LMFDB - Embedded newform 4600.2.e.m.4049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.m.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} -3.12311i q^{7} +0.561553 q^{9} +O(q^{10})$	$q+1.56155i q^{3} -3.12311i q^{7} +0.561553 q^{9} -4.00000 q^{11} -3.56155i q^{13} +5.12311i q^{17} -4.00000 q^{19} +4.87689 q^{21} -1.00000i q^{23} +5.56155i q^{27} +4.43845 q^{29} +5.56155 q^{31} -6.24621i q^{33} +1.12311i q^{37} +5.56155 q^{39} -3.56155 q^{41} +0.876894i q^{43} +8.68466i q^{47} -2.75379 q^{49} -8.00000 q^{51} -12.2462i q^{53} -6.24621i q^{57} -10.2462 q^{59} +2.87689 q^{61} -1.75379i q^{63} -10.2462i q^{67} +1.56155 q^{69} -8.68466 q^{71} -12.4384i q^{73} +12.4924i q^{77} -6.24621 q^{79} -7.00000 q^{81} -12.0000i q^{83} +6.93087i q^{87} -10.0000 q^{89} -11.1231 q^{91} +8.68466i q^{93} +0.246211i q^{97} -2.24621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 16 * q^11 - 16 * q^19 + 36 * q^21 + 26 * q^29 + 14 * q^31 + 14 * q^39 - 6 * q^41 - 44 * q^49 - 32 * q^51 - 8 * q^59 + 28 * q^61 - 2 * q^69 - 10 * q^71 + 8 * q^79 - 28 * q^81 - 40 * q^89 - 28 * q^91 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.12311i	− 1.18042i	−0.807249	−	0.590211i	$-0.799044\pi$
$7$	− 3.12311i	− 1.18042i	0.807249	−	0.590211i	$-0.200956\pi$
$8$	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
$13$	− 3.56155i	− 0.987797i	−0.869520	−	0.493899i	$-0.835571\pi$
$13$	− 3.56155i	− 0.987797i	0.869520	−	0.493899i	$-0.164429\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.12311i	1.24254i	0.783598	+	0.621268i	$0.213382\pi$
$17$	5.12311i	1.24254i	−0.783598	+	0.621268i	$0.786618\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	4.87689	1.06423
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	4.43845	0.824199	0.412099	−	0.911139i	$-0.364796\pi$
$29$	4.43845	0.824199	0.412099	+	0.911139i	$0.364796\pi$
$30$	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
$33$	− 6.24621i	− 1.08733i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.12311i	0.184637i	0.995730	+	0.0923187i	$0.0294279\pi$
$37$	1.12311i	0.184637i	−0.995730	+	0.0923187i	$0.970572\pi$
$38$	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
$43$	0.876894i	0.133725i	0.997762	+	0.0668626i	$0.0212989\pi$
$43$	0.876894i	0.133725i	−0.997762	+	0.0668626i	$0.978701\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.68466i	1.26679i	0.773830	+	0.633394i	$0.218339\pi$
$47$	8.68466i	1.26679i	−0.773830	+	0.633394i	$0.781661\pi$
$48$	0	0
$49$	−2.75379	−0.393398
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	− 12.2462i	− 1.68215i	−0.540921	−	0.841073i	$-0.681924\pi$
$53$	− 12.2462i	− 1.68215i	0.540921	−	0.841073i	$-0.318076\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.24621i	− 0.827331i
$58$	0	0
$59$	−10.2462	−1.33394	−0.666972	−	0.745083i	$-0.732410\pi$
$59$	−10.2462	−1.33394	−0.666972	+	0.745083i	$0.732410\pi$
$60$	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
$63$	− 1.75379i	− 0.220957i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.2462i	− 1.25177i	−0.779914	−	0.625887i	$-0.784737\pi$
$67$	− 10.2462i	− 1.25177i	0.779914	−	0.625887i	$-0.215263\pi$
$68$	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
$73$	− 12.4384i	− 1.45581i	−0.685678	−	0.727905i	$-0.740494\pi$
$73$	− 12.4384i	− 1.45581i	0.685678	−	0.727905i	$-0.259506\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.4924i	1.42364i
$78$	0	0
$79$	−6.24621	−0.702754	−0.351377	−	0.936234i	$-0.614286\pi$
$79$	−6.24621	−0.702754	−0.351377	+	0.936234i	$0.614286\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.93087i	0.743067i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−11.1231	−1.16602
$92$	0	0
$93$	8.68466i	0.900557i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.246211i	0.0249990i	0.999922	+	0.0124995i	$0.00397881\pi$
