LMFDB - Embedded newform 1053.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1053,2,Mod(1,1053)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1053, 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("1053.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-2,0,6,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.40824733284$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.22931361.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 10x^{2} - 11x + 2 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 10x^2 - 11*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.496094$ of defining polynomial
Character	$\chi$	$=$	1053.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.496094 q^{2} -1.75389 q^{4} -1.32688 q^{5} +3.12929 q^{7} +1.86228 q^{8} +0.658258 q^{10} -3.41470 q^{11} -1.00000 q^{13} -1.55242 q^{14} +2.58391 q^{16} -0.912179 q^{17} -1.03576 q^{19} +2.32721 q^{20} +1.69401 q^{22} +5.37222 q^{23} -3.23938 q^{25} +0.496094 q^{26} -5.48843 q^{28} -6.50423 q^{29} +2.54537 q^{31} -5.00643 q^{32} +0.452527 q^{34} -4.15219 q^{35} +7.99800 q^{37} +0.513832 q^{38} -2.47103 q^{40} -5.45361 q^{41} +4.40936 q^{43} +5.98902 q^{44} -2.66513 q^{46} -9.19730 q^{47} +2.79243 q^{49} +1.60704 q^{50} +1.75389 q^{52} -12.1878 q^{53} +4.53091 q^{55} +5.82762 q^{56} +3.22671 q^{58} -7.08032 q^{59} -14.3744 q^{61} -1.26274 q^{62} -2.68417 q^{64} +1.32688 q^{65} -9.35421 q^{67} +1.59986 q^{68} +2.05988 q^{70} -12.0670 q^{71} -5.22771 q^{73} -3.96776 q^{74} +1.81660 q^{76} -10.6856 q^{77} -12.1088 q^{79} -3.42855 q^{80} +2.70551 q^{82} +11.0036 q^{83} +1.21035 q^{85} -2.18746 q^{86} -6.35914 q^{88} -8.33216 q^{89} -3.12929 q^{91} -9.42229 q^{92} +4.56272 q^{94} +1.37432 q^{95} +14.0320 q^{97} -1.38531 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 6 q^{4} - 3 q^{5} - 6 q^{8} - 6 q^{10} - 7 q^{11} - 6 q^{13} - 13 q^{14} + 6 q^{16} - 14 q^{17} - 3 q^{19} - 17 q^{20} - 3 q^{22} - 17 q^{23} + 3 q^{25} + 2 q^{26} + 15 q^{28} - 14 q^{29}+ \cdots - 17 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 6 * q^4 - 3 * q^5 - 6 * q^8 - 6 * q^10 - 7 * q^11 - 6 * q^13 - 13 * q^14 + 6 * q^16 - 14 * q^17 - 3 * q^19 - 17 * q^20 - 3 * q^22 - 17 * q^23 + 3 * q^25 + 2 * q^26 + 15 * q^28 - 14 * q^29 + 6 * q^31 - 9 * q^32 + 3 * q^34 - 13 * q^35 - 9 * q^37 + 4 * q^38 - 6 * q^40 + 4 * q^41 - 3 * q^43 - 13 * q^44 - 9 * q^47 + 21 * q^50 - 6 * q^52 - 28 * q^53 + 15 * q^55 - 16 * q^56 + 18 * q^58 - 17 * q^59 - 6 * q^61 + 4 * q^62 - 30 * q^64 + 3 * q^65 - 12 * q^67 - 22 * q^68 + 9 * q^70 + q^71 - 9 * q^73 + 27 * q^74 - 3 * q^76 - 15 * q^77 - 6 * q^79 + 24 * q^80 + 27 * q^82 - 28 * q^83 + 27 * q^85 + 6 * q^86 - 9 * q^88 + 9 * q^89 - 44 * q^92 - 24 * q^94 - 32 * q^95 - 17 * 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 $

$2$	−0.496094	−0.350791	−0.175396	−	0.984498i	$-0.556120\pi$
$2$	−0.496094	−0.350791	−0.175396	+	0.984498i	$0.556120\pi$
$3$	0	0
$3$	0	0
$4$	−1.75389	−0.876945
$5$	−1.32688	−0.593400	−0.296700	−	0.954971i	$-0.595886\pi$
$5$	−1.32688	−0.593400	−0.296700	+	0.954971i	$0.595886\pi$
$6$	0	0
$7$	3.12929	1.18276	0.591380	−	0.806393i	$-0.298584\pi$
$7$	3.12929	1.18276	0.591380	+	0.806393i	$0.298584\pi$
$8$	1.86228	0.658416
$9$	0	0
$10$	0.658258	0.208159
$11$	−3.41470	−1.02957	−0.514786	−	0.857319i	$-0.672129\pi$
$11$	−3.41470	−1.02957	−0.514786	+	0.857319i	$0.672129\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.55242	−0.414902
$15$	0	0
$16$	2.58391	0.645979
$17$	−0.912179	−0.221236	−0.110618	−	0.993863i	$-0.535283\pi$
$17$	−0.912179	−0.221236	−0.110618	+	0.993863i	$0.535283\pi$
$18$	0	0
$19$	−1.03576	−0.237619	−0.118809	−	0.992917i	$-0.537908\pi$
$19$	−1.03576	−0.237619	−0.118809	+	0.992917i	$0.537908\pi$
$20$	2.32721	0.520379
$21$	0	0
$22$	1.69401	0.361165
$23$	5.37222	1.12019	0.560093	−	0.828430i	$-0.310765\pi$
$23$	5.37222	1.12019	0.560093	+	0.828430i	$0.310765\pi$
$24$	0	0
$25$	−3.23938	−0.647877
$26$	0.496094	0.0972920
$27$	0	0
$28$	−5.48843	−1.03722
$29$	−6.50423	−1.20781	−0.603903	−	0.797058i	$-0.706388\pi$
$29$	−6.50423	−1.20781	−0.603903	+	0.797058i	$0.706388\pi$
$30$	0	0
$31$	2.54537	0.457162	0.228581	−	0.973525i	$-0.426591\pi$
$31$	2.54537	0.457162	0.228581	+	0.973525i	$0.426591\pi$
$32$	−5.00643	−0.885020
$33$	0	0
$34$	0.452527	0.0776077
$35$	−4.15219	−0.701849
$36$	0	0
$37$	7.99800	1.31486	0.657431	−	0.753514i	$-0.271643\pi$
$37$	7.99800	1.31486	0.657431	+	0.753514i	$0.271643\pi$
$38$	0.513832	0.0833545
$39$	0	0
$40$	−2.47103	−0.390704
$41$	−5.45361	−0.851711	−0.425856	−	0.904791i	$-0.640027\pi$
$41$	−5.45361	−0.851711	−0.425856	+	0.904791i	$0.640027\pi$
$42$	0	0
$43$	4.40936	0.672421	0.336211	−	0.941787i	$-0.390855\pi$
$43$	4.40936	0.672421	0.336211	+	0.941787i	$0.390855\pi$
$44$	5.98902	0.902878
$45$	0	0
$46$	−2.66513	−0.392952
$47$	−9.19730	−1.34156	−0.670782	−	0.741655i	$-0.734041\pi$
$47$	−9.19730	−1.34156	−0.670782	+	0.741655i	$0.734041\pi$
$48$	0	0
$49$	2.79243	0.398919
$50$	1.60704	0.227270
$51$	0	0
$52$	1.75389	0.243221
$53$	−12.1878	−1.67412	−0.837060	−	0.547111i	$-0.815728\pi$
$53$	−12.1878	−1.67412	−0.837060	+	0.547111i	$0.815728\pi$
$54$	0	0
