LMFDB - Embedded newform 783.2.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [783,2,Mod(1,783)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(783, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("783.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 783 = 3^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	783.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.25228647827$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	783.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.801938 q^{2} -1.35690 q^{4} +1.69202 q^{5} -4.29590 q^{7} -2.69202 q^{8} +1.35690 q^{10} +3.96077 q^{11} -2.24698 q^{13} -3.44504 q^{14} +0.554958 q^{16} -5.80194 q^{17} -1.19806 q^{19} -2.29590 q^{20} +3.17629 q^{22} -7.04892 q^{23} -2.13706 q^{25} -1.80194 q^{26} +5.82908 q^{28} -1.00000 q^{29} +1.64310 q^{31} +5.82908 q^{32} -4.65279 q^{34} -7.26875 q^{35} -7.13706 q^{37} -0.960771 q^{38} -4.55496 q^{40} -10.8509 q^{41} +4.96615 q^{43} -5.37435 q^{44} -5.65279 q^{46} +0.740939 q^{47} +11.4547 q^{49} -1.71379 q^{50} +3.04892 q^{52} +0.445042 q^{53} +6.70171 q^{55} +11.5646 q^{56} -0.801938 q^{58} -1.18060 q^{59} +11.6136 q^{61} +1.31767 q^{62} +3.56465 q^{64} -3.80194 q^{65} -14.4819 q^{67} +7.87263 q^{68} -5.82908 q^{70} +12.7778 q^{71} +8.53750 q^{73} -5.72348 q^{74} +1.62565 q^{76} -17.0151 q^{77} +8.57002 q^{79} +0.939001 q^{80} -8.70171 q^{82} +6.17092 q^{83} -9.81700 q^{85} +3.98254 q^{86} -10.6625 q^{88} -7.51573 q^{89} +9.65279 q^{91} +9.56465 q^{92} +0.594187 q^{94} -2.02715 q^{95} +6.64071 q^{97} +9.18598 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + q^{7} - 3 q^{8} - q^{11} - 2 q^{13} - 10 q^{14} + 2 q^{16} - 13 q^{17} - 8 q^{19} + 7 q^{20} + 17 q^{22} - 12 q^{23} - q^{25} - q^{26} + 7 q^{28} - 3 q^{29} + 9 q^{31} + 7 q^{32} + 4 q^{34}+ \cdots + 13 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + q^7 - 3 * q^8 - q^11 - 2 * q^13 - 10 * q^14 + 2 * q^16 - 13 * q^17 - 8 * q^19 + 7 * q^20 + 17 * q^22 - 12 * q^23 - q^25 - q^26 + 7 * q^28 - 3 * q^29 + 9 * q^31 + 7 * q^32 + 4 * q^34 - 14 * q^35 - 16 * q^37 + 10 * q^38 - 14 * q^40 - 19 * q^41 - q^43 - 28 * q^44 + q^46 - 12 * q^47 + 12 * q^49 + 3 * q^50 + q^53 - 7 * q^55 + 13 * q^56 + 2 * q^58 + 8 * q^59 + 4 * q^61 - 13 * q^62 - 11 * q^64 - 7 * q^65 - 15 * q^67 + 7 * q^68 - 7 * q^70 - 4 * q^71 + 10 * q^73 + 13 * q^74 - 7 * q^76 - 26 * q^77 + q^79 - 7 * q^80 + q^82 + 29 * q^83 - 4 * q^86 + 8 * q^88 - 10 * q^89 + 11 * q^91 + 7 * q^92 + 15 * q^94 - 17 * q^97 + 13 * q^98

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 $

$2$	0.801938	0.567056	0.283528	−	0.958964i	$-0.408495\pi$
$2$	0.801938	0.567056	0.283528	+	0.958964i	$0.408495\pi$
$3$	0	0
$3$	0	0
$4$	−1.35690	−0.678448
$5$	1.69202	0.756695	0.378348	−	0.925664i	$-0.376492\pi$
$5$	1.69202	0.756695	0.378348	+	0.925664i	$0.376492\pi$
$6$	0	0
$7$	−4.29590	−1.62370	−0.811848	−	0.583869i	$-0.801538\pi$
$7$	−4.29590	−1.62370	−0.811848	+	0.583869i	$0.801538\pi$
$8$	−2.69202	−0.951773
$9$	0	0
$10$	1.35690	0.429088
$11$	3.96077	1.19422	0.597109	−	0.802160i	$-0.296316\pi$
$11$	3.96077	1.19422	0.597109	+	0.802160i	$0.296316\pi$
$12$	0	0
$13$	−2.24698	−0.623200	−0.311600	−	0.950213i	$-0.600865\pi$
$13$	−2.24698	−0.623200	−0.311600	+	0.950213i	$0.600865\pi$
$14$	−3.44504	−0.920726
$15$	0	0
$16$	0.554958	0.138740
$17$	−5.80194	−1.40718	−0.703588	−	0.710608i	$-0.748420\pi$
$17$	−5.80194	−1.40718	−0.703588	+	0.710608i	$0.748420\pi$
$18$	0	0
$19$	−1.19806	−0.274854	−0.137427	−	0.990512i	$-0.543883\pi$
$19$	−1.19806	−0.274854	−0.137427	+	0.990512i	$0.543883\pi$
$20$	−2.29590	−0.513378
$21$	0	0
$22$	3.17629	0.677188
$23$	−7.04892	−1.46980	−0.734900	−	0.678175i	$-0.762771\pi$
$23$	−7.04892	−1.46980	−0.734900	+	0.678175i	$0.762771\pi$
$24$	0	0
$25$	−2.13706	−0.427413
$26$	−1.80194	−0.353389
$27$	0	0
$28$	5.82908	1.10159
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.64310	0.295110	0.147555	−	0.989054i	$-0.452860\pi$
$31$	1.64310	0.295110	0.147555	+	0.989054i	$0.452860\pi$
$32$	5.82908	1.03045
$33$	0	0
$34$	−4.65279	−0.797947
$35$	−7.26875	−1.22864
$36$	0	0
$37$	−7.13706	−1.17333	−0.586663	−	0.809831i	$-0.699559\pi$
$37$	−7.13706	−1.17333	−0.586663	+	0.809831i	$0.699559\pi$
$38$	−0.960771	−0.155858
$39$	0	0
$40$	−4.55496	−0.720202
$41$	−10.8509	−1.69462	−0.847309	−	0.531100i	$-0.821779\pi$
$41$	−10.8509	−1.69462	−0.847309	+	0.531100i	$0.821779\pi$
$42$	0	0
$43$	4.96615	0.757330	0.378665	−	0.925534i	$-0.376383\pi$
$43$	4.96615	0.757330	0.378665	+	0.925534i	$0.376383\pi$
$44$	−5.37435	−0.810214
$45$	0	0
$46$	−5.65279	−0.833459
$47$	0.740939	0.108077	0.0540385	−	0.998539i	$-0.482791\pi$
$47$	0.740939	0.108077	0.0540385	+	0.998539i	$0.482791\pi$
$48$	0	0
$49$	11.4547	1.63639
$50$	−1.71379	−0.242367
$51$	0	0
$52$	3.04892	0.422809
$53$	0.445042	0.0611312	0.0305656	−	0.999533i	$-0.490269\pi$
$53$	0.445042	0.0611312	0.0305656	+	0.999533i	$0.490269\pi$
$54$	0	0
$55$	6.70171	0.903658
$56$	11.5646	1.54539
$57$	0	0
$58$	−0.801938	−0.105300
$59$	−1.18060	−0.153702	−0.0768508	−	0.997043i	$-0.524487\pi$