$97$	0.246211i	0.0249990i	−0.999922	+	0.0124995i	$0.996021\pi$
$98$	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
$103$	− 14.2462i	− 1.40372i	−0.712314	−	0.701860i	$-0.752353\pi$
$103$	− 14.2462i	− 1.40372i	0.712314	−	0.701860i	$-0.247647\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	8.24621	0.789844	0.394922	−	0.918715i	$-0.370772\pi$
$109$	8.24621	0.789844	0.394922	+	0.918715i	$0.370772\pi$
$110$	0	0
$111$	−1.75379	−0.166462
$112$	0	0
$113$	− 3.75379i	− 0.353127i	−0.984289	−	0.176563i	$-0.943502\pi$
$113$	− 3.75379i	− 0.353127i	0.984289	−	0.176563i	$-0.0564981\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	16.0000	1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 5.56155i	− 0.501468i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.80776i	0.337884i	0.985626	+	0.168942i	$0.0540351\pi$
$127$	3.80776i	0.337884i	−0.985626	+	0.168942i	$0.945965\pi$
$128$	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
$133$	12.4924i	1.08323i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.87689i	0.587533i	0.955877	+	0.293766i	$0.0949088\pi$
$137$	6.87689i	0.587533i	−0.955877	+	0.293766i	$0.905091\pi$
$138$	0	0
$139$	−12.6847	−1.07590	−0.537949	−	0.842977i	$-0.680801\pi$
$139$	−12.6847	−1.07590	−0.537949	+	0.842977i	$0.680801\pi$
$140$	0	0
$141$	−13.5616	−1.14209
$142$	0	0
$143$	14.2462i	1.19133i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.30019i	− 0.354673i
$148$	0	0
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	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
$153$	2.87689i	0.232583i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	19.1231	1.51656
$160$	0	0
$161$	−3.12311	−0.246135
$162$	0	0
$163$	12.6847i	0.993539i	0.867882	+	0.496770i	$0.165481\pi$
$163$	12.6847i	0.993539i	−0.867882	+	0.496770i	$0.834519\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	0.315342	0.0242570
$170$	0	0
$171$	−2.24621	−0.171772
$172$	0	0
$173$	22.4924i	1.71007i	0.518573	+	0.855034i	$0.326464\pi$
$173$	22.4924i	1.71007i	−0.518573	+	0.855034i	$0.673536\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 16.0000i	− 1.20263i
$178$	0	0
$179$	−4.68466	−0.350148	−0.175074	−	0.984555i	$-0.556016\pi$
$179$	−4.68466	−0.350148	−0.175074	+	0.984555i	$0.556016\pi$
$180$	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
$183$	4.49242i	0.332089i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.4924i	− 1.49855i
$188$	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
$193$	− 20.4384i	− 1.47119i	−0.677421	−	0.735596i	$-0.736902\pi$
$193$	− 20.4384i	− 1.47119i	0.677421	−	0.735596i	$-0.263098\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.5616i	− 1.10871i	−0.832279	−	0.554357i	$-0.812964\pi$
$197$	− 15.5616i	− 1.10871i	0.832279	−	0.554357i	$-0.187036\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	− 13.8617i	− 0.972903i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.561553i	− 0.0390306i
$208$	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
$213$	− 13.5616i	− 0.929222i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 17.3693i	− 1.17911i
$218$	0	0
$219$	19.4233	1.31250
$220$	0	0
$221$	18.2462	1.22737
$222$	0	0
$223$	− 12.4924i	− 0.836554i	−0.908319	−	0.418277i	$-0.862634\pi$
$223$	− 12.4924i	− 0.836554i	0.908319	−	0.418277i	$-0.137366\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.876894i	0.0582015i	0.999576	+	0.0291008i	$0.00926437\pi$
$227$	0.876894i	0.0582015i	−0.999576	+	0.0291008i	$0.990736\pi$
$228$	0	0
$229$	16.2462	1.07358	0.536790	−	0.843716i	$-0.319637\pi$
$229$	16.2462	1.07358	0.536790	+	0.843716i	$0.319637\pi$
$230$	0	0
$231$	−19.5076	−1.28350
$232$	0	0
$233$	8.43845i	0.552821i	0.961040	+	0.276411i	$0.0891449\pi$
$233$	8.43845i	0.552821i	−0.961040	+	0.276411i	$0.910855\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 9.75379i	− 0.633577i