$55$	4.53091	0.610947
$56$	5.82762	0.778748
$57$	0	0
$58$	3.22671	0.423688
$59$	−7.08032	−0.921779	−0.460890	−	0.887457i	$-0.652470\pi$
$59$	−7.08032	−0.921779	−0.460890	+	0.887457i	$0.652470\pi$
$60$	0	0
$61$	−14.3744	−1.84045	−0.920226	−	0.391387i	$-0.871995\pi$
$61$	−14.3744	−1.84045	−0.920226	+	0.391387i	$0.871995\pi$
$62$	−1.26274	−0.160369
$63$	0	0
$64$	−2.68417	−0.335521
$65$	1.32688	0.164579
$66$	0	0
$67$	−9.35421	−1.14280	−0.571399	−	0.820672i	$-0.693599\pi$
$67$	−9.35421	−1.14280	−0.571399	+	0.820672i	$0.693599\pi$
$68$	1.59986	0.194012
$69$	0	0
$70$	2.05988	0.246202
$71$	−12.0670	−1.43209	−0.716045	−	0.698054i	$-0.754049\pi$
$71$	−12.0670	−1.43209	−0.716045	+	0.698054i	$0.754049\pi$
$72$	0	0
$73$	−5.22771	−0.611857	−0.305928	−	0.952055i	$-0.598967\pi$
$73$	−5.22771	−0.611857	−0.305928	+	0.952055i	$0.598967\pi$
$74$	−3.96776	−0.461243
$75$	0	0
$76$	1.81660	0.208378
$77$	−10.6856	−1.21774
$78$	0	0
$79$	−12.1088	−1.36234	−0.681172	−	0.732123i	$-0.738530\pi$
$79$	−12.1088	−1.36234	−0.681172	+	0.732123i	$0.738530\pi$
$80$	−3.42855	−0.383323
$81$	0	0
$82$	2.70551	0.298773
$83$	11.0036	1.20780	0.603900	−	0.797060i	$-0.293613\pi$
$83$	11.0036	1.20780	0.603900	+	0.797060i	$0.293613\pi$
$84$	0	0
$85$	1.21035	0.131281
$86$	−2.18746	−0.235880
$87$	0	0
$88$	−6.35914	−0.677887
$89$	−8.33216	−0.883207	−0.441604	−	0.897210i	$-0.645590\pi$
$89$	−8.33216	−0.883207	−0.441604	+	0.897210i	$0.645590\pi$
$90$	0	0
$91$	−3.12929	−0.328038
$92$	−9.42229	−0.982342
$93$	0	0
$94$	4.56272	0.470609
$95$	1.37432	0.141003
$96$	0	0
$97$	14.0320	1.42473	0.712365	−	0.701809i	$-0.247624\pi$
$97$	14.0320	1.42473	0.712365	+	0.701809i	$0.247624\pi$
$98$	−1.38531	−0.139937
$99$	0	0
$100$	5.68153	0.568153
$101$	11.8378	1.17790	0.588951	−	0.808169i	$-0.299541\pi$
$101$	11.8378	1.17790	0.588951	+	0.808169i	$0.299541\pi$
$102$	0	0
$103$	8.20588	0.808549	0.404275	−	0.914638i	$-0.367524\pi$
$103$	8.20588	0.808549	0.404275	+	0.914638i	$0.367524\pi$
$104$	−1.86228	−0.182612
$105$	0	0
$106$	6.04628	0.587267
$107$	0.156610	0.0151401	0.00757005	−	0.999971i	$-0.497590\pi$
$107$	0.156610	0.0151401	0.00757005	+	0.999971i	$0.497590\pi$
$108$	0	0
$109$	−7.52766	−0.721019	−0.360509	−	0.932756i	$-0.617397\pi$
$109$	−7.52766	−0.721019	−0.360509	+	0.932756i	$0.617397\pi$
$110$	−2.24776	−0.214315
$111$	0	0
$112$	8.08581	0.764037
$113$	17.0734	1.60613	0.803067	−	0.595889i	$-0.203200\pi$
$113$	17.0734	1.60613	0.803067	+	0.595889i	$0.203200\pi$
$114$	0	0
$115$	−7.12830	−0.664718
$116$	11.4077	1.05918
$117$	0	0
$118$	3.51250	0.323352
$119$	−2.85447	−0.261669
$120$	0	0
$121$	0.660193	0.0600175
$122$	7.13105	0.645615
$123$	0	0
$124$	−4.46430	−0.400906
$125$	10.9327	0.977849
$126$	0	0
$127$	0.476604	0.0422917	0.0211459	−	0.999776i	$-0.493269\pi$
$127$	0.476604	0.0422917	0.0211459	+	0.999776i	$0.493269\pi$
$128$	11.3445	1.00272
$129$	0	0
$130$	−0.658258	−0.0577330
$131$	8.81100	0.769821	0.384910	−	0.922954i	$-0.374232\pi$
$131$	8.81100	0.769821	0.384910	+	0.922954i	$0.374232\pi$
$132$	0	0
$133$	−3.24117	−0.281045
$134$	4.64056	0.400884
$135$	0	0
$136$	−1.69874	−0.145665
$137$	−10.1972	−0.871206	−0.435603	−	0.900139i	$-0.643465\pi$
$137$	−10.1972	−0.871206	−0.435603	+	0.900139i	$0.643465\pi$
$138$	0	0
$139$	−10.7257	−0.909739	−0.454870	−	0.890558i	$-0.650314\pi$
$139$	−10.7257	−0.909739	−0.454870	+	0.890558i	$0.650314\pi$
$140$	7.28249	0.615483
$141$	0	0
$142$	5.98637	0.502365
$143$	3.41470	0.285552
$144$	0	0
$145$	8.63034	0.716711
$146$	2.59343	0.214634
$147$	0	0
$148$	−14.0276	−1.15306
$149$	−15.1361	−1.24000	−0.619999	−	0.784603i	$-0.712867\pi$
$149$	−15.1361	−1.24000	−0.619999	+	0.784603i	$0.712867\pi$
$150$	0	0
$151$	12.1898	0.991992	0.495996	−	0.868325i	$-0.334803\pi$
$151$	12.1898	0.991992	0.495996	+	0.868325i	$0.334803\pi$
$152$	−1.92887	−0.156452
$153$	0	0
$154$	5.30105	0.427171
$155$	−3.37741	−0.271280
$156$	0	0
$157$	−3.84432	−0.306810	−0.153405	−	0.988163i	$-0.549024\pi$
$157$	−3.84432	−0.306810	−0.153405	+	0.988163i	$0.549024\pi$
$158$	6.00709	0.477899
$159$	0	0
$160$	6.64294	0.525170
$161$	16.8112	1.32491
$162$	0	0
$163$	5.24217	0.410599	0.205299	−	0.978699i	$-0.434183\pi$
$163$	5.24217	0.410599	0.205299	+	0.978699i	$0.434183\pi$
$164$	9.56504	0.746904
$165$	0	0
$166$	−5.45881	−0.423686
$167$	17.7837	1.37615	0.688074	−	0.725641i	$-0.258457\pi$
$167$	17.7837	1.37615	0.688074	+	0.725641i	$0.258457\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.600449	−0.0460524
$171$	0	0
$172$	−7.73354	−0.589677
$173$	1.58081	0.120187	0.0600935	−	0.998193i	$-0.480860\pi$
$173$	1.58081	0.120187	0.0600935	+	0.998193i	$0.480860\pi$
$174$	0	0
$175$	−10.1370	−0.766282
$176$	−8.82330	−0.665081
$177$	0	0
$178$	4.13353	0.309822
$179$	15.2270	1.13812	0.569060	−	0.822296i	$-0.307307\pi$
$179$	15.2270	1.13812	0.569060	+	0.822296i	$0.307307\pi$
$180$	0	0
$181$	−22.9726	−1.70754	−0.853770	−	0.520650i	$-0.825690\pi$
$181$	−22.9726	−1.70754	−0.853770	+	0.520650i	$0.825690\pi$
$182$	1.55242	0.115073
$183$	0	0
$184$	10.0046	0.737549
$185$	−10.6124	−0.780239
$186$	0	0
$187$	3.11482	0.227778
$188$	16.1311	1.17648
$189$	0	0
$190$	−0.681794	−0.0494625