$59$	−1.18060	−0.153702	−0.0768508	+	0.997043i	$0.524487\pi$
$60$	0	0
$61$	11.6136	1.48696	0.743482	−	0.668756i	$-0.233173\pi$
$61$	11.6136	1.48696	0.743482	+	0.668756i	$0.233173\pi$
$62$	1.31767	0.167344
$63$	0	0
$64$	3.56465	0.445581
$65$	−3.80194	−0.471572
$66$	0	0
$67$	−14.4819	−1.76924	−0.884621	−	0.466310i	$-0.845583\pi$
$67$	−14.4819	−1.76924	−0.884621	+	0.466310i	$0.845583\pi$
$68$	7.87263	0.954696
$69$	0	0
$70$	−5.82908	−0.696709
$71$	12.7778	1.51644	0.758221	−	0.651997i	$-0.226069\pi$
$71$	12.7778	1.51644	0.758221	+	0.651997i	$0.226069\pi$
$72$	0	0
$73$	8.53750	0.999239	0.499619	−	0.866245i	$-0.333473\pi$
$73$	8.53750	0.999239	0.499619	+	0.866245i	$0.333473\pi$
$74$	−5.72348	−0.665341
$75$	0	0
$76$	1.62565	0.186474
$77$	−17.0151	−1.93905
$78$	0	0
$79$	8.57002	0.964203	0.482101	−	0.876115i	$-0.339874\pi$
$79$	8.57002	0.964203	0.482101	+	0.876115i	$0.339874\pi$
$80$	0.939001	0.104984
$81$	0	0
$82$	−8.70171	−0.960943
$83$	6.17092	0.677346	0.338673	−	0.940904i	$-0.390022\pi$
$83$	6.17092	0.677346	0.338673	+	0.940904i	$0.390022\pi$
$84$	0	0
$85$	−9.81700	−1.06480
$86$	3.98254	0.429449
$87$	0	0
$88$	−10.6625	−1.13662
$89$	−7.51573	−0.796666	−0.398333	−	0.917241i	$-0.630411\pi$
$89$	−7.51573	−0.796666	−0.398333	+	0.917241i	$0.630411\pi$
$90$	0	0
$91$	9.65279	1.01189
$92$	9.56465	0.997183
$93$	0	0
$94$	0.594187	0.0612857
$95$	−2.02715	−0.207981
$96$	0	0
$97$	6.64071	0.674262	0.337131	−	0.941458i	$-0.390543\pi$
$97$	6.64071	0.674262	0.337131	+	0.941458i	$0.390543\pi$
$98$	9.18598	0.927924
$99$	0	0
$100$	2.89977	0.289977
$101$	1.65519	0.164697	0.0823486	−	0.996604i	$-0.473758\pi$
$101$	1.65519	0.164697	0.0823486	+	0.996604i	$0.473758\pi$
$102$	0	0
$103$	−4.59419	−0.452679	−0.226339	−	0.974049i	$-0.572676\pi$
$103$	−4.59419	−0.452679	−0.226339	+	0.974049i	$0.572676\pi$
$104$	6.04892	0.593145
$105$	0	0
$106$	0.356896	0.0346648
$107$	−17.1129	−1.65437	−0.827183	−	0.561932i	$-0.810058\pi$
$107$	−17.1129	−1.65437	−0.827183	+	0.561932i	$0.810058\pi$
$108$	0	0
$109$	−10.1588	−0.973040	−0.486520	−	0.873669i	$-0.661734\pi$
$109$	−10.1588	−0.973040	−0.486520	+	0.873669i	$0.661734\pi$
$110$	5.37435	0.512425
$111$	0	0
$112$	−2.38404	−0.225271
$113$	−15.5211	−1.46010	−0.730051	−	0.683392i	$-0.760504\pi$
$113$	−15.5211	−1.46010	−0.730051	+	0.683392i	$0.760504\pi$
$114$	0	0
$115$	−11.9269	−1.11219
$116$	1.35690	0.125985
$117$	0	0
$118$	−0.946771	−0.0871573
$119$	24.9245	2.28483
$120$	0	0
$121$	4.68771	0.426155
$122$	9.31336	0.843192
$123$	0	0
$124$	−2.22952	−0.200217
$125$	−12.0761	−1.08012
$126$	0	0
$127$	12.0151	1.06616	0.533082	−	0.846063i	$-0.321034\pi$
$127$	12.0151	1.06616	0.533082	+	0.846063i	$0.321034\pi$
$128$	−8.79954	−0.777777
$129$	0	0
$130$	−3.04892	−0.267408
$131$	17.1903	1.50192	0.750961	−	0.660346i	$-0.229590\pi$
$131$	17.1903	1.50192	0.750961	+	0.660346i	$0.229590\pi$
$132$	0	0
$133$	5.14675	0.446280
$134$	−11.6136	−1.00326
$135$	0	0
$136$	15.6189	1.33931
$137$	−2.76809	−0.236494	−0.118247	−	0.992984i	$-0.537727\pi$
$137$	−2.76809	−0.236494	−0.118247	+	0.992984i	$0.537727\pi$
$138$	0	0
$139$	−0.625646	−0.0530666	−0.0265333	−	0.999648i	$-0.508447\pi$
$139$	−0.625646	−0.0530666	−0.0265333	+	0.999648i	$0.508447\pi$
$140$	9.86294	0.833570
$141$	0	0
$142$	10.2470	0.859907
$143$	−8.89977	−0.744236
$144$	0	0
$145$	−1.69202	−0.140515
$146$	6.84654	0.566624
$147$	0	0
$148$	9.68425	0.796041
$149$	13.0911	1.07247	0.536234	−	0.844070i	$-0.319847\pi$
$149$	13.0911	1.07247	0.536234	+	0.844070i	$0.319847\pi$
$150$	0	0
$151$	14.9095	1.21331	0.606657	−	0.794963i	$-0.292510\pi$
$151$	14.9095	1.21331	0.606657	+	0.794963i	$0.292510\pi$
$152$	3.22521	0.261599
$153$	0	0
$154$	−13.6450	−1.09955
$155$	2.78017	0.223308
$156$	0	0
$157$	−4.49396	−0.358657	−0.179328	−	0.983789i	$-0.557392\pi$
$157$	−4.49396	−0.358657	−0.179328	+	0.983789i	$0.557392\pi$
$158$	6.87263	0.546757
$159$	0	0
$160$	9.86294	0.779734
$161$	30.2814	2.38651
$162$	0	0
$163$	13.8834	1.08743	0.543715	−	0.839270i	$-0.317017\pi$
$163$	13.8834	1.08743	0.543715	+	0.839270i	$0.317017\pi$
$164$	14.7235	1.14971
$165$	0	0
$166$	4.94869	0.384093
$167$	16.8538	1.30419	0.652095	−	0.758138i	$-0.273890\pi$
$167$	16.8538	1.30419	0.652095	+	0.758138i	$0.273890\pi$
$168$	0	0
$169$	−7.95108	−0.611622
$170$	−7.87263	−0.603803
$171$	0	0
$172$	−6.73855	−0.513809
$173$	−8.16421	−0.620713	−0.310357	−	0.950620i	$-0.600448\pi$
$173$	−8.16421	−0.620713	−0.310357	+	0.950620i	$0.600448\pi$
$174$	0	0
$175$	9.18060	0.693988
$176$	2.19806	0.165685
$177$	0	0
$178$	−6.02715	−0.451754
$179$	−9.88769	−0.739041	−0.369520	−	0.929223i	$-0.620478\pi$
$179$	−9.88769	−0.739041	−0.369520	+	0.929223i	$0.620478\pi$
$180$	0	0
$181$	−22.9952	−1.70922	−0.854610	−	0.519270i	$-0.826204\pi$
$181$	−22.9952	−1.70922	−0.854610	+	0.519270i	$0.826204\pi$
$182$	7.74094	0.573797
$183$	0	0
$184$	18.9758	1.39892
$185$	−12.0761	−0.887850
$186$	0	0
$187$	−22.9801	−1.68047
$188$	−1.00538	−0.0733246
$189$	0	0
$190$	−1.62565	−0.117937
$191$	5.05861	0.366028	0.183014	−	0.983110i	$-0.441415\pi$
$191$	5.05861	0.366028	0.183014	+	0.983110i	$0.441415\pi$