$238$	0	0
$239$	−11.8078	−0.763781	−0.381890	−	0.924208i	$-0.624727\pi$
$239$	−11.8078	−0.763781	−0.381890	+	0.924208i	$0.624727\pi$
$240$	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
$243$	5.75379i	0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	14.2462i	0.906465i
$248$	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
$253$	4.00000i	0.251478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.0540i	1.74996i	0.484160	+	0.874979i	$0.339125\pi$
$257$	28.0540i	1.74996i	−0.484160	+	0.874979i	$0.660875\pi$
$258$	0	0
$259$	3.50758	0.217950
$260$	0	0
$261$	2.49242	0.154277
$262$	0	0
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 15.6155i	− 0.955655i
$268$	0	0
$269$	−30.3002	−1.84743	−0.923717	−	0.383074i	$-0.874865\pi$
$269$	−30.3002	−1.84743	−0.923717	+	0.383074i	$0.874865\pi$
$270$	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
$273$	− 17.3693i	− 1.05124i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.5616i	− 0.935003i	−0.883992	−	0.467502i	$-0.845154\pi$
$277$	− 15.5616i	− 0.935003i	0.883992	−	0.467502i	$-0.154846\pi$
$278$	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
$283$	− 31.1231i	− 1.85008i	−0.379874	−	0.925038i	$-0.624033\pi$
$283$	− 31.1231i	− 1.85008i	0.379874	−	0.925038i	$-0.375967\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.1231i	0.656576i
$288$	0	0
$289$	−9.24621	−0.543895
$290$	0	0
$291$	−0.384472	−0.0225381
$292$	0	0
$293$	21.1231i	1.23403i	0.786953	+	0.617013i	$0.211657\pi$
$293$	21.1231i	1.23403i	−0.786953	+	0.617013i	$0.788343\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 22.2462i	− 1.29086i
$298$	0	0
$299$	−3.56155	−0.205970
$300$	0	0
$301$	2.73863	0.157852
$302$	0	0
$303$	− 25.3693i	− 1.45743i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	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
$313$	− 6.87689i	− 0.388705i	−0.980932	−	0.194353i	$-0.937739\pi$
$313$	− 6.87689i	− 0.388705i	0.980932	−	0.194353i	$-0.0622606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 28.7386i	− 1.61412i	−0.590468	−	0.807061i	$-0.701057\pi$
$317$	− 28.7386i	− 1.61412i	0.590468	−	0.807061i	$-0.298943\pi$
$318$	0	0
$319$	−17.7538	−0.994021
$320$	0	0
$321$	18.7386	1.04589
$322$	0	0
$323$	− 20.4924i	− 1.14023i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.8769i	0.712094i
$328$	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
$333$	0.630683i	0.0345612i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.2462i	1.75656i	0.478144	+	0.878282i	$0.341310\pi$
$337$	32.2462i	1.75656i	−0.478144	+	0.878282i	$0.658690\pi$
$338$	0	0
$339$	5.86174	0.318366
$340$	0	0
$341$	−22.2462	−1.20470
$342$	0	0
$343$	− 13.2614i	− 0.716046i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.4924i	0.885360i	0.896680	+	0.442680i	$0.145972\pi$
$347$	16.4924i	0.885360i	−0.896680	+	0.442680i	$0.854028\pi$
$348$	0	0
$349$	−17.8078	−0.953228	−0.476614	−	0.879113i	$-0.658136\pi$
$349$	−17.8078	−0.953228	−0.476614	+	0.879113i	$0.658136\pi$
$350$	0	0
$351$	19.8078	1.05726
$352$	0	0
$353$	− 14.1922i	− 0.755377i	−0.925933	−	0.377688i	$-0.876719\pi$
$353$	− 14.1922i	− 0.755377i	0.925933	−	0.377688i	$-0.123281\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.9848i	1.32234i
$358$	0	0
$359$	30.2462	1.59633	0.798167	−	0.602436i	$-0.205803\pi$
$359$	30.2462	1.59633	0.798167	+	0.602436i	$0.205803\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	7.80776i	0.409801i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 12.4924i	− 0.652099i	−0.945353	−	0.326050i	$-0.894282\pi$
$367$	− 12.4924i	− 0.652099i	0.945353	−	0.326050i	$-0.105718\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−38.2462	−1.98564
$372$	0	0
$373$	− 24.7386i	− 1.28092i	−0.767992	−	0.640459i	$-0.778744\pi$