$191$	−7.07319	−0.511798	−0.255899	−	0.966704i	$-0.582371\pi$
$191$	−7.07319	−0.511798	−0.255899	+	0.966704i	$0.582371\pi$
$192$	0	0
$193$	8.01263	0.576762	0.288381	−	0.957516i	$-0.406883\pi$
$193$	8.01263	0.576762	0.288381	+	0.957516i	$0.406883\pi$
$194$	−6.96117	−0.499783
$195$	0	0
$196$	−4.89762	−0.349830
$197$	10.2986	0.733748	0.366874	−	0.930271i	$-0.380428\pi$
$197$	10.2986	0.733748	0.366874	+	0.930271i	$0.380428\pi$
$198$	0	0
$199$	−5.56945	−0.394808	−0.197404	−	0.980322i	$-0.563251\pi$
$199$	−5.56945	−0.394808	−0.197404	+	0.980322i	$0.563251\pi$
$200$	−6.03265	−0.426573
$201$	0	0
$202$	−5.87265	−0.413198
$203$	−20.3536	−1.42854
$204$	0	0
$205$	7.23630	0.505405
$206$	−4.07089	−0.283632
$207$	0	0
$208$	−2.58391	−0.179162
$209$	3.53680	0.244645
$210$	0	0
$211$	−9.43148	−0.649290	−0.324645	−	0.945836i	$-0.605245\pi$
$211$	−9.43148	−0.649290	−0.324645	+	0.945836i	$0.605245\pi$
$212$	21.3760	1.46811
$213$	0	0
$214$	−0.0776935	−0.00531102
$215$	−5.85070	−0.399014
$216$	0	0
$217$	7.96520	0.540713
$218$	3.73442	0.252927
$219$	0	0
$220$	−7.94671	−0.535767
$221$	0.912179	0.0613598
$222$	0	0
$223$	−18.6255	−1.24725	−0.623626	−	0.781723i	$-0.714341\pi$
$223$	−18.6255	−1.24725	−0.623626	+	0.781723i	$0.714341\pi$
$224$	−15.6666	−1.04677
$225$	0	0
$226$	−8.47003	−0.563418
$227$	−14.1009	−0.935910	−0.467955	−	0.883752i	$-0.655009\pi$
$227$	−14.1009	−0.935910	−0.467955	+	0.883752i	$0.655009\pi$
$228$	0	0
$229$	2.05069	0.135514	0.0677569	−	0.997702i	$-0.478416\pi$
$229$	2.05069	0.135514	0.0677569	+	0.997702i	$0.478416\pi$
$230$	3.53631	0.233177
$231$	0	0
$232$	−12.1127	−0.795239
$233$	8.01472	0.525062	0.262531	−	0.964924i	$-0.415443\pi$
$233$	8.01472	0.525062	0.262531	+	0.964924i	$0.415443\pi$
$234$	0	0
$235$	12.2037	0.796084
$236$	12.4181	0.808350
$237$	0	0
$238$	1.41609	0.0917912
$239$	−3.49522	−0.226087	−0.113043	−	0.993590i	$-0.536060\pi$
$239$	−3.49522	−0.226087	−0.113043	+	0.993590i	$0.536060\pi$
$240$	0	0
$241$	7.12783	0.459144	0.229572	−	0.973292i	$-0.426267\pi$
$241$	7.12783	0.459144	0.229572	+	0.973292i	$0.426267\pi$
$242$	−0.327518	−0.0210536
$243$	0	0
$244$	25.2111	1.61398
$245$	−3.70523	−0.236718
$246$	0	0
$247$	1.03576	0.0659035
$248$	4.74020	0.301003
$249$	0	0
$250$	−5.42364	−0.343021
$251$	−10.8057	−0.682052	−0.341026	−	0.940054i	$-0.610774\pi$
$251$	−10.8057	−0.682052	−0.341026	+	0.940054i	$0.610774\pi$
$252$	0	0
$253$	−18.3445	−1.15331
$254$	−0.236440	−0.0148356
$255$	0	0
$256$	−0.259577	−0.0162236
$257$	24.0427	1.49974	0.749870	−	0.661585i	$-0.230116\pi$
$257$	24.0427	1.49974	0.749870	+	0.661585i	$0.230116\pi$
$258$	0	0
$259$	25.0280	1.55517
$260$	−2.32721	−0.144327
$261$	0	0
$262$	−4.37108	−0.270047
$263$	−1.17408	−0.0723969	−0.0361984	−	0.999345i	$-0.511525\pi$
$263$	−1.17408	−0.0723969	−0.0361984	+	0.999345i	$0.511525\pi$
$264$	0	0
$265$	16.1717	0.993422
$266$	1.60793	0.0985883
$267$	0	0
$268$	16.4063	1.00217
$269$	−16.4749	−1.00449	−0.502247	−	0.864724i	$-0.667493\pi$
$269$	−16.4749	−1.00449	−0.502247	+	0.864724i	$0.667493\pi$
$270$	0	0
$271$	30.9493	1.88004	0.940018	−	0.341124i	$-0.110807\pi$
$271$	30.9493	1.88004	0.940018	+	0.341124i	$0.110807\pi$
$272$	−2.35699	−0.142914
$273$	0	0
$274$	5.05877	0.305612
$275$	11.0615	0.667036
$276$	0	0
$277$	−16.8419	−1.01193	−0.505967	−	0.862553i	$-0.668864\pi$
$277$	−16.8419	−1.01193	−0.505967	+	0.862553i	$0.668864\pi$
$278$	5.32094	0.319129
$279$	0	0
$280$	−7.73256	−0.462109
$281$	21.6086	1.28906	0.644530	−	0.764579i	$-0.277053\pi$
$281$	21.6086	1.28906	0.644530	+	0.764579i	$0.277053\pi$
$282$	0	0
$283$	3.91652	0.232813	0.116406	−	0.993202i	$-0.462862\pi$
$283$	3.91652	0.232813	0.116406	+	0.993202i	$0.462862\pi$
$284$	21.1642	1.25586
$285$	0	0
$286$	−1.69401	−0.100169
$287$	−17.0659	−1.00737
$288$	0	0
$289$	−16.1679	−0.951055
$290$	−4.28146	−0.251416
$291$	0	0
$292$	9.16883	0.536565
$293$	−13.3064	−0.777367	−0.388683	−	0.921371i	$-0.627070\pi$
$293$	−13.3064	−0.777367	−0.388683	+	0.921371i	$0.627070\pi$
$294$	0	0
$295$	9.39475	0.546983
$296$	14.8945	0.865727
$297$	0	0
$298$	7.50893	0.434981
$299$	−5.37222	−0.310684
$300$	0	0
$301$	13.7982	0.795312
$302$	−6.04729	−0.347982
$303$	0	0
$304$	−2.67630	−0.153497
$305$	19.0731	1.09212
$306$	0	0
$307$	2.50370	0.142894	0.0714469	−	0.997444i	$-0.477238\pi$
$307$	2.50370	0.142894	0.0714469	+	0.997444i	$0.477238\pi$
$308$	18.7413	1.06789
$309$	0	0
$310$	1.67551	0.0951626
$311$	−4.11297	−0.233225	−0.116613	−	0.993177i	$-0.537204\pi$
$311$	−4.11297	−0.233225	−0.116613	+	0.993177i	$0.537204\pi$
$312$	0	0
$313$	−3.47941	−0.196668	−0.0983340	−	0.995153i	$-0.531351\pi$
$313$	−3.47941	−0.196668	−0.0983340	+	0.995153i	$0.531351\pi$
$314$	1.90715	0.107626
$315$	0	0
$316$	21.2375	1.19470
$317$	25.0608	1.40756	0.703778	−	0.710420i	$-0.251495\pi$
$317$	25.0608	1.40756	0.703778	+	0.710420i	$0.251495\pi$
$318$	0	0
$319$	22.2100	1.24352
$320$	3.56158	0.199098
$321$	0	0
$322$	−8.33995	−0.464767
$323$	0.944794	0.0525698
$324$	0	0
$325$	3.23938	0.179689
$326$	−2.60061	−0.144034
$327$	0	0
$328$	−10.1562	−0.560781
$329$	−28.7810	−1.58675
$330$	0	0
$331$	0.270186	0.0148508	0.00742539	−	0.999972i	$-0.497636\pi$
