LMFDB - Embedded newform 5265.2.a.ba.1.6

Newspace parameters

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 31x^{5} - x^{4} - 70x^{3} + 66x^{2} - 19x + 1 $ x^8 - 3x^7 - 8x^6 + 31x^5 - x^4 - 70x^3 + 66x^2 - 19x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.0672022$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

$f(q)$	$=$	$q-0.0672022 q^{2} -1.99548 q^{4} +1.00000 q^{5} +2.46357 q^{7} +0.268505 q^{8} +O(q^{10})$	\(q-0.0672022 q^{2} -1.99548 q^{4} +1.00000 q^{5} +2.46357 q^{7} +0.268505 q^{8} -0.0672022 q^{10} -3.21214 q^{11} -1.00000 q^{13} -0.165558 q^{14} +3.97292 q^{16} +4.77678 q^{17} -3.94903 q^{19} -1.99548 q^{20} +0.215863 q^{22} -4.26977 q^{23} +1.00000 q^{25} +0.0672022 q^{26} -4.91602 q^{28} +2.30629 q^{29} -7.63305 q^{31} -0.804000 q^{32} -0.321010 q^{34} +2.46357 q^{35} +4.87293 q^{37} +0.265384 q^{38} +0.268505 q^{40} +2.53036 q^{41} -4.56607 q^{43} +6.40978 q^{44} +0.286938 q^{46} +6.26567 q^{47} -0.930808 q^{49} -0.0672022 q^{50} +1.99548 q^{52} -12.4968 q^{53} -3.21214 q^{55} +0.661482 q^{56} -0.154987 q^{58} -2.85019 q^{59} +12.7000 q^{61} +0.512957 q^{62} -7.89182 q^{64} -1.00000 q^{65} -13.5072 q^{67} -9.53198 q^{68} -0.165558 q^{70} -7.79224 q^{71} +2.26796 q^{73} -0.327472 q^{74} +7.88023 q^{76} -7.91335 q^{77} +9.13335 q^{79} +3.97292 q^{80} -0.170046 q^{82} -10.5119 q^{83} +4.77678 q^{85} +0.306850 q^{86} -0.862477 q^{88} -0.966612 q^{89} -2.46357 q^{91} +8.52026 q^{92} -0.421067 q^{94} -3.94903 q^{95} -18.7804 q^{97} +0.0625523 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8	$ 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 - 8 * q^13 - 10 * q^14 + 11 * q^16 + 2 * q^17 - 10 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 8 * q^25 + 3 * q^26 - 34 * q^28 - 14 * q^29 - 31 * q^31 - q^32 - 7 * q^34 - 11 * q^35 + q^37 - 9 * q^38 + 6 * q^40 + 12 * q^41 + 15 * q^43 - 16 * q^44 - 32 * q^46 + 18 * q^47 + 17 * q^49 - 3 * q^50 - 9 * q^52 - 2 * q^53 - 6 * q^55 - 16 * q^56 - 42 * q^58 - 24 * q^59 - 9 * q^61 + 20 * q^62 - 30 * q^64 - 8 * q^65 - 18 * q^67 + 14 * q^68 - 10 * q^70 - 10 * q^71 + 6 * q^73 + 37 * q^74 - 53 * q^76 + 34 * q^77 - 3 * q^79 + 11 * q^80 - 34 * q^82 + 10 * q^83 + 2 * q^85 - 60 * q^86 - 14 * q^88 + 13 * q^89 + 11 * q^91 - 5 * q^92 + 17 * q^94 - 10 * q^95 - 34 * q^97 + 30 * 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 $

$2$	−0.0672022	−0.0475191	−0.0237596	−	0.999718i	$-0.507564\pi$
$2$	−0.0672022	−0.0475191	−0.0237596	+	0.999718i	$0.507564\pi$
$3$	0	0
$3$	0	0
$4$	−1.99548	−0.997742
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.46357	0.931143	0.465572	−	0.885010i	$-0.345849\pi$
$7$	2.46357	0.931143	0.465572	+	0.885010i	$0.345849\pi$
$8$	0.268505	0.0949309
$9$	0	0
$10$	−0.0672022	−0.0212512
$11$	−3.21214	−0.968498	−0.484249	−	0.874930i	$-0.660907\pi$
$11$	−3.21214	−0.968498	−0.484249	+	0.874930i	$0.660907\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.165558	−0.0442471
$15$	0	0
$16$	3.97292	0.993231
$17$	4.77678	1.15854	0.579269	−	0.815136i	$-0.303338\pi$
$17$	4.77678	1.15854	0.579269	+	0.815136i	$0.303338\pi$
$18$	0	0
$19$	−3.94903	−0.905970	−0.452985	−	0.891518i	$-0.649641\pi$
$19$	−3.94903	−0.905970	−0.452985	+	0.891518i	$0.649641\pi$
$20$	−1.99548	−0.446204
$21$	0	0
$22$	0.215863	0.0460222
$23$	−4.26977	−0.890309	−0.445154	−	0.895454i	$-0.646851\pi$
$23$	−4.26977	−0.890309	−0.445154	+	0.895454i	$0.646851\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.0672022	0.0131794
$27$	0	0
$28$	−4.91602	−0.929040
$29$	2.30629	0.428267	0.214133	−	0.976804i	$-0.431307\pi$
$29$	2.30629	0.428267	0.214133	+	0.976804i	$0.431307\pi$
$30$	0	0
$31$	−7.63305	−1.37094	−0.685468	−	0.728103i	$-0.740402\pi$
$31$	−7.63305	−1.37094	−0.685468	+	0.728103i	$0.740402\pi$
$32$	−0.804000	−0.142128
$33$	0	0
$34$	−0.321010	−0.0550527
$35$	2.46357	0.416420
$36$	0	0
$37$	4.87293	0.801105	0.400553	−	0.916274i	$-0.368818\pi$
$37$	4.87293	0.801105	0.400553	+	0.916274i	$0.368818\pi$
$38$	0.265384	0.0430509
$39$	0	0
$40$	0.268505	0.0424544
$41$	2.53036	0.395176	0.197588	−	0.980285i	$-0.436689\pi$
$41$	2.53036	0.395176	0.197588	+	0.980285i	$0.436689\pi$
