LMFDB - Embedded newform 5445.2.a.bv.1.1

Newspace parameters

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.09529$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

$f(q)$	$=$	$q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} +O(q^{10})$	$q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} -1.39026 q^{10} +1.41837 q^{13} +1.77618 q^{14} -3.86111 q^{16} +5.00000 q^{17} -0.158146 q^{19} -0.0671858 q^{20} +5.00829 q^{23} +1.00000 q^{25} -1.97189 q^{26} +0.0858360 q^{28} +6.27515 q^{29} +3.04981 q^{31} -0.379898 q^{32} -6.95128 q^{34} -1.27759 q^{35} -4.69991 q^{37} +0.219863 q^{38} +2.87392 q^{40} +7.58597 q^{41} -5.41324 q^{43} -6.96281 q^{46} +8.23211 q^{47} -5.36776 q^{49} -1.39026 q^{50} -0.0952940 q^{52} -9.36648 q^{53} -3.67169 q^{56} -8.72406 q^{58} -8.09846 q^{59} +14.2917 q^{61} -4.24002 q^{62} +8.25038 q^{64} +1.41837 q^{65} -7.38362 q^{67} -0.335929 q^{68} +1.77618 q^{70} +6.77618 q^{71} -8.66107 q^{73} +6.53409 q^{74} +0.0106252 q^{76} +2.54572 q^{79} -3.86111 q^{80} -10.5464 q^{82} +10.1221 q^{83} +5.00000 q^{85} +7.52580 q^{86} -11.0447 q^{89} -1.81209 q^{91} -0.336486 q^{92} -11.4447 q^{94} -0.158146 q^{95} +6.41196 q^{97} +7.46257 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8}+O(q^{10}) $ 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8	$ 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + 5 q^{10} - 3 q^{13} + 5 q^{14} + 15 q^{16} + 20 q^{17} - 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} - 3 q^{28} + 5 q^{29} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} - 7 q^{37} - q^{38} + 15 q^{40} + 20 q^{41} + 2 q^{43} - 7 q^{46} + 20 q^{47} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} + 7 q^{61} - 12 q^{62} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} + 5 q^{70} + 25 q^{71} - 23 q^{73} - 7 q^{74} + 7 q^{76} + 15 q^{80} + 11 q^{82} + 33 q^{83} + 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} + 17 q^{94} - 3 q^{95} + 25 q^{98}+O(q^{100}) $ 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8 + 5 * q^10 - 3 * q^13 + 5 * q^14 + 15 * q^16 + 20 * q^17 - 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 - 3 * q^28 + 5 * q^29 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 - 7 * q^37 - q^38 + 15 * q^40 + 20 * q^41 + 2 * q^43 - 7 * q^46 + 20 * q^47 + 8 * q^49 + 5 * q^50 + 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 + 7 * q^61 - 12 * q^62 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 + 5 * q^70 + 25 * q^71 - 23 * q^73 - 7 * q^74 + 7 * q^76 + 15 * q^80 + 11 * q^82 + 33 * q^83 + 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 + 17 * q^94 - 3 * q^95 + 25 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.39026	−0.983060	−0.491530	−	0.870861i	$-0.663562\pi$
$2$	−1.39026	−0.983060	−0.491530	+	0.870861i	$0.663562\pi$
$3$	0	0
$3$	0	0
$4$	−0.0671858	−0.0335929
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.27759	−0.482884	−0.241442	−	0.970415i	$-0.577620\pi$
$7$	−1.27759	−0.482884	−0.241442	+	0.970415i	$0.577620\pi$
$8$	2.87392	1.01608
$9$	0	0
$10$	−1.39026	−0.439638
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.41837	0.393384	0.196692	−	0.980465i	$-0.436980\pi$
$13$	1.41837	0.393384	0.196692	+	0.980465i	$0.436980\pi$
$14$	1.77618	0.474704
$15$	0	0
$16$	−3.86111	−0.965279
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	−0.158146	−0.0362811	−0.0181406	−	0.999835i	$-0.505775\pi$
$19$	−0.158146	−0.0362811	−0.0181406	+	0.999835i	$0.505775\pi$
$20$	−0.0671858	−0.0150232
$21$	0	0
$22$	0	0
$23$	5.00829	1.04430	0.522150	−	0.852853i	$-0.325130\pi$
$23$	5.00829	1.04430	0.522150	+	0.852853i	$0.325130\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.97189	−0.386720
$27$	0	0
$28$	0.0858360	0.0162215
$29$	6.27515	1.16527	0.582633	−	0.812736i	$-0.302023\pi$
$29$	6.27515	1.16527	0.582633	+	0.812736i	$0.302023\pi$
$30$	0	0
$31$	3.04981	0.547763	0.273881	−	0.961763i	$-0.411692\pi$
$31$	3.04981	0.547763	0.273881	+	0.961763i	$0.411692\pi$
$32$	−0.379898	−0.0671570
$33$	0	0
$34$	−6.95128	−1.19214
$35$	−1.27759	−0.215952
$36$	0	0
$37$	−4.69991	−0.772661	−0.386330	−	0.922360i	$-0.626258\pi$
$37$	−4.69991	−0.772661	−0.386330	+	0.922360i	$0.626258\pi$
$38$	0.219863	0.0356665
$39$	0	0
$40$	2.87392	0.454407
$41$	7.58597	1.18473	0.592365	−	0.805670i	$-0.298194\pi$
$41$	7.58597	1.18473	0.592365	+	0.805670i	$0.298194\pi$
$42$	0	0
$43$	−5.41324	−0.825512	−0.412756	−	0.910842i	$-0.635434\pi$
$43$	−5.41324	−0.825512	−0.412756	+	0.910842i	$0.635434\pi$
$44$	0	0
$45$	0	0
$46$	−6.96281	−1.02661
$47$	8.23211	1.20078	0.600388	−	0.799709i	$-0.295013\pi$
$47$	8.23211	1.20078	0.600388	+	0.799709i	$0.295013\pi$
$48$	0	0
$49$	−5.36776	−0.766823
$50$	−1.39026	−0.196612
$51$	0	0
$52$	−0.0952940	−0.0132149
$53$	−9.36648	−1.28659	−0.643293	−	0.765620i	$-0.722432\pi$
$53$	−9.36648	−1.28659	−0.643293	+	0.765620i	$0.722432\pi$
$54$	0	0
$55$	0	0
$56$	−3.67169	−0.490651
$57$	0	0
$58$	−8.72406	−1.14553
$59$	−8.09846	−1.05433	−0.527165	−	0.849763i	$-0.676745\pi$
$59$	−8.09846	−1.05433	−0.527165	+	0.849763i	$0.676745\pi$
$60$	0	0
$61$	14.2917	1.82987	0.914934	−	0.403603i	$-0.132242\pi$
