LMFDB - Embedded newform 6030.2.a.bu.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(1,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, 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("6030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.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:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.70292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 10x + 2 $ x^4 - x^3 - 8x^2 + 10x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2010)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.88727$ of defining polynomial
Character	$\chi$	$=$	6030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.33630 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.33630 q^{7} +1.00000 q^{8} +1.00000 q^{10} +4.74553 q^{11} +0.409227 q^{13} +4.33630 q^{14} +1.00000 q^{16} -5.77453 q^{17} -5.77453 q^{19} +1.00000 q^{20} +4.74553 q^{22} +9.36530 q^{23} +1.00000 q^{25} +0.409227 q^{26} +4.33630 q^{28} +1.30730 q^{29} +2.40923 q^{31} +1.00000 q^{32} -5.77453 q^{34} +4.33630 q^{35} +3.02900 q^{37} -5.77453 q^{38} +1.00000 q^{40} -1.71653 q^{41} -9.49106 q^{43} +4.74553 q^{44} +9.36530 q^{46} +1.51115 q^{47} +11.8035 q^{49} +1.00000 q^{50} +0.409227 q^{52} +10.6726 q^{53} +4.74553 q^{55} +4.33630 q^{56} +1.30730 q^{58} -1.71653 q^{59} +6.33630 q^{61} +2.40923 q^{62} +1.00000 q^{64} +0.409227 q^{65} -1.00000 q^{67} -5.77453 q^{68} +4.33630 q^{70} -8.11084 q^{71} -8.05800 q^{73} +3.02900 q^{74} -5.77453 q^{76} +20.5781 q^{77} +7.64878 q^{79} +1.00000 q^{80} -1.71653 q^{82} -8.57806 q^{83} -5.77453 q^{85} -9.49106 q^{86} +4.74553 q^{88} +9.08700 q^{89} +1.77453 q^{91} +9.36530 q^{92} +1.51115 q^{94} -5.77453 q^{95} -4.74553 q^{97} +11.8035 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 2 q^{13} + q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{19} + 4 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 2 q^{26} + q^{28} - 2 q^{29} + 10 q^{31} + 4 q^{32} + 2 q^{34} + q^{35} + 3 q^{37} + 2 q^{38} + 4 q^{40} - 6 q^{43} + 3 q^{44} + 12 q^{46} + 14 q^{47} + 13 q^{49} + 4 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} + q^{56} - 2 q^{58} + 9 q^{61} + 10 q^{62} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + q^{70} + 9 q^{71} - 14 q^{73} + 3 q^{74} + 2 q^{76} + 23 q^{77} + 12 q^{79} + 4 q^{80} + 25 q^{83} + 2 q^{85} - 6 q^{86} + 3 q^{88} + 9 q^{89} - 18 q^{91} + 12 q^{92} + 14 q^{94} + 2 q^{95} - 3 q^{97} + 13 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 2 * q^13 + q^14 + 4 * q^16 + 2 * q^17 + 2 * q^19 + 4 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 2 * q^26 + q^28 - 2 * q^29 + 10 * q^31 + 4 * q^32 + 2 * q^34 + q^35 + 3 * q^37 + 2 * q^38 + 4 * q^40 - 6 * q^43 + 3 * q^44 + 12 * q^46 + 14 * q^47 + 13 * q^49 + 4 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 + q^56 - 2 * q^58 + 9 * q^61 + 10 * q^62 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 2 * q^68 + q^70 + 9 * q^71 - 14 * q^73 + 3 * q^74 + 2 * q^76 + 23 * q^77 + 12 * q^79 + 4 * q^80 + 25 * q^83 + 2 * q^85 - 6 * q^86 + 3 * q^88 + 9 * q^89 - 18 * q^91 + 12 * q^92 + 14 * q^94 + 2 * q^95 - 3 * q^97 + 13 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.33630	1.63897	0.819484	−	0.573101i	$-0.194260\pi$
$7$	4.33630	1.63897	0.819484	+	0.573101i	$0.194260\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	4.74553	1.43083	0.715416	−	0.698699i	$-0.246237\pi$
$11$	4.74553	1.43083	0.715416	+	0.698699i	$0.246237\pi$
$12$	0	0
$13$	0.409227	0.113499	0.0567495	−	0.998388i	$-0.481926\pi$
$13$	0.409227	0.113499	0.0567495	+	0.998388i	$0.481926\pi$
$14$	4.33630	1.15893
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.77453	−1.40053	−0.700265	−	0.713883i	$-0.746935\pi$
$17$	−5.77453	−1.40053	−0.700265	+	0.713883i	$0.746935\pi$
$18$	0	0
$19$	−5.77453	−1.32477	−0.662384	−	0.749164i	$-0.730455\pi$
$19$	−5.77453	−1.32477	−0.662384	+	0.749164i	$0.730455\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	4.74553	1.01175
$23$	9.36530	1.95280	0.976401	−	0.215968i	$-0.0692906\pi$
$23$	9.36530	1.95280	0.976401	+	0.215968i	$0.0692906\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.409227	0.0802560
$27$	0	0
$28$	4.33630	0.819484
$29$	1.30730	0.242760	0.121380	−	0.992606i	$-0.461268\pi$
$29$	1.30730	0.242760	0.121380	+	0.992606i	$0.461268\pi$
$30$	0	0
$31$	2.40923	0.432710	0.216355	−	0.976315i	$-0.430583\pi$
$31$	2.40923	0.432710	0.216355	+	0.976315i	$0.430583\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.77453	−0.990324
$35$	4.33630	0.732969
$36$	0	0
$37$	3.02900	0.497965	0.248982	−	0.968508i	$-0.419904\pi$
$37$	3.02900	0.497965	0.248982	+	0.968508i	$0.419904\pi$
$38$	−5.77453	−0.936753
$39$	0	0
$40$	1.00000	0.158114
$41$	−1.71653	−0.268077	−0.134038	−	0.990976i	$-0.542795\pi$
$41$	−1.71653	−0.268077	−0.134038	+	0.990976i	$0.542795\pi$
$42$	0	0
$43$	−9.49106	−1.44737	−0.723687	−	0.690129i	$-0.757554\pi$
$43$	−9.49106	−1.44737	−0.723687	+	0.690129i	$0.757554\pi$
$44$	4.74553	0.715416
$45$	0	0
$46$	9.36530	1.38084
$47$	1.51115	0.220424	0.110212	−	0.993908i	$-0.464847\pi$
$47$	1.51115	0.220424	0.110212	+	0.993908i	$0.464847\pi$
$48$	0	0