$42$	0	0
$43$	−4.56607	−0.696319	−0.348160	−	0.937435i	$-0.613193\pi$
$43$	−4.56607	−0.696319	−0.348160	+	0.937435i	$0.613193\pi$
$44$	6.40978	0.966311
$45$	0	0
$46$	0.286938	0.0423067
$47$	6.26567	0.913942	0.456971	−	0.889482i	$-0.348934\pi$
$47$	6.26567	0.913942	0.456971	+	0.889482i	$0.348934\pi$
$48$	0	0
$49$	−0.930808	−0.132973
$50$	−0.0672022	−0.00950382
$51$	0	0
$52$	1.99548	0.276724
$53$	−12.4968	−1.71657	−0.858286	−	0.513172i	$-0.828470\pi$
$53$	−12.4968	−1.71657	−0.858286	+	0.513172i	$0.828470\pi$
$54$	0	0
$55$	−3.21214	−0.433125
$56$	0.661482	0.0883943
$57$	0	0
$58$	−0.154987	−0.0203509
$59$	−2.85019	−0.371063	−0.185531	−	0.982638i	$-0.559401\pi$
$59$	−2.85019	−0.371063	−0.185531	+	0.982638i	$0.559401\pi$
$60$	0	0
$61$	12.7000	1.62607	0.813034	−	0.582216i	$-0.197814\pi$
$61$	12.7000	1.62607	0.813034	+	0.582216i	$0.197814\pi$
$62$	0.512957	0.0651457
$63$	0	0
$64$	−7.89182	−0.986477
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−13.5072	−1.65016	−0.825082	−	0.565013i	$-0.808871\pi$
$67$	−13.5072	−1.65016	−0.825082	+	0.565013i	$0.808871\pi$
$68$	−9.53198	−1.15592
$69$	0	0
$70$	−0.165558	−0.0197879
$71$	−7.79224	−0.924769	−0.462384	−	0.886680i	$-0.653006\pi$
$71$	−7.79224	−0.924769	−0.462384	+	0.886680i	$0.653006\pi$
$72$	0	0
$73$	2.26796	0.265444	0.132722	−	0.991153i	$-0.457628\pi$
$73$	2.26796	0.265444	0.132722	+	0.991153i	$0.457628\pi$
$74$	−0.327472	−0.0380678
$75$	0	0
$76$	7.88023	0.903924
$77$	−7.91335	−0.901810
$78$	0	0
$79$	9.13335	1.02758	0.513791	−	0.857915i	$-0.328241\pi$
$79$	9.13335	1.02758	0.513791	+	0.857915i	$0.328241\pi$
$80$	3.97292	0.444186
$81$	0	0
$82$	−0.170046	−0.0187784
$83$	−10.5119	−1.15383	−0.576915	−	0.816804i	$-0.695744\pi$
$83$	−10.5119	−1.15383	−0.576915	+	0.816804i	$0.695744\pi$
$84$	0	0
$85$	4.77678	0.518114
$86$	0.306850	0.0330885
$87$	0	0
$88$	−0.862477	−0.0919404
$89$	−0.966612	−0.102461	−0.0512303	−	0.998687i	$-0.516314\pi$
$89$	−0.966612	−0.102461	−0.0512303	+	0.998687i	$0.516314\pi$
$90$	0	0
$91$	−2.46357	−0.258253
$92$	8.52026	0.888299
$93$	0	0
$94$	−0.421067	−0.0434297
$95$	−3.94903	−0.405162
$96$	0	0
$97$	−18.7804	−1.90686	−0.953431	−	0.301612i	$-0.902475\pi$
$97$	−18.7804	−1.90686	−0.953431	+	0.301612i	$0.902475\pi$
$98$	0.0625523	0.00631874
$99$	0	0
$100$	−1.99548	−0.199548
$101$	18.5575	1.84654	0.923268	−	0.384157i	$-0.125508\pi$
$101$	18.5575	1.84654	0.923268	+	0.384157i	$0.125508\pi$
$102$	0	0
$103$	10.2563	1.01058	0.505292	−	0.862949i	$-0.331385\pi$
$103$	10.2563	1.01058	0.505292	+	0.862949i	$0.331385\pi$
$104$	−0.268505	−0.0263291
$105$	0	0
$106$	0.839814	0.0815700
$107$	13.3729	1.29281	0.646403	−	0.762996i	$-0.276273\pi$
$107$	13.3729	1.29281	0.646403	+	0.762996i	$0.276273\pi$
$108$	0	0
$109$	−1.35788	−0.130061	−0.0650306	−	0.997883i	$-0.520715\pi$
$109$	−1.35788	−0.130061	−0.0650306	+	0.997883i	$0.520715\pi$
$110$	0.215863	0.0205817
$111$	0	0
$112$	9.78759	0.924840
$113$	7.35696	0.692085	0.346042	−	0.938219i	$-0.387525\pi$
$113$	7.35696	0.692085	0.346042	+	0.938219i	$0.387525\pi$
$114$	0	0
$115$	−4.26977	−0.398158
$116$	−4.60216	−0.427300
$117$	0	0
$118$	0.191539	0.0176326
$119$	11.7679	1.07877
$120$	0	0
$121$	−0.682134	−0.0620121
$122$	−0.853468	−0.0772693
$123$	0	0
$124$	15.2316	1.36784
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.97693	−0.530367	−0.265183	−	0.964198i	$-0.585433\pi$
$127$	−5.97693	−0.530367	−0.265183	+	0.964198i	$0.585433\pi$
$128$	2.13835	0.189005
$129$	0	0
$130$	0.0672022	0.00589402
$131$	−17.8734	−1.56161	−0.780803	−	0.624778i	$-0.785190\pi$
$131$	−17.8734	−1.56161	−0.780803	+	0.624778i	$0.785190\pi$
$132$	0	0
$133$	−9.72873	−0.843588
$134$	0.907712	0.0784144
$135$	0	0
$136$	1.28259	0.109981
$137$	−10.4931	−0.896483	−0.448241	−	0.893913i	$-0.647949\pi$
$137$	−10.4931	−0.896483	−0.448241	+	0.893913i	$0.647949\pi$
$138$	0	0
$139$	−10.7069	−0.908143	−0.454072	−	0.890965i	$-0.650029\pi$
$139$	−10.7069	−0.908143	−0.454072	+	0.890965i	$0.650029\pi$
$140$	−4.91602	−0.415480
$141$	0	0
$142$	0.523655	0.0439442
$143$	3.21214	0.268613
$144$	0	0
$145$	2.30629	0.191527
$146$	−0.152412	−0.0126137
$147$	0	0
$148$	−9.72386	−0.799296
