LMFDB - Embedded newform 4600.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +3.12311 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +3.12311 q^{7} -0.561553 q^{9} -4.00000 q^{11} -3.56155 q^{13} -5.12311 q^{17} +4.00000 q^{19} +4.87689 q^{21} -1.00000 q^{23} -5.56155 q^{27} -4.43845 q^{29} +5.56155 q^{31} -6.24621 q^{33} -1.12311 q^{37} -5.56155 q^{39} -3.56155 q^{41} +0.876894 q^{43} -8.68466 q^{47} +2.75379 q^{49} -8.00000 q^{51} -12.2462 q^{53} +6.24621 q^{57} +10.2462 q^{59} +2.87689 q^{61} -1.75379 q^{63} +10.2462 q^{67} -1.56155 q^{69} -8.68466 q^{71} -12.4384 q^{73} -12.4924 q^{77} +6.24621 q^{79} -7.00000 q^{81} -12.0000 q^{83} -6.93087 q^{87} +10.0000 q^{89} -11.1231 q^{91} +8.68466 q^{93} -0.246211 q^{97} +2.24621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^7 + 3 * q^9	$ 2 q - q^{3} - 2 q^{7} + 3 q^{9} - 8 q^{11} - 3 q^{13} - 2 q^{17} + 8 q^{19} + 18 q^{21} - 2 q^{23} - 7 q^{27} - 13 q^{29} + 7 q^{31} + 4 q^{33} + 6 q^{37} - 7 q^{39} - 3 q^{41} + 10 q^{43} - 5 q^{47} + 22 q^{49} - 16 q^{51} - 8 q^{53} - 4 q^{57} + 4 q^{59} + 14 q^{61} - 20 q^{63} + 4 q^{67} + q^{69} - 5 q^{71} - 29 q^{73} + 8 q^{77} - 4 q^{79} - 14 q^{81} - 24 q^{83} + 15 q^{87} + 20 q^{89} - 14 q^{91} + 5 q^{93} + 16 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^7 + 3 * q^9 - 8 * q^11 - 3 * q^13 - 2 * q^17 + 8 * q^19 + 18 * q^21 - 2 * q^23 - 7 * q^27 - 13 * q^29 + 7 * q^31 + 4 * q^33 + 6 * q^37 - 7 * q^39 - 3 * q^41 + 10 * q^43 - 5 * q^47 + 22 * q^49 - 16 * q^51 - 8 * q^53 - 4 * q^57 + 4 * q^59 + 14 * q^61 - 20 * q^63 + 4 * q^67 + q^69 - 5 * q^71 - 29 * q^73 + 8 * q^77 - 4 * q^79 - 14 * q^81 - 24 * q^83 + 15 * q^87 + 20 * q^89 - 14 * q^91 + 5 * q^93 + 16 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.12311	1.18042	0.590211	−	0.807249i	$-0.299044\pi$
$7$	3.12311	1.18042	0.590211	+	0.807249i	$0.299044\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.87689	1.06423
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−4.43845	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$29$	−4.43845	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	−6.24621	−1.08733
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.12311	−0.184637	−0.0923187	−	0.995730i	$-0.529428\pi$
$37$	−1.12311	−0.184637	−0.0923187	+	0.995730i	$0.529428\pi$
$38$	0	0
$39$	−5.56155	−0.890561
$40$	0	0
$41$	−3.56155	−0.556221	−0.278111	−	0.960549i	$-0.589708\pi$
$41$	−3.56155	−0.556221	−0.278111	+	0.960549i	$0.589708\pi$
$42$	0	0
$43$	0.876894	0.133725	0.0668626	−	0.997762i	$-0.478701\pi$
$43$	0.876894	0.133725	0.0668626	+	0.997762i	$0.478701\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.68466	−1.26679	−0.633394	−	0.773830i	$-0.718339\pi$
$47$	−8.68466	−1.26679	−0.633394	+	0.773830i	$0.718339\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.24621	0.827331
$58$	0	0
$59$	10.2462	1.33394	0.666972	−	0.745083i	$-0.267590\pi$
$59$	10.2462	1.33394	0.666972	+	0.745083i	$0.267590\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	−1.75379	−0.220957
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.2462	1.25177	0.625887	−	0.779914i	$-0.284737\pi$
$67$	10.2462	1.25177	0.625887	+	0.779914i	$0.284737\pi$
$68$	0	0
$69$	−1.56155	−0.187989
$70$	0	0
$71$	−8.68466	−1.03068	−0.515340	−	0.856986i	$-0.672334\pi$
$71$	−8.68466	−1.03068	−0.515340	+	0.856986i	$0.672334\pi$
$72$	0	0
$73$	−12.4384	−1.45581	−0.727905	−	0.685678i	$-0.759506\pi$
$73$	−12.4384	−1.45581	−0.727905	+	0.685678i	$0.759506\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.4924	−1.42364
$78$	0	0
$79$	6.24621	0.702754	0.351377	−	0.936234i	$-0.385714\pi$
$79$	6.24621	0.702754	0.351377	+	0.936234i	$0.385714\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.93087	−0.743067
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−11.1231	−1.16602
$92$	0	0
$93$	8.68466	0.900557
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.246211	−0.0249990	−0.0124995	−	0.999922i	$-0.503979\pi$
$97$	−0.246211	−0.0249990	−0.0124995	+	0.999922i	$0.503979\pi$
$98$	0	0
$99$	2.24621	0.225753
$100$	0	0
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−8.24621	−0.789844	−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621	−0.789844	−0.394922	+	0.918715i	$0.629228\pi$