$192$	0	0
$193$	−6.31229	−0.454369	−0.227184	−	0.973852i	$-0.572952\pi$
$193$	−6.31229	−0.454369	−0.227184	+	0.973852i	$0.572952\pi$
$194$	5.32544	0.382344
$195$	0	0
$196$	−15.5429	−1.11021
$197$	4.53319	0.322976	0.161488	−	0.986875i	$-0.448371\pi$
$197$	4.53319	0.322976	0.161488	+	0.986875i	$0.448371\pi$
$198$	0	0
$199$	−4.49934	−0.318949	−0.159475	−	0.987202i	$-0.550980\pi$
$199$	−4.49934	−0.318949	−0.159475	+	0.987202i	$0.550980\pi$
$200$	5.75302	0.406800
$201$	0	0
$202$	1.32736	0.0933924
$203$	4.29590	0.301513
$204$	0	0
$205$	−18.3599	−1.28231
$206$	−3.68425	−0.256694
$207$	0	0
$208$	−1.24698	−0.0864625
$209$	−4.74525	−0.328236
$210$	0	0
$211$	−15.0411	−1.03548	−0.517738	−	0.855539i	$-0.673226\pi$
$211$	−15.0411	−1.03548	−0.517738	+	0.855539i	$0.673226\pi$
$212$	−0.603875	−0.0414743
$213$	0	0
$214$	−13.7235	−0.938118
$215$	8.40283	0.573068
$216$	0	0
$217$	−7.05861	−0.479169
$218$	−8.14675	−0.551768
$219$	0	0
$220$	−9.09352	−0.613085
$221$	13.0368	0.876952
$222$	0	0
$223$	17.0532	1.14197	0.570984	−	0.820961i	$-0.306562\pi$
$223$	17.0532	1.14197	0.570984	+	0.820961i	$0.306562\pi$
$224$	−25.0411	−1.67313
$225$	0	0
$226$	−12.4470	−0.827960
$227$	−17.3884	−1.15411	−0.577053	−	0.816707i	$-0.695797\pi$
$227$	−17.3884	−1.15411	−0.577053	+	0.816707i	$0.695797\pi$
$228$	0	0
$229$	24.6843	1.63118	0.815591	−	0.578629i	$-0.196412\pi$
$229$	24.6843	1.63118	0.815591	+	0.578629i	$0.196412\pi$
$230$	−9.56465	−0.630674
$231$	0	0
$232$	2.69202	0.176740
$233$	11.1618	0.731235	0.365617	−	0.930765i	$-0.380858\pi$
$233$	11.1618	0.731235	0.365617	+	0.930765i	$0.380858\pi$
$234$	0	0
$235$	1.25368	0.0817814
$236$	1.60196	0.104278
$237$	0	0
$238$	19.9879	1.29562
$239$	−12.4209	−0.803440	−0.401720	−	0.915763i	$-0.631587\pi$
$239$	−12.4209	−0.803440	−0.401720	+	0.915763i	$0.631587\pi$
$240$	0	0
$241$	−16.1715	−1.04170	−0.520849	−	0.853649i	$-0.674385\pi$
$241$	−16.1715	−1.04170	−0.520849	+	0.853649i	$0.674385\pi$
$242$	3.75925	0.241654
$243$	0	0
$244$	−15.7584	−1.00883
$245$	19.3817	1.23825
$246$	0	0
$247$	2.69202	0.171289
$248$	−4.42327	−0.280878
$249$	0	0
$250$	−9.68425	−0.612486
$251$	−8.75063	−0.552335	−0.276167	−	0.961110i	$-0.589064\pi$
$251$	−8.75063	−0.552335	−0.276167	+	0.961110i	$0.589064\pi$
$252$	0	0
$253$	−27.9191	−1.75526
$254$	9.63533	0.604575
$255$	0	0
$256$	−14.1860	−0.886624
$257$	−26.0368	−1.62413	−0.812067	−	0.583565i	$-0.801657\pi$
$257$	−26.0368	−1.62413	−0.812067	+	0.583565i	$0.801657\pi$
$258$	0	0
$259$	30.6601	1.90513
$260$	5.15883	0.319937
$261$	0	0
$262$	13.7855	0.851674
$263$	11.9379	0.736125	0.368062	−	0.929801i	$-0.380021\pi$
$263$	11.9379	0.736125	0.368062	+	0.929801i	$0.380021\pi$
$264$	0	0
$265$	0.753020	0.0462577
$266$	4.12737	0.253066
$267$	0	0
$268$	19.6504	1.20034
$269$	−10.7289	−0.654150	−0.327075	−	0.944998i	$-0.606063\pi$
$269$	−10.7289	−0.654150	−0.327075	+	0.944998i	$0.606063\pi$
$270$	0	0
$271$	−0.572417	−0.0347718	−0.0173859	−	0.999849i	$-0.505534\pi$
$271$	−0.572417	−0.0347718	−0.0173859	+	0.999849i	$0.505534\pi$
$272$	−3.21983	−0.195231
$273$	0	0
$274$	−2.21983	−0.134105
$275$	−8.46442	−0.510424
$276$	0	0
$277$	27.7603	1.66796	0.833978	−	0.551798i	$-0.186058\pi$
$277$	27.7603	1.66796	0.833978	+	0.551798i	$0.186058\pi$
$278$	−0.501729	−0.0300917
$279$	0	0
$280$	19.5676	1.16939
$281$	−29.6243	−1.76724	−0.883619	−	0.468206i	$-0.844901\pi$
$281$	−29.6243	−1.76724	−0.883619	+	0.468206i	$0.844901\pi$
$282$	0	0
$283$	2.00969	0.119464	0.0597318	−	0.998214i	$-0.480975\pi$
$283$	2.00969	0.119464	0.0597318	+	0.998214i	$0.480975\pi$
$284$	−17.3381	−1.02883
$285$	0	0
$286$	−7.13706	−0.422023
$287$	46.6142	2.75155
$288$	0	0
$289$	16.6625	0.980146
$290$	−1.35690	−0.0796797
$291$	0	0
$292$	−11.5845	−0.677931
$293$	−20.9729	−1.22525	−0.612624	−	0.790375i	$-0.709886\pi$
$293$	−20.9729	−1.22525	−0.612624	+	0.790375i	$0.709886\pi$
$294$	0	0
$295$	−1.99761	−0.116305
$296$	19.2131	1.11674
$297$	0	0
$298$	10.4983	0.608149
$299$	15.8388	0.915980
$300$	0	0
$301$	−21.3341	−1.22967
$302$	11.9565	0.688017
$303$	0	0
$304$	−0.664874	−0.0381332
$305$	19.6504	1.12518
$306$	0	0
$307$	−17.4644	−0.996747	−0.498374	−	0.866962i	$-0.666069\pi$
$307$	−17.4644	−0.996747	−0.498374	+	0.866962i	$0.666069\pi$
$308$	23.0877	1.31554
$309$	0	0
$310$	2.22952	0.126628
$311$	18.0054	1.02099	0.510496	−	0.859880i	$-0.329462\pi$
$311$	18.0054	1.02099	0.510496	+	0.859880i	$0.329462\pi$
$312$	0	0
$313$	22.2161	1.25573	0.627864	−	0.778323i	$-0.283929\pi$
$313$	22.2161	1.25573	0.627864	+	0.778323i	$0.283929\pi$
$314$	−3.60388	−0.203378
$315$	0	0
$316$	−11.6286	−0.654162
$317$	−16.1521	−0.907194	−0.453597	−	0.891207i	$-0.649859\pi$
$317$	−16.1521	−0.907194	−0.453597	+	0.891207i	$0.649859\pi$
$318$	0	0
$319$	−3.96077	−0.221761
$320$	6.03146	0.337169
$321$	0	0
$322$	24.2838	1.35328
$323$	6.95108	0.386769
$324$	0	0
$325$	4.80194	0.266364
$326$	11.1336	0.616633
$327$	0	0
$328$	29.2107	1.61289
$329$	−3.18300	−0.175484
$330$	0	0
$331$	−16.2687	−0.894211	−0.447106	−	0.894481i	$-0.647545\pi$
$331$	−16.2687	−0.894211	−0.447106	+	0.894481i	$0.647545\pi$