$149$	20.3190	1.66460	0.832300	−	0.554325i	$-0.187023\pi$
$149$	20.3190	1.66460	0.832300	+	0.554325i	$0.187023\pi$
$150$	0	0
$151$	−17.0220	−1.38523	−0.692615	−	0.721308i	$-0.743541\pi$
$151$	−17.0220	−1.38523	−0.692615	+	0.721308i	$0.743541\pi$
$152$	−1.06034	−0.0860046
$153$	0	0
$154$	0.531794	0.0428532
$155$	−7.63305	−0.613101
$156$	0	0
$157$	−1.30787	−0.104379	−0.0521897	−	0.998637i	$-0.516620\pi$
$157$	−1.30787	−0.104379	−0.0521897	+	0.998637i	$0.516620\pi$
$158$	−0.613781	−0.0488298
$159$	0	0
$160$	−0.804000	−0.0635618
$161$	−10.5189	−0.829005
$162$	0	0
$163$	−9.81120	−0.768473	−0.384236	−	0.923235i	$-0.625535\pi$
$163$	−9.81120	−0.768473	−0.384236	+	0.923235i	$0.625535\pi$
$164$	−5.04930	−0.394284
$165$	0	0
$166$	0.706422	0.0548290
$167$	−8.17491	−0.632593	−0.316297	−	0.948660i	$-0.602440\pi$
$167$	−8.17491	−0.632593	−0.316297	+	0.948660i	$0.602440\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.321010	−0.0246203
$171$	0	0
$172$	9.11152	0.694747
$173$	6.39771	0.486409	0.243204	−	0.969975i	$-0.421801\pi$
$173$	6.39771	0.486409	0.243204	+	0.969975i	$0.421801\pi$
$174$	0	0
$175$	2.46357	0.186229
$176$	−12.7616	−0.961942
$177$	0	0
$178$	0.0649584	0.00486884
$179$	15.6316	1.16836	0.584180	−	0.811624i	$-0.301416\pi$
$179$	15.6316	1.16836	0.584180	+	0.811624i	$0.301416\pi$
$180$	0	0
$181$	11.1033	0.825299	0.412650	−	0.910890i	$-0.364603\pi$
$181$	11.1033	0.825299	0.412650	+	0.910890i	$0.364603\pi$
$182$	0.165558	0.0122719
$183$	0	0
$184$	−1.14646	−0.0845179
$185$	4.87293	0.358265
$186$	0	0
$187$	−15.3437	−1.12204
$188$	−12.5030	−0.911878
$189$	0	0
$190$	0.265384	0.0192530
$191$	−15.5825	−1.12751	−0.563753	−	0.825943i	$-0.690643\pi$
$191$	−15.5825	−1.12751	−0.563753	+	0.825943i	$0.690643\pi$
$192$	0	0
$193$	−18.6960	−1.34577	−0.672884	−	0.739748i	$-0.734945\pi$
$193$	−18.6960	−1.34577	−0.672884	+	0.739748i	$0.734945\pi$
$194$	1.26208	0.0906124
$195$	0	0
$196$	1.85741	0.132672
$197$	10.2024	0.726893	0.363447	−	0.931615i	$-0.381600\pi$
$197$	10.2024	0.726893	0.363447	+	0.931615i	$0.381600\pi$
$198$	0	0
$199$	−16.4798	−1.16822	−0.584112	−	0.811673i	$-0.698557\pi$
$199$	−16.4798	−1.16822	−0.584112	+	0.811673i	$0.698557\pi$
$200$	0.268505	0.0189862
$201$	0	0
$202$	−1.24710	−0.0877457
$203$	5.68170	0.398777
$204$	0	0
$205$	2.53036	0.176728
$206$	−0.689246	−0.0480220
$207$	0	0
$208$	−3.97292	−0.275473
$209$	12.6849	0.877430
$210$	0	0
$211$	−25.1982	−1.73472	−0.867359	−	0.497684i	$-0.834184\pi$
$211$	−25.1982	−1.73472	−0.867359	+	0.497684i	$0.834184\pi$
$212$	24.9372	1.71270
$213$	0	0
$214$	−0.898687	−0.0614330
$215$	−4.56607	−0.311403
$216$	0	0
$217$	−18.8046	−1.27654
$218$	0.0912525	0.00618040
$219$	0	0
$220$	6.40978	0.432147
$221$	−4.77678	−0.321321
$222$	0	0
$223$	7.58438	0.507888	0.253944	−	0.967219i	$-0.418272\pi$
$223$	7.58438	0.507888	0.253944	+	0.967219i	$0.418272\pi$
$224$	−1.98071	−0.132342
$225$	0	0
$226$	−0.494404	−0.0328873
$227$	−0.301037	−0.0199806	−0.00999028	−	0.999950i	$-0.503180\pi$
$227$	−0.301037	−0.0199806	−0.00999028	+	0.999950i	$0.503180\pi$
$228$	0	0
$229$	−4.63317	−0.306168	−0.153084	−	0.988213i	$-0.548921\pi$
$229$	−4.63317	−0.306168	−0.153084	+	0.988213i	$0.548921\pi$
$230$	0.286938	0.0189201
$231$	0	0
$232$	0.619250	0.0406558
$233$	−5.62440	−0.368466	−0.184233	−	0.982883i	$-0.558980\pi$
$233$	−5.62440	−0.368466	−0.184233	+	0.982883i	$0.558980\pi$
$234$	0	0
$235$	6.26567	0.408727
$236$	5.68750	0.370225
$237$	0	0
$238$	−0.790831	−0.0512620
$239$	−2.56185	−0.165712	−0.0828562	−	0.996562i	$-0.526404\pi$
$239$	−2.56185	−0.165712	−0.0828562	+	0.996562i	$0.526404\pi$
$240$	0	0
$241$	−14.9830	−0.965139	−0.482569	−	0.875858i	$-0.660296\pi$
$241$	−14.9830	−0.965139	−0.482569	+	0.875858i	$0.660296\pi$
$242$	0.0458409	0.00294676
$243$	0	0
$244$	−25.3426	−1.62240
$245$	−0.930808	−0.0594671
$246$	0	0
$247$	3.94903	0.251271
$248$	−2.04951	−0.130144
$249$	0	0
$250$	−0.0672022	−0.00425024
$251$	7.27387	0.459123	0.229561	−	0.973294i	$-0.426271\pi$
$251$	7.27387	0.459123	0.229561	+	0.973294i	$0.426271\pi$
$252$	0	0
$253$	13.7151	0.862262
$254$	0.401663	0.0252026
$255$	0	0
$256$	15.6399	0.977496
