LMFDB - Embedded newform 4284.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4284,2,Mod(1,4284)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4284, 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("4284.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4284.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:	$34.2079122259$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	4284.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74483 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+2.74483 q^{5} -1.00000 q^{7} -3.42151 q^{11} +2.21076 q^{13} +1.00000 q^{17} -2.74483 q^{19} -8.60221 q^{23} +2.53407 q^{25} +2.95558 q^{29} -9.76855 q^{31} -2.74483 q^{35} +1.09820 q^{37} -2.21076 q^{41} -8.48965 q^{43} +1.85738 q^{47} +1.00000 q^{49} -1.74483 q^{53} -9.39145 q^{55} +0.278896 q^{59} +1.85738 q^{61} +6.06814 q^{65} +6.75919 q^{67} -8.11256 q^{71} -9.91116 q^{73} +3.42151 q^{77} +6.90180 q^{79} -10.0000 q^{83} +2.74483 q^{85} -6.64663 q^{89} -2.21076 q^{91} -7.53407 q^{95} +5.48965 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q - q^5 - 3 * q^7	$ 3 q - q^{5} - 3 q^{7} - 7 q^{11} + 5 q^{13} + 3 q^{17} + q^{19} - 3 q^{23} - 2 q^{29} + 6 q^{31} + q^{35} + 6 q^{37} - 5 q^{41} - 7 q^{43} - 8 q^{47} + 3 q^{49} + 4 q^{53} - 7 q^{55} - 16 q^{59} - 8 q^{61} + 3 q^{65} + 4 q^{67} - 20 q^{71} - 8 q^{73} + 7 q^{77} + 18 q^{79} - 30 q^{83} - q^{85} - 8 q^{89} - 5 q^{91} - 15 q^{95} - 2 q^{97}+O(q^{100}) $ 3 * q - q^5 - 3 * q^7 - 7 * q^11 + 5 * q^13 + 3 * q^17 + q^19 - 3 * q^23 - 2 * q^29 + 6 * q^31 + q^35 + 6 * q^37 - 5 * q^41 - 7 * q^43 - 8 * q^47 + 3 * q^49 + 4 * q^53 - 7 * q^55 - 16 * q^59 - 8 * q^61 + 3 * q^65 + 4 * q^67 - 20 * q^71 - 8 * q^73 + 7 * q^77 + 18 * q^79 - 30 * q^83 - q^85 - 8 * q^89 - 5 * q^91 - 15 * q^95 - 2 * 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
$5$	2.74483	1.22752	0.613762	−	0.789491i	$-0.289656\pi$
$5$	2.74483	1.22752	0.613762	+	0.789491i	$0.289656\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.42151	−1.03162	−0.515812	−	0.856702i	$-0.672510\pi$
$11$	−3.42151	−1.03162	−0.515812	+	0.856702i	$0.672510\pi$
$12$	0	0
$13$	2.21076	0.613153	0.306577	−	0.951846i	$-0.400816\pi$
$13$	2.21076	0.613153	0.306577	+	0.951846i	$0.400816\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.74483	−0.629706	−0.314853	−	0.949140i	$-0.601955\pi$
$19$	−2.74483	−0.629706	−0.314853	+	0.949140i	$0.601955\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.60221	−1.79368	−0.896842	−	0.442350i	$-0.854145\pi$
$23$	−8.60221	−1.79368	−0.896842	+	0.442350i	$0.854145\pi$
$24$	0	0
$25$	2.53407	0.506814
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.95558	0.548838	0.274419	−	0.961610i	$-0.411515\pi$
$29$	2.95558	0.548838	0.274419	+	0.961610i	$0.411515\pi$
$30$	0	0
$31$	−9.76855	−1.75448	−0.877242	−	0.480049i	$-0.840619\pi$
$31$	−9.76855	−1.75448	−0.877242	+	0.480049i	$0.840619\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.74483	−0.463960
$36$	0	0
$37$	1.09820	0.180543	0.0902713	−	0.995917i	$-0.471227\pi$
$37$	1.09820	0.180543	0.0902713	+	0.995917i	$0.471227\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.21076	−0.345262	−0.172631	−	0.984987i	$-0.555227\pi$
$41$	−2.21076	−0.345262	−0.172631	+	0.984987i	$0.555227\pi$
$42$	0	0
$43$	−8.48965	−1.29466	−0.647330	−	0.762210i	$-0.724114\pi$
$43$	−8.48965	−1.29466	−0.647330	+	0.762210i	$0.724114\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.85738	0.270927	0.135464	−	0.990782i	$-0.456748\pi$
$47$	1.85738	0.270927	0.135464	+	0.990782i	$0.456748\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.74483	−0.239670	−0.119835	−	0.992794i	$-0.538237\pi$
$53$	−1.74483	−0.239670	−0.119835	+	0.992794i	$0.538237\pi$
$54$	0	0
$55$	−9.39145	−1.26634
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.278896	0.0363092	0.0181546	−	0.999835i	$-0.494221\pi$
$59$	0.278896	0.0363092	0.0181546	+	0.999835i	$0.494221\pi$
$60$	0	0
$61$	1.85738	0.237814	0.118907	−	0.992905i	$-0.462061\pi$
$61$	1.85738	0.237814	0.118907	+	0.992905i	$0.462061\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.06814	0.752660
$66$	0	0
$67$	6.75919	0.825766	0.412883	−	0.910784i	$-0.364522\pi$
$67$	6.75919	0.825766	0.412883	+	0.910784i	$0.364522\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.11256	−0.962784	−0.481392	−	0.876506i	$-0.659869\pi$
$71$	−8.11256	−0.962784	−0.481392	+	0.876506i	$0.659869\pi$
$72$	0	0
$73$	−9.91116	−1.16001	−0.580007	−	0.814611i	$-0.696950\pi$