$332$	−8.37329	−0.459544
$333$	0	0
$334$	13.5157	0.739548
$335$	−24.5036	−1.33878
$336$	0	0
$337$	9.18060	0.500099	0.250050	−	0.968233i	$-0.419553\pi$
$337$	9.18060	0.500099	0.250050	+	0.968233i	$0.419553\pi$
$338$	−6.37627	−0.346824
$339$	0	0
$340$	13.3207	0.722414
$341$	6.50796	0.352426
$342$	0	0
$343$	−19.1371	−1.03330
$344$	−13.3690	−0.720807
$345$	0	0
$346$	−6.54719	−0.351979
$347$	−6.53750	−0.350951	−0.175476	−	0.984484i	$-0.556146\pi$
$347$	−6.53750	−0.350951	−0.175476	+	0.984484i	$0.556146\pi$
$348$	0	0
$349$	−22.1987	−1.18827	−0.594133	−	0.804367i	$-0.702505\pi$
$349$	−22.1987	−1.18827	−0.594133	+	0.804367i	$0.702505\pi$
$350$	7.36227	0.393530
$351$	0	0
$352$	23.0877	1.23058
$353$	−17.2959	−0.920568	−0.460284	−	0.887772i	$-0.652252\pi$
$353$	−17.2959	−0.920568	−0.460284	+	0.887772i	$0.652252\pi$
$354$	0	0
$355$	21.6203	1.14748
$356$	10.1981	0.540496
$357$	0	0
$358$	−7.92931	−0.419077
$359$	−27.9627	−1.47581	−0.737907	−	0.674902i	$-0.764186\pi$
$359$	−27.9627	−1.47581	−0.737907	+	0.674902i	$0.764186\pi$
$360$	0	0
$361$	−17.5646	−0.924455
$362$	−18.4407	−0.969223
$363$	0	0
$364$	−13.0978	−0.686513
$365$	14.4456	0.756119
$366$	0	0
$367$	−26.0151	−1.35798	−0.678988	−	0.734150i	$-0.737581\pi$
$367$	−26.0151	−1.35798	−0.678988	+	0.734150i	$0.737581\pi$
$368$	−3.91185	−0.203919
$369$	0	0
$370$	−9.68425	−0.503460
$371$	−1.91185	−0.0992585
$372$	0	0
$373$	5.76032	0.298258	0.149129	−	0.988818i	$-0.452353\pi$
$373$	5.76032	0.298258	0.149129	+	0.988818i	$0.452353\pi$
$374$	−18.4286	−0.952923
$375$	0	0
$376$	−1.99462	−0.102865
$377$	2.24698	0.115725
$378$	0	0
$379$	24.6122	1.26425	0.632123	−	0.774868i	$-0.282184\pi$
$379$	24.6122	1.26425	0.632123	+	0.774868i	$0.282184\pi$
$380$	2.75063	0.141104
$381$	0	0
$382$	4.05669	0.207558
$383$	−8.89307	−0.454415	−0.227207	−	0.973846i	$-0.572959\pi$
$383$	−8.89307	−0.454415	−0.227207	+	0.973846i	$0.572959\pi$
$384$	0	0
$385$	−28.7899	−1.46727
$386$	−5.06206	−0.257652
$387$	0	0
$388$	−9.01075	−0.457452
$389$	−22.4590	−1.13872	−0.569359	−	0.822089i	$-0.692809\pi$
$389$	−22.4590	−1.13872	−0.569359	+	0.822089i	$0.692809\pi$
$390$	0	0
$391$	40.8974	2.06827
$392$	−30.8364	−1.55747
$393$	0	0
$394$	3.63533	0.183146
$395$	14.5007	0.729608
$396$	0	0
$397$	9.63342	0.483487	0.241744	−	0.970340i	$-0.422281\pi$
$397$	9.63342	0.483487	0.241744	+	0.970340i	$0.422281\pi$
$398$	−3.60819	−0.180862
$399$	0	0
$400$	−1.18598	−0.0592990
$401$	−11.8116	−0.589844	−0.294922	−	0.955521i	$-0.595294\pi$
$401$	−11.8116	−0.589844	−0.294922	+	0.955521i	$0.595294\pi$
$402$	0	0
$403$	−3.69202	−0.183913
$404$	−2.24591	−0.111738
$405$	0	0
$406$	3.44504	0.170975
$407$	−28.2683	−1.40121
$408$	0	0
$409$	−12.5254	−0.619342	−0.309671	−	0.950844i	$-0.600219\pi$
$409$	−12.5254	−0.619342	−0.309671	+	0.950844i	$0.600219\pi$
$410$	−14.7235	−0.727141
$411$	0	0
$412$	6.23383	0.307119
$413$	5.07175	0.249565
$414$	0	0
$415$	10.4413	0.512544
$416$	−13.0978	−0.642174
$417$	0	0
$418$	−3.80540	−0.186128
$419$	−30.9541	−1.51221	−0.756103	−	0.654453i	$-0.772899\pi$
$419$	−30.9541	−1.51221	−0.756103	+	0.654453i	$0.772899\pi$
$420$	0	0
$421$	31.6625	1.54313	0.771567	−	0.636148i	$-0.219473\pi$
$421$	31.6625	1.54313	0.771567	+	0.636148i	$0.219473\pi$
$422$	−12.0621	−0.587172
$423$	0	0
$424$	−1.19806	−0.0581830
$425$	12.3991	0.601445
$426$	0	0
$427$	−49.8907	−2.41438
$428$	23.2204	1.12240
$429$	0	0
$430$	6.73855	0.324962
$431$	7.59658	0.365914	0.182957	−	0.983121i	$-0.441433\pi$
$431$	7.59658	0.365914	0.182957	+	0.983121i	$0.441433\pi$
$432$	0	0
$433$	18.5418	0.891063	0.445531	−	0.895266i	$-0.353015\pi$
$433$	18.5418	0.891063	0.445531	+	0.895266i	$0.353015\pi$
$434$	−5.66056	−0.271716
$435$	0	0
$436$	13.7845	0.660157
$437$	8.44504	0.403981
$438$	0	0
$439$	−14.2422	−0.679743	−0.339871	−	0.940472i	$-0.610384\pi$
$439$	−14.2422	−0.679743	−0.339871	+	0.940472i	$0.610384\pi$
$440$	−18.0411	−0.860078
$441$	0	0
$442$	10.4547	0.497281
$443$	−33.1758	−1.57623	−0.788115	−	0.615528i	$-0.788943\pi$
$443$	−33.1758	−1.57623	−0.788115	+	0.615528i	$0.788943\pi$
$444$	0	0
$445$	−12.7168	−0.602833
$446$	13.6756	0.647560
$447$	0	0
$448$	−15.3134	−0.723488
$449$	−31.5599	−1.48940	−0.744701	−	0.667398i	$-0.767408\pi$
$449$	−31.5599	−1.48940	−0.744701	+	0.667398i	$0.767408\pi$
$450$	0	0
$451$	−42.9778	−2.02374
$452$	21.0605	0.990604
$453$	0	0
$454$	−13.9444	−0.654442
$455$	16.3327	0.765690
$456$	0	0
$457$	30.3706	1.42068	0.710339	−	0.703860i	$-0.248542\pi$
$457$	30.3706	1.42068	0.710339	+	0.703860i	$0.248542\pi$
$458$	19.7952	0.924970
$459$	0	0
$460$	16.1836	0.754564
$461$	−3.40044	−0.158374	−0.0791871	−	0.996860i	$-0.525232\pi$
$461$	−3.40044	−0.158374	−0.0791871	+	0.996860i	$0.525232\pi$
$462$	0	0
$463$	−7.77538	−0.361353	−0.180676	−	0.983543i	$-0.557829\pi$
$463$	−7.77538	−0.361353	−0.180676	+	0.983543i	$0.557829\pi$
$464$	−0.554958	−0.0257633
$465$	0	0
$466$	8.95108	0.414651
$467$	−17.5415	−0.811726	−0.405863	−	0.913934i	$-0.633029\pi$
$467$	−17.5415	−0.811726	−0.405863	+	0.913934i	$0.633029\pi$
$468$	0	0
$469$	62.2127	2.87271