$331$	0.270186	0.0148508	0.00742539	+	0.999972i	$0.497636\pi$
$332$	−19.2991	−1.05917
$333$	0	0
$334$	−8.82241	−0.482741
$335$	12.4119	0.678136
$336$	0	0
$337$	−24.6622	−1.34344	−0.671718	−	0.740807i	$-0.734443\pi$
$337$	−24.6622	−1.34344	−0.671718	+	0.740807i	$0.734443\pi$
$338$	−0.496094	−0.0269840
$339$	0	0
$340$	−2.12283	−0.115127
$341$	−8.69169	−0.470681
$342$	0	0
$343$	−13.1667	−0.710934
$344$	8.21148	0.442733
$345$	0	0
$346$	−0.784232	−0.0421606
$347$	−26.1451	−1.40354	−0.701771	−	0.712403i	$-0.747607\pi$
$347$	−26.1451	−1.40354	−0.701771	+	0.712403i	$0.747607\pi$
$348$	0	0
$349$	−19.9427	−1.06751	−0.533753	−	0.845640i	$-0.679219\pi$
$349$	−19.9427	−1.06751	−0.533753	+	0.845640i	$0.679219\pi$
$350$	5.02889	0.268805
$351$	0	0
$352$	17.0955	0.911191
$353$	17.1534	0.912986	0.456493	−	0.889727i	$-0.349105\pi$
$353$	17.1534	0.912986	0.456493	+	0.889727i	$0.349105\pi$
$354$	0	0
$355$	16.0115	0.849801
$356$	14.6137	0.774525
$357$	0	0
$358$	−7.55403	−0.399243
$359$	3.15925	0.166739	0.0833694	−	0.996519i	$-0.473432\pi$
$359$	3.15925	0.166739	0.0833694	+	0.996519i	$0.473432\pi$
$360$	0	0
$361$	−17.9272	−0.943537
$362$	11.3966	0.598990
$363$	0	0
$364$	5.48843	0.287672
$365$	6.93655	0.363076
$366$	0	0
$367$	4.03943	0.210856	0.105428	−	0.994427i	$-0.466379\pi$
$367$	4.03943	0.210856	0.105428	+	0.994427i	$0.466379\pi$
$368$	13.8814	0.723616
$369$	0	0
$370$	5.26475	0.273701
$371$	−38.1391	−1.98008
$372$	0	0
$373$	11.8455	0.613335	0.306667	−	0.951817i	$-0.400786\pi$
$373$	11.8455	0.613335	0.306667	+	0.951817i	$0.400786\pi$
$374$	−1.54524	−0.0799026
$375$	0	0
$376$	−17.1280	−0.883308
$377$	6.50423	0.334985
$378$	0	0
$379$	0.705984	0.0362640	0.0181320	−	0.999836i	$-0.494228\pi$
$379$	0.705984	0.0362640	0.0181320	+	0.999836i	$0.494228\pi$
$380$	−2.41042	−0.123652
$381$	0	0
$382$	3.50897	0.179534
$383$	24.8084	1.26765	0.633826	−	0.773476i	$-0.281484\pi$
$383$	24.8084	1.26765	0.633826	+	0.773476i	$0.281484\pi$
$384$	0	0
$385$	14.1785	0.722603
$386$	−3.97502	−0.202323
$387$	0	0
$388$	−24.6105	−1.24941
$389$	10.3748	0.526025	0.263013	−	0.964792i	$-0.415284\pi$
$389$	10.3748	0.526025	0.263013	+	0.964792i	$0.415284\pi$
$390$	0	0
$391$	−4.90043	−0.247825
$392$	5.20030	0.262655
$393$	0	0
$394$	−5.10909	−0.257392
$395$	16.0669	0.808415
$396$	0	0
$397$	19.5931	0.983351	0.491675	−	0.870779i	$-0.336385\pi$
$397$	19.5931	0.983351	0.491675	+	0.870779i	$0.336385\pi$
$398$	2.76297	0.138495
$399$	0	0
$400$	−8.37029	−0.418515
$401$	22.3087	1.11404	0.557022	−	0.830498i	$-0.311944\pi$
$401$	22.3087	1.11404	0.557022	+	0.830498i	$0.311944\pi$
$402$	0	0
$403$	−2.54537	−0.126794
$404$	−20.7622	−1.03296
$405$	0	0
$406$	10.0973	0.501120
$407$	−27.3108	−1.35375
$408$	0	0
$409$	−3.49960	−0.173044	−0.0865220	−	0.996250i	$-0.527575\pi$
$409$	−3.49960	−0.173044	−0.0865220	+	0.996250i	$0.527575\pi$
$410$	−3.58989	−0.177292
$411$	0	0
$412$	−14.3922	−0.709053
$413$	−22.1564	−1.09024
$414$	0	0
$415$	−14.6004	−0.716708
$416$	5.00643	0.245460
$417$	0	0
$418$	−1.75458	−0.0858195
$419$	−21.2495	−1.03811	−0.519054	−	0.854742i	$-0.673716\pi$
$419$	−21.2495	−1.03811	−0.519054	+	0.854742i	$0.673716\pi$
$420$	0	0
$421$	2.22792	0.108582	0.0542911	−	0.998525i	$-0.482710\pi$
$421$	2.22792	0.108582	0.0542911	+	0.998525i	$0.482710\pi$
$422$	4.67890	0.227765
$423$	0	0
$424$	−22.6971	−1.10227
$425$	2.95490	0.143334
$426$	0	0
$427$	−44.9816	−2.17681
$428$	−0.274678	−0.0132770
$429$	0	0
$430$	2.90250	0.139971
$431$	10.3044	0.496343	0.248172	−	0.968716i	$-0.420170\pi$
$431$	10.3044	0.496343	0.248172	+	0.968716i	$0.420170\pi$
$432$	0	0
$433$	−10.0517	−0.483056	−0.241528	−	0.970394i	$-0.577649\pi$
$433$	−10.0517	−0.483056	−0.241528	+	0.970394i	$0.577649\pi$
$434$	−3.95149	−0.189677
$435$	0	0
$436$	13.2027	0.632294
$437$	−5.56431	−0.266177
$438$	0	0
$439$	15.7389	0.751175	0.375587	−	0.926787i	$-0.377441\pi$
$439$	15.7389	0.751175	0.375587	+	0.926787i	$0.377441\pi$
$440$	8.43783	0.402258
$441$	0	0
$442$	−0.452527	−0.0215245
$443$	−9.55924	−0.454173	−0.227087	−	0.973875i	$-0.572920\pi$
$443$	−9.55924	−0.454173	−0.227087	+	0.973875i	$0.572920\pi$
$444$	0	0
$445$	11.0558	0.524095
$446$	9.23997	0.437525
$447$	0	0
$448$	−8.39954	−0.396841
$449$	−15.7163	−0.741696	−0.370848	−	0.928694i	$-0.620933\pi$
$449$	−15.7163	−0.741696	−0.370848	+	0.928694i	$0.620933\pi$
$450$	0	0
$451$	18.6225	0.876898
$452$	−29.9450	−1.40849
$453$	0	0
$454$	6.99538	0.328309
$455$	4.15219	0.194658
$456$	0	0
$457$	32.9041	1.53919	0.769594	−	0.638533i	$-0.220458\pi$
$457$	32.9041	1.53919	0.769594	+	0.638533i	$0.220458\pi$
$458$	−1.01734	−0.0475370
$459$	0	0
$460$	12.5023	0.582921
$461$	12.7772	0.595094	0.297547	−	0.954707i	$-0.403831\pi$
$461$	12.7772	0.595094	0.297547	+	0.954707i	$0.403831\pi$
$462$	0	0
$463$	−5.27466	−0.245134	−0.122567	−	0.992460i	$-0.539113\pi$
$463$	−5.27466	−0.245134	−0.122567	+	0.992460i	$0.539113\pi$
$464$	−16.8064	−0.780216
$465$	0	0
$466$	−3.97605	−0.184187
$467$	−18.5837	−0.859953	−0.429977	−	0.902840i	$-0.641478\pi$
$467$	−18.5837	−0.859953	−0.429977	+	0.902840i	$0.641478\pi$
$468$	0	0
$469$	−29.2720	−1.35165
$470$	−6.05420	−0.279259
$471$	0	0
$472$	−13.1856	−0.606914
$473$	−15.0567	−0.692306