$73$	−9.91116	−1.16001	−0.580007	+	0.814611i	$0.696950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.42151	0.389917
$78$	0	0
$79$	6.90180	0.776513	0.388257	−	0.921551i	$-0.373077\pi$
$79$	6.90180	0.776513	0.388257	+	0.921551i	$0.373077\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	2.74483	0.297718
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.64663	−0.704541	−0.352271	−	0.935898i	$-0.614590\pi$
$89$	−6.64663	−0.704541	−0.352271	+	0.935898i	$0.614590\pi$
$90$	0	0
$91$	−2.21076	−0.231750
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.53407	−0.772979
$96$	0	0
$97$	5.48965	0.557390	0.278695	−	0.960380i	$-0.410098\pi$
$97$	5.48965	0.557390	0.278695	+	0.960380i	$0.410098\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.4359	−1.33692	−0.668460	−	0.743748i	$-0.733046\pi$
$101$	−13.4359	−1.33692	−0.668460	+	0.743748i	$0.733046\pi$
$102$	0	0
$103$	−10.6323	−1.04763	−0.523814	−	0.851833i	$-0.675491\pi$
$103$	−10.6323	−1.04763	−0.523814	+	0.851833i	$0.675491\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.02372	0.872356	0.436178	−	0.899860i	$-0.356332\pi$
$107$	9.02372	0.872356	0.436178	+	0.899860i	$0.356332\pi$
$108$	0	0
$109$	12.3026	1.17838	0.589189	−	0.807996i	$-0.299448\pi$
$109$	12.3026	1.17838	0.589189	+	0.807996i	$0.299448\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.26454	−0.401174	−0.200587	−	0.979676i	$-0.564285\pi$
$113$	−4.26454	−0.401174	−0.200587	+	0.979676i	$0.564285\pi$
$114$	0	0
$115$	−23.6116	−2.20179
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	0.706743	0.0642493
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.76855	−0.605397
$126$	0	0
$127$	21.7779	1.93248	0.966239	−	0.257649i	$-0.0829479\pi$
$127$	21.7779	1.93248	0.966239	+	0.257649i	$0.0829479\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.2345	1.06893	0.534466	−	0.845190i	$-0.320513\pi$
$131$	12.2345	1.06893	0.534466	+	0.845190i	$0.320513\pi$
$132$	0	0
$133$	2.74483	0.238007
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.7448	−1.51604	−0.758021	−	0.652230i	$-0.773834\pi$
$137$	−17.7448	−1.51604	−0.758021	+	0.652230i	$0.773834\pi$
$138$	0	0
$139$	14.1363	1.19902	0.599512	−	0.800366i	$-0.295361\pi$
$139$	14.1363	1.19902	0.599512	+	0.800366i	$0.295361\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.56413	−0.632544
$144$	0	0
$145$	8.11256	0.673711
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.42151	−0.689917	−0.344959	−	0.938618i	$-0.612107\pi$
$149$	−8.42151	−0.689917	−0.344959	+	0.938618i	$0.612107\pi$
$150$	0	0
$151$	−8.33768	−0.678510	−0.339255	−	0.940694i	$-0.610175\pi$
$151$	−8.33768	−0.678510	−0.339255	+	0.940694i	$0.610175\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−26.8130	−2.15367
$156$	0	0
$157$	7.81297	0.623543	0.311771	−	0.950157i	$-0.399078\pi$
$157$	7.81297	0.623543	0.311771	+	0.950157i	$0.399078\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.60221	0.677949
$162$	0	0
$163$	5.91116	0.462998	0.231499	−	0.972835i	$-0.425637\pi$
$163$	5.91116	0.462998	0.231499	+	0.972835i	$0.425637\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.14262	−0.552712	−0.276356	−	0.961055i	$-0.589127\pi$
$167$	−7.14262	−0.552712	−0.276356	+	0.961055i	$0.589127\pi$
$168$	0	0
$169$	−8.11256	−0.624043
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.25517	0.247486	0.123743	−	0.992314i	$-0.460510\pi$
$173$	3.25517	0.247486	0.123743	+	0.992314i	$0.460510\pi$
$174$	0	0
$175$	−2.53407	−0.191558
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.66732	−0.124622	−0.0623108	−	0.998057i	$-0.519847\pi$
$179$	−1.66732	−0.124622	−0.0623108	+	0.998057i	$0.519847\pi$
$180$	0	0
$181$	6.10756	0.453971	0.226986	−	0.973898i	$-0.427113\pi$
$181$	6.10756	0.453971	0.226986	+	0.973898i	$0.427113\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.01436	0.221620
$186$	0	0
$187$	−3.42151	−0.250206
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.15698	−0.0837159	−0.0418579	−	0.999124i	$-0.513328\pi$
$191$	−1.15698	−0.0837159	−0.0418579	+	0.999124i	$0.513328\pi$
$192$	0	0
$193$	−7.74483	−0.557485	−0.278742	−	0.960366i	$-0.589918\pi$
$193$	−7.74483	−0.557485	−0.278742	+	0.960366i	$0.589918\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.6466	−0.829788	−0.414894	−	0.909870i	$-0.636181\pi$
$197$	−11.6466	−0.829788	−0.414894	+	0.909870i	$0.636181\pi$
$198$	0	0
$199$	12.7892	0.906605	0.453303	−	0.891357i	$-0.350246\pi$
$199$	12.7892	0.906605	0.453303	+	0.891357i	$0.350246\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.95558	−0.207441
$204$	0	0