$49$	11.8035	1.68622
$50$	1.00000	0.141421
$51$	0	0
$52$	0.409227	0.0567495
$53$	10.6726	1.46600	0.732998	−	0.680231i	$-0.238121\pi$
$53$	10.6726	1.46600	0.732998	+	0.680231i	$0.238121\pi$
$54$	0	0
$55$	4.74553	0.639887
$56$	4.33630	0.579463
$57$	0	0
$58$	1.30730	0.171657
$59$	−1.71653	−0.223473	−0.111737	−	0.993738i	$-0.535641\pi$
$59$	−1.71653	−0.223473	−0.111737	+	0.993738i	$0.535641\pi$
$60$	0	0
$61$	6.33630	0.811281	0.405640	−	0.914033i	$-0.367049\pi$
$61$	6.33630	0.811281	0.405640	+	0.914033i	$0.367049\pi$
$62$	2.40923	0.305972
$63$	0	0
$64$	1.00000	0.125000
$65$	0.409227	0.0507583
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	−5.77453	−0.700265
$69$	0	0
$70$	4.33630	0.518287
$71$	−8.11084	−0.962579	−0.481290	−	0.876562i	$-0.659831\pi$
$71$	−8.11084	−0.962579	−0.481290	+	0.876562i	$0.659831\pi$
$72$	0	0
$73$	−8.05800	−0.943118	−0.471559	−	0.881835i	$-0.656309\pi$
$73$	−8.05800	−0.943118	−0.471559	+	0.881835i	$0.656309\pi$
$74$	3.02900	0.352114
$75$	0	0
$76$	−5.77453	−0.662384
$77$	20.5781	2.34509
$78$	0	0
$79$	7.64878	0.860554	0.430277	−	0.902697i	$-0.358416\pi$
$79$	7.64878	0.860554	0.430277	+	0.902697i	$0.358416\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−1.71653	−0.189559
$83$	−8.57806	−0.941565	−0.470782	−	0.882249i	$-0.656028\pi$
$83$	−8.57806	−0.941565	−0.470782	+	0.882249i	$0.656028\pi$
$84$	0	0
$85$	−5.77453	−0.626336
$86$	−9.49106	−1.02345
$87$	0	0
$88$	4.74553	0.505875
$89$	9.08700	0.963220	0.481610	−	0.876386i	$-0.340052\pi$
$89$	9.08700	0.963220	0.481610	+	0.876386i	$0.340052\pi$
$90$	0	0
$91$	1.77453	0.186021
$92$	9.36530	0.976401
$93$	0	0
$94$	1.51115	0.155863
$95$	−5.77453	−0.592454
$96$	0	0
$97$	−4.74553	−0.481836	−0.240918	−	0.970546i	$-0.577448\pi$
$97$	−4.74553	−0.481836	−0.240918	+	0.970546i	$0.577448\pi$
$98$	11.8035	1.19234
$99$	0	0
$100$	1.00000	0.100000
$101$	−8.52006	−0.847778	−0.423889	−	0.905714i	$-0.639335\pi$
$101$	−8.52006	−0.847778	−0.423889	+	0.905714i	$0.639335\pi$
$102$	0	0
$103$	−13.4911	−1.32931	−0.664657	−	0.747149i	$-0.731422\pi$
$103$	−13.4911	−1.32931	−0.664657	+	0.747149i	$0.731422\pi$
$104$	0.409227	0.0401280
$105$	0	0
$106$	10.6726	1.03662
$107$	8.81845	0.852512	0.426256	−	0.904603i	$-0.359832\pi$
$107$	8.81845	0.852512	0.426256	+	0.904603i	$0.359832\pi$
$108$	0	0
$109$	−17.8854	−1.71311	−0.856554	−	0.516058i	$-0.827399\pi$
$109$	−17.8854	−1.71311	−0.856554	+	0.516058i	$0.827399\pi$
$110$	4.74553	0.452469
$111$	0	0
$112$	4.33630	0.409742
$113$	3.86908	0.363972	0.181986	−	0.983301i	$-0.441747\pi$
$113$	3.86908	0.363972	0.181986	+	0.983301i	$0.441747\pi$
$114$	0	0
$115$	9.36530	0.873319
$116$	1.30730	0.121380
$117$	0	0
$118$	−1.71653	−0.158019
$119$	−25.0401	−2.29542
$120$	0	0
$121$	11.5201	1.04728
$122$	6.33630	0.573662
$123$	0	0
$124$	2.40923	0.216355
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.23955	−0.819877	−0.409939	−	0.912113i	$-0.634450\pi$
$127$	−9.23955	−0.819877	−0.409939	+	0.912113i	$0.634450\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.409227	0.0358916
$131$	18.9561	1.65620	0.828100	−	0.560580i	$-0.189422\pi$
$131$	18.9561	1.65620	0.828100	+	0.560580i	$0.189422\pi$
$132$	0	0
$133$	−25.0401	−2.17125
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	−5.77453	−0.495162
$137$	0.970999	0.0829581	0.0414790	−	0.999139i	$-0.486793\pi$
$137$	0.970999	0.0829581	0.0414790	+	0.999139i	$0.486793\pi$
$138$	0	0
$139$	9.41144	0.798268	0.399134	−	0.916893i	$-0.369311\pi$
$139$	9.41144	0.798268	0.399134	+	0.916893i	$0.369311\pi$
$140$	4.33630	0.366485
$141$	0	0
$142$	−8.11084	−0.680646
$143$	1.94200	0.162398
$144$	0	0
$145$	1.30730	0.108566
$146$	−8.05800	−0.666885
$147$	0	0
$148$	3.02900	0.248982
$149$	1.30730	0.107098	0.0535492	−	0.998565i	$-0.482947\pi$
$149$	1.30730	0.107098	0.0535492	+	0.998565i	$0.482947\pi$
$150$	0	0
$151$	−15.3385	−1.24823	−0.624115	−	0.781332i	$-0.714540\pi$
$151$	−15.3385	−1.24823	−0.624115	+	0.781332i	$0.714540\pi$
$152$	−5.77453	−0.468376
$153$	0	0
$154$	20.5781	1.65823
$155$	2.40923	0.193514
$156$	0	0
$157$	−7.79615	−0.622201	−0.311100	−	0.950377i	$-0.600698\pi$
$157$	−7.79615	−0.622201	−0.311100	+	0.950377i	$0.600698\pi$
$158$	7.64878	0.608504
$159$	0	0
$160$	1.00000	0.0790569
$161$	40.6108	3.20058
$162$	0	0
$163$	−6.97100	−0.546011	−0.273005	−	0.962013i	$-0.588018\pi$
$163$	−6.97100	−0.546011	−0.273005	+	0.962013i	$0.588018\pi$
$164$	−1.71653	−0.134038
$165$	0	0
$166$	−8.57806	−0.665787
$167$	−0.546851	−0.0423166	−0.0211583	−	0.999776i	$-0.506735\pi$
$167$	−0.546851	−0.0423166	−0.0211583	+	0.999776i	$0.506735\pi$
$168$	0	0
$169$	−12.8325	−0.987118
$170$	−5.77453	−0.442886
$171$	0	0
$172$	−9.49106	−0.723687
$173$	16.3147	1.24038	0.620191	−	0.784451i	$-0.287055\pi$
$173$	16.3147	1.24038	0.620191	+	0.784451i	$0.287055\pi$
$174$	0	0
$175$	4.33630	0.327794
$176$	4.74553	0.357708
$177$	0	0
$178$	9.08700	0.681100
$179$	1.92038	0.143536	0.0717679	−	0.997421i	$-0.477136\pi$
$179$	1.92038	0.143536	0.0717679	+	0.997421i	$0.477136\pi$
$180$	0	0
$181$	17.4032	1.29357	0.646785	−	0.762672i	$-0.276113\pi$
$181$	17.4032	1.29357	0.646785	+	0.762672i	$0.276113\pi$
$182$	1.77453	0.131537
$183$	0	0
$184$	9.36530	0.690419