$470$	1.00538	0.0463746
$471$	0	0
$472$	3.17821	0.146289
$473$	19.6698	0.904417
$474$	0	0
$475$	2.56033	0.117476
$476$	−33.8200	−1.55014
$477$	0	0
$478$	−9.96077	−0.455595
$479$	14.2204	0.649748	0.324874	−	0.945757i	$-0.394678\pi$
$479$	14.2204	0.649748	0.324874	+	0.945757i	$0.394678\pi$
$480$	0	0
$481$	16.0368	0.731217
$482$	−12.9685	−0.590701
$483$	0	0
$484$	−6.36073	−0.289124
$485$	11.2362	0.510211
$486$	0	0
$487$	−41.3672	−1.87453	−0.937263	−	0.348624i	$-0.886649\pi$
$487$	−41.3672	−1.87453	−0.937263	+	0.348624i	$0.886649\pi$
$488$	−31.2640	−1.41525
$489$	0	0
$490$	15.5429	0.702156
$491$	−3.49694	−0.157815	−0.0789074	−	0.996882i	$-0.525143\pi$
$491$	−3.49694	−0.157815	−0.0789074	+	0.996882i	$0.525143\pi$
$492$	0	0
$493$	5.80194	0.261306
$494$	2.15883	0.0971305
$495$	0	0
$496$	0.911854	0.0409435
$497$	−54.8920	−2.46224
$498$	0	0
$499$	−6.56571	−0.293922	−0.146961	−	0.989142i	$-0.546949\pi$
$499$	−6.56571	−0.293922	−0.146961	+	0.989142i	$0.546949\pi$
$500$	16.3860	0.732802
$501$	0	0
$502$	−7.01746	−0.313204
$503$	12.4450	0.554897	0.277448	−	0.960741i	$-0.410511\pi$
$503$	12.4450	0.554897	0.277448	+	0.960741i	$0.410511\pi$
$504$	0	0
$505$	2.80061	0.124625
$506$	−22.3894	−0.995331
$507$	0	0
$508$	−16.3032	−0.723337
$509$	21.7157	0.962532	0.481266	−	0.876575i	$-0.340177\pi$
$509$	21.7157	0.962532	0.481266	+	0.876575i	$0.340177\pi$
$510$	0	0
$511$	−36.6762	−1.62246
$512$	6.22282	0.275012
$513$	0	0
$514$	−20.8799	−0.920974
$515$	−7.77346	−0.342540
$516$	0	0
$517$	2.93469	0.129067
$518$	24.5875	1.08031
$519$	0	0
$520$	10.2349	0.448830
$521$	20.9855	0.919393	0.459696	−	0.888076i	$-0.347958\pi$
$521$	20.9855	0.919393	0.459696	+	0.888076i	$0.347958\pi$
$522$	0	0
$523$	34.3183	1.50063	0.750316	−	0.661079i	$-0.229901\pi$
$523$	34.3183	1.50063	0.750316	+	0.661079i	$0.229901\pi$
$524$	−23.3254	−1.01898
$525$	0	0
$526$	9.57348	0.417424
$527$	−9.53319	−0.415272
$528$	0	0
$529$	26.6872	1.16031
$530$	0.603875	0.0262307
$531$	0	0
$532$	−6.98361	−0.302778
$533$	24.3817	1.05609
$534$	0	0
$535$	−28.9554	−1.25185
$536$	38.9855	1.68392
$537$	0	0
$538$	−8.60388	−0.370939
$539$	45.3696	1.95421
$540$	0	0
$541$	−28.5187	−1.22612	−0.613058	−	0.790038i	$-0.710061\pi$
$541$	−28.5187	−1.22612	−0.613058	+	0.790038i	$0.710061\pi$
$542$	−0.459042	−0.0197176
$543$	0	0
$544$	−33.8200	−1.45002
$545$	−17.1890	−0.736294
$546$	0	0
$547$	10.4722	0.447758	0.223879	−	0.974617i	$-0.428128\pi$
$547$	10.4722	0.447758	0.223879	+	0.974617i	$0.428128\pi$
$548$	3.75600	0.160449
$549$	0	0
$550$	−6.78794	−0.289439
$551$	1.19806	0.0510392
$552$	0	0
$553$	−36.8159	−1.56557
$554$	22.2620	0.945824
$555$	0	0
$556$	0.848936	0.0360029
$557$	−15.6819	−0.664462	−0.332231	−	0.943198i	$-0.607801\pi$
$557$	−15.6819	−0.664462	−0.332231	+	0.943198i	$0.607801\pi$
$558$	0	0
$559$	−11.1588	−0.471968
$560$	−4.03385	−0.170461
$561$	0	0
$562$	−23.7569	−1.00212
$563$	25.6732	1.08200	0.540999	−	0.841023i	$-0.318046\pi$
$563$	25.6732	1.08200	0.540999	+	0.841023i	$0.318046\pi$
$564$	0	0
$565$	−26.2620	−1.10485
$566$	1.61165	0.0677425
$567$	0	0
$568$	−34.3980	−1.44331
$569$	33.7066	1.41305	0.706527	−	0.707686i	$-0.250261\pi$
$569$	33.7066	1.41305	0.706527	+	0.707686i	$0.250261\pi$
$570$	0	0
$571$	17.0121	0.711933	0.355967	−	0.934499i	$-0.384152\pi$
$571$	17.0121	0.711933	0.355967	+	0.934499i	$0.384152\pi$
$572$	12.0761	0.504926
$573$	0	0
$574$	37.3817	1.56028
$575$	15.0640	0.628212
$576$	0	0
$577$	45.0398	1.87503	0.937516	−	0.347942i	$-0.113119\pi$
$577$	45.0398	1.87503	0.937516	+	0.347942i	$0.113119\pi$
$578$	13.3623	0.555797
$579$	0	0
$580$	2.29590	0.0953319
$581$	−26.5096	−1.09980
$582$	0	0
$583$	1.76271	0.0730040
$584$	−22.9831	−0.951049
$585$	0	0
$586$	−16.8189	−0.694783
$587$	21.9614	0.906442	0.453221	−	0.891398i	$-0.350275\pi$
$587$	21.9614	0.906442	0.453221	+	0.891398i	$0.350275\pi$
$588$	0	0
$589$	−1.96854	−0.0811123
$590$	−1.60196	−0.0659515
$591$	0	0
$592$	−3.96077	−0.162787
$593$	41.4663	1.70282	0.851409	−	0.524502i	$-0.175748\pi$
$593$	41.4663	1.70282	0.851409	+	0.524502i	$0.175748\pi$
$594$	0	0
$595$	42.1728	1.72892
$596$	−17.7633	−0.727613
$597$	0	0
$598$	12.7017	0.519412
$599$	28.3752	1.15938	0.579690	−	0.814837i	$-0.303174\pi$
$599$	28.3752	1.15938	0.579690	+	0.814837i	$0.303174\pi$
$600$	0	0
$601$	−1.00192	−0.0408691	−0.0204346	−	0.999791i	$-0.506505\pi$
$601$	−1.00192	−0.0408691	−0.0204346	+	0.999791i	$0.506505\pi$
$602$	−17.1086	−0.697294
$603$	0	0
$604$	−20.2306	−0.823171
$605$	7.93171	0.322470
$606$	0	0
$607$	43.9571	1.78416	0.892081	−	0.451876i	$-0.149245\pi$
$607$	43.9571	1.78416	0.892081	+	0.451876i	$0.149245\pi$
$608$	−6.98361	−0.283223
$609$	0	0
$610$	15.7584	0.638039
$611$	−1.66487	−0.0673536
$612$	0	0
$613$	19.0479	0.769336	0.384668	−	0.923055i	$-0.374316\pi$
$613$	19.0479	0.769336	0.384668	+	0.923055i	$0.374316\pi$
$614$	−14.0054	−0.565211
$615$	0	0
$616$	45.8049	1.84553
$617$	−38.5297	−1.55115	−0.775574	−	0.631256i	$-0.782540\pi$
$617$	−38.5297	−1.55115	−0.775574	+	0.631256i	$0.782540\pi$
$618$	0	0
$619$	−34.1715	−1.37347	−0.686734	−	0.726908i	$-0.740956\pi$