$110$	0	0
$111$	−1.75379	−0.166462
$112$	0	0
$113$	−3.75379	−0.353127	−0.176563	−	0.984289i	$-0.556498\pi$
$113$	−3.75379	−0.353127	−0.176563	+	0.984289i	$0.556498\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−5.56155	−0.501468
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.80776	−0.337884	−0.168942	−	0.985626i	$-0.554035\pi$
$127$	−3.80776	−0.337884	−0.168942	+	0.985626i	$0.554035\pi$
$128$	0	0
$129$	1.36932	0.120562
$130$	0	0
$131$	−18.9309	−1.65400	−0.826999	−	0.562204i	$-0.809954\pi$
$131$	−18.9309	−1.65400	−0.826999	+	0.562204i	$0.809954\pi$
$132$	0	0
$133$	12.4924	1.08323
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.87689	−0.587533	−0.293766	−	0.955877i	$-0.594909\pi$
$137$	−6.87689	−0.587533	−0.293766	+	0.955877i	$0.594909\pi$
$138$	0	0
$139$	12.6847	1.07590	0.537949	−	0.842977i	$-0.319199\pi$
$139$	12.6847	1.07590	0.537949	+	0.842977i	$0.319199\pi$
$140$	0	0
$141$	−13.5616	−1.14209
$142$	0	0
$143$	14.2462	1.19133
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.30019	0.354673
$148$	0	0
$149$	12.2462	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$149$	12.2462	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$150$	0	0
$151$	3.80776	0.309871	0.154936	−	0.987925i	$-0.450483\pi$
$151$	3.80776	0.309871	0.154936	+	0.987925i	$0.450483\pi$
$152$	0	0
$153$	2.87689	0.232583
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−19.1231	−1.51656
$160$	0	0
$161$	−3.12311	−0.246135
$162$	0	0
$163$	12.6847	0.993539	0.496770	−	0.867882i	$-0.334519\pi$
$163$	12.6847	0.993539	0.496770	+	0.867882i	$0.334519\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	−2.24621	−0.171772
$172$	0	0
$173$	22.4924	1.71007	0.855034	−	0.518573i	$-0.173536\pi$
$173$	22.4924	1.71007	0.855034	+	0.518573i	$0.173536\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.0000	1.20263
$178$	0	0
$179$	4.68466	0.350148	0.175074	−	0.984555i	$-0.443984\pi$
$179$	4.68466	0.350148	0.175074	+	0.984555i	$0.443984\pi$
$180$	0	0
$181$	−5.12311	−0.380797	−0.190399	−	0.981707i	$-0.560978\pi$
$181$	−5.12311	−0.380797	−0.190399	+	0.981707i	$0.560978\pi$
$182$	0	0
$183$	4.49242	0.332089
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.4924	1.49855
$188$	0	0
$189$	−17.3693	−1.26343
$190$	0	0
$191$	−11.1231	−0.804840	−0.402420	−	0.915455i	$-0.631831\pi$
$191$	−11.1231	−0.804840	−0.402420	+	0.915455i	$0.631831\pi$
$192$	0	0
$193$	−20.4384	−1.47119	−0.735596	−	0.677421i	$-0.763098\pi$
$193$	−20.4384	−1.47119	−0.735596	+	0.677421i	$0.763098\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.5616	1.10871	0.554357	−	0.832279i	$-0.312964\pi$
$197$	15.5616	1.10871	0.554357	+	0.832279i	$0.312964\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	−13.8617	−0.972903
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.561553	0.0390306
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	2.24621	0.154636	0.0773178	−	0.997006i	$-0.475364\pi$
$211$	2.24621	0.154636	0.0773178	+	0.997006i	$0.475364\pi$
$212$	0	0
$213$	−13.5616	−0.929222
$214$	0	0
$215$	0	0
$216$	0	0
$217$	17.3693	1.17911
$218$	0	0
$219$	−19.4233	−1.31250
$220$	0	0
$221$	18.2462	1.22737
$222$	0	0
$223$	−12.4924	−0.836554	−0.418277	−	0.908319i	$-0.637366\pi$
$223$	−12.4924	−0.836554	−0.418277	+	0.908319i	$0.637366\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.876894	−0.0582015	−0.0291008	−	0.999576i	$-0.509264\pi$
$227$	−0.876894	−0.0582015	−0.0291008	+	0.999576i	$0.509264\pi$
$228$	0	0
$229$	−16.2462	−1.07358	−0.536790	−	0.843716i	$-0.680363\pi$
$229$	−16.2462	−1.07358	−0.536790	+	0.843716i	$0.680363\pi$
$230$	0	0
$231$	−19.5076	−1.28350
$232$	0	0
$233$	8.43845	0.552821	0.276411	−	0.961040i	$-0.410855\pi$
$233$	8.43845	0.552821	0.276411	+	0.961040i	$0.410855\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.75379	0.633577
$238$	0	0
$239$	11.8078	0.763781	0.381890	−	0.924208i	$-0.375273\pi$
$239$	11.8078	0.763781	0.381890	+	0.924208i	$0.375273\pi$
$240$	0	0
$241$	14.4924	0.933539	0.466769	−	0.884379i	$-0.345418\pi$
$241$	14.4924	0.933539	0.466769	+	0.884379i	$0.345418\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.2462	−0.906465