$185$	3.02900	0.222697
$186$	0	0
$187$	−27.4032	−2.00392
$188$	1.51115	0.110212
$189$	0	0
$190$	−5.77453	−0.418929
$191$	2.50894	0.181540	0.0907702	−	0.995872i	$-0.471067\pi$
$191$	2.50894	0.181540	0.0907702	+	0.995872i	$0.471067\pi$
$192$	0	0
$193$	25.9382	1.86707	0.933536	−	0.358483i	$-0.116706\pi$
$193$	25.9382	1.86707	0.933536	+	0.358483i	$0.116706\pi$
$194$	−4.74553	−0.340709
$195$	0	0
$196$	11.8035	0.843109
$197$	1.18155	0.0841817	0.0420909	−	0.999114i	$-0.486598\pi$
$197$	1.18155	0.0841817	0.0420909	+	0.999114i	$0.486598\pi$
$198$	0	0
$199$	−16.5781	−1.17519	−0.587594	−	0.809156i	$-0.699925\pi$
$199$	−16.5781	−1.17519	−0.587594	+	0.809156i	$0.699925\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−8.52006	−0.599470
$203$	5.66886	0.397876
$204$	0	0
$205$	−1.71653	−0.119888
$206$	−13.4911	−0.939967
$207$	0	0
$208$	0.409227	0.0283748
$209$	−27.4032	−1.89552
$210$	0	0
$211$	−25.0617	−1.72532	−0.862661	−	0.505783i	$-0.831204\pi$
$211$	−25.0617	−1.72532	−0.862661	+	0.505783i	$0.831204\pi$
$212$	10.6726	0.732998
$213$	0	0
$214$	8.81845	0.602817
$215$	−9.49106	−0.647285
$216$	0	0
$217$	10.4471	0.709198
$218$	−17.8854	−1.21135
$219$	0	0
$220$	4.74553	0.319944
$221$	−2.36309	−0.158959
$222$	0	0
$223$	−25.8586	−1.73162	−0.865809	−	0.500374i	$-0.833196\pi$
$223$	−25.8586	−1.73162	−0.865809	+	0.500374i	$0.833196\pi$
$224$	4.33630	0.289731
$225$	0	0
$226$	3.86908	0.257367
$227$	−17.7127	−1.17564	−0.587818	−	0.808993i	$-0.700013\pi$
$227$	−17.7127	−1.17564	−0.587818	+	0.808993i	$0.700013\pi$
$228$	0	0
$229$	−15.8274	−1.04590	−0.522951	−	0.852363i	$-0.675169\pi$
$229$	−15.8274	−1.04590	−0.522951	+	0.852363i	$0.675169\pi$
$230$	9.36530	0.617530
$231$	0	0
$232$	1.30730	0.0858287
$233$	22.4322	1.46958	0.734792	−	0.678293i	$-0.237280\pi$
$233$	22.4322	1.46958	0.734792	+	0.678293i	$0.237280\pi$
$234$	0	0
$235$	1.51115	0.0985766
$236$	−1.71653	−0.111737
$237$	0	0
$238$	−25.0401	−1.62311
$239$	−9.23955	−0.597657	−0.298828	−	0.954307i	$-0.596596\pi$
$239$	−9.23955	−0.597657	−0.298828	+	0.954307i	$0.596596\pi$
$240$	0	0
$241$	6.17859	0.397998	0.198999	−	0.980000i	$-0.436231\pi$
$241$	6.17859	0.397998	0.198999	+	0.980000i	$0.436231\pi$
$242$	11.5201	0.740538
$243$	0	0
$244$	6.33630	0.405640
$245$	11.8035	0.754100
$246$	0	0
$247$	−2.36309	−0.150360
$248$	2.40923	0.152986
$249$	0	0
$250$	1.00000	0.0632456
$251$	6.97100	0.440006	0.220003	−	0.975499i	$-0.429393\pi$
$251$	6.97100	0.440006	0.220003	+	0.975499i	$0.429393\pi$
$252$	0	0
$253$	44.4433	2.79413
$254$	−9.23955	−0.579741
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.7090	−0.917522	−0.458761	−	0.888560i	$-0.651707\pi$
$257$	−14.7090	−0.917522	−0.458761	+	0.888560i	$0.651707\pi$
$258$	0	0
$259$	13.1347	0.816149
$260$	0.409227	0.0253792
$261$	0	0
$262$	18.9561	1.17111
$263$	−8.79836	−0.542530	−0.271265	−	0.962505i	$-0.587442\pi$
$263$	−8.79836	−0.542530	−0.271265	+	0.962505i	$0.587442\pi$
$264$	0	0
$265$	10.6726	0.655613
$266$	−25.0401	−1.53531
$267$	0	0
$268$	−1.00000	−0.0610847
$269$	−28.8564	−1.75940	−0.879702	−	0.475526i	$-0.842258\pi$
$269$	−28.8564	−1.75940	−0.879702	+	0.475526i	$0.842258\pi$
$270$	0	0
$271$	13.8124	0.839046	0.419523	−	0.907745i	$-0.362197\pi$
$271$	13.8124	0.839046	0.419523	+	0.907745i	$0.362197\pi$
$272$	−5.77453	−0.350132
$273$	0	0
$274$	0.970999	0.0586602
$275$	4.74553	0.286166
$276$	0	0
$277$	−3.28051	−0.197107	−0.0985535	−	0.995132i	$-0.531422\pi$
$277$	−3.28051	−0.197107	−0.0985535	+	0.995132i	$0.531422\pi$
$278$	9.41144	0.564461
$279$	0	0
$280$	4.33630	0.259144
$281$	−0.363093	−0.0216603	−0.0108302	−	0.999941i	$-0.503447\pi$
$281$	−0.363093	−0.0216603	−0.0108302	+	0.999941i	$0.503447\pi$
$282$	0	0
$283$	12.1570	0.722657	0.361328	−	0.932439i	$-0.382323\pi$
$283$	12.1570	0.722657	0.361328	+	0.932439i	$0.382323\pi$
$284$	−8.11084	−0.481290
$285$	0	0
$286$	1.94200	0.114833
$287$	−7.44340	−0.439370
$288$	0	0
$289$	16.3452	0.961483
$290$	1.30730	0.0767675
$291$	0	0
$292$	−8.05800	−0.471559
$293$	0.871287	0.0509012	0.0254506	−	0.999676i	$-0.491898\pi$
$293$	0.871287	0.0509012	0.0254506	+	0.999676i	$0.491898\pi$
$294$	0	0
$295$	−1.71653	−0.0999402
$296$	3.02900	0.176057
$297$	0	0
$298$	1.30730	0.0757300
$299$	3.83253	0.221641
$300$	0	0
$301$	−41.1561	−2.37220
$302$	−15.3385	−0.882632
$303$	0	0
$304$	−5.77453	−0.331192
$305$	6.33630	0.362816
$306$	0	0
$307$	26.4583	1.51005	0.755026	−	0.655694i	$-0.227624\pi$
$307$	26.4583	1.51005	0.755026	+	0.655694i	$0.227624\pi$
$308$	20.5781	1.17254
$309$	0	0
$310$	2.40923	0.136835
$311$	26.1637	1.48361	0.741803	−	0.670618i	$-0.233971\pi$
$311$	26.1637	1.48361	0.741803	+	0.670618i	$0.233971\pi$
$312$	0	0
$313$	12.4404	0.703175	0.351588	−	0.936155i	$-0.385642\pi$
$313$	12.4404	0.703175	0.351588	+	0.936155i	$0.385642\pi$
$314$	−7.79615	−0.439962
$315$	0	0
$316$	7.64878	0.430277
$317$	31.2558	1.75550	0.877751	−	0.479116i	$-0.159043\pi$
$317$	31.2558	1.75550	0.877751	+	0.479116i	$0.159043\pi$
$318$	0	0
$319$	6.20385	0.347349
$320$	1.00000	0.0559017
$321$	0	0
$322$	40.6108	2.26315
$323$	33.3452	1.85538
$324$	0	0
$325$	0.409227	0.0226998
$326$	−6.97100	−0.386088
$327$	0	0
$328$	−1.71653	−0.0947795