$619$	−34.1715	−1.37347	−0.686734	+	0.726908i	$0.740956\pi$
$620$	−3.77240	−0.151503
$621$	0	0
$622$	14.4392	0.578959
$623$	32.2868	1.29354
$624$	0	0
$625$	−9.74764	−0.389906
$626$	17.8159	0.712068
$627$	0	0
$628$	6.09783	0.243330
$629$	41.4088	1.65108
$630$	0	0
$631$	12.0175	0.478407	0.239204	−	0.970969i	$-0.423114\pi$
$631$	12.0175	0.478407	0.239204	+	0.970969i	$0.423114\pi$
$632$	−23.0707	−0.917703
$633$	0	0
$634$	−12.9530	−0.514429
$635$	20.3297	0.806761
$636$	0	0
$637$	−25.7385	−1.01980
$638$	−3.17629	−0.125751
$639$	0	0
$640$	−14.8890	−0.588540
$641$	0.263373	0.0104026	0.00520131	−	0.999986i	$-0.498344\pi$
$641$	0.263373	0.0104026	0.00520131	+	0.999986i	$0.498344\pi$
$642$	0	0
$643$	23.3515	0.920894	0.460447	−	0.887687i	$-0.347689\pi$
$643$	23.3515	0.920894	0.460447	+	0.887687i	$0.347689\pi$
$644$	−41.0887	−1.61912
$645$	0	0
$646$	5.57434	0.219319
$647$	21.7380	0.854607	0.427304	−	0.904108i	$-0.359464\pi$
$647$	21.7380	0.854607	0.427304	+	0.904108i	$0.359464\pi$
$648$	0	0
$649$	−4.67610	−0.183553
$650$	3.85086	0.151043
$651$	0	0
$652$	−18.8383	−0.737765
$653$	−12.0339	−0.470921	−0.235461	−	0.971884i	$-0.575660\pi$
$653$	−12.0339	−0.470921	−0.235461	+	0.971884i	$0.575660\pi$
$654$	0	0
$655$	29.0863	1.13650
$656$	−6.02177	−0.235111
$657$	0	0
$658$	−2.55257	−0.0995094
$659$	−29.8877	−1.16426	−0.582130	−	0.813096i	$-0.697780\pi$
$659$	−29.8877	−1.16426	−0.582130	+	0.813096i	$0.697780\pi$
$660$	0	0
$661$	−23.1696	−0.901192	−0.450596	−	0.892728i	$-0.648789\pi$
$661$	−23.1696	−0.901192	−0.450596	+	0.892728i	$0.648789\pi$
$662$	−13.0465	−0.507068
$663$	0	0
$664$	−16.6122	−0.644680
$665$	8.70841	0.337698
$666$	0	0
$667$	7.04892	0.272935
$668$	−22.8689	−0.884824
$669$	0	0
$670$	−19.6504	−0.759161
$671$	45.9987	1.77576
$672$	0	0
$673$	−34.6504	−1.33567	−0.667837	−	0.744307i	$-0.732780\pi$
$673$	−34.6504	−1.33567	−0.667837	+	0.744307i	$0.732780\pi$
$674$	7.36227	0.283584
$675$	0	0
$676$	10.7888	0.414954
$677$	9.32304	0.358314	0.179157	−	0.983821i	$-0.442663\pi$
$677$	9.32304	0.358314	0.179157	+	0.983821i	$0.442663\pi$
$678$	0	0
$679$	−28.5278	−1.09480
$680$	26.4276	1.01345
$681$	0	0
$682$	5.21898	0.199845
$683$	18.8049	0.719550	0.359775	−	0.933039i	$-0.382853\pi$
$683$	18.8049	0.719550	0.359775	+	0.933039i	$0.382853\pi$
$684$	0	0
$685$	−4.68366	−0.178953
$686$	−15.3467	−0.585941
$687$	0	0
$688$	2.75600	0.105072
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	5.52840	0.210310	0.105155	−	0.994456i	$-0.466466\pi$
$691$	5.52840	0.210310	0.105155	+	0.994456i	$0.466466\pi$
$692$	11.0780	0.421122
$693$	0	0
$694$	−5.24267	−0.199009
$695$	−1.05861	−0.0401552
$696$	0	0
$697$	62.9560	2.38463
$698$	−17.8019	−0.673813
$699$	0	0
$700$	−12.4571	−0.470835
$701$	14.1787	0.535522	0.267761	−	0.963485i	$-0.413716\pi$
$701$	14.1787	0.535522	0.267761	+	0.963485i	$0.413716\pi$
$702$	0	0
$703$	8.55065	0.322494
$704$	14.1188	0.532120
$705$	0	0
$706$	−13.8702	−0.522013
$707$	−7.11051	−0.267418
$708$	0	0
$709$	−22.9584	−0.862220	−0.431110	−	0.902299i	$-0.641878\pi$
$709$	−22.9584	−0.862220	−0.431110	+	0.902299i	$0.641878\pi$
$710$	17.3381	0.650688
$711$	0	0
$712$	20.2325	0.758245
$713$	−11.5821	−0.433753
$714$	0	0
$715$	−15.0586	−0.563160
$716$	13.4166	0.501401
$717$	0	0
$718$	−22.4243	−0.836869
$719$	10.8562	0.404869	0.202435	−	0.979296i	$-0.435115\pi$
$719$	10.8562	0.404869	0.202435	+	0.979296i	$0.435115\pi$
$720$	0	0
$721$	19.7362	0.735013
$722$	−14.0858	−0.524217
$723$	0	0
$724$	31.2021	1.15962
$725$	2.13706	0.0793685
$726$	0	0
$727$	−23.8049	−0.882876	−0.441438	−	0.897292i	$-0.645531\pi$
$727$	−23.8049	−0.882876	−0.441438	+	0.897292i	$0.645531\pi$
$728$	−25.9855	−0.963088
$729$	0	0
$730$	11.5845	0.428762
$731$	−28.8133	−1.06570
$732$	0	0
$733$	15.9202	0.588027	0.294013	−	0.955801i	$-0.405009\pi$
$733$	15.9202	0.588027	0.294013	+	0.955801i	$0.405009\pi$
$734$	−20.8625	−0.770048
$735$	0	0
$736$	−41.0887	−1.51455
$737$	−57.3594	−2.11286
$738$	0	0
$739$	−8.79523	−0.323538	−0.161769	−	0.986829i	$-0.551720\pi$
$739$	−8.79523	−0.323538	−0.161769	+	0.986829i	$0.551720\pi$
$740$	16.3860	0.602360
$741$	0	0
$742$	−1.53319	−0.0562851
$743$	1.60925	0.0590377	0.0295189	−	0.999564i	$-0.490602\pi$
$743$	1.60925	0.0590377	0.0295189	+	0.999564i	$0.490602\pi$
$744$	0	0
$745$	22.1505	0.811531
$746$	4.61941	0.169129
$747$	0	0
$748$	31.1817	1.14011
$749$	73.5153	2.68619
$750$	0	0
$751$	−26.6708	−0.973233	−0.486616	−	0.873616i	$-0.661769\pi$
$751$	−26.6708	−0.973233	−0.486616	+	0.873616i	$0.661769\pi$
$752$	0.411190	0.0149946
$753$	0	0
$754$	1.80194	0.0656227
$755$	25.2271	0.918109
$756$	0	0
$757$	−30.0398	−1.09182	−0.545908	−	0.837845i	$-0.683815\pi$
$757$	−30.0398	−1.09182	−0.545908	+	0.837845i	$0.683815\pi$
$758$	19.7375	0.716898
$759$	0	0
$760$	5.45712	0.197951
$761$	−8.17151	−0.296217	−0.148108	−	0.988971i	$-0.547318\pi$
$761$	−8.17151	−0.296217	−0.148108	+	0.988971i	$0.547318\pi$
$762$	0	0
$763$	43.6413	1.57992
$764$	−6.86400	−0.248331
$765$	0	0
$766$	−7.13169	−0.257678
$767$	2.65279	0.0957868
$768$	0	0
$769$	−22.0774	−0.796131	−0.398066	−	0.917357i	$-0.630318\pi$