$373$	− 24.7386i	− 1.28092i	0.767992	−	0.640459i	$-0.221256\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.8078i	− 0.814141i
$378$	0	0
$379$	−2.24621	−0.115380	−0.0576901	−	0.998335i	$-0.518374\pi$
$379$	−2.24621	−0.115380	−0.0576901	+	0.998335i	$0.518374\pi$
$380$	0	0
$381$	−5.94602	−0.304624
$382$	0	0
$383$	7.61553i	0.389135i	0.980889	+	0.194568i	$0.0623304\pi$
$383$	7.61553i	0.389135i	−0.980889	+	0.194568i	$0.937670\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.492423i	0.0250312i
$388$	0	0
$389$	33.6155	1.70437	0.852187	−	0.523237i	$-0.175276\pi$
$389$	33.6155	1.70437	0.852187	+	0.523237i	$0.175276\pi$
$390$	0	0
$391$	5.12311	0.259087
$392$	0	0
$393$	− 29.5616i	− 1.49118i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 24.9309i	− 1.25124i	−0.780126	−	0.625622i	$-0.784845\pi$
$397$	− 24.9309i	− 1.25124i	0.780126	−	0.625622i	$-0.215155\pi$
$398$	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
$403$	− 19.8078i	− 0.986695i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.49242i	− 0.222681i
$408$	0	0
$409$	14.6847	0.726110	0.363055	−	0.931768i	$-0.381734\pi$
$409$	14.6847	0.726110	0.363055	+	0.931768i	$0.381734\pi$
$410$	0	0
$411$	−10.7386	−0.529698
$412$	0	0
$413$	32.0000i	1.57462i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 19.8078i	− 0.969990i
$418$	0	0
$419$	0.876894	0.0428391	0.0214195	−	0.999771i	$-0.493181\pi$
$419$	0.876894	0.0428391	0.0214195	+	0.999771i	$0.493181\pi$
$420$	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
$423$	4.87689i	0.237123i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.98485i	− 0.434807i
$428$	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
$433$	24.7386i	1.18886i	0.804146	+	0.594431i	$0.202623\pi$
$433$	24.7386i	1.18886i	−0.804146	+	0.594431i	$0.797377\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	32.3002	1.54160	0.770802	−	0.637075i	$-0.219856\pi$
$439$	32.3002	1.54160	0.770802	+	0.637075i	$0.219856\pi$
$440$	0	0
$441$	−1.54640	−0.0736380
$442$	0	0
$443$	3.31534i	0.157517i	0.996894	+	0.0787583i	$0.0250955\pi$
$443$	3.31534i	0.157517i	−0.996894	+	0.0787583i	$0.974904\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 19.1231i	− 0.904492i
$448$	0	0
$449$	32.7386	1.54503	0.772516	−	0.634996i	$-0.218998\pi$
$449$	32.7386	1.54503	0.772516	+	0.634996i	$0.218998\pi$
$450$	0	0
$451$	14.2462	0.670828
$452$	0	0
$453$	5.94602i	0.279369i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9.12311i	− 0.426761i	−0.976969	−	0.213380i	$-0.931553\pi$
$457$	− 9.12311i	− 0.426761i	0.976969	−	0.213380i	$-0.0684474\pi$
$458$	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
$463$	6.24621i	0.290286i	0.989411	+	0.145143i	$0.0463642\pi$
$463$	6.24621i	0.290286i	−0.989411	+	0.145143i	$0.953636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.6307i	0.491929i	0.969279	+	0.245965i	$0.0791047\pi$
$467$	10.6307i	0.491929i	−0.969279	+	0.245965i	$0.920895\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	3.12311	0.143905
$472$	0	0
$473$	− 3.50758i	− 0.161279i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.87689i	− 0.314871i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	− 4.87689i	− 0.221906i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	34.0540i	1.54313i	0.636149	+	0.771566i	$0.280526\pi$
$487$	34.0540i	1.54313i	−0.636149	+	0.771566i	$0.719474\pi$
$488$	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
$493$	22.7386i	1.02410i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.1231i	1.21664i
$498$	0	0
$499$	−38.0540	−1.70353	−0.851765	−	0.523924i	$-0.824468\pi$
$499$	−38.0540	−1.70353	−0.851765	+	0.523924i	$0.824468\pi$
$500$	0	0
$501$	12.4924	0.558120
$502$	0	0
$503$	15.6155i	0.696262i	0.937446	+	0.348131i	$0.113184\pi$
$503$	15.6155i	0.696262i	−0.937446	+	0.348131i	$0.886816\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.492423i	0.0218693i