$257$	9.06613	0.565530	0.282765	−	0.959189i	$-0.408748\pi$
$257$	9.06613	0.565530	0.282765	+	0.959189i	$0.408748\pi$
$258$	0	0
$259$	12.0048	0.745944
$260$	1.99548	0.123755
$261$	0	0
$262$	1.20113	0.0742061
$263$	−27.5384	−1.69809	−0.849045	−	0.528320i	$-0.822822\pi$
$263$	−27.5384	−1.69809	−0.849045	+	0.528320i	$0.822822\pi$
$264$	0	0
$265$	−12.4968	−0.767674
$266$	0.653792	0.0400866
$267$	0	0
$268$	26.9534	1.64644
$269$	6.82403	0.416069	0.208034	−	0.978122i	$-0.433293\pi$
$269$	6.82403	0.416069	0.208034	+	0.978122i	$0.433293\pi$
$270$	0	0
$271$	1.91734	0.116470	0.0582352	−	0.998303i	$-0.481453\pi$
$271$	1.91734	0.116470	0.0582352	+	0.998303i	$0.481453\pi$
$272$	18.9778	1.15070
$273$	0	0
$274$	0.705157	0.0426001
$275$	−3.21214	−0.193700
$276$	0	0
$277$	−27.2990	−1.64024	−0.820119	−	0.572193i	$-0.806093\pi$
$277$	−27.2990	−1.64024	−0.820119	+	0.572193i	$0.806093\pi$
$278$	0.719524	0.0431542
$279$	0	0
$280$	0.661482	0.0395311
$281$	0.656824	0.0391828	0.0195914	−	0.999808i	$-0.493763\pi$
$281$	0.656824	0.0391828	0.0195914	+	0.999808i	$0.493763\pi$
$282$	0	0
$283$	−5.01978	−0.298395	−0.149198	−	0.988807i	$-0.547669\pi$
$283$	−5.01978	−0.298395	−0.149198	+	0.988807i	$0.547669\pi$
$284$	15.5493	0.922680
$285$	0	0
$286$	−0.215863	−0.0127643
$287$	6.23373	0.367966
$288$	0	0
$289$	5.81760	0.342212
$290$	−0.154987	−0.00910118
$291$	0	0
$292$	−4.52567	−0.264845
$293$	12.2237	0.714114	0.357057	−	0.934083i	$-0.383780\pi$
$293$	12.2237	0.714114	0.357057	+	0.934083i	$0.383780\pi$
$294$	0	0
$295$	−2.85019	−0.165944
$296$	1.30841	0.0760497
$297$	0	0
$298$	−1.36548	−0.0791004
$299$	4.26977	0.246927
$300$	0	0
$301$	−11.2489	−0.648373
$302$	1.14391	0.0658249
$303$	0	0
$304$	−15.6892	−0.899838
$305$	12.7000	0.727200
$306$	0	0
$307$	−1.01733	−0.0580619	−0.0290309	−	0.999579i	$-0.509242\pi$
$307$	−1.01733	−0.0580619	−0.0290309	+	0.999579i	$0.509242\pi$
$308$	15.7910	0.899774
$309$	0	0
$310$	0.512957	0.0291340
$311$	22.2892	1.26390	0.631952	−	0.775008i	$-0.282254\pi$
$311$	22.2892	1.26390	0.631952	+	0.775008i	$0.282254\pi$
$312$	0	0
$313$	−23.7748	−1.34383	−0.671915	−	0.740628i	$-0.734528\pi$
$313$	−23.7748	−1.34383	−0.671915	+	0.740628i	$0.734528\pi$
$314$	0.0878917	0.00496001
$315$	0	0
$316$	−18.2255	−1.02526
$317$	−28.0373	−1.57473	−0.787366	−	0.616485i	$-0.788556\pi$
$317$	−28.0373	−1.57473	−0.787366	+	0.616485i	$0.788556\pi$
$318$	0	0
$319$	−7.40812	−0.414775
$320$	−7.89182	−0.441166
$321$	0	0
$322$	0.706893	0.0393936
$323$	−18.8636	−1.04960
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0.659334	0.0365172
$327$	0	0
$328$	0.679416	0.0375145
$329$	15.4359	0.851011
$330$	0	0
$331$	−17.8604	−0.981694	−0.490847	−	0.871246i	$-0.663313\pi$
$331$	−17.8604	−0.981694	−0.490847	+	0.871246i	$0.663313\pi$
$332$	20.9763	1.15122
$333$	0	0
$334$	0.549372	0.0300603
$335$	−13.5072	−0.737976
$336$	0	0
$337$	9.77161	0.532294	0.266147	−	0.963932i	$-0.414249\pi$
$337$	9.77161	0.532294	0.266147	+	0.963932i	$0.414249\pi$
$338$	−0.0672022	−0.00365532
$339$	0	0
$340$	−9.53198	−0.516944
$341$	24.5184	1.32775
$342$	0	0
$343$	−19.5381	−1.05496
$344$	−1.22601	−0.0661023
$345$	0	0
$346$	−0.429940	−0.0231137
$347$	−21.0145	−1.12812	−0.564059	−	0.825735i	$-0.690761\pi$
$347$	−21.0145	−1.12812	−0.564059	+	0.825735i	$0.690761\pi$
$348$	0	0
$349$	−9.23952	−0.494580	−0.247290	−	0.968941i	$-0.579540\pi$
$349$	−9.23952	−0.494580	−0.247290	+	0.968941i	$0.579540\pi$
$350$	−0.165558	−0.00884942
$351$	0	0
$352$	2.58256	0.137651
$353$	−4.50725	−0.239896	−0.119948	−	0.992780i	$-0.538273\pi$
$353$	−4.50725	−0.239896	−0.119948	+	0.992780i	$0.538273\pi$
$354$	0	0
$355$	−7.79224	−0.413569
$356$	1.92886	0.102229
$357$	0	0
$358$	−1.05048	−0.0555195
$359$	−25.2752	−1.33397	−0.666986	−	0.745070i	$-0.732416\pi$
$359$	−25.2752	−1.33397	−0.666986	+	0.745070i	$0.732416\pi$
$360$	0	0
$361$	−3.40514	−0.179218
$362$	−0.746164	−0.0392175
$363$	0	0
$364$	4.91602	0.257669
$365$	2.26796	0.118710
$366$	0	0
$367$	−11.1067	−0.579766	−0.289883	−	0.957062i	$-0.593616\pi$
$367$	−11.1067	−0.579766	−0.289883	+	0.957062i	$0.593616\pi$
$368$	−16.9635	−0.884282
$369$	0	0
$370$	−0.327472	−0.0170244