$61$	14.2917	1.82987	0.914934	+	0.403603i	$0.132242\pi$
$62$	−4.24002	−0.538484
$63$	0	0
$64$	8.25038	1.03130
$65$	1.41837	0.175927
$66$	0	0
$67$	−7.38362	−0.902053	−0.451026	−	0.892511i	$-0.648942\pi$
$67$	−7.38362	−0.902053	−0.451026	+	0.892511i	$0.648942\pi$
$68$	−0.335929	−0.0407374
$69$	0	0
$70$	1.77618	0.212294
$71$	6.77618	0.804185	0.402092	−	0.915599i	$-0.368283\pi$
$71$	6.77618	0.804185	0.402092	+	0.915599i	$0.368283\pi$
$72$	0	0
$73$	−8.66107	−1.01370	−0.506851	−	0.862034i	$-0.669190\pi$
$73$	−8.66107	−1.01370	−0.506851	+	0.862034i	$0.669190\pi$
$74$	6.53409	0.759572
$75$	0	0
$76$	0.0106252	0.00121879
$77$	0	0
$78$	0	0
$79$	2.54572	0.286416	0.143208	−	0.989693i	$-0.454258\pi$
$79$	2.54572	0.286416	0.143208	+	0.989693i	$0.454258\pi$
$80$	−3.86111	−0.431686
$81$	0	0
$82$	−10.5464	−1.16466
$83$	10.1221	1.11105	0.555524	−	0.831501i	$-0.312518\pi$
$83$	10.1221	1.11105	0.555524	+	0.831501i	$0.312518\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	7.52580	0.811527
$87$	0	0
$88$	0	0
$89$	−11.0447	−1.17073	−0.585367	−	0.810768i	$-0.699050\pi$
$89$	−11.0447	−1.17073	−0.585367	+	0.810768i	$0.699050\pi$
$90$	0	0
$91$	−1.81209	−0.189959
$92$	−0.336486	−0.0350811
$93$	0	0
$94$	−11.4447	−1.18044
$95$	−0.158146	−0.0162254
$96$	0	0
$97$	6.41196	0.651036	0.325518	−	0.945536i	$-0.394461\pi$
$97$	6.41196	0.651036	0.325518	+	0.945536i	$0.394461\pi$
$98$	7.46257	0.753833
$99$	0	0
$100$	−0.0671858	−0.00671858
$101$	8.77143	0.872790	0.436395	−	0.899755i	$-0.356255\pi$
$101$	8.77143	0.872790	0.436395	+	0.899755i	$0.356255\pi$
$102$	0	0
$103$	11.5666	1.13969	0.569847	−	0.821751i	$-0.307002\pi$
$103$	11.5666	1.13969	0.569847	+	0.821751i	$0.307002\pi$
$104$	4.07627	0.399711
$105$	0	0
$106$	13.0218	1.26479
$107$	15.3467	1.48362	0.741809	−	0.670611i	$-0.233968\pi$
$107$	15.3467	1.48362	0.741809	+	0.670611i	$0.233968\pi$
$108$	0	0
$109$	−14.4004	−1.37931	−0.689656	−	0.724137i	$-0.742238\pi$
$109$	−14.4004	−1.37931	−0.689656	+	0.724137i	$0.742238\pi$
$110$	0	0
$111$	0	0
$112$	4.93293	0.466118
$113$	−19.3645	−1.82166	−0.910831	−	0.412780i	$-0.864558\pi$
$113$	−19.3645	−1.82166	−0.910831	+	0.412780i	$0.864558\pi$
$114$	0	0
$115$	5.00829	0.467026
$116$	−0.421601	−0.0391446
$117$	0	0
$118$	11.2589	1.03647
$119$	−6.38796	−0.585583
$120$	0	0
$121$	0	0
$122$	−19.8692	−1.79887
$123$	0	0
$124$	−0.204904	−0.0184009
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.02415	0.534557	0.267278	−	0.963619i	$-0.413876\pi$
$127$	6.02415	0.534557	0.267278	+	0.963619i	$0.413876\pi$
$128$	−10.7104	−0.946671
$129$	0	0
$130$	−1.97189	−0.172946
$131$	−18.7278	−1.63626	−0.818130	−	0.575034i	$-0.804989\pi$
$131$	−18.7278	−1.63626	−0.818130	+	0.575034i	$0.804989\pi$
$132$	0	0
$133$	0.202046	0.0175196
$134$	10.2651	0.886772
$135$	0	0
$136$	14.3696	1.23218
$137$	−3.03719	−0.259485	−0.129742	−	0.991548i	$-0.541415\pi$
$137$	−3.03719	−0.259485	−0.129742	+	0.991548i	$0.541415\pi$
$138$	0	0
$139$	2.21192	0.187612	0.0938062	−	0.995590i	$-0.470097\pi$
$139$	2.21192	0.187612	0.0938062	+	0.995590i	$0.470097\pi$
$140$	0.0858360	0.00725446
$141$	0	0
$142$	−9.42063	−0.790562
$143$	0	0
$144$	0	0
$145$	6.27515	0.521122
$146$	12.0411	0.996529
$147$	0	0
$148$	0.315767	0.0259559
$149$	1.48122	0.121346	0.0606730	−	0.998158i	$-0.480675\pi$
$149$	1.48122	0.121346	0.0606730	+	0.998158i	$0.480675\pi$
$150$	0	0
$151$	9.03395	0.735173	0.367586	−	0.929989i	$-0.380184\pi$
$151$	9.03395	0.735173	0.367586	+	0.929989i	$0.380184\pi$
$152$	−0.454498	−0.0368647
$153$	0	0
$154$	0	0
$155$	3.04981	0.244967
$156$	0	0
$157$	−17.6377	−1.40764	−0.703820	−	0.710379i	$-0.748524\pi$
$157$	−17.6377	−1.40764	−0.703820	+	0.710379i	$0.748524\pi$
$158$	−3.53921	−0.281564
$159$	0	0
$160$	−0.379898	−0.0300335
$161$	−6.39855	−0.504276
$162$	0	0
$163$	23.5040	1.84098	0.920489	−	0.390770i	$-0.127791\pi$
$163$	23.5040	1.84098	0.920489	+	0.390770i	$0.127791\pi$
$164$	−0.509669	−0.0397985
$165$	0	0
$166$	−14.0724	−1.09223
$167$	0.602731	0.0466407	0.0233203	−	0.999728i	$-0.492576\pi$
$167$	0.602731	0.0466407	0.0233203	+	0.999728i	$0.492576\pi$
$168$	0	0
$169$	−10.9882	−0.845249
$170$	−6.95128	−0.533139
$171$	0	0
$172$	0.363693	0.0277313
$173$	9.66389	0.734732	0.367366	−	0.930076i	$-0.380260\pi$
$173$	9.66389	0.734732	0.367366	+	0.930076i	$0.380260\pi$
$174$	0	0
$175$	−1.27759	−0.0965768
$176$	0	0
$177$	0	0
$178$	15.3550	1.15090
$179$	1.19571	0.0893717	0.0446859	−	0.999001i	$-0.485771\pi$
$179$	1.19571	0.0893717	0.0446859	+	0.999001i	$0.485771\pi$
$180$	0	0
$181$	15.5741	1.15761	0.578806	−	0.815466i	$-0.303519\pi$
$181$	15.5741	1.15761	0.578806	+	0.815466i	$0.303519\pi$
$182$	2.51927	0.186741
$183$	0	0
$184$	14.3934	1.06110
$185$	−4.69991	−0.345544
$186$	0	0
$187$	0	0
$188$	−0.553081	−0.0403376
$189$	0	0
$190$	0.219863	0.0159506
$191$	−18.4016	−1.33149	−0.665747	−	0.746178i	$-0.731887\pi$
$191$	−18.4016	−1.33149	−0.665747	+	0.746178i	$0.731887\pi$
$192$	0	0
$193$	−1.56799	−0.112866	−0.0564331	−	0.998406i	$-0.517973\pi$
$193$	−1.56799	−0.112866	−0.0564331	+	0.998406i	$0.517973\pi$
$194$	−8.91428	−0.640008
$195$	0	0
$196$	0.360637	0.0257598