$474$	0	0
$475$	3.35521	0.153948
$476$	5.00643	0.229469
$477$	0	0
$478$	1.73396	0.0793093
$479$	−1.08236	−0.0494541	−0.0247271	−	0.999694i	$-0.507872\pi$
$479$	−1.08236	−0.0494541	−0.0247271	+	0.999694i	$0.507872\pi$
$480$	0	0
$481$	−7.99800	−0.364677
$482$	−3.53607	−0.161064
$483$	0	0
$484$	−1.15791	−0.0526321
$485$	−18.6188	−0.845434
$486$	0	0
$487$	17.4217	0.789451	0.394726	−	0.918799i	$-0.370840\pi$
$487$	17.4217	0.789451	0.394726	+	0.918799i	$0.370840\pi$
$488$	−26.7692	−1.21178
$489$	0	0
$490$	1.83814	0.0830388
$491$	−16.8097	−0.758612	−0.379306	−	0.925271i	$-0.623837\pi$
$491$	−16.8097	−0.758612	−0.379306	+	0.925271i	$0.623837\pi$
$492$	0	0
$493$	5.93302	0.267210
$494$	−0.513832	−0.0231184
$495$	0	0
$496$	6.57702	0.295317
$497$	−37.7611	−1.69382
$498$	0	0
$499$	−20.8840	−0.934898	−0.467449	−	0.884020i	$-0.654827\pi$
$499$	−20.8840	−0.934898	−0.467449	+	0.884020i	$0.654827\pi$
$500$	−19.1747	−0.857521
$501$	0	0
$502$	5.36066	0.239258
$503$	2.96223	0.132079	0.0660397	−	0.997817i	$-0.478964\pi$
$503$	2.96223	0.132079	0.0660397	+	0.997817i	$0.478964\pi$
$504$	0	0
$505$	−15.7073	−0.698967
$506$	9.10062	0.404572
$507$	0	0
$508$	−0.835911	−0.0370876
$509$	20.9766	0.929771	0.464885	−	0.885371i	$-0.346096\pi$
$509$	20.9766	0.929771	0.464885	+	0.885371i	$0.346096\pi$
$510$	0	0
$511$	−16.3590	−0.723679
$512$	−22.5601	−0.997027
$513$	0	0
$514$	−11.9274	−0.526096
$515$	−10.8882	−0.479793
$516$	0	0
$517$	31.4060	1.38124
$518$	−12.4163	−0.545539
$519$	0	0
$520$	2.47103	0.108362
$521$	34.1685	1.49695	0.748474	−	0.663164i	$-0.230787\pi$
$521$	34.1685	1.49695	0.748474	+	0.663164i	$0.230787\pi$
$522$	0	0
$523$	19.3302	0.845250	0.422625	−	0.906305i	$-0.361109\pi$
$523$	19.3302	0.845250	0.422625	+	0.906305i	$0.361109\pi$
$524$	−15.4535	−0.675091
$525$	0	0
$526$	0.582454	0.0253962
$527$	−2.32184	−0.101141
$528$	0	0
$529$	5.86078	0.254817
$530$	−8.02270	−0.348484
$531$	0	0
$532$	5.68467	0.246462
$533$	5.45361	0.236222
$534$	0	0
$535$	−0.207804	−0.00898413
$536$	−17.4202	−0.752437
$537$	0	0
$538$	8.17311	0.352368
$539$	−9.53533	−0.410716
$540$	0	0
$541$	8.72119	0.374953	0.187477	−	0.982269i	$-0.439969\pi$
$541$	8.72119	0.374953	0.187477	+	0.982269i	$0.439969\pi$
$542$	−15.3538	−0.659501
$543$	0	0
$544$	4.56676	0.195798
$545$	9.98831	0.427852
$546$	0	0
$547$	−21.7198	−0.928670	−0.464335	−	0.885660i	$-0.653707\pi$
$547$	−21.7198	−0.928670	−0.464335	+	0.885660i	$0.653707\pi$
$548$	17.8848	0.764001
$549$	0	0
$550$	−5.48756	−0.233990
$551$	6.73679	0.286997
$552$	0	0
$553$	−37.8918	−1.61133
$554$	8.35519	0.354978
$555$	0	0
$556$	18.8116	0.797791
$557$	18.3186	0.776183	0.388091	−	0.921621i	$-0.373135\pi$
$557$	18.3186	0.776183	0.388091	+	0.921621i	$0.373135\pi$
$558$	0	0
$559$	−4.40936	−0.186496
$560$	−10.7289	−0.453379
$561$	0	0
$562$	−10.7199	−0.452191
$563$	−44.9062	−1.89257	−0.946286	−	0.323330i	$-0.895198\pi$
$563$	−44.9062	−1.89257	−0.946286	+	0.323330i	$0.895198\pi$
$564$	0	0
$565$	−22.6544	−0.953079
$566$	−1.94296	−0.0816688
$567$	0	0
$568$	−22.4722	−0.942911
$569$	34.2914	1.43757	0.718786	−	0.695231i	$-0.244698\pi$
$569$	34.2914	1.43757	0.718786	+	0.695231i	$0.244698\pi$
$570$	0	0
$571$	15.8328	0.662583	0.331292	−	0.943528i	$-0.392516\pi$
$571$	15.8328	0.662583	0.331292	+	0.943528i	$0.392516\pi$
$572$	−5.98902	−0.250413
$573$	0	0
$574$	8.46630	0.353377
$575$	−17.4027	−0.725743
$576$	0	0
$577$	17.1698	0.714789	0.357395	−	0.933953i	$-0.383665\pi$
$577$	17.1698	0.714789	0.357395	+	0.933953i	$0.383665\pi$
$578$	8.02081	0.333622
$579$	0	0
$580$	−15.1367	−0.628516
$581$	34.4334	1.42854
$582$	0	0
$583$	41.6176	1.72363
$584$	−9.73547	−0.402856
$585$	0	0
$586$	6.60121	0.272694
$587$	−14.6558	−0.604910	−0.302455	−	0.953164i	$-0.597806\pi$
$587$	−14.6558	−0.604910	−0.302455	+	0.953164i	$0.597806\pi$
$588$	0	0
$589$	−2.63638	−0.108630
$590$	−4.66068	−0.191877
$591$	0	0
$592$	20.6661	0.849373
$593$	20.3341	0.835021	0.417510	−	0.908672i	$-0.362903\pi$
$593$	20.3341	0.835021	0.417510	+	0.908672i	$0.362903\pi$
$594$	0	0
$595$	3.78754	0.155274
$596$	26.5471	1.08741
$597$	0	0
$598$	2.66513	0.108985
$599$	−45.9476	−1.87737	−0.938684	−	0.344780i	$-0.887954\pi$
$599$	−45.9476	−1.87737	−0.938684	+	0.344780i	$0.887954\pi$
$600$	0	0
$601$	24.1036	0.983205	0.491602	−	0.870820i	$-0.336411\pi$
$601$	24.1036	0.983205	0.491602	+	0.870820i	$0.336411\pi$
$602$	−6.84518	−0.278989
$603$	0	0
$604$	−21.3796	−0.869923
$605$	−0.875998	−0.0356144
$606$	0	0
$607$	−36.7210	−1.49046	−0.745229	−	0.666809i	$-0.767660\pi$
$607$	−36.7210	−1.49046	−0.745229	+	0.666809i	$0.767660\pi$
$608$	5.18543	0.210297
$609$	0	0
$610$	−9.46206	−0.383108
$611$	9.19730	0.372083
$612$	0	0
$613$	12.6282	0.510047	0.255024	−	0.966935i	$-0.417917\pi$
$613$	12.6282	0.510047	0.255024	+	0.966935i	$0.417917\pi$
$614$	−1.24207	−0.0501259
$615$	0	0
$616$	−19.8996	−0.801777
$617$	10.3329	0.415988	0.207994	−	0.978130i	$-0.433307\pi$
$617$	10.3329	0.415988	0.207994	+	0.978130i	$0.433307\pi$
$618$	0	0
$619$	−36.6505	−1.47311	−0.736554	−	0.676378i	$-0.763548\pi$
$619$	−36.6505	−1.47311	−0.736554	+	0.676378i	$0.763548\pi$
$620$	5.92360	0.237898
$621$	0	0