$508$	0	0
$509$	4.43845	0.196731	0.0983654	−	0.995150i	$-0.468639\pi$
$509$	4.43845	0.196731	0.0983654	+	0.995150i	$0.468639\pi$
$510$	0	0
$511$	−38.8466	−1.71847
$512$	0	0
$513$	− 22.2462i	− 0.982194i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 34.7386i	− 1.52780i
$518$	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
$523$	− 33.8617i	− 1.48067i	−0.672238	−	0.740335i	$-0.734667\pi$
$523$	− 33.8617i	− 1.48067i	0.672238	−	0.740335i	$-0.265333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.4924i	1.24115i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−5.75379	−0.249693
$532$	0	0
$533$	12.6847i	0.549434i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7.31534i	− 0.315680i
$538$	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
$543$	− 8.00000i	− 0.343313i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.0540i	1.62707i	0.581516	+	0.813535i	$0.302460\pi$
$547$	38.0540i	1.62707i	−0.581516	+	0.813535i	$0.697540\pi$
$548$	0	0
$549$	1.61553	0.0689491
$550$	0	0
$551$	−17.7538	−0.756337
$552$	0	0
$553$	19.5076i	0.829547i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.2462i	− 1.02734i	−0.857986	−	0.513672i	$-0.828285\pi$
$557$	− 24.2462i	− 1.02734i	0.857986	−	0.513672i	$-0.171715\pi$
$558$	0	0
$559$	3.12311	0.132093
$560$	0	0
$561$	32.0000	1.35104
$562$	0	0
$563$	− 43.2311i	− 1.82197i	−0.412438	−	0.910986i	$-0.635322\pi$
$563$	− 43.2311i	− 1.82197i	0.412438	−	0.910986i	$-0.364678\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	21.8617i	0.918107i
$568$	0	0
$569$	−28.7386	−1.20479	−0.602393	−	0.798200i	$-0.705786\pi$
$569$	−28.7386	−1.20479	−0.602393	+	0.798200i	$0.705786\pi$
$570$	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
$573$	− 17.3693i	− 0.725614i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.4233i	1.22491i	0.790506	+	0.612454i	$0.209817\pi$
$577$	29.4233i	1.22491i	−0.790506	+	0.612454i	$0.790183\pi$
$578$	0	0
$579$	31.9157	1.32637
$580$	0	0
$581$	−37.4773	−1.55482
$582$	0	0
$583$	48.9848i	2.02874i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.17708i	0.378779i	0.981902	+	0.189389i	$0.0606508\pi$
$587$	9.17708i	0.378779i	−0.981902	+	0.189389i	$0.939349\pi$
$588$	0	0
$589$	−22.2462	−0.916639
$590$	0	0
$591$	24.3002	0.999576
$592$	0	0
$593$	38.9848i	1.60092i	0.599389	+	0.800458i	$0.295410\pi$
$593$	38.9848i	1.60092i	−0.599389	+	0.800458i	$0.704590\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 37.4773i	− 1.53384i
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	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
$603$	− 5.75379i	− 0.234312i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	21.6458	0.877134
$610$	0	0
$611$	30.9309	1.25133
$612$	0	0
$613$	5.12311i	0.206920i	0.994634	+	0.103460i	$0.0329914\pi$
$613$	5.12311i	0.206920i	−0.994634	+	0.103460i	$0.967009\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.61553i	− 0.226073i	−0.993591	−	0.113036i	$-0.963942\pi$
$617$	− 5.61553i	− 0.226073i	0.993591	−	0.113036i	$-0.0360576\pi$
$618$	0	0
$619$	−36.9848	−1.48655	−0.743273	−	0.668988i	$-0.766728\pi$
$619$	−36.9848	−1.48655	−0.743273	+	0.668988i	$0.766728\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	31.2311i	1.25125i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	24.9848i	0.997799i
$628$	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
$633$	3.50758i	0.139414i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.80776i	0.388598i
$638$	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
$643$	46.3542i	1.82803i	0.405681	+	0.914015i	$0.367034\pi$
$643$	46.3542i	1.82803i	−0.405681	+	0.914015i	$0.632966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.56155i	− 0.218647i	−0.994006	−	0.109324i	$-0.965132\pi$