$197$	7.97000	0.567839	0.283920	−	0.958848i	$-0.408365\pi$
$197$	7.97000	0.567839	0.283920	+	0.958848i	$0.408365\pi$
$198$	0	0
$199$	3.53141	0.250335	0.125167	−	0.992136i	$-0.460053\pi$
$199$	3.53141	0.250335	0.125167	+	0.992136i	$0.460053\pi$
$200$	2.87392	0.203217
$201$	0	0
$202$	−12.1945	−0.858005
$203$	−8.01707	−0.562688
$204$	0	0
$205$	7.58597	0.529827
$206$	−16.0806	−1.12039
$207$	0	0
$208$	−5.47647	−0.379725
$209$	0	0
$210$	0	0
$211$	−20.2550	−1.39441	−0.697207	−	0.716870i	$-0.745574\pi$
$211$	−20.2550	−1.39441	−0.697207	+	0.716870i	$0.745574\pi$
$212$	0.629295	0.0432201
$213$	0	0
$214$	−21.3358	−1.45849
$215$	−5.41324	−0.369180
$216$	0	0
$217$	−3.89642	−0.264506
$218$	20.0203	1.35595
$219$	0	0
$220$	0	0
$221$	7.09183	0.477048
$222$	0	0
$223$	22.5500	1.51006	0.755029	−	0.655691i	$-0.227623\pi$
$223$	22.5500	1.51006	0.755029	+	0.655691i	$0.227623\pi$
$224$	0.485354	0.0324291
$225$	0	0
$226$	26.9217	1.79080
$227$	18.8062	1.24821	0.624105	−	0.781341i	$-0.285464\pi$
$227$	18.8062	1.24821	0.624105	+	0.781341i	$0.285464\pi$
$228$	0	0
$229$	−23.8320	−1.57486	−0.787431	−	0.616403i	$-0.788589\pi$
$229$	−23.8320	−1.57486	−0.787431	+	0.616403i	$0.788589\pi$
$230$	−6.96281	−0.459114
$231$	0	0
$232$	18.0343	1.18401
$233$	−10.0918	−0.661137	−0.330569	−	0.943782i	$-0.607241\pi$
$233$	−10.0918	−0.661137	−0.330569	+	0.943782i	$0.607241\pi$
$234$	0	0
$235$	8.23211	0.537004
$236$	0.544102	0.0354180
$237$	0	0
$238$	8.88090	0.575663
$239$	0.167227	0.0108170	0.00540850	−	0.999985i	$-0.498278\pi$
$239$	0.167227	0.0108170	0.00540850	+	0.999985i	$0.498278\pi$
$240$	0	0
$241$	−0.965256	−0.0621776	−0.0310888	−	0.999517i	$-0.509897\pi$
$241$	−0.965256	−0.0621776	−0.0310888	+	0.999517i	$0.509897\pi$
$242$	0	0
$243$	0	0
$244$	−0.960201	−0.0614706
$245$	−5.36776	−0.342934
$246$	0	0
$247$	−0.224308	−0.0142724
$248$	8.76492	0.556573
$249$	0	0
$250$	−1.39026	−0.0879276
$251$	23.5102	1.48395	0.741975	−	0.670428i	$-0.233889\pi$
$251$	23.5102	1.48395	0.741975	+	0.670428i	$0.233889\pi$
$252$	0	0
$253$	0	0
$254$	−8.37512	−0.525502
$255$	0	0
$256$	−1.61062	−0.100664
$257$	6.58273	0.410620	0.205310	−	0.978697i	$-0.434180\pi$
$257$	6.58273	0.410620	0.205310	+	0.978697i	$0.434180\pi$
$258$	0	0
$259$	6.00457	0.373106
$260$	−0.0952940	−0.00590988
$261$	0	0
$262$	26.0365	1.60854
$263$	12.4538	0.767936	0.383968	−	0.923346i	$-0.374557\pi$
$263$	12.4538	0.767936	0.383968	+	0.923346i	$0.374557\pi$
$264$	0	0
$265$	−9.36648	−0.575378
$266$	−0.280895	−0.0172228
$267$	0	0
$268$	0.496074	0.0303026
$269$	−20.7729	−1.26655	−0.633274	−	0.773927i	$-0.718290\pi$
$269$	−20.7729	−1.26655	−0.633274	+	0.773927i	$0.718290\pi$
$270$	0	0
$271$	−14.3903	−0.874146	−0.437073	−	0.899426i	$-0.643985\pi$
$271$	−14.3903	−0.874146	−0.437073	+	0.899426i	$0.643985\pi$
$272$	−19.3056	−1.17057
$273$	0	0
$274$	4.22247	0.255089
$275$	0	0
$276$	0	0
$277$	17.4872	1.05070	0.525352	−	0.850885i	$-0.323934\pi$
$277$	17.4872	1.05070	0.525352	+	0.850885i	$0.323934\pi$
$278$	−3.07513	−0.184434
$279$	0	0
$280$	−3.67169	−0.219426
$281$	−18.8860	−1.12664	−0.563322	−	0.826238i	$-0.690477\pi$
$281$	−18.8860	−1.12664	−0.563322	+	0.826238i	$0.690477\pi$
$282$	0	0
$283$	13.5813	0.807326	0.403663	−	0.914908i	$-0.367737\pi$
$283$	13.5813	0.807326	0.403663	+	0.914908i	$0.367737\pi$
$284$	−0.455263	−0.0270149
$285$	0	0
$286$	0	0
$287$	−9.69177	−0.572087
$288$	0	0
$289$	8.00000	0.470588
$290$	−8.72406	−0.512295
$291$	0	0
$292$	0.581901	0.0340532
$293$	19.9023	1.16271	0.581353	−	0.813651i	$-0.302523\pi$
$293$	19.9023	1.16271	0.581353	+	0.813651i	$0.302523\pi$
$294$	0	0
$295$	−8.09846	−0.471511
$296$	−13.5072	−0.785088
$297$	0	0
$298$	−2.05927	−0.119290
$299$	7.10358	0.410811
$300$	0	0
$301$	6.91591	0.398626
$302$	−12.5595	−0.722719
$303$	0	0
$304$	0.610619	0.0350214
$305$	14.2917	0.818342
$306$	0	0
$307$	14.9354	0.852410	0.426205	−	0.904627i	$-0.359850\pi$
$307$	14.9354	0.852410	0.426205	+	0.904627i	$0.359850\pi$
$308$	0	0
$309$	0	0
$310$	−4.24002	−0.240817
$311$	4.99700	0.283354	0.141677	−	0.989913i	$-0.454751\pi$
$311$	4.99700	0.283354	0.141677	+	0.989913i	$0.454751\pi$
$312$	0	0
$313$	9.39648	0.531120	0.265560	−	0.964094i	$-0.414443\pi$
$313$	9.39648	0.531120	0.265560	+	0.964094i	$0.414443\pi$
$314$	24.5209	1.38379
$315$	0	0
$316$	−0.171037	−0.00962155
$317$	−2.15253	−0.120898	−0.0604492	−	0.998171i	$-0.519253\pi$
$317$	−2.15253	−0.120898	−0.0604492	+	0.998171i	$0.519253\pi$
$318$	0	0
$319$	0	0
$320$	8.25038	0.461210
$321$	0	0
$322$	8.89563	0.495734
$323$	−0.790729	−0.0439973
$324$	0	0
$325$	1.41837	0.0786767
$326$	−32.6766	−1.80979
$327$	0	0
$328$	21.8015	1.20378
$329$	−10.5173	−0.579836
$330$	0	0
$331$	−19.5116	−1.07245	−0.536227	−	0.844074i	$-0.680151\pi$
$331$	−19.5116	−1.07245	−0.536227	+	0.844074i	$0.680151\pi$
$332$	−0.680063	−0.0373233
$333$	0	0
$334$	−0.837950	−0.0458506
$335$	−7.38362	−0.403410
$336$	0	0
$337$	−31.6047	−1.72162	−0.860809	−	0.508929i	$-0.830042\pi$
$337$	−31.6047	−1.72162	−0.860809	+	0.508929i	$0.830042\pi$
$338$	15.2765	0.830931
$339$	0	0
$340$	−0.335929	−0.0182183
$341$	0	0
$342$	0	0
$343$	15.8009	0.853171
$344$	−15.5572	−0.838789
$345$	0	0
$346$	−13.4353	−0.722286