$248$	0	0
$249$	−18.7386	−1.18751
$250$	0	0
$251$	−10.2462	−0.646735	−0.323368	−	0.946273i	$-0.604815\pi$
$251$	−10.2462	−0.646735	−0.323368	+	0.946273i	$0.604815\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.0540	−1.74996	−0.874979	−	0.484160i	$-0.839125\pi$
$257$	−28.0540	−1.74996	−0.874979	+	0.484160i	$0.839125\pi$
$258$	0	0
$259$	−3.50758	−0.217950
$260$	0	0
$261$	2.49242	0.154277
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.6155	0.955655
$268$	0	0
$269$	30.3002	1.84743	0.923717	−	0.383074i	$-0.125135\pi$
$269$	30.3002	1.84743	0.923717	+	0.383074i	$0.125135\pi$
$270$	0	0
$271$	14.2462	0.865396	0.432698	−	0.901539i	$-0.357562\pi$
$271$	14.2462	0.865396	0.432698	+	0.901539i	$0.357562\pi$
$272$	0	0
$273$	−17.3693	−1.05124
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.5616	0.935003	0.467502	−	0.883992i	$-0.345154\pi$
$277$	15.5616	0.935003	0.467502	+	0.883992i	$0.345154\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	−2.49242	−0.148685	−0.0743427	−	0.997233i	$-0.523686\pi$
$281$	−2.49242	−0.148685	−0.0743427	+	0.997233i	$0.523686\pi$
$282$	0	0
$283$	−31.1231	−1.85008	−0.925038	−	0.379874i	$-0.875967\pi$
$283$	−31.1231	−1.85008	−0.925038	+	0.379874i	$0.875967\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.1231	−0.656576
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	−0.384472	−0.0225381
$292$	0	0
$293$	21.1231	1.23403	0.617013	−	0.786953i	$-0.288343\pi$
$293$	21.1231	1.23403	0.617013	+	0.786953i	$0.288343\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	22.2462	1.29086
$298$	0	0
$299$	3.56155	0.205970
$300$	0	0
$301$	2.73863	0.157852
$302$	0	0
$303$	−25.3693	−1.45743
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	−22.2462	−1.26554
$310$	0	0
$311$	−14.9309	−0.846652	−0.423326	−	0.905977i	$-0.639137\pi$
$311$	−14.9309	−0.846652	−0.423326	+	0.905977i	$0.639137\pi$
$312$	0	0
$313$	−6.87689	−0.388705	−0.194353	−	0.980932i	$-0.562261\pi$
$313$	−6.87689	−0.388705	−0.194353	+	0.980932i	$0.562261\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.7386	1.61412	0.807061	−	0.590468i	$-0.201057\pi$
$317$	28.7386	1.61412	0.807061	+	0.590468i	$0.201057\pi$
$318$	0	0
$319$	17.7538	0.994021
$320$	0	0
$321$	18.7386	1.04589
$322$	0	0
$323$	−20.4924	−1.14023
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.8769	−0.712094
$328$	0	0
$329$	−27.1231	−1.49535
$330$	0	0
$331$	6.43845	0.353889	0.176945	−	0.984221i	$-0.443379\pi$
$331$	6.43845	0.353889	0.176945	+	0.984221i	$0.443379\pi$
$332$	0	0
$333$	0.630683	0.0345612
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−32.2462	−1.75656	−0.878282	−	0.478144i	$-0.841310\pi$
$337$	−32.2462	−1.75656	−0.878282	+	0.478144i	$0.841310\pi$
$338$	0	0
$339$	−5.86174	−0.318366
$340$	0	0
$341$	−22.2462	−1.20470
$342$	0	0
$343$	−13.2614	−0.716046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.4924	−0.885360	−0.442680	−	0.896680i	$-0.645972\pi$
$347$	−16.4924	−0.885360	−0.442680	+	0.896680i	$0.645972\pi$
$348$	0	0
$349$	17.8078	0.953228	0.476614	−	0.879113i	$-0.341864\pi$
$349$	17.8078	0.953228	0.476614	+	0.879113i	$0.341864\pi$
$350$	0	0
$351$	19.8078	1.05726
$352$	0	0
$353$	−14.1922	−0.755377	−0.377688	−	0.925933i	$-0.623281\pi$
$353$	−14.1922	−0.755377	−0.377688	+	0.925933i	$0.623281\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−24.9848	−1.32234
$358$	0	0
$359$	−30.2462	−1.59633	−0.798167	−	0.602436i	$-0.794197\pi$
$359$	−30.2462	−1.59633	−0.798167	+	0.602436i	$0.794197\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	7.80776	0.409801
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.4924	0.652099	0.326050	−	0.945353i	$-0.394282\pi$
$367$	12.4924	0.652099	0.326050	+	0.945353i	$0.394282\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−38.2462	−1.98564
$372$	0	0
$373$	−24.7386	−1.28092	−0.640459	−	0.767992i	$-0.721256\pi$
$373$	−24.7386	−1.28092	−0.640459	+	0.767992i	$0.721256\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.8078	0.814141
$378$	0	0
$379$	2.24621	0.115380	0.0576901	−	0.998335i	$-0.481626\pi$
$379$	2.24621	0.115380	0.0576901	+	0.998335i	$0.481626\pi$
$380$	0	0
$381$	−5.94602	−0.304624
$382$	0	0
$383$	7.61553	0.389135	0.194568	−	0.980889i	$-0.437670\pi$