$205$	−6.06814	−0.423817
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.39145	0.649621
$210$	0	0
$211$	27.1456	1.86878	0.934392	−	0.356248i	$-0.115944\pi$
$211$	27.1456	1.86878	0.934392	+	0.356248i	$0.115944\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−23.3026	−1.58923
$216$	0	0
$217$	9.76855	0.663132
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.21076	0.148712
$222$	0	0
$223$	17.5040	1.17216	0.586078	−	0.810255i	$-0.300671\pi$
$223$	17.5040	1.17216	0.586078	+	0.810255i	$0.300671\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.7685	−1.24571	−0.622856	−	0.782336i	$-0.714028\pi$
$227$	−18.7685	−1.24571	−0.622856	+	0.782336i	$0.714028\pi$
$228$	0	0
$229$	−21.7385	−1.43652	−0.718260	−	0.695775i	$-0.755061\pi$
$229$	−21.7385	−1.43652	−0.718260	+	0.695775i	$0.755061\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.2883	0.936055	0.468027	−	0.883714i	$-0.344965\pi$
$233$	14.2883	0.936055	0.468027	+	0.883714i	$0.344965\pi$
$234$	0	0
$235$	5.09820	0.332570
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.5197	−0.874517	−0.437259	−	0.899336i	$-0.644051\pi$
$239$	−13.5197	−0.874517	−0.437259	+	0.899336i	$0.644051\pi$
$240$	0	0
$241$	−0.843024	−0.0543039	−0.0271520	−	0.999631i	$-0.508644\pi$
$241$	−0.843024	−0.0543039	−0.0271520	+	0.999631i	$0.508644\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.74483	0.175361
$246$	0	0
$247$	−6.06814	−0.386107
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.01436	0.0640259	0.0320129	−	0.999487i	$-0.489808\pi$
$251$	1.01436	0.0640259	0.0320129	+	0.999487i	$0.489808\pi$
$252$	0	0
$253$	29.4326	1.85041
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.3945	−1.33455	−0.667276	−	0.744811i	$-0.732540\pi$
$257$	−21.3945	−1.33455	−0.667276	+	0.744811i	$0.732540\pi$
$258$	0	0
$259$	−1.09820	−0.0682387
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.6259	0.840211	0.420106	−	0.907475i	$-0.361993\pi$
$263$	13.6259	0.840211	0.420106	+	0.907475i	$0.361993\pi$
$264$	0	0
$265$	−4.78924	−0.294201
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.19006	0.194501	0.0972507	−	0.995260i	$-0.468995\pi$
$269$	3.19006	0.194501	0.0972507	+	0.995260i	$0.468995\pi$
$270$	0	0
$271$	0.294592	0.0178952	0.00894761	−	0.999960i	$-0.497152\pi$
$271$	0.294592	0.0178952	0.00894761	+	0.999960i	$0.497152\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.67035	−0.522842
$276$	0	0
$277$	17.6259	1.05904	0.529520	−	0.848298i	$-0.322372\pi$
$277$	17.6259	1.05904	0.529520	+	0.848298i	$0.322372\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.3120	0.794126	0.397063	−	0.917791i	$-0.370029\pi$
$281$	13.3120	0.794126	0.397063	+	0.917791i	$0.370029\pi$
$282$	0	0
$283$	−26.7479	−1.58999	−0.794997	−	0.606613i	$-0.792528\pi$
$283$	−26.7479	−1.58999	−0.794997	+	0.606613i	$0.792528\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.21076	0.130497
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.3534	0.663271	0.331636	−	0.943408i	$-0.392400\pi$
$293$	11.3534	0.663271	0.331636	+	0.943408i	$0.392400\pi$
$294$	0	0
$295$	0.765522	0.0445704
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−19.0174	−1.09980
$300$	0	0
$301$	8.48965	0.489335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.09820	0.291922
$306$	0	0
$307$	16.2726	0.928724	0.464362	−	0.885645i	$-0.346284\pi$
$307$	16.2726	0.928724	0.464362	+	0.885645i	$0.346284\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−10.8717	−0.616480	−0.308240	−	0.951309i	$-0.599740\pi$
$311$	−10.8717	−0.616480	−0.308240	+	0.951309i	$0.599740\pi$
$312$	0	0
$313$	0.456568	0.0258068	0.0129034	−	0.999917i	$-0.495893\pi$
$313$	0.456568	0.0258068	0.0129034	+	0.999917i	$0.495893\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.6860	−0.881016	−0.440508	−	0.897749i	$-0.645202\pi$
$317$	−15.6860	−0.881016	−0.440508	+	0.897749i	$0.645202\pi$
$318$	0	0
$319$	−10.1126	−0.566195
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.74483	−0.152726
$324$	0	0
$325$	5.60221	0.310755
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.85738	−0.102401
$330$	0	0
$331$	−25.3564	−1.39371	−0.696857	−	0.717210i	$-0.745419\pi$
$331$	−25.3564	−1.39371	−0.696857	+	0.717210i	$0.745419\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.5528	1.01365
$336$	0	0
$337$	−12.3026	−0.670166	−0.335083	−	0.942189i	$-0.608764\pi$
$337$	−12.3026	−0.670166	−0.335083	+	0.942189i	$0.608764\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.4232	1.80997