$622$	2.04042	0.0818133
$623$	−26.0737	−1.04462
$624$	0	0
$625$	1.69054	0.0676216
$626$	1.72611	0.0689894
$627$	0	0
$628$	6.74252	0.269056
$629$	−7.29561	−0.290895
$630$	0	0
$631$	5.45904	0.217321	0.108660	−	0.994079i	$-0.465344\pi$
$631$	5.45904	0.217321	0.108660	+	0.994079i	$0.465344\pi$
$632$	−22.5500	−0.896990
$633$	0	0
$634$	−12.4325	−0.493758
$635$	−0.632397	−0.0250959
$636$	0	0
$637$	−2.79243	−0.110640
$638$	−11.0183	−0.436217
$639$	0	0
$640$	−15.0528	−0.595012
$641$	6.62539	0.261687	0.130844	−	0.991403i	$-0.458231\pi$
$641$	6.62539	0.261687	0.130844	+	0.991403i	$0.458231\pi$
$642$	0	0
$643$	31.4582	1.24059	0.620294	−	0.784369i	$-0.287013\pi$
$643$	31.4582	1.24059	0.620294	+	0.784369i	$0.287013\pi$
$644$	−29.4851	−1.16187
$645$	0	0
$646$	−0.468707	−0.0184410
$647$	20.2079	0.794456	0.397228	−	0.917720i	$-0.369972\pi$
$647$	20.2079	0.794456	0.397228	+	0.917720i	$0.369972\pi$
$648$	0	0
$649$	24.1772	0.949038
$650$	−1.60704	−0.0630333
$651$	0	0
$652$	−9.19420	−0.360073
$653$	−2.59372	−0.101500	−0.0507500	−	0.998711i	$-0.516161\pi$
$653$	−2.59372	−0.101500	−0.0507500	+	0.998711i	$0.516161\pi$
$654$	0	0
$655$	−11.6912	−0.456811
$656$	−14.0917	−0.550187
$657$	0	0
$658$	14.2781	0.556617
$659$	−15.9077	−0.619678	−0.309839	−	0.950789i	$-0.600275\pi$
$659$	−15.9077	−0.619678	−0.309839	+	0.950789i	$0.600275\pi$
$660$	0	0
$661$	37.1737	1.44589	0.722945	−	0.690905i	$-0.242788\pi$
$661$	37.1737	1.44589	0.722945	+	0.690905i	$0.242788\pi$
$662$	−0.134038	−0.00520953
$663$	0	0
$664$	20.4918	0.795235
$665$	4.30066	0.166772
$666$	0	0
$667$	−34.9422	−1.35297
$668$	−31.1907	−1.20681
$669$	0	0
$670$	−6.15748	−0.237884
$671$	49.0843	1.89488
$672$	0	0
$673$	−40.3983	−1.55724	−0.778621	−	0.627495i	$-0.784080\pi$
$673$	−40.3983	−1.55724	−0.778621	+	0.627495i	$0.784080\pi$
$674$	12.2348	0.471265
$675$	0	0
$676$	−1.75389	−0.0674573
$677$	−0.802538	−0.0308440	−0.0154220	−	0.999881i	$-0.504909\pi$
$677$	−0.802538	−0.0308440	−0.0154220	+	0.999881i	$0.504909\pi$
$678$	0	0
$679$	43.9100	1.68511
$680$	2.25402	0.0864378
$681$	0	0
$682$	4.31189	0.165111
$683$	−50.3853	−1.92794	−0.963971	−	0.266008i	$-0.914295\pi$
$683$	−50.3853	−1.92794	−0.963971	+	0.266008i	$0.914295\pi$
$684$	0	0
$685$	13.5305	0.516974
$686$	6.53191	0.249389
$687$	0	0
$688$	11.3934	0.434370
$689$	12.1878	0.464317
$690$	0	0
$691$	−17.4429	−0.663558	−0.331779	−	0.943357i	$-0.607649\pi$
$691$	−17.4429	−0.663558	−0.331779	+	0.943357i	$0.607649\pi$
$692$	−2.77257	−0.105397
$693$	0	0
$694$	12.9704	0.492350
$695$	14.2317	0.539839
$696$	0	0
$697$	4.97467	0.188429
$698$	9.89344	0.374472
$699$	0	0
$700$	17.7791	0.671988
$701$	−29.9544	−1.13136	−0.565680	−	0.824625i	$-0.691386\pi$
$701$	−29.9544	−1.13136	−0.565680	+	0.824625i	$0.691386\pi$
$702$	0	0
$703$	−8.28397	−0.312436
$704$	9.16564	0.345443
$705$	0	0
$706$	−8.50972	−0.320268
$707$	37.0438	1.39318
$708$	0	0
$709$	3.43733	0.129092	0.0645458	−	0.997915i	$-0.479440\pi$
$709$	3.43733	0.129092	0.0645458	+	0.997915i	$0.479440\pi$
$710$	−7.94320	−0.298103
$711$	0	0
$712$	−15.5168	−0.581518
$713$	13.6743	0.512107
$714$	0	0
$715$	−4.53091	−0.169446
$716$	−26.7065	−0.998070
$717$	0	0
$718$	−1.56729	−0.0584906
$719$	−34.0368	−1.26936	−0.634680	−	0.772775i	$-0.718868\pi$
$719$	−34.0368	−1.26936	−0.634680	+	0.772775i	$0.718868\pi$
$720$	0	0
$721$	25.6785	0.956319
$722$	8.89358	0.330985
$723$	0	0
$724$	40.2914	1.49742
$725$	21.0697	0.782509
$726$	0	0
$727$	−37.1392	−1.37742	−0.688708	−	0.725039i	$-0.741822\pi$
$727$	−37.1392	−1.37742	−0.688708	+	0.725039i	$0.741822\pi$
$728$	−5.82762	−0.215986
$729$	0	0
$730$	−3.44118	−0.127364
$731$	−4.02213	−0.148764
$732$	0	0
$733$	27.3998	1.01204	0.506018	−	0.862523i	$-0.331117\pi$
$733$	27.3998	1.01204	0.506018	+	0.862523i	$0.331117\pi$
$734$	−2.00393	−0.0739666
$735$	0	0
$736$	−26.8957	−0.991387
$737$	31.9418	1.17659
$738$	0	0
$739$	−22.9544	−0.844390	−0.422195	−	0.906505i	$-0.638740\pi$
$739$	−22.9544	−0.844390	−0.422195	+	0.906505i	$0.638740\pi$
$740$	18.6130	0.684227
$741$	0	0
$742$	18.9206	0.694595
$743$	−17.7792	−0.652256	−0.326128	−	0.945326i	$-0.605744\pi$
$743$	−17.7792	−0.652256	−0.326128	+	0.945326i	$0.605744\pi$
$744$	0	0
$745$	20.0838	0.735814
$746$	−5.87646	−0.215153
$747$	0	0
$748$	−5.46306	−0.199749
$749$	0.490079	0.0179071
$750$	0	0
$751$	27.9757	1.02085	0.510424	−	0.859923i	$-0.329489\pi$
$751$	27.9757	1.02085	0.510424	+	0.859923i	$0.329489\pi$
$752$	−23.7650	−0.866622
$753$	0	0
$754$	−3.22671	−0.117510
$755$	−16.1744	−0.588648
$756$	0	0
$757$	25.6113	0.930860	0.465430	−	0.885085i	$-0.345900\pi$
$757$	25.6113	0.930860	0.465430	+	0.885085i	$0.345900\pi$
$758$	−0.350234	−0.0127211
$759$	0	0
$760$	2.55938	0.0928385
$761$	4.89702	0.177517	0.0887583	−	0.996053i	$-0.471710\pi$
$761$	4.89702	0.177517	0.0887583	+	0.996053i	$0.471710\pi$
$762$	0	0
$763$	−23.5562	−0.852791
$764$	12.4056	0.448819
$765$	0	0
$766$	−12.3073	−0.444681
$767$	7.08032	0.255656
$768$	0	0
$769$	−20.0246	−0.722106	−0.361053	−	0.932545i	$-0.617583\pi$
$769$	−20.0246	−0.722106	−0.361053	+	0.932545i	$0.617583\pi$
$770$	−7.03387	−0.253483
$771$	0	0
$772$	−14.0533	−0.505789
$773$	−0.827218	−0.0297530	−0.0148765	−	0.999889i	$-0.504736\pi$