$647$	− 5.56155i	− 0.218647i	0.994006	−	0.109324i	$-0.0348685\pi$
$648$	0	0
$649$	40.9848	1.60880
$650$	0	0
$651$	27.1231	1.06304
$652$	0	0
$653$	− 50.7926i	− 1.98767i	−0.110876	−	0.993834i	$-0.535366\pi$
$653$	− 50.7926i	− 1.98767i	0.110876	−	0.993834i	$-0.464634\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.98485i	− 0.272505i
$658$	0	0
$659$	2.24621	0.0875000	0.0437500	−	0.999043i	$-0.486070\pi$
$659$	2.24621	0.0875000	0.0437500	+	0.999043i	$0.486070\pi$
$660$	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
$663$	28.4924i	1.10655i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.43845i	− 0.171857i
$668$	0	0
$669$	19.5076	0.754207
$670$	0	0
$671$	−11.5076	−0.444245
$672$	0	0
$673$	0.438447i	0.0169009i	0.999964	+	0.00845045i	$0.00268989\pi$
$673$	0.438447i	0.0169009i	−0.999964	+	0.00845045i	$0.997310\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.7386i	− 1.10452i	−0.833673	−	0.552258i	$-0.813766\pi$
$677$	− 28.7386i	− 1.10452i	0.833673	−	0.552258i	$-0.186234\pi$
$678$	0	0
$679$	0.768944	0.0295094
$680$	0	0
$681$	−1.36932	−0.0524723
$682$	0	0
$683$	14.4384i	0.552472i	0.961090	+	0.276236i	$0.0890871\pi$
$683$	14.4384i	0.552472i	−0.961090	+	0.276236i	$0.910913\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	25.3693i	0.967900i
$688$	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
$693$	7.01515i	0.266484i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.2462i	− 0.691125i
$698$	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
$703$	− 4.49242i	− 0.169435i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50.7386i	1.90822i
$708$	0	0
$709$	−8.73863	−0.328186	−0.164093	−	0.986445i	$-0.552470\pi$
$709$	−8.73863	−0.328186	−0.164093	+	0.986445i	$0.552470\pi$
$710$	0	0
$711$	−3.50758	−0.131544
$712$	0	0
$713$	− 5.56155i	− 0.208282i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 18.4384i	− 0.688596i
$718$	0	0
$719$	10.7386	0.400483	0.200242	−	0.979747i	$-0.435827\pi$
$719$	10.7386	0.400483	0.200242	+	0.979747i	$0.435827\pi$
$720$	0	0
$721$	−44.4924	−1.65698
$722$	0	0
$723$	22.6307i	0.841644i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.8769i	1.07098i	0.844540	+	0.535492i	$0.179874\pi$
$727$	28.8769i	1.07098i	−0.844540	+	0.535492i	$0.820126\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	−4.49242	−0.166158
$732$	0	0
$733$	− 2.87689i	− 0.106261i	−0.998588	−	0.0531303i	$-0.983080\pi$
$733$	− 2.87689i	− 0.106261i	0.998588	−	0.0531303i	$-0.0169199\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.9848i	1.50970i
$738$	0	0
$739$	42.5464	1.56509	0.782547	−	0.622591i	$-0.213920\pi$
$739$	42.5464	1.56509	0.782547	+	0.622591i	$0.213920\pi$
$740$	0	0
$741$	−22.2462	−0.817235
$742$	0	0
$743$	30.2462i	1.10963i	0.831975	+	0.554813i	$0.187210\pi$
$743$	30.2462i	1.10963i	−0.831975	+	0.554813i	$0.812790\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6.73863i	− 0.246554i
$748$	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
$753$	− 16.0000i	− 0.583072i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.1231i	1.20388i	0.798541	+	0.601940i	$0.205605\pi$
$757$	33.1231i	1.20388i	−0.798541	+	0.601940i	$0.794395\pi$
$758$	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
$763$	− 25.7538i	− 0.932350i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.4924i	1.31767i
$768$	0	0
$769$	−5.50758	−0.198608	−0.0993042	−	0.995057i	$-0.531662\pi$
$769$	−5.50758	−0.198608	−0.0993042	+	0.995057i	$0.531662\pi$
$770$	0	0
$771$	−43.8078	−1.57770
$772$	0	0
$773$	− 12.2462i	− 0.440466i	−0.975447	−	0.220233i	$-0.929318\pi$
$773$	− 12.2462i	− 0.440466i	0.975447	−	0.220233i	$-0.0706817\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.47727i	0.196496i
$778$	0	0
$779$	14.2462	0.510423
$780$	0	0
$781$	34.7386	1.24305
$782$	0	0