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.08384	−0.111866	−0.0559331	−	0.998435i	$-0.517813\pi$
$347$	−2.08384	−0.111866	−0.0559331	+	0.998435i	$0.517813\pi$
$348$	0	0
$349$	−0.608545	−0.0325747	−0.0162873	−	0.999867i	$-0.505185\pi$
$349$	−0.608545	−0.0325747	−0.0162873	+	0.999867i	$0.505185\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−33.0805	−1.76070	−0.880349	−	0.474326i	$-0.842692\pi$
$353$	−33.0805	−1.76070	−0.880349	+	0.474326i	$0.842692\pi$
$354$	0	0
$355$	−22.2676	−1.18184
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.70541	0.248342	0.124171	−	0.992261i	$-0.460373\pi$
$359$	4.70541	0.248342	0.124171	+	0.992261i	$0.460373\pi$
$360$	0	0
$361$	−11.4659	−0.603470
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−27.2044	−1.42394
$366$	0	0
$367$	−19.4295	−1.01421	−0.507107	−	0.861883i	$-0.669285\pi$
$367$	−19.4295	−1.01421	−0.507107	+	0.861883i	$0.669285\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.74483	0.0905869
$372$	0	0
$373$	−6.22012	−0.322065	−0.161033	−	0.986949i	$-0.551482\pi$
$373$	−6.22012	−0.322065	−0.161033	+	0.986949i	$0.551482\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.53407	0.336522
$378$	0	0
$379$	11.9586	0.614272	0.307136	−	0.951666i	$-0.400629\pi$
$379$	11.9586	0.614272	0.307136	+	0.951666i	$0.400629\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	35.1694	1.79707	0.898535	−	0.438901i	$-0.144632\pi$
$383$	35.1694	1.79707	0.898535	+	0.438901i	$0.144632\pi$
$384$	0	0
$385$	9.39145	0.478633
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.44221	−0.174527	−0.0872634	−	0.996185i	$-0.527812\pi$
$389$	−3.44221	−0.174527	−0.0872634	+	0.996185i	$0.527812\pi$
$390$	0	0
$391$	−8.60221	−0.435032
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.9442	0.953189
$396$	0	0
$397$	0.789244	0.0396110	0.0198055	−	0.999804i	$-0.493695\pi$
$397$	0.789244	0.0396110	0.0198055	+	0.999804i	$0.493695\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.1363	1.35512	0.677561	−	0.735467i	$-0.263037\pi$
$401$	27.1363	1.35512	0.677561	+	0.735467i	$0.263037\pi$
$402$	0	0
$403$	−21.5959	−1.07577
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.75750	−0.186252
$408$	0	0
$409$	−32.7272	−1.61825	−0.809127	−	0.587634i	$-0.800060\pi$
$409$	−32.7272	−1.61825	−0.809127	+	0.587634i	$0.800060\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.278896	−0.0137236
$414$	0	0
$415$	−27.4483	−1.34738
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.7305	0.524218	0.262109	−	0.965038i	$-0.415582\pi$
$419$	10.7305	0.524218	0.262109	+	0.965038i	$0.415582\pi$
$420$	0	0
$421$	−23.3564	−1.13832	−0.569161	−	0.822226i	$-0.692732\pi$
$421$	−23.3564	−1.13832	−0.569161	+	0.822226i	$0.692732\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.53407	0.122920
$426$	0	0
$427$	−1.85738	−0.0898851
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.6052	−1.57054	−0.785270	−	0.619154i	$-0.787476\pi$
$431$	−32.6052	−1.57054	−0.785270	+	0.619154i	$0.787476\pi$
$432$	0	0
$433$	−1.36273	−0.0654888	−0.0327444	−	0.999464i	$-0.510425\pi$
$433$	−1.36273	−0.0654888	−0.0327444	+	0.999464i	$0.510425\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.6116	1.12949
$438$	0	0
$439$	28.3263	1.35194	0.675971	−	0.736928i	$-0.263724\pi$
$439$	28.3263	1.35194	0.675971	+	0.736928i	$0.263724\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.9286	−1.61199	−0.805997	−	0.591920i	$-0.798370\pi$
$443$	−33.9286	−1.61199	−0.805997	+	0.591920i	$0.798370\pi$
$444$	0	0
$445$	−18.2438	−0.864841
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.69105	−0.362963	−0.181481	−	0.983394i	$-0.558089\pi$
$449$	−7.69105	−0.362963	−0.181481	+	0.983394i	$0.558089\pi$
$450$	0	0
$451$	7.56413	0.356181
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.06814	−0.284479
$456$	0	0
$457$	−19.4452	−0.909610	−0.454805	−	0.890591i	$-0.650291\pi$
$457$	−19.4452	−0.909610	−0.454805	+	0.890591i	$0.650291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.3263	−0.667244	−0.333622	−	0.942707i	$-0.608271\pi$
$461$	−14.3263	−0.667244	−0.333622	+	0.942707i	$0.608271\pi$
$462$	0	0
$463$	−3.55477	−0.165204	−0.0826020	−	0.996583i	$-0.526323\pi$
$463$	−3.55477	−0.165204	−0.0826020	+	0.996583i	$0.526323\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.48965	−0.439129	−0.219564	−	0.975598i	$-0.570464\pi$
$467$	−9.48965	−0.439129	−0.219564	+	0.975598i	$0.570464\pi$
$468$	0	0
$469$	−6.75919	−0.312110
$470$	0	0
$471$	0	0
$472$	0	0
$473$	29.0474	1.33560
$474$	0	0
$475$	−6.95558	−0.319144