$773$	−0.827218	−0.0297530	−0.0148765	+	0.999889i	$0.504736\pi$
$774$	0	0
$775$	−8.24544	−0.296185
$776$	26.1315	0.938065
$777$	0	0
$778$	−5.14690	−0.184525
$779$	5.64861	0.202382
$780$	0	0
$781$	41.2052	1.47444
$782$	2.43107	0.0869350
$783$	0	0
$784$	7.21541	0.257693
$785$	5.10096	0.182061
$786$	0	0
$787$	−4.20205	−0.149787	−0.0748934	−	0.997192i	$-0.523862\pi$
$787$	−4.20205	−0.149787	−0.0748934	+	0.997192i	$0.523862\pi$
$788$	−18.0627	−0.643457
$789$	0	0
$790$	−7.97070	−0.283585
$791$	53.4277	1.89967
$792$	0	0
$793$	14.3744	0.510450
$794$	−9.72003	−0.344951
$795$	0	0
$796$	9.76821	0.346225
$797$	−13.4875	−0.477753	−0.238876	−	0.971050i	$-0.576779\pi$
$797$	−13.4875	−0.477753	−0.238876	+	0.971050i	$0.576779\pi$
$798$	0	0
$799$	8.38959	0.296802
$800$	16.2178	0.573384
$801$	0	0
$802$	−11.0672	−0.390797
$803$	17.8511	0.629950
$804$	0	0
$805$	−22.3065	−0.786201
$806$	1.26274	0.0444782
$807$	0	0
$808$	22.0453	0.775550
$809$	3.02951	0.106512	0.0532559	−	0.998581i	$-0.483040\pi$
$809$	3.02951	0.106512	0.0532559	+	0.998581i	$0.483040\pi$
$810$	0	0
$811$	−31.9363	−1.12143	−0.560717	−	0.828008i	$-0.689474\pi$
$811$	−31.9363	−1.12143	−0.560717	+	0.828008i	$0.689474\pi$
$812$	35.6980	1.25275
$813$	0	0
$814$	13.5487	0.474882
$815$	−6.95574	−0.243649
$816$	0	0
$817$	−4.56702	−0.159780
$818$	1.73613	0.0607023
$819$	0	0
$820$	−12.6917	−0.443213
$821$	23.8708	0.833097	0.416548	−	0.909114i	$-0.363240\pi$
$821$	23.8708	0.833097	0.416548	+	0.909114i	$0.363240\pi$
$822$	0	0
$823$	−18.8072	−0.655576	−0.327788	−	0.944751i	$-0.606303\pi$
$823$	−18.8072	−0.655576	−0.327788	+	0.944751i	$0.606303\pi$
$824$	15.2817	0.532362
$825$	0	0
$826$	10.9916	0.382448
$827$	−53.8381	−1.87213	−0.936067	−	0.351822i	$-0.885562\pi$
$827$	−53.8381	−1.87213	−0.936067	+	0.351822i	$0.885562\pi$
$828$	0	0
$829$	31.1172	1.08074	0.540372	−	0.841426i	$-0.318283\pi$
$829$	31.1172	1.08074	0.540372	+	0.841426i	$0.318283\pi$
$830$	7.24319	0.251415
$831$	0	0
$832$	2.68417	0.0930569
$833$	−2.54720	−0.0882553
$834$	0	0
$835$	−23.5969	−0.816605
$836$	−6.20315	−0.214541
$837$	0	0
$838$	10.5418	0.364159
$839$	52.0832	1.79811	0.899056	−	0.437835i	$-0.144254\pi$
$839$	52.0832	1.79811	0.899056	+	0.437835i	$0.144254\pi$
$840$	0	0
$841$	13.3050	0.458793
$842$	−1.10526	−0.0380897
$843$	0	0
$844$	16.5418	0.569392
$845$	−1.32688	−0.0456461
$846$	0	0
$847$	2.06593	0.0709863
$848$	−31.4922	−1.08145
$849$	0	0
$850$	−1.46591	−0.0502802
$851$	42.9670	1.47289
$852$	0	0
$853$	−51.9182	−1.77765	−0.888823	−	0.458250i	$-0.848476\pi$
$853$	−51.9182	−1.77765	−0.888823	+	0.458250i	$0.848476\pi$
$854$	22.3151	0.763607
$855$	0	0
$856$	0.291653	0.00996849
$857$	−28.9273	−0.988138	−0.494069	−	0.869423i	$-0.664491\pi$
$857$	−28.9273	−0.988138	−0.494069	+	0.869423i	$0.664491\pi$
$858$	0	0
$859$	−6.27173	−0.213989	−0.106994	−	0.994260i	$-0.534123\pi$
$859$	−6.27173	−0.213989	−0.106994	+	0.994260i	$0.534123\pi$
$860$	10.2615	0.349914
$861$	0	0
$862$	−5.11193	−0.174113
$863$	5.95115	0.202580	0.101290	−	0.994857i	$-0.467703\pi$
$863$	5.95115	0.202580	0.101290	+	0.994857i	$0.467703\pi$
$864$	0	0
$865$	−2.09755	−0.0713189
$866$	4.98661	0.169452
$867$	0	0
$868$	−13.9701	−0.474176
$869$	41.3479	1.40263
$870$	0	0
$871$	9.35421	0.316955
$872$	−14.0186	−0.474730
$873$	0	0
$874$	2.76042	0.0933726
$875$	34.2115	1.15656
$876$	0	0
$877$	−54.5515	−1.84207	−0.921036	−	0.389476i	$-0.872656\pi$
$877$	−54.5515	−1.84207	−0.921036	+	0.389476i	$0.872656\pi$
$878$	−7.80795	−0.263506
$879$	0	0
$880$	11.7075	0.394659
$881$	34.6855	1.16858	0.584292	−	0.811543i	$-0.301372\pi$
$881$	34.6855	1.16858	0.584292	+	0.811543i	$0.301372\pi$
$882$	0	0
$883$	29.1891	0.982290	0.491145	−	0.871078i	$-0.336578\pi$
$883$	29.1891	0.982290	0.491145	+	0.871078i	$0.336578\pi$
$884$	−1.59986	−0.0538092
$885$	0	0
$886$	4.74228	0.159320
$887$	2.41418	0.0810601	0.0405300	−	0.999178i	$-0.487095\pi$
$887$	2.41418	0.0810601	0.0405300	+	0.999178i	$0.487095\pi$
$888$	0	0
$889$	1.49143	0.0500210
$890$	−5.48471	−0.183848
$891$	0	0
$892$	32.6670	1.09377
$893$	9.52615	0.318780
$894$	0	0
$895$	−20.2045	−0.675360
$896$	35.5001	1.18597
$897$	0	0
$898$	7.79674	0.260181
$899$	−16.5557	−0.552163
$900$	0	0
$901$	11.1174	0.370376
$902$	−9.23849	−0.307608
$903$	0	0
$904$	31.7956	1.05750
$905$	30.4819	1.01325
$906$	0	0
$907$	46.4429	1.54211	0.771056	−	0.636768i	$-0.219729\pi$
$907$	46.4429	1.54211	0.771056	+	0.636768i	$0.219729\pi$
$908$	24.7315	0.820742
$909$	0	0
$910$	−2.05988	−0.0682843
$911$	0.144960	0.00480273	0.00240137	−	0.999997i	$-0.499236\pi$
$911$	0.144960	0.00480273	0.00240137	+	0.999997i	$0.499236\pi$
$912$	0	0
$913$	−37.5740	−1.24352
$914$	−16.3235	−0.539934
$915$	0	0
$916$	−3.59669	−0.118838
$917$	27.5722	0.910513
$918$	0	0
$919$	−37.8928	−1.24997	−0.624983	−	0.780638i	$-0.714894\pi$
$919$	−37.8928	−1.24997	−0.624983	+	0.780638i	$0.714894\pi$
$920$	−13.2749	−0.437661
$921$	0	0
$922$	−6.33870	−0.208754
$923$	12.0670	0.397190
$924$	0	0
$925$	−25.9086	−0.851869
$926$	2.61673	0.0859910
$927$	0	0
$928$	32.5630	1.06893
$929$	−51.1602	−1.67851	−0.839257	−	0.543736i	$-0.817009\pi$