$783$	24.6847i	0.882158i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 29.7538i	− 1.06061i	−0.847808	−	0.530304i	$-0.822078\pi$
$787$	− 29.7538i	− 1.06061i	0.847808	−	0.530304i	$-0.177922\pi$
$788$	0	0
$789$	−12.4924	−0.444742
$790$	0	0
$791$	−11.7235	−0.416839
$792$	0	0
$793$	− 10.2462i	− 0.363854i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.8769i	0.668654i	0.942457	+	0.334327i	$0.108509\pi$
$797$	18.8769i	0.668654i	−0.942457	+	0.334327i	$0.891491\pi$
$798$	0	0
$799$	−44.4924	−1.57403
$800$	0	0
$801$	−5.61553	−0.198415
$802$	0	0
$803$	49.7538i	1.75577i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 47.3153i	− 1.66558i
$808$	0	0
$809$	−22.4924	−0.790791	−0.395396	−	0.918511i	$-0.629393\pi$
$809$	−22.4924	−0.790791	−0.395396	+	0.918511i	$0.629393\pi$
$810$	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
$813$	22.2462i	0.780209i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.50758i	− 0.122715i
$818$	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
$823$	21.5616i	0.751588i	0.926703	+	0.375794i	$0.122630\pi$
$823$	21.5616i	0.751588i	−0.926703	+	0.375794i	$0.877370\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−17.5076	−0.608063	−0.304032	−	0.952662i	$-0.598333\pi$
$829$	−17.5076	−0.608063	−0.304032	+	0.952662i	$0.598333\pi$
$830$	0	0
$831$	24.3002	0.842964
$832$	0	0
$833$	− 14.1080i	− 0.488812i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30.9309i	1.06913i
$838$	0	0
$839$	31.6155	1.09149	0.545745	−	0.837952i	$-0.316247\pi$
$839$	31.6155	1.09149	0.545745	+	0.837952i	$0.316247\pi$
$840$	0	0
$841$	−9.30019	−0.320696
$842$	0	0
$843$	− 3.89205i	− 0.134049i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 15.6155i	− 0.536556i
$848$	0	0
$849$	48.6004	1.66796
$850$	0	0
$851$	1.12311	0.0384996
$852$	0	0
$853$	42.9848i	1.47177i	0.677105	+	0.735887i	$0.263234\pi$
$853$	42.9848i	1.47177i	−0.677105	+	0.735887i	$0.736766\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11.1771i	− 0.381802i	−0.981609	−	0.190901i	$-0.938859\pi$
$857$	− 11.1771i	− 0.381802i	0.981609	−	0.190901i	$-0.0611409\pi$
$858$	0	0
$859$	23.4233	0.799192	0.399596	−	0.916691i	$-0.369150\pi$
$859$	23.4233	0.799192	0.399596	+	0.916691i	$0.369150\pi$
$860$	0	0
$861$	−17.3693	−0.591945
$862$	0	0
$863$	43.4233i	1.47815i	0.673625	+	0.739073i	$0.264736\pi$
$863$	43.4233i	1.47815i	−0.673625	+	0.739073i	$0.735264\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14.4384i	− 0.490355i
$868$	0	0
$869$	24.9848	0.847553
$870$	0	0
$871$	−36.4924	−1.23650
$872$	0	0
$873$	0.138261i	0.00467941i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.73863i	0.295083i	0.989056	+	0.147541i	$0.0471359\pi$
$877$	8.73863i	0.295083i	−0.989056	+	0.147541i	$0.952864\pi$
$878$	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
$883$	7.50758i	0.252650i	0.991989	+	0.126325i	$0.0403182\pi$
$883$	7.50758i	0.252650i	−0.991989	+	0.126325i	$0.959682\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37.5616i	− 1.26119i	−0.776111	−	0.630597i	$-0.782810\pi$
$887$	− 37.5616i	− 1.26119i	0.776111	−	0.630597i	$-0.217190\pi$
$888$	0	0
$889$	11.8920	0.398847
$890$	0	0
$891$	28.0000	0.938035
$892$	0	0
$893$	− 34.7386i	− 1.16248i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.56155i	− 0.185695i
$898$	0	0
$899$	24.6847	0.823279
$900$	0	0
$901$	62.7386	2.09013
$902$	0	0
$903$	4.27652i	0.142314i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	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
$913$	48.0000i	1.58857i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	59.1231i	1.95242i
$918$	0	0
$919$	−21.8617	−0.721152	−0.360576	−	0.932730i	$-0.617420\pi$
$919$	−21.8617	−0.721152	−0.360576	+	0.932730i	$0.617420\pi$
$920$	0	0
$921$	−18.7386	−0.617459
$922$	0	0
$923$	30.9309i	1.01810i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	24.0540	0.789185	0.394593	−	0.918856i	$-0.370886\pi$