$347$	9.38313	0.503713	0.251856	−	0.967765i	$-0.418959\pi$
$347$	9.38313	0.503713	0.251856	+	0.967765i	$0.418959\pi$
$348$	0	0
$349$	24.8131	1.32822	0.664108	−	0.747637i	$-0.268812\pi$
$349$	24.8131	1.32822	0.664108	+	0.747637i	$0.268812\pi$
$350$	1.77618	0.0949408
$351$	0	0
$352$	0	0
$353$	19.4788	1.03675	0.518375	−	0.855153i	$-0.326537\pi$
$353$	19.4788	1.03675	0.518375	+	0.855153i	$0.326537\pi$
$354$	0	0
$355$	6.77618	0.359642
$356$	0.742046	0.0393284
$357$	0	0
$358$	−1.66235	−0.0878578
$359$	29.7701	1.57121	0.785603	−	0.618732i	$-0.212353\pi$
$359$	29.7701	1.57121	0.785603	+	0.618732i	$0.212353\pi$
$360$	0	0
$361$	−18.9750	−0.998684
$362$	−21.6520	−1.13800
$363$	0	0
$364$	0.121747	0.00638126
$365$	−8.66107	−0.453341
$366$	0	0
$367$	−6.43597	−0.335955	−0.167977	−	0.985791i	$-0.553724\pi$
$367$	−6.43597	−0.335955	−0.167977	+	0.985791i	$0.553724\pi$
$368$	−19.3376	−1.00804
$369$	0	0
$370$	6.53409	0.339691
$371$	11.9665	0.621272
$372$	0	0
$373$	20.8924	1.08177	0.540883	−	0.841098i	$-0.318090\pi$
$373$	20.8924	1.08177	0.540883	+	0.841098i	$0.318090\pi$
$374$	0	0
$375$	0	0
$376$	23.6584	1.22009
$377$	8.90045	0.458396
$378$	0	0
$379$	−14.1991	−0.729359	−0.364680	−	0.931133i	$-0.618822\pi$
$379$	−14.1991	−0.729359	−0.364680	+	0.931133i	$0.618822\pi$
$380$	0.0106252	0.000545059 0
$381$	0	0
$382$	25.5830	1.30894
$383$	26.4783	1.35298	0.676489	−	0.736453i	$-0.263501\pi$
$383$	26.4783	1.35298	0.676489	+	0.736453i	$0.263501\pi$
$384$	0	0
$385$	0	0
$386$	2.17990	0.110954
$387$	0	0
$388$	−0.430793	−0.0218702
$389$	36.4965	1.85044	0.925222	−	0.379427i	$-0.123879\pi$
$389$	36.4965	1.85044	0.925222	+	0.379427i	$0.123879\pi$
$390$	0	0
$391$	25.0415	1.26640
$392$	−15.4265	−0.779157
$393$	0	0
$394$	−11.0804	−0.558220
$395$	2.54572	0.128089
$396$	0	0
$397$	−8.03969	−0.403500	−0.201750	−	0.979437i	$-0.564663\pi$
$397$	−8.03969	−0.403500	−0.201750	+	0.979437i	$0.564663\pi$
$398$	−4.90956	−0.246094
$399$	0	0
$400$	−3.86111	−0.193056
$401$	28.0902	1.40276	0.701379	−	0.712788i	$-0.252568\pi$
$401$	28.0902	1.40276	0.701379	+	0.712788i	$0.252568\pi$
$402$	0	0
$403$	4.32575	0.215481
$404$	−0.589316	−0.0293196
$405$	0	0
$406$	11.1458	0.553156
$407$	0	0
$408$	0	0
$409$	5.97950	0.295667	0.147834	−	0.989012i	$-0.452770\pi$
$409$	5.97950	0.295667	0.147834	+	0.989012i	$0.452770\pi$
$410$	−10.5464	−0.520852
$411$	0	0
$412$	−0.777114	−0.0382857
$413$	10.3465	0.509119
$414$	0	0
$415$	10.1221	0.496876
$416$	−0.538833	−0.0264185
$417$	0	0
$418$	0	0
$419$	15.4707	0.755795	0.377897	−	0.925847i	$-0.376647\pi$
$419$	15.4707	0.755795	0.377897	+	0.925847i	$0.376647\pi$
$420$	0	0
$421$	34.5746	1.68506	0.842532	−	0.538647i	$-0.181064\pi$
$421$	34.5746	1.68506	0.842532	+	0.538647i	$0.181064\pi$
$422$	28.1597	1.37079
$423$	0	0
$424$	−26.9185	−1.30728
$425$	5.00000	0.242536
$426$	0	0
$427$	−18.2590	−0.883614
$428$	−1.03108	−0.0498390
$429$	0	0
$430$	7.52580	0.362926
$431$	3.36052	0.161870	0.0809352	−	0.996719i	$-0.474209\pi$
$431$	3.36052	0.161870	0.0809352	+	0.996719i	$0.474209\pi$
$432$	0	0
$433$	3.67779	0.176744	0.0883718	−	0.996088i	$-0.471834\pi$
$433$	3.67779	0.176744	0.0883718	+	0.996088i	$0.471834\pi$
$434$	5.41702	0.260025
$435$	0	0
$436$	0.967505	0.0463351
$437$	−0.792040	−0.0378884
$438$	0	0
$439$	1.55432	0.0741835	0.0370917	−	0.999312i	$-0.488191\pi$
$439$	1.55432	0.0741835	0.0370917	+	0.999312i	$0.488191\pi$
$440$	0	0
$441$	0	0
$442$	−9.85946	−0.468967
$443$	−21.2059	−1.00752	−0.503761	−	0.863843i	$-0.668051\pi$
$443$	−21.2059	−1.00752	−0.503761	+	0.863843i	$0.668051\pi$
$444$	0	0
$445$	−11.0447	−0.523569
$446$	−31.3503	−1.48448
$447$	0	0
$448$	−10.5406	−0.497997
$449$	−15.1106	−0.713113	−0.356557	−	0.934274i	$-0.616049\pi$
$449$	−15.1106	−0.713113	−0.356557	+	0.934274i	$0.616049\pi$
$450$	0	0
$451$	0	0
$452$	1.30102	0.0611949
$453$	0	0
$454$	−26.1454	−1.22707
$455$	−1.81209	−0.0849521
$456$	0	0
$457$	−41.3042	−1.93213	−0.966064	−	0.258303i	$-0.916837\pi$
$457$	−41.3042	−1.93213	−0.966064	+	0.258303i	$0.916837\pi$
$458$	33.1326	1.54818
$459$	0	0
$460$	−0.336486	−0.0156887
$461$	−16.3158	−0.759901	−0.379951	−	0.925007i	$-0.624059\pi$
$461$	−16.3158	−0.759901	−0.379951	+	0.925007i	$0.624059\pi$
$462$	0	0
$463$	8.47904	0.394054	0.197027	−	0.980398i	$-0.436871\pi$
$463$	8.47904	0.394054	0.197027	+	0.980398i	$0.436871\pi$
$464$	−24.2291	−1.12481
$465$	0	0
$466$	14.0302	0.649938
$467$	36.5669	1.69212	0.846058	−	0.533090i	$-0.178969\pi$
$467$	36.5669	1.69212	0.846058	+	0.533090i	$0.178969\pi$
$468$	0	0
$469$	9.43325	0.435587
$470$	−11.4447	−0.527907
$471$	0	0
$472$	−23.2743	−1.07129
$473$	0	0
$474$	0	0
$475$	−0.158146	−0.00725623
$476$	0.429180	0.0196714
$477$	0	0
$478$	−0.232488	−0.0106338
$479$	−38.9513	−1.77973	−0.889866	−	0.456222i	$-0.849202\pi$
$479$	−38.9513	−1.77973	−0.889866	+	0.456222i	$0.849202\pi$
$480$	0	0
$481$	−6.66619	−0.303952
$482$	1.34195	0.0611243
$483$	0	0
$484$	0	0
$485$	6.41196	0.291152
$486$	0	0
$487$	8.27209	0.374844	0.187422	−	0.982279i	$-0.439987\pi$
$487$	8.27209	0.374844	0.187422	+	0.982279i	$0.439987\pi$
$488$	41.0733	1.85930
$489$	0	0
$490$	7.46257	0.337124