$383$	7.61553	0.389135	0.194568	+	0.980889i	$0.437670\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.492423	−0.0250312
$388$	0	0
$389$	−33.6155	−1.70437	−0.852187	−	0.523237i	$-0.824724\pi$
$389$	−33.6155	−1.70437	−0.852187	+	0.523237i	$0.824724\pi$
$390$	0	0
$391$	5.12311	0.259087
$392$	0	0
$393$	−29.5616	−1.49118
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.9309	1.25124	0.625622	−	0.780126i	$-0.284845\pi$
$397$	24.9309	1.25124	0.625622	+	0.780126i	$0.284845\pi$
$398$	0	0
$399$	19.5076	0.976600
$400$	0	0
$401$	20.7386	1.03564	0.517819	−	0.855490i	$-0.326744\pi$
$401$	20.7386	1.03564	0.517819	+	0.855490i	$0.326744\pi$
$402$	0	0
$403$	−19.8078	−0.986695
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.49242	0.222681
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	0	0
$411$	−10.7386	−0.529698
$412$	0	0
$413$	32.0000	1.57462
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.8078	0.969990
$418$	0	0
$419$	−0.876894	−0.0428391	−0.0214195	−	0.999771i	$-0.506819\pi$
$419$	−0.876894	−0.0428391	−0.0214195	+	0.999771i	$0.506819\pi$
$420$	0	0
$421$	−6.49242	−0.316421	−0.158211	−	0.987405i	$-0.550573\pi$
$421$	−6.49242	−0.316421	−0.158211	+	0.987405i	$0.550573\pi$
$422$	0	0
$423$	4.87689	0.237123
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.98485	0.434807
$428$	0	0
$429$	22.2462	1.07406
$430$	0	0
$431$	7.61553	0.366827	0.183414	−	0.983036i	$-0.441285\pi$
$431$	7.61553	0.366827	0.183414	+	0.983036i	$0.441285\pi$
$432$	0	0
$433$	24.7386	1.18886	0.594431	−	0.804146i	$-0.297377\pi$
$433$	24.7386	1.18886	0.594431	+	0.804146i	$0.297377\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−32.3002	−1.54160	−0.770802	−	0.637075i	$-0.780144\pi$
$439$	−32.3002	−1.54160	−0.770802	+	0.637075i	$0.780144\pi$
$440$	0	0
$441$	−1.54640	−0.0736380
$442$	0	0
$443$	3.31534	0.157517	0.0787583	−	0.996894i	$-0.474904\pi$
$443$	3.31534	0.157517	0.0787583	+	0.996894i	$0.474904\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	19.1231	0.904492
$448$	0	0
$449$	−32.7386	−1.54503	−0.772516	−	0.634996i	$-0.781002\pi$
$449$	−32.7386	−1.54503	−0.772516	+	0.634996i	$0.781002\pi$
$450$	0	0
$451$	14.2462	0.670828
$452$	0	0
$453$	5.94602	0.279369
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.12311	0.426761	0.213380	−	0.976969i	$-0.431553\pi$
$457$	9.12311	0.426761	0.213380	+	0.976969i	$0.431553\pi$
$458$	0	0
$459$	28.4924	1.32991
$460$	0	0
$461$	27.5616	1.28367	0.641835	−	0.766843i	$-0.278174\pi$
$461$	27.5616	1.28367	0.641835	+	0.766843i	$0.278174\pi$
$462$	0	0
$463$	6.24621	0.290286	0.145143	−	0.989411i	$-0.453636\pi$
$463$	6.24621	0.290286	0.145143	+	0.989411i	$0.453636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.6307	−0.491929	−0.245965	−	0.969279i	$-0.579105\pi$
$467$	−10.6307	−0.491929	−0.245965	+	0.969279i	$0.579105\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	3.12311	0.143905
$472$	0	0
$473$	−3.50758	−0.161279
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.87689	0.314871
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	−4.87689	−0.221906
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−34.0540	−1.54313	−0.771566	−	0.636149i	$-0.780526\pi$
$487$	−34.0540	−1.54313	−0.771566	+	0.636149i	$0.780526\pi$
$488$	0	0
$489$	19.8078	0.895738
$490$	0	0
$491$	6.43845	0.290563	0.145282	−	0.989390i	$-0.453591\pi$
$491$	6.43845	0.290563	0.145282	+	0.989390i	$0.453591\pi$
$492$	0	0
$493$	22.7386	1.02410
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.1231	−1.21664
$498$	0	0
$499$	38.0540	1.70353	0.851765	−	0.523924i	$-0.175532\pi$
$499$	38.0540	1.70353	0.851765	+	0.523924i	$0.175532\pi$
$500$	0	0
$501$	12.4924	0.558120
$502$	0	0
$503$	15.6155	0.696262	0.348131	−	0.937446i	$-0.386816\pi$
$503$	15.6155	0.696262	0.348131	+	0.937446i	$0.386816\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.492423	−0.0218693
$508$	0	0
$509$	−4.43845	−0.196731	−0.0983654	−	0.995150i	$-0.531361\pi$
$509$	−4.43845	−0.196731	−0.0983654	+	0.995150i	$0.531361\pi$
$510$	0	0
$511$	−38.8466	−1.71847
$512$	0	0
$513$	−22.2462	−0.982194
$514$	0	0
$515$	0	0
$516$	0	0
$517$	34.7386	1.52780
$518$	0	0
$519$	35.1231	1.54173
$520$	0	0