$929$	24.0540	0.789185	0.394593	+	0.918856i	$0.370886\pi$
$930$	0	0
$931$	11.0152	0.361007
$932$	0	0
$933$	− 23.3153i	− 0.763310i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.6307i	0.543301i	0.962396	+	0.271650i	$0.0875694\pi$
$937$	16.6307i	0.543301i	−0.962396	+	0.271650i	$0.912431\pi$
$938$	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
$943$	3.56155i	0.115980i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 55.8078i	− 1.81351i	−0.421659	−	0.906754i	$-0.638552\pi$
$947$	− 55.8078i	− 1.81351i	0.421659	−	0.906754i	$-0.361448\pi$
$948$	0	0
$949$	−44.3002	−1.43804
$950$	0	0
$951$	44.8769	1.45523
$952$	0	0
$953$	− 46.4924i	− 1.50604i	−0.657999	−	0.753019i	$-0.728597\pi$
$953$	− 46.4924i	− 1.50604i	0.657999	−	0.753019i	$-0.271403\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 27.7235i	− 0.896173i
$958$	0	0
$959$	21.4773	0.693537
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	− 6.73863i	− 0.217149i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 45.1771i	− 1.45280i	−0.687274	−	0.726398i	$-0.741193\pi$
$967$	− 45.1771i	− 1.45280i	0.687274	−	0.726398i	$-0.258807\pi$
$968$	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
$973$	39.6155i	1.27002i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 26.1080i	− 0.835267i	−0.908615	−	0.417634i	$-0.862860\pi$
$977$	− 26.1080i	− 0.835267i	0.908615	−	0.417634i	$-0.137140\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	4.63068	0.147846
$982$	0	0
$983$	− 28.8769i	− 0.921030i	−0.887652	−	0.460515i	$-0.847665\pi$
$983$	− 28.8769i	− 0.921030i	0.887652	−	0.460515i	$-0.152335\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	42.3542i	1.34815i
$988$	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
$993$	10.0540i	0.319053i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0.738634i	0.0233928i	0.999932	+	0.0116964i	$0.00372316\pi$
$997$	0.738634i	0.0233928i	−0.999932	+	0.0116964i	$0.996277\pi$
$998$	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.e.m.4049.3		4
5.2	odd	4		4600.2.a.r.1.2		2
5.3	odd	4		920.2.a.f.1.1	✓	2
5.4	even	2	inner	4600.2.e.m.4049.2		4
15.8	even	4		8280.2.a.bb.1.1		2
20.3	even	4		1840.2.a.k.1.2		2
20.7	even	4		9200.2.a.bx.1.1		2
40.3	even	4		7360.2.a.bm.1.1		2
40.13	odd	4		7360.2.a.bj.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.f.1.1	✓	2	5.3	odd	4
1840.2.a.k.1.2		2	20.3	even	4
4600.2.a.r.1.2		2	5.2	odd	4
4600.2.e.m.4049.2		4	5.4	even	2	inner
4600.2.e.m.4049.3		4	1.1	even	1	trivial
7360.2.a.bj.1.2		2	40.13	odd	4
7360.2.a.bm.1.1		2	40.3	even	4
8280.2.a.bb.1.1		2	15.8	even	4
9200.2.a.bx.1.1		2	20.7	even	4

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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} -3.12311i q^{7} +0.561553 q^{9} +O(q^{10})\)	\(q+1.56155i q^{3} -3.12311i q^{7} +0.561553 q^{9} -4.00000 q^{11} -3.56155i q^{13} +5.12311i q^{17} -4.00000 q^{19} +4.87689 q^{21} -1.00000i q^{23} +5.56155i q^{27} +4.43845 q^{29} +5.56155 q^{31} -6.24621i q^{33} +1.12311i q^{37} +5.56155 q^{39} -3.56155 q^{41} +0.876894i q^{43} +8.68466i q^{47} -2.75379 q^{49} -8.00000 q^{51} -12.2462i q^{53} -6.24621i q^{57} -10.2462 q^{59} +2.87689 q^{61} -1.75379i q^{63} -10.2462i q^{67} +1.56155 q^{69} -8.68466 q^{71} -12.4384i q^{73} +12.4924i q^{77} -6.24621 q^{79} -7.00000 q^{81} -12.0000i q^{83} +6.93087i q^{87} -10.0000 q^{89} -11.1231 q^{91} +8.68466i q^{93} +0.246211i q^{97} -2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 16 * q^11 - 16 * q^19 + 36 * q^21 + 26 * q^29 + 14 * q^31 + 14 * q^39 - 6 * q^41 - 44 * q^49 - 32 * q^51 - 8 * q^59 + 28 * q^61 - 2 * q^69 - 10 * q^71 + 8 * q^79 - 28 * q^81 - 40 * q^89 - 28 * q^91 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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