$491$	−12.1322	−0.547518	−0.273759	−	0.961798i	$-0.588267\pi$
$491$	−12.1322	−0.547518	−0.273759	+	0.961798i	$0.588267\pi$
$492$	0	0
$493$	31.3757	1.41309
$494$	0.311846	0.0140306
$495$	0	0
$496$	−11.7757	−0.528744
$497$	−8.65719	−0.388328
$498$	0	0
$499$	−28.3581	−1.26948	−0.634740	−	0.772725i	$-0.718893\pi$
$499$	−28.3581	−1.26948	−0.634740	+	0.772725i	$0.718893\pi$
$500$	−0.0671858	−0.00300464
$501$	0	0
$502$	−32.6852	−1.45881
$503$	−15.2800	−0.681300	−0.340650	−	0.940190i	$-0.610647\pi$
$503$	−15.2800	−0.681300	−0.340650	+	0.940190i	$0.610647\pi$
$504$	0	0
$505$	8.77143	0.390324
$506$	0	0
$507$	0	0
$508$	−0.404737	−0.0179573
$509$	22.0494	0.977321	0.488661	−	0.872474i	$-0.337486\pi$
$509$	22.0494	0.977321	0.488661	+	0.872474i	$0.337486\pi$
$510$	0	0
$511$	11.0653	0.489500
$512$	23.6599	1.04563
$513$	0	0
$514$	−9.15169	−0.403664
$515$	11.5666	0.509687
$516$	0	0
$517$	0	0
$518$	−8.34789	−0.366785
$519$	0	0
$520$	4.07627	0.178756
$521$	23.9380	1.04874	0.524371	−	0.851490i	$-0.324300\pi$
$521$	23.9380	1.04874	0.524371	+	0.851490i	$0.324300\pi$
$522$	0	0
$523$	25.3421	1.10813	0.554066	−	0.832473i	$-0.313075\pi$
$523$	25.3421	1.10813	0.554066	+	0.832473i	$0.313075\pi$
$524$	1.25824	0.0549667
$525$	0	0
$526$	−17.3140	−0.754927
$527$	15.2491	0.664260
$528$	0	0
$529$	2.08298	0.0905642
$530$	13.0218	0.565632
$531$	0	0
$532$	−0.0135746	−0.000588534 0
$533$	10.7597	0.466053
$534$	0	0
$535$	15.3467	0.663494
$536$	−21.2199	−0.916561
$537$	0	0
$538$	28.8797	1.24509
$539$	0	0
$540$	0	0
$541$	35.0634	1.50749	0.753747	−	0.657165i	$-0.228245\pi$
$541$	35.0634	1.50749	0.753747	+	0.657165i	$0.228245\pi$
$542$	20.0062	0.859338
$543$	0	0
$544$	−1.89949	−0.0814399
$545$	−14.4004	−0.616847
$546$	0	0
$547$	1.10787	0.0473689	0.0236845	−	0.999719i	$-0.492460\pi$
$547$	1.10787	0.0473689	0.0236845	+	0.999719i	$0.492460\pi$
$548$	0.204056	0.00871684
$549$	0	0
$550$	0	0
$551$	−0.992388	−0.0422771
$552$	0	0
$553$	−3.25239	−0.138306
$554$	−24.3117	−1.03291
$555$	0	0
$556$	−0.148609	−0.00630244
$557$	−5.41988	−0.229648	−0.114824	−	0.993386i	$-0.536630\pi$
$557$	−5.41988	−0.229648	−0.114824	+	0.993386i	$0.536630\pi$
$558$	0	0
$559$	−7.67795	−0.324743
$560$	4.93293	0.208454
$561$	0	0
$562$	26.2564	1.10756
$563$	32.7448	1.38003	0.690015	−	0.723795i	$-0.257604\pi$
$563$	32.7448	1.38003	0.690015	+	0.723795i	$0.257604\pi$
$564$	0	0
$565$	−19.3645	−0.814672
$566$	−18.8815	−0.793650
$567$	0	0
$568$	19.4742	0.817119
$569$	−3.51736	−0.147455	−0.0737277	−	0.997278i	$-0.523490\pi$
$569$	−3.51736	−0.147455	−0.0737277	+	0.997278i	$0.523490\pi$
$570$	0	0
$571$	36.0252	1.50761	0.753804	−	0.657099i	$-0.228217\pi$
$571$	36.0252	1.50761	0.753804	+	0.657099i	$0.228217\pi$
$572$	0	0
$573$	0	0
$574$	13.4740	0.562396
$575$	5.00829	0.208860
$576$	0	0
$577$	15.5501	0.647357	0.323679	−	0.946167i	$-0.395080\pi$
$577$	15.5501	0.647357	0.323679	+	0.946167i	$0.395080\pi$
$578$	−11.1221	−0.462617
$579$	0	0
$580$	−0.421601	−0.0175060
$581$	−12.9319	−0.536507
$582$	0	0
$583$	0	0
$584$	−24.8912	−1.03001
$585$	0	0
$586$	−27.6694	−1.14301
$587$	24.2604	1.00133	0.500667	−	0.865640i	$-0.333088\pi$
$587$	24.2604	1.00133	0.500667	+	0.865640i	$0.333088\pi$
$588$	0	0
$589$	−0.482315	−0.0198735
$590$	11.2589	0.463523
$591$	0	0
$592$	18.1469	0.745833
$593$	27.4019	1.12526	0.562630	−	0.826709i	$-0.309790\pi$
$593$	27.4019	1.12526	0.562630	+	0.826709i	$0.309790\pi$
$594$	0	0
$595$	−6.38796	−0.261881
$596$	−0.0995167	−0.00407636
$597$	0	0
$598$	−9.87581	−0.403852
$599$	−10.6773	−0.436262	−0.218131	−	0.975920i	$-0.569996\pi$
$599$	−10.6773	−0.436262	−0.218131	+	0.975920i	$0.569996\pi$
$600$	0	0
$601$	−11.0663	−0.451403	−0.225701	−	0.974197i	$-0.572467\pi$
$601$	−11.0663	−0.451403	−0.225701	+	0.974197i	$0.572467\pi$
$602$	−9.61489	−0.391874
$603$	0	0
$604$	−0.606953	−0.0246966
$605$	0	0
$606$	0	0
$607$	0.514518	0.0208836	0.0104418	−	0.999945i	$-0.496676\pi$
$607$	0.514518	0.0208836	0.0104418	+	0.999945i	$0.496676\pi$
$608$	0.0600792	0.00243653
$609$	0	0
$610$	−19.8692	−0.804479
$611$	11.6761	0.472366
$612$	0	0
$613$	−19.0977	−0.771348	−0.385674	−	0.922635i	$-0.626031\pi$
$613$	−19.0977	−0.771348	−0.385674	+	0.922635i	$0.626031\pi$
$614$	−20.7641	−0.837970
$615$	0	0
$616$	0	0
$617$	4.70745	0.189515	0.0947573	−	0.995500i	$-0.469792\pi$
$617$	4.70745	0.189515	0.0947573	+	0.995500i	$0.469792\pi$
$618$	0	0
$619$	37.3691	1.50199	0.750996	−	0.660307i	$-0.229574\pi$
$619$	37.3691	1.50199	0.750996	+	0.660307i	$0.229574\pi$
$620$	−0.204904	−0.00822915
$621$	0	0
$622$	−6.94711	−0.278554
$623$	14.1106	0.565329
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.0635	−0.522123
$627$	0	0
$628$	1.18500	0.0472867
$629$	−23.4996	−0.936989
$630$	0	0
$631$	35.5669	1.41590	0.707949	−	0.706264i	$-0.249621\pi$
$631$	35.5669	1.41590	0.707949	+	0.706264i	$0.249621\pi$
$632$	7.31621	0.291023
$633$	0	0
$634$	2.99257	0.118850
$635$	6.02415	0.239061
$636$	0	0
$637$	−7.61344	−0.301656
$638$	0	0
$639$	0	0
$640$	−10.7104	−0.423364
$641$	−12.2098	−0.482260	−0.241130	−	0.970493i	$-0.577518\pi$
$641$	−12.2098	−0.482260	−0.241130	+	0.970493i	$0.577518\pi$
$642$	0	0
$643$	−20.9596	−0.826565	−0.413282	−	0.910603i	$-0.635618\pi$