$521$	39.8617	1.74637	0.873187	−	0.487385i	$-0.162049\pi$
$521$	39.8617	1.74637	0.873187	+	0.487385i	$0.162049\pi$
$522$	0	0
$523$	−33.8617	−1.48067	−0.740335	−	0.672238i	$-0.765333\pi$
$523$	−33.8617	−1.48067	−0.740335	+	0.672238i	$0.765333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.4924	−1.24115
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.75379	−0.249693
$532$	0	0
$533$	12.6847	0.549434
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.31534	0.315680
$538$	0	0
$539$	−11.0152	−0.474456
$540$	0	0
$541$	21.3153	0.916418	0.458209	−	0.888844i	$-0.348491\pi$
$541$	21.3153	0.916418	0.458209	+	0.888844i	$0.348491\pi$
$542$	0	0
$543$	−8.00000	−0.343313
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−38.0540	−1.62707	−0.813535	−	0.581516i	$-0.802460\pi$
$547$	−38.0540	−1.62707	−0.813535	+	0.581516i	$0.802460\pi$
$548$	0	0
$549$	−1.61553	−0.0689491
$550$	0	0
$551$	−17.7538	−0.756337
$552$	0	0
$553$	19.5076	0.829547
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.2462	1.02734	0.513672	−	0.857986i	$-0.328285\pi$
$557$	24.2462	1.02734	0.513672	+	0.857986i	$0.328285\pi$
$558$	0	0
$559$	−3.12311	−0.132093
$560$	0	0
$561$	32.0000	1.35104
$562$	0	0
$563$	−43.2311	−1.82197	−0.910986	−	0.412438i	$-0.864678\pi$
$563$	−43.2311	−1.82197	−0.910986	+	0.412438i	$0.864678\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.8617	−0.918107
$568$	0	0
$569$	28.7386	1.20479	0.602393	−	0.798200i	$-0.294214\pi$
$569$	28.7386	1.20479	0.602393	+	0.798200i	$0.294214\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	−17.3693	−0.725614
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−29.4233	−1.22491	−0.612454	−	0.790506i	$-0.709817\pi$
$577$	−29.4233	−1.22491	−0.612454	+	0.790506i	$0.709817\pi$
$578$	0	0
$579$	−31.9157	−1.32637
$580$	0	0
$581$	−37.4773	−1.55482
$582$	0	0
$583$	48.9848	2.02874
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.17708	−0.378779	−0.189389	−	0.981902i	$-0.560651\pi$
$587$	−9.17708	−0.378779	−0.189389	+	0.981902i	$0.560651\pi$
$588$	0	0
$589$	22.2462	0.916639
$590$	0	0
$591$	24.3002	0.999576
$592$	0	0
$593$	38.9848	1.60092	0.800458	−	0.599389i	$-0.204590\pi$
$593$	38.9848	1.60092	0.800458	+	0.599389i	$0.204590\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	37.4773	1.53384
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−27.1771	−1.10858	−0.554288	−	0.832325i	$-0.687009\pi$
$601$	−27.1771	−1.10858	−0.554288	+	0.832325i	$0.687009\pi$
$602$	0	0
$603$	−5.75379	−0.234312
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−21.6458	−0.877134
$610$	0	0
$611$	30.9309	1.25133
$612$	0	0
$613$	5.12311	0.206920	0.103460	−	0.994634i	$-0.467009\pi$
$613$	5.12311	0.206920	0.103460	+	0.994634i	$0.467009\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.61553	0.226073	0.113036	−	0.993591i	$-0.463942\pi$
$617$	5.61553	0.226073	0.113036	+	0.993591i	$0.463942\pi$
$618$	0	0
$619$	36.9848	1.48655	0.743273	−	0.668988i	$-0.233272\pi$
$619$	36.9848	1.48655	0.743273	+	0.668988i	$0.233272\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	31.2311	1.25125
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−24.9848	−0.997799
$628$	0	0
$629$	5.75379	0.229419
$630$	0	0
$631$	−6.63068	−0.263963	−0.131982	−	0.991252i	$-0.542134\pi$
$631$	−6.63068	−0.263963	−0.131982	+	0.991252i	$0.542134\pi$
$632$	0	0
$633$	3.50758	0.139414
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.80776	−0.388598
$638$	0	0
$639$	4.87689	0.192927
$640$	0	0
$641$	40.2462	1.58963	0.794815	−	0.606852i	$-0.207568\pi$
$641$	40.2462	1.58963	0.794815	+	0.606852i	$0.207568\pi$
$642$	0	0
$643$	46.3542	1.82803	0.914015	−	0.405681i	$-0.132966\pi$
$643$	46.3542	1.82803	0.914015	+	0.405681i	$0.132966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.56155	0.218647	0.109324	−	0.994006i	$-0.465132\pi$
$647$	5.56155	0.218647	0.109324	+	0.994006i	$0.465132\pi$
$648$	0	0
$649$	−40.9848	−1.60880
$650$	0	0
$651$	27.1231	1.06304
$652$	0	0
$653$	−50.7926	−1.98767	−0.993834	−	0.110876i	$-0.964634\pi$
$653$	−50.7926	−1.98767	−0.993834	+	0.110876i	$0.964634\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.98485	0.272505
$658$	0	0
$659$	−2.24621	−0.0875000	−0.0437500	−	0.999043i	$-0.513930\pi$