$329$	6.55281	0.361268
$330$	0	0
$331$	28.6041	1.57222	0.786112	−	0.618084i	$-0.212091\pi$
$331$	28.6041	1.57222	0.786112	+	0.618084i	$0.212091\pi$
$332$	−8.57806	−0.470782
$333$	0	0
$334$	−0.546851	−0.0299224
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−17.5751	−0.957377	−0.478689	−	0.877985i	$-0.658888\pi$
$337$	−17.5751	−0.957377	−0.478689	+	0.877985i	$0.658888\pi$
$338$	−12.8325	−0.697998
$339$	0	0
$340$	−5.77453	−0.313168
$341$	11.4331	0.619135
$342$	0	0
$343$	20.8296	1.12469
$344$	−9.49106	−0.511724
$345$	0	0
$346$	16.3147	0.877083
$347$	34.1122	1.83124	0.915620	−	0.402046i	$-0.131701\pi$
$347$	34.1122	1.83124	0.915620	+	0.402046i	$0.131701\pi$
$348$	0	0
$349$	−12.9821	−0.694917	−0.347459	−	0.937695i	$-0.612955\pi$
$349$	−12.9821	−0.694917	−0.347459	+	0.937695i	$0.612955\pi$
$350$	4.33630	0.231785
$351$	0	0
$352$	4.74553	0.252938
$353$	−3.55286	−0.189100	−0.0945498	−	0.995520i	$-0.530141\pi$
$353$	−3.55286	−0.189100	−0.0945498	+	0.995520i	$0.530141\pi$
$354$	0	0
$355$	−8.11084	−0.430478
$356$	9.08700	0.481610
$357$	0	0
$358$	1.92038	0.101495
$359$	22.0230	1.16233	0.581165	−	0.813786i	$-0.302597\pi$
$359$	22.0230	1.16233	0.581165	+	0.813786i	$0.302597\pi$
$360$	0	0
$361$	14.3452	0.755011
$362$	17.4032	0.914693
$363$	0	0
$364$	1.77453	0.0930107
$365$	−8.05800	−0.421775
$366$	0	0
$367$	−2.34300	−0.122304	−0.0611519	−	0.998128i	$-0.519477\pi$
$367$	−2.34300	−0.122304	−0.0611519	+	0.998128i	$0.519477\pi$
$368$	9.36530	0.488200
$369$	0	0
$370$	3.02900	0.157470
$371$	46.2797	2.40272
$372$	0	0
$373$	16.9360	0.876912	0.438456	−	0.898753i	$-0.355525\pi$
$373$	16.9360	0.876912	0.438456	+	0.898753i	$0.355525\pi$
$374$	−27.4032	−1.41699
$375$	0	0
$376$	1.51115	0.0779316
$377$	0.534983	0.0275530
$378$	0	0
$379$	7.93378	0.407531	0.203765	−	0.979020i	$-0.434682\pi$
$379$	7.93378	0.407531	0.203765	+	0.979020i	$0.434682\pi$
$380$	−5.77453	−0.296227
$381$	0	0
$382$	2.50894	0.128368
$383$	−26.3742	−1.34766	−0.673830	−	0.738887i	$-0.735352\pi$
$383$	−26.3742	−1.34766	−0.673830	+	0.738887i	$0.735352\pi$
$384$	0	0
$385$	20.5781	1.04876
$386$	25.9382	1.32022
$387$	0	0
$388$	−4.74553	−0.240918
$389$	−39.4277	−1.99907	−0.999533	−	0.0305693i	$-0.990268\pi$
$389$	−39.4277	−1.99907	−0.999533	+	0.0305693i	$0.990268\pi$
$390$	0	0
$391$	−54.0802	−2.73496
$392$	11.8035	0.596168
$393$	0	0
$394$	1.18155	0.0595255
$395$	7.64878	0.384852
$396$	0	0
$397$	−23.8073	−1.19485	−0.597426	−	0.801924i	$-0.703810\pi$
$397$	−23.8073	−1.19485	−0.597426	+	0.801924i	$0.703810\pi$
$398$	−16.5781	−0.830983
$399$	0	0
$400$	1.00000	0.0500000
$401$	17.8109	0.889435	0.444717	−	0.895671i	$-0.353304\pi$
$401$	17.8109	0.889435	0.444717	+	0.895671i	$0.353304\pi$
$402$	0	0
$403$	0.985920	0.0491122
$404$	−8.52006	−0.423889
$405$	0	0
$406$	5.66886	0.281341
$407$	14.3742	0.712503
$408$	0	0
$409$	9.18155	0.453998	0.226999	−	0.973895i	$-0.427109\pi$
$409$	9.18155	0.453998	0.226999	+	0.973895i	$0.427109\pi$
$410$	−1.71653	−0.0847734
$411$	0	0
$412$	−13.4911	−0.664657
$413$	−7.44340	−0.366266
$414$	0	0
$415$	−8.57806	−0.421081
$416$	0.409227	0.0200640
$417$	0	0
$418$	−27.4032	−1.34034
$419$	12.3013	0.600958	0.300479	−	0.953789i	$-0.402854\pi$
$419$	12.3013	0.600958	0.300479	+	0.953789i	$0.402854\pi$
$420$	0	0
$421$	32.8363	1.60034	0.800171	−	0.599772i	$-0.204742\pi$
$421$	32.8363	1.60034	0.800171	+	0.599772i	$0.204742\pi$
$422$	−25.0617	−1.21999
$423$	0	0
$424$	10.6726	0.518308
$425$	−5.77453	−0.280106
$426$	0	0
$427$	27.4761	1.32966
$428$	8.81845	0.426256
$429$	0	0
$430$	−9.49106	−0.457700
$431$	41.0051	1.97515	0.987573	−	0.157158i	$-0.0502332\pi$
$431$	41.0051	1.97515	0.987573	+	0.157158i	$0.0502332\pi$
$432$	0	0
$433$	23.5274	1.13066	0.565328	−	0.824866i	$-0.308749\pi$
$433$	23.5274	1.13066	0.565328	+	0.824866i	$0.308749\pi$
$434$	10.4471	0.501479
$435$	0	0
$436$	−17.8854	−0.856554
$437$	−54.0802	−2.58701
$438$	0	0
$439$	−22.0691	−1.05330	−0.526651	−	0.850082i	$-0.676553\pi$
$439$	−22.0691	−1.05330	−0.526651	+	0.850082i	$0.676553\pi$
$440$	4.74553	0.226234
$441$	0	0
$442$	−2.36309	−0.112401
$443$	−7.54906	−0.358667	−0.179333	−	0.983788i	$-0.557394\pi$
$443$	−7.54906	−0.358667	−0.179333	+	0.983788i	$0.557394\pi$
$444$	0	0
$445$	9.08700	0.430765
$446$	−25.8586	−1.22444
$447$	0	0
$448$	4.33630	0.204871
$449$	−27.4398	−1.29496	−0.647481	−	0.762081i	$-0.724178\pi$
$449$	−27.4398	−1.29496	−0.647481	+	0.762081i	$0.724178\pi$
$450$	0	0
$451$	−8.14585	−0.383573
$452$	3.86908	0.181986
$453$	0	0
$454$	−17.7127	−0.831300
$455$	1.77453	0.0831913
$456$	0	0
$457$	−14.3891	−0.673095	−0.336548	−	0.941666i	$-0.609259\pi$
$457$	−14.3891	−0.673095	−0.336548	+	0.941666i	$0.609259\pi$
$458$	−15.8274	−0.739564
$459$	0	0
$460$	9.36530	0.436660
$461$	14.5365	0.677033	0.338517	−	0.940960i	$-0.390075\pi$
$461$	14.5365	0.677033	0.338517	+	0.940960i	$0.390075\pi$
$462$	0	0
$463$	11.6124	0.539674	0.269837	−	0.962906i	$-0.413030\pi$
$463$	11.6124	0.539674	0.269837	+	0.962906i	$0.413030\pi$
$464$	1.30730	0.0606900
$465$	0	0
$466$	22.4322	1.03915
$467$	−8.88316	−0.411063	−0.205532	−	0.978650i	$-0.565892\pi$
$467$	−8.88316	−0.411063	−0.205532	+	0.978650i	$0.565892\pi$
$468$	0	0
$469$	−4.33630	−0.200232