$643$	−20.9596	−0.826565	−0.413282	+	0.910603i	$0.635618\pi$
$644$	0.429892	0.0169401
$645$	0	0
$646$	1.09932	0.0432520
$647$	−14.8766	−0.584859	−0.292430	−	0.956287i	$-0.594464\pi$
$647$	−14.8766	−0.584859	−0.292430	+	0.956287i	$0.594464\pi$
$648$	0	0
$649$	0	0
$650$	−1.97189	−0.0773440
$651$	0	0
$652$	−1.57914	−0.0618438
$653$	12.1308	0.474713	0.237357	−	0.971423i	$-0.423719\pi$
$653$	12.1308	0.474713	0.237357	+	0.971423i	$0.423719\pi$
$654$	0	0
$655$	−18.7278	−0.731757
$656$	−29.2903	−1.14359
$657$	0	0
$658$	14.6217	0.570014
$659$	7.49994	0.292156	0.146078	−	0.989273i	$-0.453335\pi$
$659$	7.49994	0.292156	0.146078	+	0.989273i	$0.453335\pi$
$660$	0	0
$661$	9.65248	0.375438	0.187719	−	0.982223i	$-0.439891\pi$
$661$	9.65248	0.375438	0.187719	+	0.982223i	$0.439891\pi$
$662$	27.1261	1.05429
$663$	0	0
$664$	29.0902	1.12892
$665$	0.202046	0.00783500
$666$	0	0
$667$	31.4278	1.21689
$668$	−0.0404949	−0.00156680
$669$	0	0
$670$	10.2651	0.396577
$671$	0	0
$672$	0	0
$673$	−24.4496	−0.942463	−0.471231	−	0.882010i	$-0.656190\pi$
$673$	−24.4496	−0.942463	−0.471231	+	0.882010i	$0.656190\pi$
$674$	43.9386	1.69245
$675$	0	0
$676$	0.738254	0.0283944
$677$	46.7394	1.79634	0.898171	−	0.439646i	$-0.144896\pi$
$677$	46.7394	1.79634	0.898171	+	0.439646i	$0.144896\pi$
$678$	0	0
$679$	−8.19187	−0.314375
$680$	14.3696	0.551049
$681$	0	0
$682$	0	0
$683$	28.1941	1.07882	0.539408	−	0.842045i	$-0.318648\pi$
$683$	28.1941	1.07882	0.539408	+	0.842045i	$0.318648\pi$
$684$	0	0
$685$	−3.03719	−0.116045
$686$	−21.9674	−0.838718
$687$	0	0
$688$	20.9011	0.796849
$689$	−13.2851	−0.506122
$690$	0	0
$691$	11.5971	0.441175	0.220587	−	0.975367i	$-0.429203\pi$
$691$	11.5971	0.441175	0.220587	+	0.975367i	$0.429203\pi$
$692$	−0.649276	−0.0246818
$693$	0	0
$694$	−13.0450	−0.495180
$695$	2.21192	0.0839028
$696$	0	0
$697$	37.9298	1.43670
$698$	−34.4966	−1.30572
$699$	0	0
$700$	0.0858360	0.00324429
$701$	−36.7019	−1.38621	−0.693107	−	0.720835i	$-0.743759\pi$
$701$	−36.7019	−1.38621	−0.693107	+	0.720835i	$0.743759\pi$
$702$	0	0
$703$	0.743272	0.0280330
$704$	0	0
$705$	0	0
$706$	−27.0805	−1.01919
$707$	−11.2063	−0.421456
$708$	0	0
$709$	36.4905	1.37043	0.685214	−	0.728341i	$-0.259708\pi$
$709$	36.4905	1.37043	0.685214	+	0.728341i	$0.259708\pi$
$710$	−9.42063	−0.353550
$711$	0	0
$712$	−31.7415	−1.18956
$713$	15.2744	0.572029
$714$	0	0
$715$	0	0
$716$	−0.0803349	−0.00300225
$717$	0	0
$718$	−41.3881	−1.54459
$719$	9.32888	0.347909	0.173954	−	0.984754i	$-0.444346\pi$
$719$	9.32888	0.347909	0.173954	+	0.984754i	$0.444346\pi$
$720$	0	0
$721$	−14.7774	−0.550340
$722$	26.3801	0.981766
$723$	0	0
$724$	−1.04636	−0.0388875
$725$	6.27515	0.233053
$726$	0	0
$727$	−8.46883	−0.314091	−0.157046	−	0.987591i	$-0.550197\pi$
$727$	−8.46883	−0.314091	−0.157046	+	0.987591i	$0.550197\pi$
$728$	−5.20780	−0.193014
$729$	0	0
$730$	12.0411	0.445661
$731$	−27.0662	−1.00108
$732$	0	0
$733$	−29.0470	−1.07288	−0.536438	−	0.843940i	$-0.680230\pi$
$733$	−29.0470	−1.07288	−0.536438	+	0.843940i	$0.680230\pi$
$734$	8.94765	0.330264
$735$	0	0
$736$	−1.90264	−0.0701322
$737$	0	0
$738$	0	0
$739$	13.7551	0.505990	0.252995	−	0.967468i	$-0.418584\pi$
$739$	13.7551	0.505990	0.252995	+	0.967468i	$0.418584\pi$
$740$	0.315767	0.0116078
$741$	0	0
$742$	−16.6366	−0.610747
$743$	−34.9135	−1.28085	−0.640427	−	0.768019i	$-0.721242\pi$
$743$	−34.9135	−1.28085	−0.640427	+	0.768019i	$0.721242\pi$
$744$	0	0
$745$	1.48122	0.0542676
$746$	−29.0458	−1.06344
$747$	0	0
$748$	0	0
$749$	−19.6068	−0.716416
$750$	0	0
$751$	22.0992	0.806412	0.403206	−	0.915109i	$-0.367896\pi$
$751$	22.0992	0.806412	0.403206	+	0.915109i	$0.367896\pi$
$752$	−31.7851	−1.15908
$753$	0	0
$754$	−12.3739	−0.450631
$755$	9.03395	0.328779
$756$	0	0
$757$	−45.0300	−1.63664	−0.818321	−	0.574761i	$-0.805095\pi$
$757$	−45.0300	−1.63664	−0.818321	+	0.574761i	$0.805095\pi$
$758$	19.7404	0.717004
$759$	0	0
$760$	−0.454498	−0.0164864
$761$	33.6001	1.21800	0.609001	−	0.793169i	$-0.291570\pi$
$761$	33.6001	1.21800	0.609001	+	0.793169i	$0.291570\pi$
$762$	0	0
$763$	18.3979	0.666048
$764$	1.23633	0.0447287
$765$	0	0
$766$	−36.8116	−1.33006
$767$	−11.4866	−0.414756
$768$	0	0
$769$	−24.6086	−0.887408	−0.443704	−	0.896173i	$-0.646336\pi$
$769$	−24.6086	−0.887408	−0.443704	+	0.896173i	$0.646336\pi$
$770$	0	0
$771$	0	0
$772$	0.105346	0.00379150
$773$	23.6816	0.851768	0.425884	−	0.904778i	$-0.359963\pi$
$773$	23.6816	0.851768	0.425884	+	0.904778i	$0.359963\pi$
$774$	0	0
$775$	3.04981	0.109553
$776$	18.4275	0.661507
$777$	0	0
$778$	−50.7394	−1.81910
$779$	−1.19969	−0.0429833
$780$	0	0
$781$	0	0
$782$	−34.8141	−1.24495
$783$	0	0
$784$	20.7255	0.740198
$785$	−17.6377	−0.629515
$786$	0	0
$787$	18.4210	0.656639	0.328320	−	0.944567i	$-0.393518\pi$
$787$	18.4210	0.656639	0.328320	+	0.944567i	$0.393518\pi$
$788$	−0.535471	−0.0190754
$789$	0	0
$790$	−3.53921	−0.125919
$791$	24.7399	0.879651
$792$	0	0
$793$	20.2709	0.719840
$794$	11.1772	0.396665
$795$	0	0
$796$	−0.237260	−0.00840947
$797$	6.50665	0.230477	0.115239	−	0.993338i	$-0.463237\pi$
$797$	6.50665	0.230477	0.115239	+	0.993338i	$0.463237\pi$
$798$	0	0