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.1012	−1.87796	−0.938981	−	0.343968i	$-0.888229\pi$
$479$	−41.1012	−1.87796	−0.938981	+	0.343968i	$0.888229\pi$
$480$	0	0
$481$	2.42785	0.110700
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.0681	0.684209
$486$	0	0
$487$	30.9793	1.40381	0.701903	−	0.712272i	$-0.252334\pi$
$487$	30.9793	1.40381	0.701903	+	0.712272i	$0.252334\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.89047	0.220704	0.110352	−	0.993893i	$-0.464802\pi$
$491$	4.89047	0.220704	0.110352	+	0.993893i	$0.464802\pi$
$492$	0	0
$493$	2.95558	0.133113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.11256	0.363898
$498$	0	0
$499$	−0.873081	−0.0390845	−0.0195422	−	0.999809i	$-0.506221\pi$
$499$	−0.873081	−0.0390845	−0.0195422	+	0.999809i	$0.506221\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7.30262	−0.325608	−0.162804	−	0.986658i	$-0.552054\pi$
$503$	−7.30262	−0.325608	−0.162804	+	0.986658i	$0.552054\pi$
$504$	0	0
$505$	−36.8791	−1.64110
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.7829	1.18713	0.593566	−	0.804785i	$-0.297720\pi$
$509$	26.7829	1.18713	0.593566	+	0.804785i	$0.297720\pi$
$510$	0	0
$511$	9.91116	0.438444
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−29.1837	−1.28599
$516$	0	0
$517$	−6.35506	−0.279495
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.2582	−0.624663	−0.312332	−	0.949973i	$-0.601110\pi$
$521$	−14.2582	−0.624663	−0.312332	+	0.949973i	$0.601110\pi$
$522$	0	0
$523$	16.0949	0.703780	0.351890	−	0.936041i	$-0.385539\pi$
$523$	16.0949	0.703780	0.351890	+	0.936041i	$0.385539\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.76855	−0.425525
$528$	0	0
$529$	50.9980	2.21731
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.88744	−0.211699
$534$	0	0
$535$	24.7685	1.07084
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.42151	−0.147375
$540$	0	0
$541$	24.8304	1.06754	0.533770	−	0.845630i	$-0.320775\pi$
$541$	24.8304	1.06754	0.533770	+	0.845630i	$0.320775\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	33.7685	1.44649
$546$	0	0
$547$	−2.10622	−0.0900556	−0.0450278	−	0.998986i	$-0.514338\pi$
$547$	−2.10622	−0.0900556	−0.0450278	+	0.998986i	$0.514338\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.11256	−0.345607
$552$	0	0
$553$	−6.90180	−0.293494
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−42.0648	−1.78234	−0.891172	−	0.453665i	$-0.850116\pi$
$557$	−42.0648	−1.78234	−0.891172	+	0.453665i	$0.850116\pi$
$558$	0	0
$559$	−18.7685	−0.793825
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.64663	0.364412	0.182206	−	0.983260i	$-0.441676\pi$
$563$	8.64663	0.364412	0.182206	+	0.983260i	$0.441676\pi$
$564$	0	0
$565$	−11.7054	−0.492450
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.3120	−0.558067	−0.279034	−	0.960281i	$-0.590014\pi$
$569$	−13.3120	−0.558067	−0.279034	+	0.960281i	$0.590014\pi$
$570$	0	0
$571$	6.36140	0.266216	0.133108	−	0.991102i	$-0.457504\pi$
$571$	6.36140	0.266216	0.133108	+	0.991102i	$0.457504\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−21.7986	−0.909065
$576$	0	0
$577$	2.01436	0.0838589	0.0419295	−	0.999121i	$-0.486650\pi$
$577$	2.01436	0.0838589	0.0419295	+	0.999121i	$0.486650\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.0000	0.414870
$582$	0	0
$583$	5.96994	0.247250
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.0411	1.48758	0.743788	−	0.668416i	$-0.233027\pi$
$587$	36.0411	1.48758	0.743788	+	0.668416i	$0.233027\pi$
$588$	0	0
$589$	26.8130	1.10481
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.7004	0.767934	0.383967	−	0.923347i	$-0.374558\pi$
$593$	18.7004	0.767934	0.383967	+	0.923347i	$0.374558\pi$
$594$	0	0
$595$	−2.74483	−0.112527
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.9586	1.14236	0.571179	−	0.820826i	$-0.306486\pi$
$599$	27.9586	1.14236	0.571179	+	0.820826i	$0.306486\pi$
$600$	0	0
$601$	20.8367	0.849946	0.424973	−	0.905206i	$-0.360284\pi$
$601$	20.8367	0.849946	0.424973	+	0.905206i	$0.360284\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.93989	0.0788676
$606$	0	0
$607$	32.0411	1.30051	0.650254	−	0.759717i	$-0.274662\pi$
$607$	32.0411	1.30051	0.650254	+	0.759717i	$0.274662\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.10622	0.166120
$612$	0	0
$613$	5.64663	0.228065	0.114033	−	0.993477i	$-0.463623\pi$
$613$	5.64663	0.228065	0.114033	+	0.993477i	$0.463623\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.4927	−0.744487	−0.372244	−	0.928135i	$-0.621411\pi$
$617$	−18.4927	−0.744487	−0.372244	+	0.928135i	$0.621411\pi$