$470$	1.51115	0.0697042
$471$	0	0
$472$	−1.71653	−0.0790097
$473$	−45.0401	−2.07095
$474$	0	0
$475$	−5.77453	−0.264954
$476$	−25.0401	−1.14771
$477$	0	0
$478$	−9.23955	−0.422607
$479$	−24.5319	−1.12089	−0.560446	−	0.828191i	$-0.689370\pi$
$479$	−24.5319	−1.12089	−0.560446	+	0.828191i	$0.689370\pi$
$480$	0	0
$481$	1.23955	0.0565185
$482$	6.17859	0.281427
$483$	0	0
$484$	11.5201	0.523639
$485$	−4.74553	−0.215483
$486$	0	0
$487$	−12.9553	−0.587062	−0.293531	−	0.955950i	$-0.594830\pi$
$487$	−12.9553	−0.587062	−0.293531	+	0.955950i	$0.594830\pi$
$488$	6.33630	0.286831
$489$	0	0
$490$	11.8035	0.533229
$491$	−27.6911	−1.24968	−0.624841	−	0.780752i	$-0.714836\pi$
$491$	−27.6911	−1.24968	−0.624841	+	0.780752i	$0.714836\pi$
$492$	0	0
$493$	−7.54906	−0.339993
$494$	−2.36309	−0.106321
$495$	0	0
$496$	2.40923	0.108177
$497$	−35.1710	−1.57764
$498$	0	0
$499$	32.7737	1.46715	0.733576	−	0.679607i	$-0.237850\pi$
$499$	32.7737	1.46715	0.733576	+	0.679607i	$0.237850\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	6.97100	0.311131
$503$	−20.5677	−0.917070	−0.458535	−	0.888676i	$-0.651626\pi$
$503$	−20.5677	−0.917070	−0.458535	+	0.888676i	$0.651626\pi$
$504$	0	0
$505$	−8.52006	−0.379138
$506$	44.4433	1.97575
$507$	0	0
$508$	−9.23955	−0.409939
$509$	1.97991	0.0877580	0.0438790	−	0.999037i	$-0.486028\pi$
$509$	1.97991	0.0877580	0.0438790	+	0.999037i	$0.486028\pi$
$510$	0	0
$511$	−34.9419	−1.54574
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.7090	−0.648786
$515$	−13.4911	−0.594487
$516$	0	0
$517$	7.17121	0.315389
$518$	13.1347	0.577104
$519$	0	0
$520$	0.409227	0.0179458
$521$	−22.6407	−0.991905	−0.495952	−	0.868350i	$-0.665181\pi$
$521$	−22.6407	−0.991905	−0.495952	+	0.868350i	$0.665181\pi$
$522$	0	0
$523$	−23.8519	−1.04297	−0.521485	−	0.853261i	$-0.674622\pi$
$523$	−23.8519	−1.04297	−0.521485	+	0.853261i	$0.674622\pi$
$524$	18.9561	0.828100
$525$	0	0
$526$	−8.79836	−0.383627
$527$	−13.9122	−0.606023
$528$	0	0
$529$	64.7089	2.81343
$530$	10.6726	0.463588
$531$	0	0
$532$	−25.0401	−1.08563
$533$	−0.702450	−0.0304265
$534$	0	0
$535$	8.81845	0.381255
$536$	−1.00000	−0.0431934
$537$	0	0
$538$	−28.8564	−1.24409
$539$	56.0140	2.41269
$540$	0	0
$541$	−4.49919	−0.193435	−0.0967175	−	0.995312i	$-0.530834\pi$
$541$	−4.49919	−0.193435	−0.0967175	+	0.995312i	$0.530834\pi$
$542$	13.8124	0.593295
$543$	0	0
$544$	−5.77453	−0.247581
$545$	−17.8854	−0.766125
$546$	0	0
$547$	5.68015	0.242866	0.121433	−	0.992600i	$-0.461251\pi$
$547$	5.68015	0.242866	0.121433	+	0.992600i	$0.461251\pi$
$548$	0.970999	0.0414790
$549$	0	0
$550$	4.74553	0.202350
$551$	−7.54906	−0.321601
$552$	0	0
$553$	33.1674	1.41042
$554$	−3.28051	−0.139376
$555$	0	0
$556$	9.41144	0.399134
$557$	20.6412	0.874597	0.437299	−	0.899316i	$-0.355935\pi$
$557$	20.6412	0.874597	0.437299	+	0.899316i	$0.355935\pi$
$558$	0	0
$559$	−3.88400	−0.164276
$560$	4.33630	0.183242
$561$	0	0
$562$	−0.363093	−0.0153162
$563$	2.50894	0.105739	0.0528696	−	0.998601i	$-0.483163\pi$
$563$	2.50894	0.105739	0.0528696	+	0.998601i	$0.483163\pi$
$564$	0	0
$565$	3.86908	0.162773
$566$	12.1570	0.510996
$567$	0	0
$568$	−8.11084	−0.340323
$569$	−41.6138	−1.74454	−0.872270	−	0.489025i	$-0.837353\pi$
$569$	−41.6138	−1.74454	−0.872270	+	0.489025i	$0.837353\pi$
$570$	0	0
$571$	11.5491	0.483313	0.241657	−	0.970362i	$-0.422309\pi$
$571$	11.5491	0.483313	0.241657	+	0.970362i	$0.422309\pi$
$572$	1.94200	0.0811990
$573$	0	0
$574$	−7.44340	−0.310681
$575$	9.36530	0.390560
$576$	0	0
$577$	−40.6211	−1.69108	−0.845540	−	0.533912i	$-0.820721\pi$
$577$	−40.6211	−1.69108	−0.845540	+	0.533912i	$0.820721\pi$
$578$	16.3452	0.679871
$579$	0	0
$580$	1.30730	0.0542828
$581$	−37.1971	−1.54320
$582$	0	0
$583$	50.6472	2.09759
$584$	−8.05800	−0.333442
$585$	0	0
$586$	0.871287	0.0359926
$587$	−7.82220	−0.322857	−0.161428	−	0.986884i	$-0.551610\pi$
$587$	−7.82220	−0.322857	−0.161428	+	0.986884i	$0.551610\pi$
$588$	0	0
$589$	−13.9122	−0.573240
$590$	−1.71653	−0.0706684
$591$	0	0
$592$	3.02900	0.124491
$593$	12.4041	0.509374	0.254687	−	0.967024i	$-0.418028\pi$
$593$	12.4041	0.509374	0.254687	+	0.967024i	$0.418028\pi$
$594$	0	0
$595$	−25.0401	−1.02655
$596$	1.30730	0.0535492
$597$	0	0
$598$	3.83253	0.156724
$599$	−20.5134	−0.838153	−0.419077	−	0.907951i	$-0.637646\pi$
$599$	−20.5134	−0.838153	−0.419077	+	0.907951i	$0.637646\pi$
$600$	0	0
$601$	20.4984	0.836149	0.418074	−	0.908413i	$-0.362705\pi$
$601$	20.4984	0.836149	0.418074	+	0.908413i	$0.362705\pi$
$602$	−41.1561	−1.67740
$603$	0	0
$604$	−15.3385	−0.624115
$605$	11.5201	0.468357
$606$	0	0
$607$	8.96430	0.363850	0.181925	−	0.983312i	$-0.441767\pi$
$607$	8.96430	0.363850	0.181925	+	0.983312i	$0.441767\pi$
$608$	−5.77453	−0.234188
$609$	0	0
$610$	6.33630	0.256549
$611$	0.618403	0.0250179
$612$	0	0
$613$	−28.6904	−1.15880	−0.579398	−	0.815045i	$-0.696712\pi$
$613$	−28.6904	−1.15880	−0.579398	+	0.815045i	$0.696712\pi$
$614$	26.4583	1.06777
$615$	0	0
$616$	20.5781	0.829114
$617$	2.85483	0.114931	0.0574656	−	0.998347i	$-0.481698\pi$
$617$	2.85483	0.114931	0.0574656	+	0.998347i	$0.481698\pi$
$618$	0	0
$619$	−25.7745	−1.03597	−0.517983	−	0.855391i	$-0.673317\pi$