$371$	−30.7868	−1.59837
$372$	0	0
$373$	−7.73645	−0.400578	−0.200289	−	0.979737i	$-0.564188\pi$
$373$	−7.73645	−0.400578	−0.200289	+	0.979737i	$0.564188\pi$
$374$	1.03113	0.0533185
$375$	0	0
$376$	1.68237	0.0867614
$377$	−2.30629	−0.118780
$378$	0	0
$379$	26.0870	1.34000	0.669999	−	0.742362i	$-0.266295\pi$
$379$	26.0870	1.34000	0.669999	+	0.742362i	$0.266295\pi$
$380$	7.88023	0.404247
$381$	0	0
$382$	1.04718	0.0535781
$383$	0.842569	0.0430533	0.0215266	−	0.999768i	$-0.493147\pi$
$383$	0.842569	0.0430533	0.0215266	+	0.999768i	$0.493147\pi$
$384$	0	0
$385$	−7.91335	−0.403302
$386$	1.25641	0.0639497
$387$	0	0
$388$	37.4760	1.90256
$389$	8.75451	0.443871	0.221936	−	0.975061i	$-0.428762\pi$
$389$	8.75451	0.443871	0.221936	+	0.975061i	$0.428762\pi$
$390$	0	0
$391$	−20.3958	−1.03146
$392$	−0.249927	−0.0126232
$393$	0	0
$394$	−0.685626	−0.0345413
$395$	9.13335	0.459549
$396$	0	0
$397$	−6.72904	−0.337721	−0.168860	−	0.985640i	$-0.554009\pi$
$397$	−6.72904	−0.337721	−0.168860	+	0.985640i	$0.554009\pi$
$398$	1.10748	0.0555130
$399$	0	0
$400$	3.97292	0.198646
$401$	28.8942	1.44291	0.721453	−	0.692463i	$-0.243474\pi$
$401$	28.8942	1.44291	0.721453	+	0.692463i	$0.243474\pi$
$402$	0	0
$403$	7.63305	0.380229
$404$	−37.0311	−1.84237
$405$	0	0
$406$	−0.381823	−0.0189496
$407$	−15.6526	−0.775869
$408$	0	0
$409$	−18.3885	−0.909254	−0.454627	−	0.890682i	$-0.650227\pi$
$409$	−18.3885	−0.909254	−0.454627	+	0.890682i	$0.650227\pi$
$410$	−0.170046	−0.00839797
$411$	0	0
$412$	−20.4663	−1.00830
$413$	−7.02164	−0.345513
$414$	0	0
$415$	−10.5119	−0.516008
$416$	0.804000	0.0394193
$417$	0	0
$418$	−0.852450	−0.0416947
$419$	25.2007	1.23113	0.615567	−	0.788085i	$-0.288927\pi$
$419$	25.2007	1.23113	0.615567	+	0.788085i	$0.288927\pi$
$420$	0	0
$421$	−17.8645	−0.870664	−0.435332	−	0.900270i	$-0.643369\pi$
$421$	−17.8645	−0.870664	−0.435332	+	0.900270i	$0.643369\pi$
$422$	1.69338	0.0824322
$423$	0	0
$424$	−3.35546	−0.162956
$425$	4.77678	0.231708
$426$	0	0
$427$	31.2874	1.51410
$428$	−26.6854	−1.28989
$429$	0	0
$430$	0.306850	0.0147976
$431$	−28.4290	−1.36937	−0.684687	−	0.728837i	$-0.740061\pi$
$431$	−28.4290	−1.36937	−0.684687	+	0.728837i	$0.740061\pi$
$432$	0	0
$433$	−20.4163	−0.981146	−0.490573	−	0.871400i	$-0.663212\pi$
$433$	−20.4163	−0.981146	−0.490573	+	0.871400i	$0.663212\pi$
$434$	1.26371	0.0606599
$435$	0	0
$436$	2.70963	0.129768
$437$	16.8615	0.806593
$438$	0	0
$439$	9.82504	0.468924	0.234462	−	0.972125i	$-0.424667\pi$
$439$	9.82504	0.468924	0.234462	+	0.972125i	$0.424667\pi$
$440$	−0.862477	−0.0411170
$441$	0	0
$442$	0.321010	0.0152689
$443$	19.6477	0.933489	0.466745	−	0.884392i	$-0.345427\pi$
$443$	19.6477	0.933489	0.466745	+	0.884392i	$0.345427\pi$
$444$	0	0
$445$	−0.966612	−0.0458218
$446$	−0.509687	−0.0241344
$447$	0	0
$448$	−19.4421	−0.918551
$449$	−27.3713	−1.29173	−0.645866	−	0.763451i	$-0.723504\pi$
$449$	−27.3713	−1.29173	−0.645866	+	0.763451i	$0.723504\pi$
$450$	0	0
$451$	−8.12789	−0.382727
$452$	−14.6807	−0.690522
$453$	0	0
$454$	0.0202304	0.000949459 0
$455$	−2.46357	−0.115494
$456$	0	0
$457$	−24.4583	−1.14411	−0.572056	−	0.820215i	$-0.693854\pi$
$457$	−24.4583	−1.14411	−0.572056	+	0.820215i	$0.693854\pi$
$458$	0.311359	0.0145488
$459$	0	0
$460$	8.52026	0.397259
$461$	−8.76073	−0.408028	−0.204014	−	0.978968i	$-0.565399\pi$
$461$	−8.76073	−0.408028	−0.204014	+	0.978968i	$0.565399\pi$
$462$	0	0
$463$	15.1032	0.701905	0.350952	−	0.936393i	$-0.385858\pi$
$463$	15.1032	0.701905	0.350952	+	0.936393i	$0.385858\pi$
$464$	9.16270	0.425368
$465$	0	0
$466$	0.377972	0.0175092
$467$	19.5484	0.904591	0.452296	−	0.891868i	$-0.350605\pi$
$467$	19.5484	0.904591	0.452296	+	0.891868i	$0.350605\pi$
$468$	0	0
$469$	−33.2759	−1.53654
$470$	−0.421067	−0.0194224
$471$	0	0
$472$	−0.765290	−0.0352253
$473$	14.6669	0.674384
$474$	0	0
$475$	−3.94903	−0.181194
$476$	−23.4827	−1.07633
$477$	0	0
$478$	0.172162	0.00787451
$479$	−1.30713	−0.0597242	−0.0298621	−	0.999554i	$-0.509507\pi$
$479$	−1.30713	−0.0597242	−0.0298621	+	0.999554i	$0.509507\pi$
$480$	0	0
$481$	−4.87293	−0.222187
$482$	1.00689	0.0458626
$483$	0	0
$484$	1.36119	0.0618721
$485$	−18.7804	−0.852774