$618$	0	0
$619$	4.98564	0.200390	0.100195	−	0.994968i	$-0.468053\pi$
$619$	4.98564	0.200390	0.100195	+	0.994968i	$0.468053\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.64663	0.266292
$624$	0	0
$625$	−31.2488	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.09820	0.0437880
$630$	0	0
$631$	43.7415	1.74132	0.870661	−	0.491883i	$-0.163691\pi$
$631$	43.7415	1.74132	0.870661	+	0.491883i	$0.163691\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	59.7766	2.37216
$636$	0	0
$637$	2.21076	0.0875933
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.64163	0.380821	0.190411	−	0.981705i	$-0.439018\pi$
$641$	9.64163	0.380821	0.190411	+	0.981705i	$0.439018\pi$
$642$	0	0
$643$	1.73349	0.0683622	0.0341811	−	0.999416i	$-0.489118\pi$
$643$	1.73349	0.0683622	0.0341811	+	0.999416i	$0.489118\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.24581	−0.363490	−0.181745	−	0.983346i	$-0.558175\pi$
$647$	−9.24581	−0.363490	−0.181745	+	0.983346i	$0.558175\pi$
$648$	0	0
$649$	−0.954247	−0.0374575
$650$	0	0
$651$	0	0
$652$	0	0
$653$	43.8855	1.71737	0.858686	−	0.512503i	$-0.171282\pi$
$653$	43.8855	1.71737	0.858686	+	0.512503i	$0.171282\pi$
$654$	0	0
$655$	33.5815	1.31214
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−26.1663	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$659$	−26.1663	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$660$	0	0
$661$	−2.41215	−0.0938218	−0.0469109	−	0.998899i	$-0.514938\pi$
$661$	−2.41215	−0.0938218	−0.0469109	+	0.998899i	$0.514938\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.53407	0.292159
$666$	0	0
$667$	−25.4245	−0.984442
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.35506	−0.245334
$672$	0	0
$673$	−50.1837	−1.93444	−0.967220	−	0.253939i	$-0.918274\pi$
$673$	−50.1837	−1.93444	−0.967220	+	0.253939i	$0.918274\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.5828	0.906363	0.453181	−	0.891418i	$-0.350289\pi$
$677$	23.5828	0.906363	0.453181	+	0.891418i	$0.350289\pi$
$678$	0	0
$679$	−5.48965	−0.210674
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.80361	0.260333	0.130166	−	0.991492i	$-0.458449\pi$
$683$	6.80361	0.260333	0.130166	+	0.991492i	$0.458449\pi$
$684$	0	0
$685$	−48.7065	−1.86098
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.85738	−0.146955
$690$	0	0
$691$	19.3183	0.734903	0.367452	−	0.930043i	$-0.380230\pi$
$691$	19.3183	0.734903	0.367452	+	0.930043i	$0.380230\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	38.8016	1.47183
$696$	0	0
$697$	−2.21076	−0.0837384
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.2645	−1.02977	−0.514884	−	0.857260i	$-0.672165\pi$
$701$	−27.2645	−1.02977	−0.514884	+	0.857260i	$0.672165\pi$
$702$	0	0
$703$	−3.01436	−0.113689
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.4359	0.505308
$708$	0	0
$709$	47.9586	1.80112	0.900562	−	0.434728i	$-0.143156\pi$
$709$	47.9586	1.80112	0.900562	+	0.434728i	$0.143156\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	84.0311	3.14699
$714$	0	0
$715$	−20.7622	−0.776463
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.2345	0.903794	0.451897	−	0.892070i	$-0.350748\pi$
$719$	24.2345	0.903794	0.451897	+	0.892070i	$0.350748\pi$
$720$	0	0
$721$	10.6323	0.395966
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.48965	0.278159
$726$	0	0
$727$	−24.2852	−0.900689	−0.450345	−	0.892855i	$-0.648699\pi$
$727$	−24.2852	−0.900689	−0.450345	+	0.892855i	$0.648699\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.48965	−0.314001
$732$	0	0
$733$	−6.44023	−0.237875	−0.118938	−	0.992902i	$-0.537949\pi$
$733$	−6.44023	−0.237875	−0.118938	+	0.992902i	$0.537949\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23.1266	−0.851881
$738$	0	0
$739$	−28.9112	−1.06351	−0.531757	−	0.846897i	$-0.678468\pi$
$739$	−28.9112	−1.06351	−0.531757	+	0.846897i	$0.678468\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.1126	−0.884604	−0.442302	−	0.896866i	$-0.645838\pi$
$743$	−24.1126	−0.884604	−0.442302	+	0.896866i	$0.645838\pi$
$744$	0	0
$745$	−23.1156	−0.846890
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.02372	−0.329720
$750$	0	0
$751$	18.6640	0.681060	0.340530	−	0.940234i	$-0.389394\pi$
$751$	18.6640	0.681060	0.340530	+	0.940234i	$0.389394\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.8855	−0.832887
$756$	0	0
$757$	1.39779	0.0508035	0.0254018	−	0.999677i	$-0.491913\pi$
$757$	1.39779	0.0508035	0.0254018	+	0.999677i	$0.491913\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.9526	0.650780	0.325390	−	0.945580i	$-0.394504\pi$