$659$	−2.24621	−0.0875000	−0.0437500	+	0.999043i	$0.513930\pi$
$660$	0	0
$661$	−22.4924	−0.874854	−0.437427	−	0.899254i	$-0.644110\pi$
$661$	−22.4924	−0.874854	−0.437427	+	0.899254i	$0.644110\pi$
$662$	0	0
$663$	28.4924	1.10655
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.43845	0.171857
$668$	0	0
$669$	−19.5076	−0.754207
$670$	0	0
$671$	−11.5076	−0.444245
$672$	0	0
$673$	0.438447	0.0169009	0.00845045	−	0.999964i	$-0.497310\pi$
$673$	0.438447	0.0169009	0.00845045	+	0.999964i	$0.497310\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.7386	1.10452	0.552258	−	0.833673i	$-0.313766\pi$
$677$	28.7386	1.10452	0.552258	+	0.833673i	$0.313766\pi$
$678$	0	0
$679$	−0.768944	−0.0295094
$680$	0	0
$681$	−1.36932	−0.0524723
$682$	0	0
$683$	14.4384	0.552472	0.276236	−	0.961090i	$-0.410913\pi$
$683$	14.4384	0.552472	0.276236	+	0.961090i	$0.410913\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−25.3693	−0.967900
$688$	0	0
$689$	43.6155	1.66162
$690$	0	0
$691$	−10.2462	−0.389784	−0.194892	−	0.980825i	$-0.562436\pi$
$691$	−10.2462	−0.389784	−0.194892	+	0.980825i	$0.562436\pi$
$692$	0	0
$693$	7.01515	0.266484
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.2462	0.691125
$698$	0	0
$699$	13.1771	0.498403
$700$	0	0
$701$	−26.9848	−1.01920	−0.509602	−	0.860410i	$-0.670207\pi$
$701$	−26.9848	−1.01920	−0.509602	+	0.860410i	$0.670207\pi$
$702$	0	0
$703$	−4.49242	−0.169435
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−50.7386	−1.90822
$708$	0	0
$709$	8.73863	0.328186	0.164093	−	0.986445i	$-0.447530\pi$
$709$	8.73863	0.328186	0.164093	+	0.986445i	$0.447530\pi$
$710$	0	0
$711$	−3.50758	−0.131544
$712$	0	0
$713$	−5.56155	−0.208282
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18.4384	0.688596
$718$	0	0
$719$	−10.7386	−0.400483	−0.200242	−	0.979747i	$-0.564173\pi$
$719$	−10.7386	−0.400483	−0.200242	+	0.979747i	$0.564173\pi$
$720$	0	0
$721$	−44.4924	−1.65698
$722$	0	0
$723$	22.6307	0.841644
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−28.8769	−1.07098	−0.535492	−	0.844540i	$-0.679874\pi$
$727$	−28.8769	−1.07098	−0.535492	+	0.844540i	$0.679874\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−4.49242	−0.166158
$732$	0	0
$733$	−2.87689	−0.106261	−0.0531303	−	0.998588i	$-0.516920\pi$
$733$	−2.87689	−0.106261	−0.0531303	+	0.998588i	$0.516920\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40.9848	−1.50970
$738$	0	0
$739$	−42.5464	−1.56509	−0.782547	−	0.622591i	$-0.786080\pi$
$739$	−42.5464	−1.56509	−0.782547	+	0.622591i	$0.786080\pi$
$740$	0	0
$741$	−22.2462	−0.817235
$742$	0	0
$743$	30.2462	1.10963	0.554813	−	0.831975i	$-0.312790\pi$
$743$	30.2462	1.10963	0.554813	+	0.831975i	$0.312790\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.73863	0.246554
$748$	0	0
$749$	37.4773	1.36939
$750$	0	0
$751$	8.38447	0.305954	0.152977	−	0.988230i	$-0.451114\pi$
$751$	8.38447	0.305954	0.152977	+	0.988230i	$0.451114\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.1231	−1.20388	−0.601940	−	0.798541i	$-0.705605\pi$
$757$	−33.1231	−1.20388	−0.601940	+	0.798541i	$0.705605\pi$
$758$	0	0
$759$	6.24621	0.226723
$760$	0	0
$761$	42.3002	1.53338	0.766690	−	0.642017i	$-0.221902\pi$
$761$	42.3002	1.53338	0.766690	+	0.642017i	$0.221902\pi$
$762$	0	0
$763$	−25.7538	−0.932350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.4924	−1.31767
$768$	0	0
$769$	5.50758	0.198608	0.0993042	−	0.995057i	$-0.468338\pi$
$769$	5.50758	0.198608	0.0993042	+	0.995057i	$0.468338\pi$
$770$	0	0
$771$	−43.8078	−1.57770
$772$	0	0
$773$	−12.2462	−0.440466	−0.220233	−	0.975447i	$-0.570682\pi$
$773$	−12.2462	−0.440466	−0.220233	+	0.975447i	$0.570682\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.47727	−0.196496
$778$	0	0
$779$	−14.2462	−0.510423
$780$	0	0
$781$	34.7386	1.24305
$782$	0	0
$783$	24.6847	0.882158
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29.7538	1.06061	0.530304	−	0.847808i	$-0.322078\pi$
$787$	29.7538	1.06061	0.530304	+	0.847808i	$0.322078\pi$
$788$	0	0
$789$	12.4924	0.444742
$790$	0	0
$791$	−11.7235	−0.416839
$792$	0	0
$793$	−10.2462	−0.363854
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.8769	−0.668654	−0.334327	−	0.942457i	$-0.608509\pi$