$799$	41.1606	1.45616
$800$	−0.379898	−0.0134314
$801$	0	0
$802$	−39.0526	−1.37900
$803$	0	0
$804$	0	0
$805$	−6.39855	−0.225519
$806$	−6.01390	−0.211831
$807$	0	0
$808$	25.2084	0.886828
$809$	11.8505	0.416642	0.208321	−	0.978060i	$-0.433200\pi$
$809$	11.8505	0.416642	0.208321	+	0.978060i	$0.433200\pi$
$810$	0	0
$811$	−7.51391	−0.263849	−0.131925	−	0.991260i	$-0.542116\pi$
$811$	−7.51391	−0.263849	−0.131925	+	0.991260i	$0.542116\pi$
$812$	0.538633	0.0189023
$813$	0	0
$814$	0	0
$815$	23.5040	0.823310
$816$	0	0
$817$	0.856081	0.0299505
$818$	−8.31305	−0.290659
$819$	0	0
$820$	−0.509669	−0.0177984
$821$	−37.3365	−1.30305	−0.651527	−	0.758625i	$-0.725871\pi$
$821$	−37.3365	−1.30305	−0.651527	+	0.758625i	$0.725871\pi$
$822$	0	0
$823$	−43.0498	−1.50062	−0.750311	−	0.661085i	$-0.770096\pi$
$823$	−43.0498	−1.50062	−0.750311	+	0.661085i	$0.770096\pi$
$824$	33.2416	1.15803
$825$	0	0
$826$	−14.3843	−0.500495
$827$	−20.0304	−0.696524	−0.348262	−	0.937397i	$-0.613228\pi$
$827$	−20.0304	−0.696524	−0.348262	+	0.937397i	$0.613228\pi$
$828$	0	0
$829$	−47.9207	−1.66435	−0.832177	−	0.554510i	$-0.812906\pi$
$829$	−47.9207	−1.66435	−0.832177	+	0.554510i	$0.812906\pi$
$830$	−14.0724	−0.488458
$831$	0	0
$832$	11.7021	0.405696
$833$	−26.8388	−0.929909
$834$	0	0
$835$	0.602731	0.0208584
$836$	0	0
$837$	0	0
$838$	−21.5083	−0.742992
$839$	−2.22093	−0.0766750	−0.0383375	−	0.999265i	$-0.512206\pi$
$839$	−2.22093	−0.0766750	−0.0383375	+	0.999265i	$0.512206\pi$
$840$	0	0
$841$	10.3775	0.357843
$842$	−48.0676	−1.65652
$843$	0	0
$844$	1.36085	0.0468424
$845$	−10.9882	−0.378007
$846$	0	0
$847$	0	0
$848$	36.1651	1.24191
$849$	0	0
$850$	−6.95128	−0.238427
$851$	−23.5385	−0.806890
$852$	0	0
$853$	6.37161	0.218160	0.109080	−	0.994033i	$-0.465210\pi$
$853$	6.37161	0.218160	0.109080	+	0.994033i	$0.465210\pi$
$854$	25.3847	0.868646
$855$	0	0
$856$	44.1051	1.50748
$857$	35.9060	1.22653	0.613263	−	0.789878i	$-0.289856\pi$
$857$	35.9060	1.22653	0.613263	+	0.789878i	$0.289856\pi$
$858$	0	0
$859$	56.4697	1.92672	0.963360	−	0.268212i	$-0.0864328\pi$
$859$	56.4697	1.92672	0.963360	+	0.268212i	$0.0864328\pi$
$860$	0.363693	0.0124018
$861$	0	0
$862$	−4.67198	−0.159128
$863$	31.0742	1.05778	0.528888	−	0.848691i	$-0.322609\pi$
$863$	31.0742	1.05778	0.528888	+	0.848691i	$0.322609\pi$
$864$	0	0
$865$	9.66389	0.328582
$866$	−5.11308	−0.173749
$867$	0	0
$868$	0.261784	0.00888552
$869$	0	0
$870$	0	0
$871$	−10.4727	−0.354853
$872$	−41.3857	−1.40150
$873$	0	0
$874$	1.10114	0.0372466
$875$	−1.27759	−0.0431905
$876$	0	0
$877$	−16.8224	−0.568053	−0.284026	−	0.958816i	$-0.591670\pi$
$877$	−16.8224	−0.568053	−0.284026	+	0.958816i	$0.591670\pi$
$878$	−2.16090	−0.0729268
$879$	0	0
$880$	0	0
$881$	−26.5633	−0.894940	−0.447470	−	0.894299i	$-0.647675\pi$
$881$	−26.5633	−0.894940	−0.447470	+	0.894299i	$0.647675\pi$
$882$	0	0
$883$	−54.3742	−1.82984	−0.914919	−	0.403638i	$-0.867746\pi$
$883$	−54.3742	−1.82984	−0.914919	+	0.403638i	$0.867746\pi$
$884$	−0.476470	−0.0160254
$885$	0	0
$886$	29.4816	0.990455
$887$	−41.2262	−1.38424	−0.692120	−	0.721783i	$-0.743323\pi$
$887$	−41.2262	−1.38424	−0.692120	+	0.721783i	$0.743323\pi$
$888$	0	0
$889$	−7.69640	−0.258129
$890$	15.3550	0.514699
$891$	0	0
$892$	−1.51504	−0.0507273
$893$	−1.30187	−0.0435655
$894$	0	0
$895$	1.19571	0.0399682
$896$	13.6835	0.457132
$897$	0	0
$898$	21.0076	0.701033
$899$	19.1380	0.638289
$900$	0	0
$901$	−46.8324	−1.56021
$902$	0	0
$903$	0	0
$904$	−55.6521	−1.85096
$905$	15.5741	0.517699
$906$	0	0
$907$	42.4046	1.40802	0.704010	−	0.710190i	$-0.251391\pi$
$907$	42.4046	1.40802	0.704010	+	0.710190i	$0.251391\pi$
$908$	−1.26351	−0.0419310
$909$	0	0
$910$	2.51927	0.0835130
$911$	8.01199	0.265449	0.132725	−	0.991153i	$-0.457627\pi$
$911$	8.01199	0.265449	0.132725	+	0.991153i	$0.457627\pi$
$912$	0	0
$913$	0	0
$914$	57.4234	1.89940
$915$	0	0
$916$	1.60117	0.0529042
$917$	23.9265	0.790123
$918$	0	0
$919$	−41.2336	−1.36017	−0.680086	−	0.733132i	$-0.738058\pi$
$919$	−41.2336	−1.36017	−0.680086	+	0.733132i	$0.738058\pi$
$920$	14.3934	0.474537
$921$	0	0
$922$	22.6831	0.747028
$923$	9.61110	0.316353
$924$	0	0
$925$	−4.69991	−0.154532
$926$	−11.7880	−0.387379
$927$	0	0
$928$	−2.38391	−0.0782558
$929$	15.1939	0.498495	0.249247	−	0.968440i	$-0.419817\pi$
$929$	15.1939	0.498495	0.249247	+	0.968440i	$0.419817\pi$
$930$	0	0
$931$	0.848889	0.0278212
$932$	0.678027	0.0222095
$933$	0	0
$934$	−50.8375	−1.66345
$935$	0	0
$936$	0	0
$937$	−25.5213	−0.833746	−0.416873	−	0.908965i	$-0.636874\pi$
$937$	−25.5213	−0.833746	−0.416873	+	0.908965i	$0.636874\pi$
$938$	−13.1146	−0.428208
$939$	0	0
$940$	−0.553081	−0.0180395
$941$	32.3673	1.05514	0.527572	−	0.849511i	$-0.323103\pi$
$941$	32.3673	1.05514	0.527572	+	0.849511i	$0.323103\pi$
$942$	0	0
$943$	37.9927	1.23721
$944$	31.2691	1.01772
$945$	0	0
$946$	0	0
$947$	40.1516	1.30475	0.652375	−	0.757896i	$-0.273773\pi$
$947$	40.1516	1.30475	0.652375	+	0.757896i	$0.273773\pi$
$948$	0	0
$949$	−12.2846	−0.398774
$950$	0.219863	0.00713331
$951$	0	0
$952$	−18.3585	−0.595001
$953$	35.6513	1.15486	0.577429	−	0.816441i	$-0.304056\pi$
$953$	35.6513	1.15486	0.577429	+	0.816441i	$0.304056\pi$