$619$	−25.7745	−1.03597	−0.517983	+	0.855391i	$0.673317\pi$
$620$	2.40923	0.0967569
$621$	0	0
$622$	26.1637	1.04907
$623$	39.4040	1.57869
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.4404	0.497220
$627$	0	0
$628$	−7.79615	−0.311100
$629$	−17.4911	−0.697414
$630$	0	0
$631$	−29.1398	−1.16004	−0.580019	−	0.814603i	$-0.696955\pi$
$631$	−29.1398	−1.16004	−0.580019	+	0.814603i	$0.696955\pi$
$632$	7.64878	0.304252
$633$	0	0
$634$	31.2558	1.24133
$635$	−9.23955	−0.366660
$636$	0	0
$637$	4.83032	0.191384
$638$	6.20385	0.245613
$639$	0	0
$640$	1.00000	0.0395285
$641$	−31.0535	−1.22654	−0.613270	−	0.789873i	$-0.710146\pi$
$641$	−31.0535	−1.22654	−0.613270	+	0.789873i	$0.710146\pi$
$642$	0	0
$643$	−28.1273	−1.10923	−0.554616	−	0.832106i	$-0.687135\pi$
$643$	−28.1273	−1.10923	−0.554616	+	0.832106i	$0.687135\pi$
$644$	40.6108	1.60029
$645$	0	0
$646$	33.3452	1.31195
$647$	−37.2641	−1.46500	−0.732501	−	0.680766i	$-0.761647\pi$
$647$	−37.2641	−1.46500	−0.732501	+	0.680766i	$0.761647\pi$
$648$	0	0
$649$	−8.14585	−0.319752
$650$	0.409227	0.0160512
$651$	0	0
$652$	−6.97100	−0.273005
$653$	11.1963	0.438145	0.219073	−	0.975709i	$-0.429697\pi$
$653$	11.1963	0.438145	0.219073	+	0.975709i	$0.429697\pi$
$654$	0	0
$655$	18.9561	0.740675
$656$	−1.71653	−0.0670192
$657$	0	0
$658$	6.55281	0.255455
$659$	18.8727	0.735174	0.367587	−	0.929989i	$-0.380184\pi$
$659$	18.8727	0.735174	0.367587	+	0.929989i	$0.380184\pi$
$660$	0	0
$661$	13.5290	0.526216	0.263108	−	0.964766i	$-0.415252\pi$
$661$	13.5290	0.526216	0.263108	+	0.964766i	$0.415252\pi$
$662$	28.6041	1.11173
$663$	0	0
$664$	−8.57806	−0.332893
$665$	−25.0401	−0.971014
$666$	0	0
$667$	12.2433	0.474062
$668$	−0.546851	−0.0211583
$669$	0	0
$670$	−1.00000	−0.0386334
$671$	30.0691	1.16081
$672$	0	0
$673$	−43.2358	−1.66662	−0.833308	−	0.552809i	$-0.813556\pi$
$673$	−43.2358	−1.66662	−0.833308	+	0.552809i	$0.813556\pi$
$674$	−17.5751	−0.676968
$675$	0	0
$676$	−12.8325	−0.493559
$677$	−3.63691	−0.139778	−0.0698888	−	0.997555i	$-0.522264\pi$
$677$	−3.63691	−0.139778	−0.0698888	+	0.997555i	$0.522264\pi$
$678$	0	0
$679$	−20.5781	−0.789714
$680$	−5.77453	−0.221443
$681$	0	0
$682$	11.4331	0.437794
$683$	−42.9523	−1.64352	−0.821762	−	0.569831i	$-0.807009\pi$
$683$	−42.9523	−1.64352	−0.821762	+	0.569831i	$0.807009\pi$
$684$	0	0
$685$	0.970999	0.0371000
$686$	20.8296	0.795277
$687$	0	0
$688$	−9.49106	−0.361843
$689$	4.36752	0.166389
$690$	0	0
$691$	43.0683	1.63839	0.819197	−	0.573512i	$-0.194419\pi$
$691$	43.0683	1.63839	0.819197	+	0.573512i	$0.194419\pi$
$692$	16.3147	0.620191
$693$	0	0
$694$	34.1122	1.29488
$695$	9.41144	0.356996
$696$	0	0
$697$	9.91216	0.375450
$698$	−12.9821	−0.491381
$699$	0	0
$700$	4.33630	0.163897
$701$	−42.5379	−1.60663	−0.803317	−	0.595552i	$-0.796933\pi$
$701$	−42.5379	−1.60663	−0.803317	+	0.595552i	$0.796933\pi$
$702$	0	0
$703$	−17.4911	−0.659688
$704$	4.74553	0.178854
$705$	0	0
$706$	−3.55286	−0.133714
$707$	−36.9456	−1.38948
$708$	0	0
$709$	−22.8319	−0.857468	−0.428734	−	0.903431i	$-0.641040\pi$
$709$	−22.8319	−0.857468	−0.428734	+	0.903431i	$0.641040\pi$
$710$	−8.11084	−0.304394
$711$	0	0
$712$	9.08700	0.340550
$713$	22.5631	0.844996
$714$	0	0
$715$	1.94200	0.0726266
$716$	1.92038	0.0717679
$717$	0	0
$718$	22.0230	0.821891
$719$	11.3869	0.424661	0.212330	−	0.977198i	$-0.431895\pi$
$719$	11.3869	0.424661	0.212330	+	0.977198i	$0.431895\pi$
$720$	0	0
$721$	−58.5013	−2.17870
$722$	14.3452	0.533874
$723$	0	0
$724$	17.4032	0.646785
$725$	1.30730	0.0485520
$726$	0	0
$727$	8.38761	0.311079	0.155540	−	0.987830i	$-0.450288\pi$
$727$	8.38761	0.311079	0.155540	+	0.987830i	$0.450288\pi$
$728$	1.77453	0.0657685
$729$	0	0
$730$	−8.05800	−0.298240
$731$	54.8064	2.02709
$732$	0	0
$733$	−18.8244	−0.695295	−0.347648	−	0.937625i	$-0.613019\pi$
$733$	−18.8244	−0.695295	−0.347648	+	0.937625i	$0.613019\pi$
$734$	−2.34300	−0.0864819
$735$	0	0
$736$	9.36530	0.345210
$737$	−4.74553	−0.174804
$738$	0	0
$739$	−1.86238	−0.0685086	−0.0342543	−	0.999413i	$-0.510906\pi$
$739$	−1.86238	−0.0685086	−0.0342543	+	0.999413i	$0.510906\pi$
$740$	3.02900	0.111348
$741$	0	0
$742$	46.2797	1.69898
$743$	−3.21504	−0.117948	−0.0589741	−	0.998260i	$-0.518783\pi$
$743$	−3.21504	−0.117948	−0.0589741	+	0.998260i	$0.518783\pi$
$744$	0	0
$745$	1.30730	0.0478959
$746$	16.9360	0.620071
$747$	0	0
$748$	−27.4032	−1.00196
$749$	38.2395	1.39724
$750$	0	0
$751$	14.9821	0.546705	0.273353	−	0.961914i	$-0.411867\pi$
$751$	14.9821	0.546705	0.273353	+	0.961914i	$0.411867\pi$
$752$	1.51115	0.0551060
$753$	0	0
$754$	0.534983	0.0194829
$755$	−15.3385	−0.558226
$756$	0	0
$757$	47.8749	1.74004	0.870021	−	0.493015i	$-0.164105\pi$
$757$	47.8749	1.74004	0.870021	+	0.493015i	$0.164105\pi$
$758$	7.93378	0.288168
$759$	0	0
$760$	−5.77453	−0.209464
$761$	−10.3266	−0.374337	−0.187169	−	0.982328i	$-0.559931\pi$
$761$	−10.3266	−0.374337	−0.187169	+	0.982328i	$0.559931\pi$
$762$	0	0
$763$	−77.5564	−2.80773
$764$	2.50894	0.0907702
$765$	0	0
$766$	−26.3742	−0.952939
$767$	−0.702450	−0.0253640
$768$	0	0
$769$	−9.60264	−0.346280	−0.173140	−	0.984897i	$-0.555391\pi$
$769$	−9.60264	−0.346280	−0.173140	+	0.984897i	$0.555391\pi$
$770$	20.5781	0.741582
$771$	0	0