$769$	−22.0774	−0.796131	−0.398066	+	0.917357i	$0.630318\pi$
$770$	−23.0877	−0.832022
$771$	0	0
$772$	8.56512	0.308265
$773$	9.95944	0.358216	0.179108	−	0.983829i	$-0.442679\pi$
$773$	9.95944	0.358216	0.179108	+	0.983829i	$0.442679\pi$
$774$	0	0
$775$	−3.51142	−0.126134
$776$	−17.8769	−0.641745
$777$	0	0
$778$	−18.0108	−0.645717
$779$	13.0000	0.465773
$780$	0	0
$781$	50.6098	1.81096
$782$	32.7972	1.17282
$783$	0	0
$784$	6.35690	0.227032
$785$	−7.60388	−0.271394
$786$	0	0
$787$	7.44504	0.265387	0.132694	−	0.991157i	$-0.457637\pi$
$787$	7.44504	0.265387	0.132694	+	0.991157i	$0.457637\pi$
$788$	−6.15106	−0.219123
$789$	0	0
$790$	11.6286	0.413728
$791$	66.6771	2.37076
$792$	0	0
$793$	−26.0954	−0.926676
$794$	7.72540	0.274164
$795$	0	0
$796$	6.10513	0.216391
$797$	−32.1099	−1.13739	−0.568696	−	0.822548i	$-0.692552\pi$
$797$	−32.1099	−1.13739	−0.568696	+	0.822548i	$0.692552\pi$
$798$	0	0
$799$	−4.29888	−0.152083
$800$	−12.4571	−0.440426
$801$	0	0
$802$	−9.47219	−0.334475
$803$	33.8151	1.19331
$804$	0	0
$805$	51.2368	1.80586
$806$	−2.96077	−0.104289
$807$	0	0
$808$	−4.45580	−0.156754
$809$	−2.34422	−0.0824185	−0.0412093	−	0.999151i	$-0.513121\pi$
$809$	−2.34422	−0.0824185	−0.0412093	+	0.999151i	$0.513121\pi$
$810$	0	0
$811$	−23.3351	−0.819407	−0.409704	−	0.912219i	$-0.634368\pi$
$811$	−23.3351	−0.819407	−0.409704	+	0.912219i	$0.634368\pi$
$812$	−5.82908	−0.204561
$813$	0	0
$814$	−22.6694	−0.794562
$815$	23.4910	0.822853
$816$	0	0
$817$	−5.94975	−0.208156
$818$	−10.0446	−0.351201
$819$	0	0
$820$	24.9124	0.869980
$821$	28.1987	0.984140	0.492070	−	0.870556i	$-0.336240\pi$
$821$	28.1987	0.984140	0.492070	+	0.870556i	$0.336240\pi$
$822$	0	0
$823$	−30.4722	−1.06219	−0.531097	−	0.847311i	$-0.678220\pi$
$823$	−30.4722	−1.06219	−0.531097	+	0.847311i	$0.678220\pi$
$824$	12.3676	0.430847
$825$	0	0
$826$	4.06723	0.141517
$827$	18.7657	0.652547	0.326274	−	0.945275i	$-0.394207\pi$
$827$	18.7657	0.652547	0.326274	+	0.945275i	$0.394207\pi$
$828$	0	0
$829$	−29.0465	−1.00883	−0.504414	−	0.863462i	$-0.668291\pi$
$829$	−29.0465	−1.00883	−0.504414	+	0.863462i	$0.668291\pi$
$830$	8.37329	0.290641
$831$	0	0
$832$	−8.00969	−0.277686
$833$	−66.4596	−2.30269
$834$	0	0
$835$	28.5171	0.986873
$836$	6.43881	0.222691
$837$	0	0
$838$	−24.8232	−0.857504
$839$	−49.6872	−1.71539	−0.857697	−	0.514156i	$-0.828105\pi$
$839$	−49.6872	−1.71539	−0.857697	+	0.514156i	$0.828105\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	25.3913	0.875043
$843$	0	0
$844$	20.4093	0.702516
$845$	−13.4534	−0.462811
$846$	0	0
$847$	−20.1379	−0.691947
$848$	0.246980	0.00848131
$849$	0	0
$850$	9.94331	0.341053
$851$	50.3086	1.72456
$852$	0	0
$853$	−1.67935	−0.0574998	−0.0287499	−	0.999587i	$-0.509153\pi$
$853$	−1.67935	−0.0574998	−0.0287499	+	0.999587i	$0.509153\pi$
$854$	−40.0092	−1.36909
$855$	0	0
$856$	46.0683	1.57458
$857$	48.6926	1.66331	0.831654	−	0.555294i	$-0.187394\pi$
$857$	48.6926	1.66331	0.831654	+	0.555294i	$0.187394\pi$
$858$	0	0
$859$	41.9971	1.43292	0.716462	−	0.697626i	$-0.245760\pi$
$859$	41.9971	1.43292	0.716462	+	0.697626i	$0.245760\pi$
$860$	−11.4018	−0.388797
$861$	0	0
$862$	6.09198	0.207494
$863$	6.57865	0.223940	0.111970	−	0.993712i	$-0.464284\pi$
$863$	6.57865	0.223940	0.111970	+	0.993712i	$0.464284\pi$
$864$	0	0
$865$	−13.8140	−0.469691
$866$	14.8694	0.505282
$867$	0	0
$868$	9.57779	0.325091
$869$	33.9439	1.15147
$870$	0	0
$871$	32.5405	1.10259
$872$	27.3478	0.926113
$873$	0	0
$874$	6.77240	0.229080
$875$	51.8775	1.75378
$876$	0	0
$877$	36.0890	1.21864	0.609319	−	0.792925i	$-0.291443\pi$
$877$	36.0890	1.21864	0.609319	+	0.792925i	$0.291443\pi$
$878$	−11.4214	−0.385452
$879$	0	0
$880$	3.71917	0.125373
$881$	26.0653	0.878163	0.439081	−	0.898447i	$-0.355304\pi$
$881$	26.0653	0.878163	0.439081	+	0.898447i	$0.355304\pi$
$882$	0	0
$883$	11.2567	0.378817	0.189409	−	0.981898i	$-0.439343\pi$
$883$	11.2567	0.378817	0.189409	+	0.981898i	$0.439343\pi$
$884$	−17.6896	−0.594967
$885$	0	0
$886$	−26.6049	−0.893810
$887$	13.7095	0.460319	0.230160	−	0.973153i	$-0.426075\pi$
$887$	13.7095	0.460319	0.230160	+	0.973153i	$0.426075\pi$
$888$	0	0
$889$	−51.6155	−1.73113
$890$	−10.1981	−0.341840
$891$	0	0
$892$	−23.1395	−0.774766
$893$	−0.887691	−0.0297054
$894$	0	0
$895$	−16.7302	−0.559228
$896$	37.8019	1.26287
$897$	0	0
$898$	−25.3090	−0.844574
$899$	−1.64310	−0.0548006
$900$	0	0
$901$	−2.58211	−0.0860224
$902$	−34.4655	−1.14758
$903$	0	0
$904$	41.7832	1.38969
$905$	−38.9084	−1.29336
$906$	0	0
$907$	−2.36599	−0.0785615	−0.0392808	−	0.999228i	$-0.512507\pi$
$907$	−2.36599	−0.0785615	−0.0392808	+	0.999228i	$0.512507\pi$
$908$	23.5942	0.783001
$909$	0	0
$910$	13.0978	0.434189
$911$	46.9807	1.55654	0.778271	−	0.627929i	$-0.216097\pi$
$911$	46.9807	1.55654	0.778271	+	0.627929i	$0.216097\pi$
$912$	0	0
$913$	24.4416	0.808898
$914$	24.3554	0.805604
$915$	0	0
$916$	−33.4940	−1.10667
$917$	−73.8477	−2.43867
$918$	0	0
$919$	14.4125	0.475425	0.237713	−	0.971336i	$-0.423602\pi$
$919$	14.4125	0.475425	0.237713	+	0.971336i	$0.423602\pi$
$920$	32.1075	1.05855
$921$	0	0