$797$	−18.8769	−0.668654	−0.334327	+	0.942457i	$0.608509\pi$
$798$	0	0
$799$	44.4924	1.57403
$800$	0	0
$801$	−5.61553	−0.198415
$802$	0	0
$803$	49.7538	1.75577
$804$	0	0
$805$	0	0
$806$	0	0
$807$	47.3153	1.66558
$808$	0	0
$809$	22.4924	0.790791	0.395396	−	0.918511i	$-0.370607\pi$
$809$	22.4924	0.790791	0.395396	+	0.918511i	$0.370607\pi$
$810$	0	0
$811$	−20.6847	−0.726337	−0.363168	−	0.931724i	$-0.618305\pi$
$811$	−20.6847	−0.726337	−0.363168	+	0.931724i	$0.618305\pi$
$812$	0	0
$813$	22.2462	0.780209
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.50758	0.122715
$818$	0	0
$819$	6.24621	0.218260
$820$	0	0
$821$	−48.2462	−1.68380	−0.841902	−	0.539630i	$-0.818564\pi$
$821$	−48.2462	−1.68380	−0.841902	+	0.539630i	$0.818564\pi$
$822$	0	0
$823$	21.5616	0.751588	0.375794	−	0.926703i	$-0.377370\pi$
$823$	21.5616	0.751588	0.375794	+	0.926703i	$0.377370\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	17.5076	0.608063	0.304032	−	0.952662i	$-0.401667\pi$
$829$	17.5076	0.608063	0.304032	+	0.952662i	$0.401667\pi$
$830$	0	0
$831$	24.3002	0.842964
$832$	0	0
$833$	−14.1080	−0.488812
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−30.9309	−1.06913
$838$	0	0
$839$	−31.6155	−1.09149	−0.545745	−	0.837952i	$-0.683753\pi$
$839$	−31.6155	−1.09149	−0.545745	+	0.837952i	$0.683753\pi$
$840$	0	0
$841$	−9.30019	−0.320696
$842$	0	0
$843$	−3.89205	−0.134049
$844$	0	0
$845$	0	0
$846$	0	0
$847$	15.6155	0.536556
$848$	0	0
$849$	−48.6004	−1.66796
$850$	0	0
$851$	1.12311	0.0384996
$852$	0	0
$853$	42.9848	1.47177	0.735887	−	0.677105i	$-0.236766\pi$
$853$	42.9848	1.47177	0.735887	+	0.677105i	$0.236766\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.1771	0.381802	0.190901	−	0.981609i	$-0.438859\pi$
$857$	11.1771	0.381802	0.190901	+	0.981609i	$0.438859\pi$
$858$	0	0
$859$	−23.4233	−0.799192	−0.399596	−	0.916691i	$-0.630850\pi$
$859$	−23.4233	−0.799192	−0.399596	+	0.916691i	$0.630850\pi$
$860$	0	0
$861$	−17.3693	−0.591945
$862$	0	0
$863$	43.4233	1.47815	0.739073	−	0.673625i	$-0.235264\pi$
$863$	43.4233	1.47815	0.739073	+	0.673625i	$0.235264\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.4384	0.490355
$868$	0	0
$869$	−24.9848	−0.847553
$870$	0	0
$871$	−36.4924	−1.23650
$872$	0	0
$873$	0.138261	0.00467941
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.73863	−0.295083	−0.147541	−	0.989056i	$-0.547136\pi$
$877$	−8.73863	−0.295083	−0.147541	+	0.989056i	$0.547136\pi$
$878$	0	0
$879$	32.9848	1.11255
$880$	0	0
$881$	−50.1080	−1.68818	−0.844090	−	0.536202i	$-0.819859\pi$
$881$	−50.1080	−1.68818	−0.844090	+	0.536202i	$0.819859\pi$
$882$	0	0
$883$	7.50758	0.252650	0.126325	−	0.991989i	$-0.459682\pi$
$883$	7.50758	0.252650	0.126325	+	0.991989i	$0.459682\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.5616	1.26119	0.630597	−	0.776111i	$-0.282810\pi$
$887$	37.5616	1.26119	0.630597	+	0.776111i	$0.282810\pi$
$888$	0	0
$889$	−11.8920	−0.398847
$890$	0	0
$891$	28.0000	0.938035
$892$	0	0
$893$	−34.7386	−1.16248
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.56155	0.185695
$898$	0	0
$899$	−24.6847	−0.823279
$900$	0	0
$901$	62.7386	2.09013
$902$	0	0
$903$	4.27652	0.142314
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	0	0
$909$	9.12311	0.302594
$910$	0	0
$911$	12.4924	0.413892	0.206946	−	0.978352i	$-0.433647\pi$
$911$	12.4924	0.413892	0.206946	+	0.978352i	$0.433647\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−59.1231	−1.95242
$918$	0	0
$919$	21.8617	0.721152	0.360576	−	0.932730i	$-0.382580\pi$
$919$	21.8617	0.721152	0.360576	+	0.932730i	$0.382580\pi$
$920$	0	0
$921$	−18.7386	−0.617459
$922$	0	0
$923$	30.9309	1.01810
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−24.0540	−0.789185	−0.394593	−	0.918856i	$-0.629114\pi$
$929$	−24.0540	−0.789185	−0.394593	+	0.918856i	$0.629114\pi$
$930$	0	0
$931$	11.0152	0.361007
$932$	0	0
$933$	−23.3153	−0.763310
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.6307	−0.543301	−0.271650	−	0.962396i	$-0.587569\pi$
$937$	−16.6307	−0.543301	−0.271650	+	0.962396i	$0.587569\pi$
$938$	0	0
$939$	−10.7386	−0.350442
$940$	0	0
$941$	46.6004	1.51913	0.759564	−	0.650432i	$-0.225412\pi$