$761$	17.9526	0.650780	0.325390	+	0.945580i	$0.394504\pi$
$762$	0	0
$763$	−12.3026	−0.445385
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.616572	0.0222631
$768$	0	0
$769$	−3.33901	−0.120408	−0.0602039	−	0.998186i	$-0.519175\pi$
$769$	−3.33901	−0.120408	−0.0602039	+	0.998186i	$0.519175\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.1299	1.22757	0.613784	−	0.789474i	$-0.289646\pi$
$773$	34.1299	1.22757	0.613784	+	0.789474i	$0.289646\pi$
$774$	0	0
$775$	−24.7542	−0.889197
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.06814	0.217414
$780$	0	0
$781$	27.7572	0.993231
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.4452	0.765413
$786$	0	0
$787$	49.1280	1.75122	0.875612	−	0.483016i	$-0.160459\pi$
$787$	49.1280	1.75122	0.875612	+	0.483016i	$0.160459\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.26454	0.151629
$792$	0	0
$793$	4.10622	0.145816
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.1443	1.74078	0.870390	−	0.492362i	$-0.163866\pi$
$797$	49.1443	1.74078	0.870390	+	0.492362i	$0.163866\pi$
$798$	0	0
$799$	1.85738	0.0657095
$800$	0	0
$801$	0	0
$802$	0	0
$803$	33.9112	1.19670
$804$	0	0
$805$	23.6116	0.832199
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43.1236	1.51615	0.758073	−	0.652170i	$-0.226141\pi$
$809$	43.1236	1.51615	0.758073	+	0.652170i	$0.226141\pi$
$810$	0	0
$811$	4.51035	0.158380	0.0791899	−	0.996860i	$-0.474767\pi$
$811$	4.51035	0.158380	0.0791899	+	0.996860i	$0.474767\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.2251	0.568341
$816$	0	0
$817$	23.3026	0.815255
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.3170	−0.848668	−0.424334	−	0.905506i	$-0.639492\pi$
$821$	−24.3170	−0.848668	−0.424334	+	0.905506i	$0.639492\pi$
$822$	0	0
$823$	−15.5384	−0.541636	−0.270818	−	0.962631i	$-0.587294\pi$
$823$	−15.5384	−0.541636	−0.270818	+	0.962631i	$0.587294\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−51.2124	−1.78083	−0.890416	−	0.455148i	$-0.849586\pi$
$827$	−51.2124	−1.78083	−0.890416	+	0.455148i	$0.849586\pi$
$828$	0	0
$829$	−1.33465	−0.0463543	−0.0231771	−	0.999731i	$-0.507378\pi$
$829$	−1.33465	−0.0463543	−0.0231771	+	0.999731i	$0.507378\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−19.6052	−0.678467
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38.4733	−1.32825	−0.664123	−	0.747623i	$-0.731195\pi$
$839$	−38.4733	−1.32825	−0.664123	+	0.747623i	$0.731195\pi$
$840$	0	0
$841$	−20.2645	−0.698777
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.2676	−0.766027
$846$	0	0
$847$	−0.706743	−0.0242840
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.44693	−0.323836
$852$	0	0
$853$	38.4339	1.31595	0.657976	−	0.753039i	$-0.271413\pi$
$853$	38.4339	1.31595	0.657976	+	0.753039i	$0.271413\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.8016	1.25712	0.628560	−	0.777761i	$-0.283645\pi$
$857$	36.8016	1.25712	0.628560	+	0.777761i	$0.283645\pi$
$858$	0	0
$859$	31.2519	1.06630	0.533150	−	0.846021i	$-0.321008\pi$
$859$	31.2519	1.06630	0.533150	+	0.846021i	$0.321008\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.3420	1.54346	0.771730	−	0.635950i	$-0.219391\pi$
$863$	45.3420	1.54346	0.771730	+	0.635950i	$0.219391\pi$
$864$	0	0
$865$	8.93489	0.303795
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.6146	−0.801070
$870$	0	0
$871$	14.9429	0.506321
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.76855	0.228819
$876$	0	0
$877$	−30.6767	−1.03588	−0.517939	−	0.855418i	$-0.673301\pi$
$877$	−30.6767	−1.03588	−0.517939	+	0.855418i	$0.673301\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.5992	0.694004	0.347002	−	0.937864i	$-0.387200\pi$
$881$	20.5992	0.694004	0.347002	+	0.937864i	$0.387200\pi$
$882$	0	0
$883$	32.9823	1.10994	0.554972	−	0.831869i	$-0.312729\pi$
$883$	32.9823	1.10994	0.554972	+	0.831869i	$0.312729\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.9997	−1.37664	−0.688318	−	0.725409i	$-0.741651\pi$
$887$	−40.9997	−1.37664	−0.688318	+	0.725409i	$0.741651\pi$
$888$	0	0
$889$	−21.7779	−0.730408
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.09820	−0.170605
$894$	0	0
$895$	−4.57652	−0.152976
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−28.8717	−0.962927
$900$	0	0
$901$	−1.74483	−0.0581286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.7642	0.557260
$906$	0	0
$907$	12.2852	0.407924	0.203962	−	0.978979i	$-0.434618\pi$
$907$	12.2852	0.407924	0.203962	+	0.978979i	$0.434618\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.5371	−1.47558	−0.737790	−	0.675030i	$-0.764130\pi$