$486$	0	0
$487$	30.3533	1.37544	0.687719	−	0.725977i	$-0.258612\pi$
$487$	30.3533	1.37544	0.687719	+	0.725977i	$0.258612\pi$
$488$	3.41002	0.154364
$489$	0	0
$490$	0.0625523	0.00282583
$491$	−8.37717	−0.378056	−0.189028	−	0.981972i	$-0.560534\pi$
$491$	−8.37717	−0.378056	−0.189028	+	0.981972i	$0.560534\pi$
$492$	0	0
$493$	11.0166	0.496163
$494$	−0.265384	−0.0119402
$495$	0	0
$496$	−30.3255	−1.36166
$497$	−19.1967	−0.861092
$498$	0	0
$499$	12.8603	0.575704	0.287852	−	0.957675i	$-0.407059\pi$
$499$	12.8603	0.575704	0.287852	+	0.957675i	$0.407059\pi$
$500$	−1.99548	−0.0892408
$501$	0	0
$502$	−0.488820	−0.0218171
$503$	−5.22447	−0.232947	−0.116474	−	0.993194i	$-0.537159\pi$
$503$	−5.22447	−0.232947	−0.116474	+	0.993194i	$0.537159\pi$
$504$	0	0
$505$	18.5575	0.825796
$506$	−0.921686	−0.0409739
$507$	0	0
$508$	11.9269	0.529169
$509$	−10.0224	−0.444234	−0.222117	−	0.975020i	$-0.571297\pi$
$509$	−10.0224	−0.444234	−0.222117	+	0.975020i	$0.571297\pi$
$510$	0	0
$511$	5.58728	0.247167
$512$	−5.32773	−0.235455
$513$	0	0
$514$	−0.609264	−0.0268735
$515$	10.2563	0.451947
$516$	0	0
$517$	−20.1262	−0.885151
$518$	−0.806751	−0.0354466
$519$	0	0
$520$	−0.268505	−0.0117747
$521$	−20.3192	−0.890202	−0.445101	−	0.895480i	$-0.646832\pi$
$521$	−20.3192	−0.890202	−0.445101	+	0.895480i	$0.646832\pi$
$522$	0	0
$523$	−0.572523	−0.0250347	−0.0125173	−	0.999922i	$-0.503984\pi$
$523$	−0.572523	−0.0250347	−0.0125173	+	0.999922i	$0.503984\pi$
$524$	35.6661	1.55808
$525$	0	0
$526$	1.85064	0.0806918
$527$	−36.4614	−1.58828
$528$	0	0
$529$	−4.76905	−0.207350
$530$	0.839814	0.0364792
$531$	0	0
$532$	19.4135	0.841683
$533$	−2.53036	−0.109602
$534$	0	0
$535$	13.3729	0.578160
$536$	−3.62675	−0.156652
$537$	0	0
$538$	−0.458590	−0.0197712
$539$	2.98989	0.128784
$540$	0	0
$541$	−23.5766	−1.01364	−0.506819	−	0.862052i	$-0.669179\pi$
$541$	−23.5766	−1.01364	−0.506819	+	0.862052i	$0.669179\pi$
$542$	−0.128850	−0.00553457
$543$	0	0
$544$	−3.84053	−0.164661
$545$	−1.35788	−0.0581652
$546$	0	0
$547$	−3.75708	−0.160641	−0.0803205	−	0.996769i	$-0.525594\pi$
$547$	−3.75708	−0.160641	−0.0803205	+	0.996769i	$0.525594\pi$
$548$	20.9387	0.894458
$549$	0	0
$550$	0.215863	0.00920443
$551$	−9.10760	−0.387997
$552$	0	0
$553$	22.5007	0.956826
$554$	1.83455	0.0779427
$555$	0	0
$556$	21.3654	0.906093
$557$	39.9225	1.69157	0.845785	−	0.533523i	$-0.179132\pi$
$557$	39.9225	1.69157	0.845785	+	0.533523i	$0.179132\pi$
$558$	0	0
$559$	4.56607	0.193124
$560$	9.78759	0.413601
$561$	0	0
$562$	−0.0441400	−0.00186193
$563$	−33.5570	−1.41426	−0.707130	−	0.707084i	$-0.750010\pi$
$563$	−33.5570	−1.41426	−0.707130	+	0.707084i	$0.750010\pi$
$564$	0	0
$565$	7.35696	0.309510
$566$	0.337340	0.0141795
$567$	0	0
$568$	−2.09226	−0.0877892
$569$	−15.4337	−0.647016	−0.323508	−	0.946225i	$-0.604862\pi$
$569$	−15.4337	−0.647016	−0.323508	+	0.946225i	$0.604862\pi$
$570$	0	0
$571$	39.9382	1.67136	0.835681	−	0.549215i	$-0.185073\pi$
$571$	39.9382	1.67136	0.835681	+	0.549215i	$0.185073\pi$
$572$	−6.40978	−0.268006
$573$	0	0
$574$	−0.418921	−0.0174854
$575$	−4.26977	−0.178062
$576$	0	0
$577$	25.4359	1.05891	0.529455	−	0.848338i	$-0.322396\pi$
$577$	25.4359	1.05891	0.529455	+	0.848338i	$0.322396\pi$
$578$	−0.390956	−0.0162616
$579$	0	0
$580$	−4.60216	−0.191094
$581$	−25.8968	−1.07438
$582$	0	0
$583$	40.1416	1.66250
$584$	0.608958	0.0251989
$585$	0	0
$586$	−0.821457	−0.0339341
$587$	40.9279	1.68927	0.844637	−	0.535339i	$-0.179816\pi$
$587$	40.9279	1.68927	0.844637	+	0.535339i	$0.179816\pi$
$588$	0	0
$589$	30.1431	1.24203
$590$	0.191539	0.00788553
$591$	0	0
$592$	19.3598	0.795683
$593$	1.00808	0.0413971	0.0206985	−	0.999786i	$-0.493411\pi$
$593$	1.00808	0.0413971	0.0206985	+	0.999786i	$0.493411\pi$
$594$	0	0
$595$	11.7679	0.482439
$596$	−40.5463	−1.66084
$597$	0	0
$598$	−0.286938	−0.0117338
$599$	−1.87952	−0.0767949	−0.0383975	−	0.999263i	$-0.512225\pi$
$599$	−1.87952	−0.0767949	−0.0383975	+	0.999263i	$0.512225\pi$
$600$	0	0
$601$	−2.94528	−0.120140	−0.0600702	−	0.998194i	$-0.519132\pi$
$601$	−2.94528	−0.120140	−0.0600702	+	0.998194i	$0.519132\pi$
$602$	0.755947	0.0308101
$603$	0	0
$604$	33.9671	1.38210