$941$	46.6004	1.51913	0.759564	+	0.650432i	$0.225412\pi$
$942$	0	0
$943$	3.56155	0.115980
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.8078	1.81351	0.906754	−	0.421659i	$-0.138552\pi$
$947$	55.8078	1.81351	0.906754	+	0.421659i	$0.138552\pi$
$948$	0	0
$949$	44.3002	1.43804
$950$	0	0
$951$	44.8769	1.45523
$952$	0	0
$953$	−46.4924	−1.50604	−0.753019	−	0.657999i	$-0.771403\pi$
$953$	−46.4924	−1.50604	−0.753019	+	0.657999i	$0.771403\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	27.7235	0.896173
$958$	0	0
$959$	−21.4773	−0.693537
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	−6.73863	−0.217149
$964$	0	0
$965$	0	0
$966$	0	0
$967$	45.1771	1.45280	0.726398	−	0.687274i	$-0.241193\pi$
$967$	45.1771	1.45280	0.726398	+	0.687274i	$0.241193\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	0	0
$971$	−21.3693	−0.685774	−0.342887	−	0.939377i	$-0.611405\pi$
$971$	−21.3693	−0.685774	−0.342887	+	0.939377i	$0.611405\pi$
$972$	0	0
$973$	39.6155	1.27002
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.1080	0.835267	0.417634	−	0.908615i	$-0.362860\pi$
$977$	26.1080	0.835267	0.417634	+	0.908615i	$0.362860\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	4.63068	0.147846
$982$	0	0
$983$	−28.8769	−0.921030	−0.460515	−	0.887652i	$-0.652335\pi$
$983$	−28.8769	−0.921030	−0.460515	+	0.887652i	$0.652335\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−42.3542	−1.34815
$988$	0	0
$989$	−0.876894	−0.0278836
$990$	0	0
$991$	20.4924	0.650963	0.325482	−	0.945548i	$-0.394474\pi$
$991$	20.4924	0.650963	0.325482	+	0.945548i	$0.394474\pi$
$992$	0	0
$993$	10.0540	0.319053
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−0.738634	−0.0233928	−0.0116964	−	0.999932i	$-0.503723\pi$
$997$	−0.738634	−0.0233928	−0.0116964	+	0.999932i	$0.503723\pi$
$998$	0	0
$999$	6.24621	0.197621

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.r.1.2		2
4.3	odd	2		9200.2.a.bx.1.1		2
5.2	odd	4		4600.2.e.m.4049.2		4
5.3	odd	4		4600.2.e.m.4049.3		4
5.4	even	2		920.2.a.f.1.1	✓	2
15.14	odd	2		8280.2.a.bb.1.1		2
20.19	odd	2		1840.2.a.k.1.2		2
40.19	odd	2		7360.2.a.bm.1.1		2
40.29	even	2		7360.2.a.bj.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.f.1.1	✓	2	5.4	even	2
1840.2.a.k.1.2		2	20.19	odd	2
4600.2.a.r.1.2		2	1.1	even	1	trivial
4600.2.e.m.4049.2		4	5.2	odd	4
4600.2.e.m.4049.3		4	5.3	odd	4
7360.2.a.bj.1.2		2	40.29	even	2
7360.2.a.bm.1.1		2	40.19	odd	2
8280.2.a.bb.1.1		2	15.14	odd	2
9200.2.a.bx.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +3.12311 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +3.12311 q^{7} -0.561553 q^{9} -4.00000 q^{11} -3.56155 q^{13} -5.12311 q^{17} +4.00000 q^{19} +4.87689 q^{21} -1.00000 q^{23} -5.56155 q^{27} -4.43845 q^{29} +5.56155 q^{31} -6.24621 q^{33} -1.12311 q^{37} -5.56155 q^{39} -3.56155 q^{41} +0.876894 q^{43} -8.68466 q^{47} +2.75379 q^{49} -8.00000 q^{51} -12.2462 q^{53} +6.24621 q^{57} +10.2462 q^{59} +2.87689 q^{61} -1.75379 q^{63} +10.2462 q^{67} -1.56155 q^{69} -8.68466 q^{71} -12.4384 q^{73} -12.4924 q^{77} +6.24621 q^{79} -7.00000 q^{81} -12.0000 q^{83} -6.93087 q^{87} +10.0000 q^{89} -11.1231 q^{91} +8.68466 q^{93} -0.246211 q^{97} +2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^7 + 3 * q^9	\( 2 q - q^{3} - 2 q^{7} + 3 q^{9} - 8 q^{11} - 3 q^{13} - 2 q^{17} + 8 q^{19} + 18 q^{21} - 2 q^{23} - 7 q^{27} - 13 q^{29} + 7 q^{31} + 4 q^{33} + 6 q^{37} - 7 q^{39} - 3 q^{41} + 10 q^{43} - 5 q^{47} + 22 q^{49} - 16 q^{51} - 8 q^{53} - 4 q^{57} + 4 q^{59} + 14 q^{61} - 20 q^{63} + 4 q^{67} + q^{69} - 5 q^{71} - 29 q^{73} + 8 q^{77} - 4 q^{79} - 14 q^{81} - 24 q^{83} + 15 q^{87} + 20 q^{89} - 14 q^{91} + 5 q^{93} + 16 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^7 + 3 * q^9 - 8 * q^11 - 3 * q^13 - 2 * q^17 + 8 * q^19 + 18 * q^21 - 2 * q^23 - 7 * q^27 - 13 * q^29 + 7 * q^31 + 4 * q^33 + 6 * q^37 - 7 * q^39 - 3 * q^41 + 10 * q^43 - 5 * q^47 + 22 * q^49 - 16 * q^51 - 8 * q^53 - 4 * q^57 + 4 * q^59 + 14 * q^61 - 20 * q^63 + 4 * q^67 + q^69 - 5 * q^71 - 29 * q^73 + 8 * q^77 - 4 * q^79 - 14 * q^81 - 24 * q^83 + 15 * q^87 + 20 * q^89 - 14 * q^91 + 5 * q^93 + 16 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more