$922$	−2.72694	−0.0898069
$923$	−28.7114	−0.945047
$924$	0	0
$925$	15.2524	0.501494
$926$	−6.23537	−0.204907
$927$	0	0
$928$	−5.82908	−0.191349
$929$	−41.0447	−1.34663	−0.673317	−	0.739354i	$-0.735131\pi$
$929$	−41.0447	−1.34663	−0.673317	+	0.739354i	$0.735131\pi$
$930$	0	0
$931$	−13.7235	−0.449769
$932$	−15.1454	−0.496105
$933$	0	0
$934$	−14.0672	−0.460294
$935$	−38.8829	−1.27161
$936$	0	0
$937$	−27.6819	−0.904327	−0.452163	−	0.891935i	$-0.649348\pi$
$937$	−27.6819	−0.904327	−0.452163	+	0.891935i	$0.649348\pi$
$938$	49.8907	1.62899
$939$	0	0
$940$	−1.70112	−0.0554844
$941$	−40.8447	−1.33150	−0.665750	−	0.746175i	$-0.731888\pi$
$941$	−40.8447	−1.33150	−0.665750	+	0.746175i	$0.731888\pi$
$942$	0	0
$943$	76.4868	2.49075
$944$	−0.655186	−0.0213245
$945$	0	0
$946$	15.7739	0.512855
$947$	−20.3720	−0.662000	−0.331000	−	0.943631i	$-0.607386\pi$
$947$	−20.3720	−0.662000	−0.331000	+	0.943631i	$0.607386\pi$
$948$	0	0
$949$	−19.1836	−0.622726
$950$	2.05323	0.0666156
$951$	0	0
$952$	−67.0974	−2.17464
$953$	−39.9226	−1.29322	−0.646610	−	0.762821i	$-0.723814\pi$
$953$	−39.9226	−1.29322	−0.646610	+	0.762821i	$0.723814\pi$
$954$	0	0
$955$	8.55927	0.276971
$956$	16.8538	0.545092
$957$	0	0
$958$	11.4039	0.368443
$959$	11.8914	0.383994
$960$	0	0
$961$	−28.3002	−0.912910
$962$	12.8605	0.414641
$963$	0	0
$964$	21.9430	0.706738
$965$	−10.6805	−0.343818
$966$	0	0
$967$	−12.6082	−0.405452	−0.202726	−	0.979236i	$-0.564980\pi$
$967$	−12.6082	−0.405452	−0.202726	+	0.979236i	$0.564980\pi$
$968$	−12.6194	−0.405603
$969$	0	0
$970$	9.01075	0.289318
$971$	2.29398	0.0736173	0.0368086	−	0.999322i	$-0.488281\pi$
$971$	2.29398	0.0736173	0.0368086	+	0.999322i	$0.488281\pi$
$972$	0	0
$973$	2.68771	0.0861640
$974$	−33.1739	−1.06296
$975$	0	0
$976$	6.44504	0.206301
$977$	−22.5399	−0.721115	−0.360558	−	0.932737i	$-0.617414\pi$
$977$	−22.5399	−0.721115	−0.360558	+	0.932737i	$0.617414\pi$
$978$	0	0
$979$	−29.7681	−0.951392
$980$	−26.2989	−0.840087
$981$	0	0
$982$	−2.80433	−0.0894898
$983$	−0.909935	−0.0290224	−0.0145112	−	0.999895i	$-0.504619\pi$
$983$	−0.909935	−0.0290224	−0.0145112	+	0.999895i	$0.504619\pi$
$984$	0	0
$985$	7.67025	0.244395
$986$	4.65279	0.148175
$987$	0	0
$988$	−3.65279	−0.116211
$989$	−35.0060	−1.11313
$990$	0	0
$991$	55.2605	1.75541	0.877704	−	0.479203i	$-0.159074\pi$
$991$	55.2605	1.75541	0.877704	+	0.479203i	$0.159074\pi$
$992$	9.57779	0.304095
$993$	0	0
$994$	−44.0200	−1.39623
$995$	−7.61297	−0.241347
$996$	0	0
$997$	13.6649	0.432771	0.216385	−	0.976308i	$-0.430573\pi$
$997$	13.6649	0.432771	0.216385	+	0.976308i	$0.430573\pi$
$998$	−5.26529	−0.166670
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	783.2.a.e.1.3	✓	3
3.2	odd	2		783.2.a.f.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
783.2.a.e.1.3	✓	3	1.1	even	1	trivial
783.2.a.f.1.1	yes	3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 783 = 3^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	783.a (trivial)

\(f(q)\)	\(=\)	\(q+0.801938 q^{2} -1.35690 q^{4} +1.69202 q^{5} -4.29590 q^{7} -2.69202 q^{8} +1.35690 q^{10} +3.96077 q^{11} -2.24698 q^{13} -3.44504 q^{14} +0.554958 q^{16} -5.80194 q^{17} -1.19806 q^{19} -2.29590 q^{20} +3.17629 q^{22} -7.04892 q^{23} -2.13706 q^{25} -1.80194 q^{26} +5.82908 q^{28} -1.00000 q^{29} +1.64310 q^{31} +5.82908 q^{32} -4.65279 q^{34} -7.26875 q^{35} -7.13706 q^{37} -0.960771 q^{38} -4.55496 q^{40} -10.8509 q^{41} +4.96615 q^{43} -5.37435 q^{44} -5.65279 q^{46} +0.740939 q^{47} +11.4547 q^{49} -1.71379 q^{50} +3.04892 q^{52} +0.445042 q^{53} +6.70171 q^{55} +11.5646 q^{56} -0.801938 q^{58} -1.18060 q^{59} +11.6136 q^{61} +1.31767 q^{62} +3.56465 q^{64} -3.80194 q^{65} -14.4819 q^{67} +7.87263 q^{68} -5.82908 q^{70} +12.7778 q^{71} +8.53750 q^{73} -5.72348 q^{74} +1.62565 q^{76} -17.0151 q^{77} +8.57002 q^{79} +0.939001 q^{80} -8.70171 q^{82} +6.17092 q^{83} -9.81700 q^{85} +3.98254 q^{86} -10.6625 q^{88} -7.51573 q^{89} +9.65279 q^{91} +9.56465 q^{92} +0.594187 q^{94} -2.02715 q^{95} +6.64071 q^{97} +9.18598 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + q^{7} - 3 q^{8} - q^{11} - 2 q^{13} - 10 q^{14} + 2 q^{16} - 13 q^{17} - 8 q^{19} + 7 q^{20} + 17 q^{22} - 12 q^{23} - q^{25} - q^{26} + 7 q^{28} - 3 q^{29} + 9 q^{31} + 7 q^{32} + 4 q^{34}+ \cdots + 13 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + q^7 - 3 * q^8 - q^11 - 2 * q^13 - 10 * q^14 + 2 * q^16 - 13 * q^17 - 8 * q^19 + 7 * q^20 + 17 * q^22 - 12 * q^23 - q^25 - q^26 + 7 * q^28 - 3 * q^29 + 9 * q^31 + 7 * q^32 + 4 * q^34 - 14 * q^35 - 16 * q^37 + 10 * q^38 - 14 * q^40 - 19 * q^41 - q^43 - 28 * q^44 + q^46 - 12 * q^47 + 12 * q^49 + 3 * q^50 + q^53 - 7 * q^55 + 13 * q^56 + 2 * q^58 + 8 * q^59 + 4 * q^61 - 13 * q^62 - 11 * q^64 - 7 * q^65 - 15 * q^67 + 7 * q^68 - 7 * q^70 - 4 * q^71 + 10 * q^73 + 13 * q^74 - 7 * q^76 - 26 * q^77 + q^79 - 7 * q^80 + q^82 + 29 * q^83 - 4 * q^86 + 8 * q^88 - 10 * q^89 + 11 * q^91 + 7 * q^92 + 15 * q^94 - 17 * q^97 + 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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