$772$	25.9382	0.933536
$773$	32.7151	1.17668	0.588340	−	0.808613i	$-0.299782\pi$
$773$	32.7151	1.17668	0.588340	+	0.808613i	$0.299782\pi$
$774$	0	0
$775$	2.40923	0.0865420
$776$	−4.74553	−0.170355
$777$	0	0
$778$	−39.4277	−1.41355
$779$	9.91216	0.355140
$780$	0	0
$781$	−38.4902	−1.37729
$782$	−54.0802	−1.93391
$783$	0	0
$784$	11.8035	0.421555
$785$	−7.79615	−0.278257
$786$	0	0
$787$	−15.3795	−0.548219	−0.274110	−	0.961698i	$-0.588383\pi$
$787$	−15.3795	−0.548219	−0.274110	+	0.961698i	$0.588383\pi$
$788$	1.18155	0.0420909
$789$	0	0
$790$	7.64878	0.272131
$791$	16.7775	0.596539
$792$	0	0
$793$	2.59299	0.0920796
$794$	−23.8073	−0.844889
$795$	0	0
$796$	−16.5781	−0.587594
$797$	24.5260	0.868756	0.434378	−	0.900731i	$-0.356968\pi$
$797$	24.5260	0.868756	0.434378	+	0.900731i	$0.356968\pi$
$798$	0	0
$799$	−8.72619	−0.308710
$800$	1.00000	0.0353553
$801$	0	0
$802$	17.8109	0.628925
$803$	−38.2395	−1.34944
$804$	0	0
$805$	40.6108	1.43134
$806$	0.985920	0.0347276
$807$	0	0
$808$	−8.52006	−0.299735
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−36.9329	−1.29689	−0.648445	−	0.761261i	$-0.724581\pi$
$811$	−36.9329	−1.29689	−0.648445	+	0.761261i	$0.724581\pi$
$812$	5.66886	0.198938
$813$	0	0
$814$	14.3742	0.503816
$815$	−6.97100	−0.244183
$816$	0	0
$817$	54.8064	1.91743
$818$	9.18155	0.321025
$819$	0	0
$820$	−1.71653	−0.0599438
$821$	−45.4991	−1.58793	−0.793965	−	0.607963i	$-0.791987\pi$
$821$	−45.4991	−1.58793	−0.793965	+	0.607963i	$0.791987\pi$
$822$	0	0
$823$	28.9241	1.00823	0.504116	−	0.863636i	$-0.331818\pi$
$823$	28.9241	1.00823	0.504116	+	0.863636i	$0.331818\pi$
$824$	−13.4911	−0.469983
$825$	0	0
$826$	−7.44340	−0.258989
$827$	31.0767	1.08064	0.540321	−	0.841459i	$-0.318303\pi$
$827$	31.0767	1.08064	0.540321	+	0.841459i	$0.318303\pi$
$828$	0	0
$829$	10.2217	0.355013	0.177507	−	0.984120i	$-0.443197\pi$
$829$	10.2217	0.355013	0.177507	+	0.984120i	$0.443197\pi$
$830$	−8.57806	−0.297749
$831$	0	0
$832$	0.409227	0.0141874
$833$	−68.1599	−2.36160
$834$	0	0
$835$	−0.546851	−0.0189246
$836$	−27.4032	−0.947760
$837$	0	0
$838$	12.3013	0.424941
$839$	55.7870	1.92598	0.962991	−	0.269533i	$-0.0868695\pi$
$839$	55.7870	1.92598	0.962991	+	0.269533i	$0.0868695\pi$
$840$	0	0
$841$	−27.2910	−0.941068
$842$	32.8363	1.13161
$843$	0	0
$844$	−25.0617	−0.862661
$845$	−12.8325	−0.441453
$846$	0	0
$847$	49.9545	1.71646
$848$	10.6726	0.366499
$849$	0	0
$850$	−5.77453	−0.198065
$851$	28.3675	0.972426
$852$	0	0
$853$	31.9880	1.09525	0.547624	−	0.836725i	$-0.315533\pi$
$853$	31.9880	1.09525	0.547624	+	0.836725i	$0.315533\pi$
$854$	27.4761	0.940214
$855$	0	0
$856$	8.81845	0.301409
$857$	−29.4329	−1.00541	−0.502704	−	0.864458i	$-0.667662\pi$
$857$	−29.4329	−1.00541	−0.502704	+	0.864458i	$0.667662\pi$
$858$	0	0
$859$	−14.5306	−0.495776	−0.247888	−	0.968789i	$-0.579737\pi$
$859$	−14.5306	−0.495776	−0.247888	+	0.968789i	$0.579737\pi$
$860$	−9.49106	−0.323643
$861$	0	0
$862$	41.0051	1.39664
$863$	20.9442	0.712949	0.356475	−	0.934305i	$-0.383979\pi$
$863$	20.9442	0.712949	0.356475	+	0.934305i	$0.383979\pi$
$864$	0	0
$865$	16.3147	0.554716
$866$	23.5274	0.799495
$867$	0	0
$868$	10.4471	0.354599
$869$	36.2975	1.23131
$870$	0	0
$871$	−0.409227	−0.0138661
$872$	−17.8854	−0.605675
$873$	0	0
$874$	−54.0802	−1.82929
$875$	4.33630	0.146594
$876$	0	0
$877$	−42.5789	−1.43779	−0.718893	−	0.695121i	$-0.755351\pi$
$877$	−42.5789	−1.43779	−0.718893	+	0.695121i	$0.755351\pi$
$878$	−22.0691	−0.744797
$879$	0	0
$880$	4.74553	0.159972
$881$	40.1851	1.35387	0.676936	−	0.736042i	$-0.263308\pi$
$881$	40.1851	1.35387	0.676936	+	0.736042i	$0.263308\pi$
$882$	0	0
$883$	−38.1637	−1.28431	−0.642155	−	0.766575i	$-0.721959\pi$
$883$	−38.1637	−1.28431	−0.642155	+	0.766575i	$0.721959\pi$
$884$	−2.36309	−0.0794794
$885$	0	0
$886$	−7.54906	−0.253616
$887$	−22.4353	−0.753303	−0.376651	−	0.926355i	$-0.622925\pi$
$887$	−22.4353	−0.753303	−0.376651	+	0.926355i	$0.622925\pi$
$888$	0	0
$889$	−40.0655	−1.34375
$890$	9.08700	0.304597
$891$	0	0
$892$	−25.8586	−0.865809
$893$	−8.72619	−0.292011
$894$	0	0
$895$	1.92038	0.0641911
$896$	4.33630	0.144866
$897$	0	0
$898$	−27.4398	−0.915677
$899$	3.14959	0.105045
$900$	0	0
$901$	−61.6293	−2.05317
$902$	−8.14585	−0.271227
$903$	0	0
$904$	3.86908	0.128684
$905$	17.4032	0.578503
$906$	0	0
$907$	45.2945	1.50398	0.751990	−	0.659174i	$-0.229094\pi$
$907$	45.2945	1.50398	0.751990	+	0.659174i	$0.229094\pi$
$908$	−17.7127	−0.587818
$909$	0	0
$910$	1.77453	0.0588252
$911$	6.75528	0.223813	0.111906	−	0.993719i	$-0.464304\pi$
$911$	6.75528	0.223813	0.111906	+	0.993719i	$0.464304\pi$
$912$	0	0
$913$	−40.7075	−1.34722
$914$	−14.3891	−0.475950
$915$	0	0
$916$	−15.8274	−0.522951
$917$	82.1993	2.71446
$918$	0	0
$919$	−1.48511	−0.0489891	−0.0244946	−	0.999700i	$-0.507798\pi$
$919$	−1.48511	−0.0489891	−0.0244946	+	0.999700i	$0.507798\pi$
$920$	9.36530	0.308765
$921$	0	0
$922$	14.5365	0.478735
$923$	−3.31917	−0.109252
$924$	0	0
$925$	3.02900	0.0995929
$926$	11.6124	0.381607
$927$	0	0
$928$	1.30730	0.0429143
$929$	−30.3490	−0.995717	−0.497859	−	0.867258i	$-0.665880\pi$
$929$	−30.3490	−0.995717	−0.497859	+	0.867258i	$0.665880\pi$
$930$	0	0
$931$	−68.1599	−2.23385