$929$	−51.1602	−1.67851	−0.839257	+	0.543736i	$0.817009\pi$
$930$	0	0
$931$	−2.89228	−0.0947906
$932$	−14.0569	−0.460451
$933$	0	0
$934$	9.21928	0.301664
$935$	−4.13300	−0.135164
$936$	0	0
$937$	−33.4925	−1.09415	−0.547076	−	0.837083i	$-0.684259\pi$
$937$	−33.4925	−1.09415	−0.547076	+	0.837083i	$0.684259\pi$
$938$	14.5217	0.474149
$939$	0	0
$940$	−21.4040	−0.698122
$941$	−16.8675	−0.549866	−0.274933	−	0.961463i	$-0.588656\pi$
$941$	−16.8675	−0.549866	−0.274933	+	0.961463i	$0.588656\pi$
$942$	0	0
$943$	−29.2980	−0.954075
$944$	−18.2949	−0.595450
$945$	0	0
$946$	7.46952	0.242855
$947$	16.0139	0.520381	0.260191	−	0.965557i	$-0.416215\pi$
$947$	16.0139	0.520381	0.260191	+	0.965557i	$0.416215\pi$
$948$	0	0
$949$	5.22771	0.169699
$950$	−1.66450	−0.0540035
$951$	0	0
$952$	−5.31583	−0.172287
$953$	35.3365	1.14466	0.572331	−	0.820023i	$-0.306039\pi$
$953$	35.3365	1.14466	0.572331	+	0.820023i	$0.306039\pi$
$954$	0	0
$955$	9.38528	0.303701
$956$	6.13023	0.198266
$957$	0	0
$958$	0.536950	0.0173481
$959$	−31.9100	−1.03043
$960$	0	0
$961$	−24.5211	−0.791003
$962$	3.96776	0.127926
$963$	0	0
$964$	−12.5014	−0.402644
$965$	−10.6318	−0.342250
$966$	0	0
$967$	53.3379	1.71523	0.857616	−	0.514290i	$-0.171945\pi$
$967$	53.3379	1.71523	0.857616	+	0.514290i	$0.171945\pi$
$968$	1.22947	0.0395165
$969$	0	0
$970$	9.23665	0.296571
$971$	50.1657	1.60989	0.804946	−	0.593348i	$-0.202194\pi$
$971$	50.1657	1.60989	0.804946	+	0.593348i	$0.202194\pi$
$972$	0	0
$973$	−33.5637	−1.07600
$974$	−8.64279	−0.276933
$975$	0	0
$976$	−37.1422	−1.18889
$977$	−22.9432	−0.734016	−0.367008	−	0.930218i	$-0.619618\pi$
$977$	−22.9432	−0.734016	−0.367008	+	0.930218i	$0.619618\pi$
$978$	0	0
$979$	28.4519	0.909325
$980$	6.49857	0.207589
$981$	0	0
$982$	8.33919	0.266114
$983$	−0.258313	−0.00823892	−0.00411946	−	0.999992i	$-0.501311\pi$
$983$	−0.258313	−0.00823892	−0.00411946	+	0.999992i	$0.501311\pi$
$984$	0	0
$985$	−13.6651	−0.435406
$986$	−2.94334	−0.0937349
$987$	0	0
$988$	−1.81660	−0.0577938
$989$	23.6881	0.753237
$990$	0	0
$991$	−23.1558	−0.735568	−0.367784	−	0.929911i	$-0.619883\pi$
$991$	−23.1558	−0.735568	−0.367784	+	0.929911i	$0.619883\pi$
$992$	−12.7432	−0.404598
$993$	0	0
$994$	18.7331	0.594176
$995$	7.39000	0.234279
$996$	0	0
$997$	41.7866	1.32339	0.661697	−	0.749772i	$-0.269837\pi$
$997$	41.7866	1.32339	0.661697	+	0.749772i	$0.269837\pi$
$998$	10.3605	0.327954
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.a.l.1.3		6
3.2	odd	2		1053.2.a.m.1.4		6
9.2	odd	6		351.2.e.c.118.3		12
9.4	even	3		117.2.e.c.79.4	yes	12
9.5	odd	6		351.2.e.c.235.3		12
9.7	even	3		117.2.e.c.40.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.e.c.40.4	✓	12	9.7	even	3
117.2.e.c.79.4	yes	12	9.4	even	3
351.2.e.c.118.3		12	9.2	odd	6
351.2.e.c.235.3		12	9.5	odd	6
1053.2.a.l.1.3		6	1.1	even	1	trivial
1053.2.a.m.1.4		6	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:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.a (trivial)

\(f(q)\)	\(=\)	\(q-0.496094 q^{2} -1.75389 q^{4} -1.32688 q^{5} +3.12929 q^{7} +1.86228 q^{8} +0.658258 q^{10} -3.41470 q^{11} -1.00000 q^{13} -1.55242 q^{14} +2.58391 q^{16} -0.912179 q^{17} -1.03576 q^{19} +2.32721 q^{20} +1.69401 q^{22} +5.37222 q^{23} -3.23938 q^{25} +0.496094 q^{26} -5.48843 q^{28} -6.50423 q^{29} +2.54537 q^{31} -5.00643 q^{32} +0.452527 q^{34} -4.15219 q^{35} +7.99800 q^{37} +0.513832 q^{38} -2.47103 q^{40} -5.45361 q^{41} +4.40936 q^{43} +5.98902 q^{44} -2.66513 q^{46} -9.19730 q^{47} +2.79243 q^{49} +1.60704 q^{50} +1.75389 q^{52} -12.1878 q^{53} +4.53091 q^{55} +5.82762 q^{56} +3.22671 q^{58} -7.08032 q^{59} -14.3744 q^{61} -1.26274 q^{62} -2.68417 q^{64} +1.32688 q^{65} -9.35421 q^{67} +1.59986 q^{68} +2.05988 q^{70} -12.0670 q^{71} -5.22771 q^{73} -3.96776 q^{74} +1.81660 q^{76} -10.6856 q^{77} -12.1088 q^{79} -3.42855 q^{80} +2.70551 q^{82} +11.0036 q^{83} +1.21035 q^{85} -2.18746 q^{86} -6.35914 q^{88} -8.33216 q^{89} -3.12929 q^{91} -9.42229 q^{92} +4.56272 q^{94} +1.37432 q^{95} +14.0320 q^{97} -1.38531 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 6 q^{4} - 3 q^{5} - 6 q^{8} - 6 q^{10} - 7 q^{11} - 6 q^{13} - 13 q^{14} + 6 q^{16} - 14 q^{17} - 3 q^{19} - 17 q^{20} - 3 q^{22} - 17 q^{23} + 3 q^{25} + 2 q^{26} + 15 q^{28} - 14 q^{29}+ \cdots - 17 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 6 * q^4 - 3 * q^5 - 6 * q^8 - 6 * q^10 - 7 * q^11 - 6 * q^13 - 13 * q^14 + 6 * q^16 - 14 * q^17 - 3 * q^19 - 17 * q^20 - 3 * q^22 - 17 * q^23 + 3 * q^25 + 2 * q^26 + 15 * q^28 - 14 * q^29 + 6 * q^31 - 9 * q^32 + 3 * q^34 - 13 * q^35 - 9 * q^37 + 4 * q^38 - 6 * q^40 + 4 * q^41 - 3 * q^43 - 13 * q^44 - 9 * q^47 + 21 * q^50 - 6 * q^52 - 28 * q^53 + 15 * q^55 - 16 * q^56 + 18 * q^58 - 17 * q^59 - 6 * q^61 + 4 * q^62 - 30 * q^64 + 3 * q^65 - 12 * q^67 - 22 * q^68 + 9 * q^70 + q^71 - 9 * q^73 + 27 * q^74 - 3 * q^76 - 15 * q^77 - 6 * q^79 + 24 * q^80 + 27 * q^82 - 28 * q^83 + 27 * q^85 + 6 * q^86 - 9 * q^88 + 9 * q^89 - 44 * q^92 - 24 * q^94 - 32 * q^95 - 17 * q^98

Introduction

L-functions

Modular forms

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Database

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