$911$	−44.5371	−1.47558	−0.737790	+	0.675030i	$0.764130\pi$
$912$	0	0
$913$	34.2151	1.13236
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.2345	−0.404018
$918$	0	0
$919$	30.7809	1.01537	0.507685	−	0.861543i	$-0.330501\pi$
$919$	30.7809	1.01537	0.507685	+	0.861543i	$0.330501\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−17.9349	−0.590334
$924$	0	0
$925$	2.78291	0.0915015
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.1280	−0.922849	−0.461424	−	0.887180i	$-0.652661\pi$
$929$	−28.1280	−0.922849	−0.461424	+	0.887180i	$0.652661\pi$
$930$	0	0
$931$	−2.74483	−0.0899580
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.39145	−0.307133
$936$	0	0
$937$	−58.2262	−1.90217	−0.951083	−	0.308935i	$-0.900027\pi$
$937$	−58.2262	−1.90217	−0.951083	+	0.308935i	$0.900027\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.91116	0.127500	0.0637501	−	0.997966i	$-0.479694\pi$
$941$	3.91116	0.127500	0.0637501	+	0.997966i	$0.479694\pi$
$942$	0	0
$943$	19.0174	0.619291
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.6577	1.32120	0.660599	−	0.750739i	$-0.270303\pi$
$947$	40.6577	1.32120	0.660599	+	0.750739i	$0.270303\pi$
$948$	0	0
$949$	−21.9112	−0.711266
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.8003	1.19208	0.596039	−	0.802955i	$-0.296740\pi$
$953$	36.8003	1.19208	0.596039	+	0.802955i	$0.296740\pi$
$954$	0	0
$955$	−3.17570	−0.102763
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.7448	0.573010
$960$	0	0
$961$	64.4245	2.07821
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−21.2582	−0.684326
$966$	0	0
$967$	14.5321	0.467321	0.233660	−	0.972318i	$-0.424930\pi$
$967$	14.5321	0.467321	0.233660	+	0.972318i	$0.424930\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.93989	−0.126437	−0.0632185	−	0.998000i	$-0.520136\pi$
$971$	−3.93989	−0.126437	−0.0632185	+	0.998000i	$0.520136\pi$
$972$	0	0
$973$	−14.1363	−0.453188
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.2706	1.70428	0.852139	−	0.523316i	$-0.175305\pi$
$977$	53.2706	1.70428	0.852139	+	0.523316i	$0.175305\pi$
$978$	0	0
$979$	22.7415	0.726822
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.6383	1.04100	0.520500	−	0.853861i	$-0.325745\pi$
$983$	32.6383	1.04100	0.520500	+	0.853861i	$0.325745\pi$
$984$	0	0
$985$	−31.9680	−1.01858
$986$	0	0
$987$	0	0
$988$	0	0
$989$	73.0298	2.32221
$990$	0	0
$991$	−42.1423	−1.33870	−0.669348	−	0.742949i	$-0.733426\pi$
$991$	−42.1423	−1.33870	−0.669348	+	0.742949i	$0.733426\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	35.1043	1.11288
$996$	0	0
$997$	35.5371	1.12547	0.562736	−	0.826637i	$-0.309749\pi$
$997$	35.5371	1.12547	0.562736	+	0.826637i	$0.309749\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4284.2.a.t.1.3	✓	3
3.2	odd	2		4284.2.a.u.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4284.2.a.t.1.3	✓	3	1.1	even	1	trivial
4284.2.a.u.1.1	yes	3	3.2	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:	\( N \)	\(=\)	\( 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4284.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74483 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+2.74483 q^{5} -1.00000 q^{7} -3.42151 q^{11} +2.21076 q^{13} +1.00000 q^{17} -2.74483 q^{19} -8.60221 q^{23} +2.53407 q^{25} +2.95558 q^{29} -9.76855 q^{31} -2.74483 q^{35} +1.09820 q^{37} -2.21076 q^{41} -8.48965 q^{43} +1.85738 q^{47} +1.00000 q^{49} -1.74483 q^{53} -9.39145 q^{55} +0.278896 q^{59} +1.85738 q^{61} +6.06814 q^{65} +6.75919 q^{67} -8.11256 q^{71} -9.91116 q^{73} +3.42151 q^{77} +6.90180 q^{79} -10.0000 q^{83} +2.74483 q^{85} -6.64663 q^{89} -2.21076 q^{91} -7.53407 q^{95} +5.48965 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q - q^5 - 3 * q^7	\( 3 q - q^{5} - 3 q^{7} - 7 q^{11} + 5 q^{13} + 3 q^{17} + q^{19} - 3 q^{23} - 2 q^{29} + 6 q^{31} + q^{35} + 6 q^{37} - 5 q^{41} - 7 q^{43} - 8 q^{47} + 3 q^{49} + 4 q^{53} - 7 q^{55} - 16 q^{59} - 8 q^{61} + 3 q^{65} + 4 q^{67} - 20 q^{71} - 8 q^{73} + 7 q^{77} + 18 q^{79} - 30 q^{83} - q^{85} - 8 q^{89} - 5 q^{91} - 15 q^{95} - 2 q^{97}+O(q^{100}) \) 3 * q - q^5 - 3 * q^7 - 7 * q^11 + 5 * q^13 + 3 * q^17 + q^19 - 3 * q^23 - 2 * q^29 + 6 * q^31 + q^35 + 6 * q^37 - 5 * q^41 - 7 * q^43 - 8 * q^47 + 3 * q^49 + 4 * q^53 - 7 * q^55 - 16 * q^59 - 8 * q^61 + 3 * q^65 + 4 * q^67 - 20 * q^71 - 8 * q^73 + 7 * q^77 + 18 * q^79 - 30 * q^83 - q^85 - 8 * q^89 - 5 * q^91 - 15 * q^95 - 2 * q^97

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