$605$	−0.682134	−0.0277327
$606$	0	0
$607$	1.34069	0.0544171	0.0272085	−	0.999630i	$-0.491338\pi$
$607$	1.34069	0.0544171	0.0272085	+	0.999630i	$0.491338\pi$
$608$	3.17502	0.128764
$609$	0	0
$610$	−0.853468	−0.0345559
$611$	−6.26567	−0.253482
$612$	0	0
$613$	−20.1796	−0.815046	−0.407523	−	0.913195i	$-0.633607\pi$
$613$	−20.1796	−0.815046	−0.407523	+	0.913195i	$0.633607\pi$
$614$	0.0683665	0.00275905
$615$	0	0
$616$	−2.12478	−0.0856097
$617$	7.13478	0.287235	0.143618	−	0.989633i	$-0.454126\pi$
$617$	7.13478	0.287235	0.143618	+	0.989633i	$0.454126\pi$
$618$	0	0
$619$	−30.8954	−1.24179	−0.620895	−	0.783894i	$-0.713231\pi$
$619$	−30.8954	−1.24179	−0.620895	+	0.783894i	$0.713231\pi$
$620$	15.2316	0.611717
$621$	0	0
$622$	−1.49788	−0.0600596
$623$	−2.38132	−0.0954055
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.59772	0.0638576
$627$	0	0
$628$	2.60983	0.104144
$629$	23.2769	0.928112
$630$	0	0
$631$	11.8799	0.472932	0.236466	−	0.971640i	$-0.424011\pi$
$631$	11.8799	0.472932	0.236466	+	0.971640i	$0.424011\pi$
$632$	2.45235	0.0975494
$633$	0	0
$634$	1.88417	0.0748299
$635$	−5.97693	−0.237187
$636$	0	0
$637$	0.930808	0.0368800
$638$	0.497842	0.0197098
$639$	0	0
$640$	2.13835	0.0845256
$641$	−25.1307	−0.992605	−0.496302	−	0.868150i	$-0.665309\pi$
$641$	−25.1307	−0.992605	−0.496302	+	0.868150i	$0.665309\pi$
$642$	0	0
$643$	−37.8971	−1.49452	−0.747258	−	0.664534i	$-0.768630\pi$
$643$	−37.8971	−1.49452	−0.747258	+	0.664534i	$0.768630\pi$
$644$	20.9903	0.827133
$645$	0	0
$646$	1.26768	0.0498761
$647$	5.59065	0.219791	0.109896	−	0.993943i	$-0.464948\pi$
$647$	5.59065	0.219791	0.109896	+	0.993943i	$0.464948\pi$
$648$	0	0
$649$	9.15521	0.359373
$650$	0.0672022	0.00263589
$651$	0	0
$652$	19.5781	0.766737
$653$	8.58496	0.335955	0.167978	−	0.985791i	$-0.446276\pi$
$653$	8.58496	0.335955	0.167978	+	0.985791i	$0.446276\pi$
$654$	0	0
$655$	−17.8734	−0.698371
$656$	10.0529	0.392501
$657$	0	0
$658$	−1.03733	−0.0404393
$659$	19.7197	0.768172	0.384086	−	0.923297i	$-0.374517\pi$
$659$	19.7197	0.768172	0.384086	+	0.923297i	$0.374517\pi$
$660$	0	0
$661$	35.3485	1.37490	0.687448	−	0.726234i	$-0.258731\pi$
$661$	35.3485	1.37490	0.687448	+	0.726234i	$0.258731\pi$
$662$	1.20025	0.0466492
$663$	0	0
$664$	−2.82250	−0.109534
$665$	−9.72873	−0.377264
$666$	0	0
$667$	−9.84732	−0.381290
$668$	16.3129	0.631165
$669$	0	0
$670$	0.907712	0.0350680
$671$	−40.7942	−1.57484
$672$	0	0
$673$	17.0655	0.657826	0.328913	−	0.944360i	$-0.393318\pi$
$673$	17.0655	0.657826	0.328913	+	0.944360i	$0.393318\pi$
$674$	−0.656674	−0.0252941
$675$	0	0
$676$	−1.99548	−0.0767494
$677$	11.0885	0.426164	0.213082	−	0.977034i	$-0.431650\pi$
$677$	11.0885	0.426164	0.213082	+	0.977034i	$0.431650\pi$
$678$	0	0
$679$	−46.2669	−1.77556
$680$	1.28259	0.0491851
$681$	0	0
$682$	−1.64769	−0.0630934
$683$	42.5037	1.62636	0.813179	−	0.582014i	$-0.197735\pi$
$683$	42.5037	1.62636	0.813179	+	0.582014i	$0.197735\pi$
$684$	0	0
$685$	−10.4931	−0.400919
$686$	1.31300	0.0501308
$687$	0	0
$688$	−18.1407	−0.691606
$689$	12.4968	0.476091
$690$	0	0
$691$	1.31266	0.0499361	0.0249681	−	0.999688i	$-0.492052\pi$
$691$	1.31266	0.0499361	0.0249681	+	0.999688i	$0.492052\pi$
$692$	−12.7665	−0.485310
$693$	0	0
$694$	1.41222	0.0536072
$695$	−10.7069	−0.406134
$696$	0	0
$697$	12.0870	0.457827
$698$	0.620916	0.0235020
$699$	0	0
$700$	−4.91602	−0.185808
$701$	−21.2054	−0.800916	−0.400458	−	0.916315i	$-0.631149\pi$
$701$	−21.2054	−0.800916	−0.400458	+	0.916315i	$0.631149\pi$
$702$	0	0
$703$	−19.2434	−0.725778
$704$	25.3496	0.955401
$705$	0	0
$706$	0.302897	0.0113997
$707$	45.7176	1.71939
$708$	0	0
$709$	49.2060	1.84797	0.923985	−	0.382429i	$-0.124912\pi$
$709$	49.2060	1.84797	0.923985	+	0.382429i	$0.124912\pi$
$710$	0.523655	0.0196524
$711$	0	0
$712$	−0.259540	−0.00972669
$713$	32.5914	1.22056
$714$	0	0
$715$	3.21214	0.120127
$716$	−31.1926	−1.16572
$717$	0	0
$718$	1.69855	0.0633892
$719$	31.1080	1.16013	0.580067	−	0.814569i	$-0.303026\pi$
$719$	31.1080	1.16013	0.580067	+	0.814569i	$0.303026\pi$
$720$	0	0
$721$	25.2671	0.940998
$722$	0.228833	0.00851629
$723$	0	0
$724$	−22.1564	−0.823436
$725$	2.30629	0.0856533