$932$	22.4322	0.734792
$933$	0	0
$934$	−8.88316	−0.290666
$935$	−27.4032	−0.896181
$936$	0	0
$937$	−53.1992	−1.73794	−0.868971	−	0.494863i	$-0.835218\pi$
$937$	−53.1992	−1.73794	−0.868971	+	0.494863i	$0.835218\pi$
$938$	−4.33630	−0.141585
$939$	0	0
$940$	1.51115	0.0492883
$941$	−32.2395	−1.05098	−0.525489	−	0.850801i	$-0.676118\pi$
$941$	−32.2395	−1.05098	−0.525489	+	0.850801i	$0.676118\pi$
$942$	0	0
$943$	−16.0758	−0.523501
$944$	−1.71653	−0.0558683
$945$	0	0
$946$	−45.0401	−1.46438
$947$	27.7060	0.900325	0.450163	−	0.892947i	$-0.351366\pi$
$947$	27.7060	0.900325	0.450163	+	0.892947i	$0.351366\pi$
$948$	0	0
$949$	−3.29755	−0.107043
$950$	−5.77453	−0.187351
$951$	0	0
$952$	−25.0401	−0.811555
$953$	50.3771	1.63187	0.815937	−	0.578140i	$-0.196221\pi$
$953$	50.3771	1.63187	0.815937	+	0.578140i	$0.196221\pi$
$954$	0	0
$955$	2.50894	0.0811873
$956$	−9.23955	−0.298828
$957$	0	0
$958$	−24.5319	−0.792591
$959$	4.21055	0.135966
$960$	0	0
$961$	−25.1956	−0.812762
$962$	1.23955	0.0399646
$963$	0	0
$964$	6.17859	0.198999
$965$	25.9382	0.834980
$966$	0	0
$967$	2.23675	0.0719291	0.0359646	−	0.999353i	$-0.488550\pi$
$967$	2.23675	0.0719291	0.0359646	+	0.999353i	$0.488550\pi$
$968$	11.5201	0.370269
$969$	0	0
$970$	−4.74553	−0.152370
$971$	−37.2878	−1.19662	−0.598312	−	0.801263i	$-0.704162\pi$
$971$	−37.2878	−1.19662	−0.598312	+	0.801263i	$0.704162\pi$
$972$	0	0
$973$	40.8109	1.30834
$974$	−12.9553	−0.415116
$975$	0	0
$976$	6.33630	0.202820
$977$	−13.5261	−0.432738	−0.216369	−	0.976312i	$-0.569421\pi$
$977$	−13.5261	−0.432738	−0.216369	+	0.976312i	$0.569421\pi$
$978$	0	0
$979$	43.1227	1.37821
$980$	11.8035	0.377050
$981$	0	0
$982$	−27.6911	−0.883659
$983$	59.9924	1.91346	0.956730	−	0.290976i	$-0.0939800\pi$
$983$	59.9924	1.91346	0.956730	+	0.290976i	$0.0939800\pi$
$984$	0	0
$985$	1.18155	0.0376472
$986$	−7.54906	−0.240411
$987$	0	0
$988$	−2.36309	−0.0751800
$989$	−88.8867	−2.82643
$990$	0	0
$991$	3.37353	0.107164	0.0535818	−	0.998563i	$-0.482936\pi$
$991$	3.37353	0.107164	0.0535818	+	0.998563i	$0.482936\pi$
$992$	2.40923	0.0764930
$993$	0	0
$994$	−35.1710	−1.11556
$995$	−16.5781	−0.525560
$996$	0	0
$997$	−10.0647	−0.318752	−0.159376	−	0.987218i	$-0.550948\pi$
$997$	−10.0647	−0.318752	−0.159376	+	0.987218i	$0.550948\pi$
$998$	32.7737	1.03743
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.a.bu.1.4		4
3.2	odd	2		2010.2.a.r.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2010.2.a.r.1.4	✓	4	3.2	odd	2
6030.2.a.bu.1.4		4	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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.33630 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.33630 q^{7} +1.00000 q^{8} +1.00000 q^{10} +4.74553 q^{11} +0.409227 q^{13} +4.33630 q^{14} +1.00000 q^{16} -5.77453 q^{17} -5.77453 q^{19} +1.00000 q^{20} +4.74553 q^{22} +9.36530 q^{23} +1.00000 q^{25} +0.409227 q^{26} +4.33630 q^{28} +1.30730 q^{29} +2.40923 q^{31} +1.00000 q^{32} -5.77453 q^{34} +4.33630 q^{35} +3.02900 q^{37} -5.77453 q^{38} +1.00000 q^{40} -1.71653 q^{41} -9.49106 q^{43} +4.74553 q^{44} +9.36530 q^{46} +1.51115 q^{47} +11.8035 q^{49} +1.00000 q^{50} +0.409227 q^{52} +10.6726 q^{53} +4.74553 q^{55} +4.33630 q^{56} +1.30730 q^{58} -1.71653 q^{59} +6.33630 q^{61} +2.40923 q^{62} +1.00000 q^{64} +0.409227 q^{65} -1.00000 q^{67} -5.77453 q^{68} +4.33630 q^{70} -8.11084 q^{71} -8.05800 q^{73} +3.02900 q^{74} -5.77453 q^{76} +20.5781 q^{77} +7.64878 q^{79} +1.00000 q^{80} -1.71653 q^{82} -8.57806 q^{83} -5.77453 q^{85} -9.49106 q^{86} +4.74553 q^{88} +9.08700 q^{89} +1.77453 q^{91} +9.36530 q^{92} +1.51115 q^{94} -5.77453 q^{95} -4.74553 q^{97} +11.8035 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 2 q^{13} + q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{19} + 4 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 2 q^{26} + q^{28} - 2 q^{29} + 10 q^{31} + 4 q^{32} + 2 q^{34} + q^{35} + 3 q^{37} + 2 q^{38} + 4 q^{40} - 6 q^{43} + 3 q^{44} + 12 q^{46} + 14 q^{47} + 13 q^{49} + 4 q^{50} + 2 q^{52} + 10 q^{53} + 3 q^{55} + q^{56} - 2 q^{58} + 9 q^{61} + 10 q^{62} + 4 q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + q^{70} + 9 q^{71} - 14 q^{73} + 3 q^{74} + 2 q^{76} + 23 q^{77} + 12 q^{79} + 4 q^{80} + 25 q^{83} + 2 q^{85} - 6 q^{86} + 3 q^{88} + 9 q^{89} - 18 q^{91} + 12 q^{92} + 14 q^{94} + 2 q^{95} - 3 q^{97} + 13 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + q^7 + 4 * q^8 + 4 * q^10 + 3 * q^11 + 2 * q^13 + q^14 + 4 * q^16 + 2 * q^17 + 2 * q^19 + 4 * q^20 + 3 * q^22 + 12 * q^23 + 4 * q^25 + 2 * q^26 + q^28 - 2 * q^29 + 10 * q^31 + 4 * q^32 + 2 * q^34 + q^35 + 3 * q^37 + 2 * q^38 + 4 * q^40 - 6 * q^43 + 3 * q^44 + 12 * q^46 + 14 * q^47 + 13 * q^49 + 4 * q^50 + 2 * q^52 + 10 * q^53 + 3 * q^55 + q^56 - 2 * q^58 + 9 * q^61 + 10 * q^62 + 4 * q^64 + 2 * q^65 - 4 * q^67 + 2 * q^68 + q^70 + 9 * q^71 - 14 * q^73 + 3 * q^74 + 2 * q^76 + 23 * q^77 + 12 * q^79 + 4 * q^80 + 25 * q^83 + 2 * q^85 - 6 * q^86 + 3 * q^88 + 9 * q^89 - 18 * q^91 + 12 * q^92 + 14 * q^94 + 2 * q^95 - 3 * q^97 + 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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