701 q^{77} -3.71458 q^{79} +5.90125 q^{83} -3.83814 q^{85} -9.88073 q^{89} -6.93380 q^{91} +4.40288 q^{95} -8.96090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - 9 * q^5 + 2 * q^7	\( 9 q - 9 q^{5} + 2 q^{7} - 7 q^{11} + 6 q^{13} - 7 q^{17} + 2 q^{19} - 19 q^{23} + 22 q^{25} - 13 q^{29} + 12 q^{31} - 4 q^{35} + 15 q^{37} - 18 q^{41} - 6 q^{43} - 25 q^{47} + 19 q^{49} - 17 q^{53} - 3 q^{55} - 3 q^{59} + 14 q^{61} - 14 q^{65} - 4 q^{67} - 17 q^{71} - 20 q^{73} - 14 q^{77} - 8 q^{79} + q^{83} + 5 q^{85} - 36 q^{89} - 41 q^{91} - 5 q^{95} + 31 q^{97}+O(q^{100}) \) 9 * q - 9 * q^5 + 2 * q^7 - 7 * q^11 + 6 * q^13 - 7 * q^17 + 2 * q^19 - 19 * q^23 + 22 * q^25 - 13 * q^29 + 12 * q^31 - 4 * q^35 + 15 * q^37 - 18 * q^41 - 6 * q^43 - 25 * q^47 + 19 * q^49 - 17 * q^53 - 3 * q^55 - 3 * q^59 + 14 * q^61 - 14 * q^65 - 4 * q^67 - 17 * q^71 - 20 * q^73 - 14 * q^77 - 8 * q^79 + q^83 + 5 * q^85 - 36 * q^89 - 41 * q^91 - 5 * q^95 + 31 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more