$954$	0	0
$955$	−18.4016	−0.595462
$956$	−0.0112353	−0.000363374 0
$957$	0	0
$958$	54.1524	1.74958
$959$	3.88029	0.125301
$960$	0	0
$961$	−21.6986	−0.699956
$962$	9.26772	0.298803
$963$	0	0
$964$	0.0648515	0.00208873
$965$	−1.56799	−0.0504753
$966$	0	0
$967$	20.0622	0.645156	0.322578	−	0.946543i	$-0.395451\pi$
$967$	20.0622	0.645156	0.322578	+	0.946543i	$0.395451\pi$
$968$	0	0
$969$	0	0
$970$	−8.91428	−0.286220
$971$	41.6307	1.33599	0.667996	−	0.744165i	$-0.267152\pi$
$971$	41.6307	1.33599	0.667996	+	0.744165i	$0.267152\pi$
$972$	0	0
$973$	−2.82593	−0.0905950
$974$	−11.5003	−0.368494
$975$	0	0
$976$	−55.1820	−1.76633
$977$	−19.0722	−0.610173	−0.305087	−	0.952325i	$-0.598685\pi$
$977$	−19.0722	−0.610173	−0.305087	+	0.952325i	$0.598685\pi$
$978$	0	0
$979$	0	0
$980$	0.360637	0.0115201
$981$	0	0
$982$	16.8668	0.538243
$983$	−26.5563	−0.847015	−0.423507	−	0.905893i	$-0.639201\pi$
$983$	−26.5563	−0.847015	−0.423507	+	0.905893i	$0.639201\pi$
$984$	0	0
$985$	7.97000	0.253945
$986$	−43.6203	−1.38915
$987$	0	0
$988$	0.0150703	0.000479452 0
$989$	−27.1111	−0.862082
$990$	0	0
$991$	−24.2494	−0.770307	−0.385153	−	0.922853i	$-0.625851\pi$
$991$	−24.2494	−0.770307	−0.385153	+	0.922853i	$0.625851\pi$
$992$	−1.15862	−0.0367861
$993$	0	0
$994$	12.0357	0.381750
$995$	3.53141	0.111953
$996$	0	0
$997$	−16.0616	−0.508675	−0.254338	−	0.967116i	$-0.581857\pi$
$997$	−16.0616	−0.508675	−0.254338	+	0.967116i	$0.581857\pi$
$998$	39.4250	1.24798
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bv.1.1		4
3.2	odd	2		1815.2.a.o.1.4		4
11.2	odd	10		495.2.n.d.136.1		8
11.6	odd	10		495.2.n.d.91.1		8
11.10	odd	2		5445.2.a.be.1.4		4
15.14	odd	2		9075.2.a.dj.1.1		4
33.2	even	10		165.2.m.a.136.2	yes	8
33.17	even	10		165.2.m.a.91.2	✓	8
33.32	even	2		1815.2.a.x.1.1		4
165.2	odd	20		825.2.bx.h.499.2		16
165.17	odd	20		825.2.bx.h.124.3		16
165.68	odd	20		825.2.bx.h.499.3		16
165.83	odd	20		825.2.bx.h.124.2		16
165.134	even	10		825.2.n.k.301.1		8
165.149	even	10		825.2.n.k.751.1		8
165.164	even	2		9075.2.a.cl.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.a.91.2	✓	8	33.17	even	10
165.2.m.a.136.2	yes	8	33.2	even	10
495.2.n.d.91.1		8	11.6	odd	10
495.2.n.d.136.1		8	11.2	odd	10
825.2.n.k.301.1		8	165.134	even	10
825.2.n.k.751.1		8	165.149	even	10
825.2.bx.h.124.2		16	165.83	odd	20
825.2.bx.h.124.3		16	165.17	odd	20
825.2.bx.h.499.2		16	165.2	odd	20
825.2.bx.h.499.3		16	165.68	odd	20
1815.2.a.o.1.4		4	3.2	odd	2
1815.2.a.x.1.1		4	33.32	even	2
5445.2.a.be.1.4		4	11.10	odd	2
5445.2.a.bv.1.1		4	1.1	even	1	trivial
9075.2.a.cl.1.4		4	165.164	even	2
9075.2.a.dj.1.1		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} +O(q^{10})\)	\(q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} -1.39026 q^{10} +1.41837 q^{13} +1.77618 q^{14} -3.86111 q^{16} +5.00000 q^{17} -0.158146 q^{19} -0.0671858 q^{20} +5.00829 q^{23} +1.00000 q^{25} -1.97189 q^{26} +0.0858360 q^{28} +6.27515 q^{29} +3.04981 q^{31} -0.379898 q^{32} -6.95128 q^{34} -1.27759 q^{35} -4.69991 q^{37} +0.219863 q^{38} +2.87392 q^{40} +7.58597 q^{41} -5.41324 q^{43} -6.96281 q^{46} +8.23211 q^{47} -5.36776 q^{49} -1.39026 q^{50} -0.0952940 q^{52} -9.36648 q^{53} -3.67169 q^{56} -8.72406 q^{58} -8.09846 q^{59} +14.2917 q^{61} -4.24002 q^{62} +8.25038 q^{64} +1.41837 q^{65} -7.38362 q^{67} -0.335929 q^{68} +1.77618 q^{70} +6.77618 q^{71} -8.66107 q^{73} +6.53409 q^{74} +0.0106252 q^{76} +2.54572 q^{79} -3.86111 q^{80} -10.5464 q^{82} +10.1221 q^{83} +5.00000 q^{85} +7.52580 q^{86} -11.0447 q^{89} -1.81209 q^{91} -0.336486 q^{92} -11.4447 q^{94} -0.158146 q^{95} +6.41196 q^{97} +7.46257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8}+O(q^{10}) \) 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8	\( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + 5 q^{10} - 3 q^{13} + 5 q^{14} + 15 q^{16} + 20 q^{17} - 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} - 3 q^{28} + 5 q^{29} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} - 7 q^{37} - q^{38} + 15 q^{40} + 20 q^{41} + 2 q^{43} - 7 q^{46} + 20 q^{47} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} + 7 q^{61} - 12 q^{62} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} + 5 q^{70} + 25 q^{71} - 23 q^{73} - 7 q^{74} + 7 q^{76} + 15 q^{80} + 11 q^{82} + 33 q^{83} + 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} + 17 q^{94} - 3 q^{95} + 25 q^{98}+O(q^{100}) \) 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8 + 5 * q^10 - 3 * q^13 + 5 * q^14 + 15 * q^16 + 20 * q^17 - 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 - 3 * q^28 + 5 * q^29 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 - 7 * q^37 - q^38 + 15 * q^40 + 20 * q^41 + 2 * q^43 - 7 * q^46 + 20 * q^47 + 8 * q^49 + 5 * q^50 + 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 + 7 * q^61 - 12 * q^62 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 + 5 * q^70 + 25 * q^71 - 23 * q^73 - 7 * q^74 + 7 * q^76 + 15 * q^80 + 11 * q^82 + 33 * q^83 + 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 + 17 * q^94 - 3 * q^95 + 25 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more