$726$	0	0
$727$	30.0940	1.11612	0.558062	−	0.829799i	$-0.311545\pi$
$727$	30.0940	1.11612	0.558062	+	0.829799i	$0.311545\pi$
$728$	−0.661482	−0.0245162
$729$	0	0
$730$	−0.152412	−0.00564101
$731$	−21.8111	−0.806713
$732$	0	0
$733$	−12.9997	−0.480156	−0.240078	−	0.970754i	$-0.577173\pi$
$733$	−12.9997	−0.480156	−0.240078	+	0.970754i	$0.577173\pi$
$734$	0.746396	0.0275500
$735$	0	0
$736$	3.43290	0.126538
$737$	43.3870	1.59818
$738$	0	0
$739$	43.9402	1.61637	0.808183	−	0.588932i	$-0.200451\pi$
$739$	43.9402	1.61637	0.808183	+	0.588932i	$0.200451\pi$
$740$	−9.72386	−0.357456
$741$	0	0
$742$	2.06894	0.0759533
$743$	48.8508	1.79216	0.896081	−	0.443890i	$-0.146402\pi$
$743$	48.8508	1.79216	0.896081	+	0.443890i	$0.146402\pi$
$744$	0	0
$745$	20.3190	0.744432
$746$	0.519907	0.0190351
$747$	0	0
$748$	30.6181	1.11951
$749$	32.9451	1.20379
$750$	0	0
$751$	−33.4142	−1.21930	−0.609651	−	0.792670i	$-0.708690\pi$
$751$	−33.4142	−1.21930	−0.609651	+	0.792670i	$0.708690\pi$
$752$	24.8930	0.907756
$753$	0	0
$754$	0.154987	0.00564431
$755$	−17.0220	−0.619493
$756$	0	0
$757$	45.6922	1.66071	0.830356	−	0.557233i	$-0.188137\pi$
$757$	45.6922	1.66071	0.830356	+	0.557233i	$0.188137\pi$
$758$	−1.75310	−0.0636755
$759$	0	0
$760$	−1.06034	−0.0384624
$761$	4.87858	0.176848	0.0884242	−	0.996083i	$-0.471817\pi$
$761$	4.87858	0.176848	0.0884242	+	0.996083i	$0.471817\pi$
$762$	0	0
$763$	−3.34524	−0.121106
$764$	31.0945	1.12496
$765$	0	0
$766$	−0.0566225	−0.00204585
$767$	2.85019	0.102914
$768$	0	0
$769$	11.2596	0.406030	0.203015	−	0.979176i	$-0.434926\pi$
$769$	11.2596	0.406030	0.203015	+	0.979176i	$0.434926\pi$
$770$	0.531794	0.0191645
$771$	0	0
$772$	37.3076	1.34273
$773$	−17.8461	−0.641880	−0.320940	−	0.947100i	$-0.603999\pi$
$773$	−17.8461	−0.641880	−0.320940	+	0.947100i	$0.603999\pi$
$774$	0	0
$775$	−7.63305	−0.274187
$776$	−5.04264	−0.181020
$777$	0	0
$778$	−0.588323	−0.0210924
$779$	−9.99248	−0.358018
$780$	0	0
$781$	25.0298	0.895636
$782$	1.37064	0.0490140
$783$	0	0
$784$	−3.69803	−0.132072
$785$	−1.30787	−0.0466799
$786$	0	0
$787$	24.7113	0.880863	0.440432	−	0.897786i	$-0.354825\pi$
$787$	24.7113	0.880863	0.440432	+	0.897786i	$0.354825\pi$
$788$	−20.3588	−0.725252
$789$	0	0
$790$	−0.613781	−0.0218374
$791$	18.1244	0.644430
$792$	0	0
$793$	−12.7000	−0.450990
$794$	0.452206	0.0160482
$795$	0	0
$796$	32.8852	1.16559
$797$	−39.7377	−1.40758	−0.703791	−	0.710407i	$-0.748511\pi$
$797$	−39.7377	−1.40758	−0.703791	+	0.710407i	$0.748511\pi$
$798$	0	0
$799$	29.9297	1.05884
$800$	−0.804000	−0.0284257
$801$	0	0
$802$	−1.94175	−0.0685657
$803$	−7.28500	−0.257082
$804$	0	0
$805$	−10.5189	−0.370742
$806$	−0.512957	−0.0180682
$807$	0	0
$808$	4.98277	0.175293
$809$	−40.0717	−1.40885	−0.704423	−	0.709780i	$-0.748794\pi$
$809$	−40.0717	−1.40885	−0.704423	+	0.709780i	$0.748794\pi$
$810$	0	0
$811$	49.6910	1.74489	0.872444	−	0.488714i	$-0.162534\pi$
$811$	49.6910	1.74489	0.872444	+	0.488714i	$0.162534\pi$
$812$	−11.3377	−0.397877
$813$	0	0
$814$	1.05189	0.0368686
$815$	−9.81120	−0.343671
$816$	0	0
$817$	18.0316	0.630845
$818$	1.23575	0.0432070
$819$	0	0
$820$	−5.04930	−0.176329
$821$	−18.8802	−0.658924	−0.329462	−	0.944169i	$-0.606867\pi$
$821$	−18.8802	−0.658924	−0.329462	+	0.944169i	$0.606867\pi$
$822$	0	0
$823$	−28.4726	−0.992494	−0.496247	−	0.868182i	$-0.665289\pi$
$823$	−28.4726	−0.992494	−0.496247	+	0.868182i	$0.665289\pi$
$824$	2.75387	0.0959356
$825$	0	0
$826$	0.471870	0.0164185
$827$	15.9961	0.556239	0.278120	−	0.960546i	$-0.410289\pi$
$827$	15.9961	0.556239	0.278120	+	0.960546i	$0.410289\pi$
$828$	0	0
$829$	20.7729	0.721474	0.360737	−	0.932668i	$-0.382525\pi$
$829$	20.7729	0.721474	0.360737	+	0.932668i	$0.382525\pi$
$830$	0.706422	0.0245203
$831$	0	0
$832$	7.89182	0.273600
$833$	−4.44626	−0.154054
$834$	0	0
$835$	−8.17491	−0.282904
$836$	−25.3124	−0.875449
$837$	0	0
$838$	−1.69354	−0.0585024
$839$	17.1840	0.593259	0.296630	−	0.954993i	$-0.404137\pi$
$839$	17.1840	0.593259	0.296630	+	0.954993i	$0.404137\pi$
$840$	0	0
$841$	−23.6810	−0.816588
$842$	1.20054	0.0413732
$843$	0	0
$844$	50.2826	1.73080
$845$	1.00000	0.0344010
$846$	0	0
$847$	−1.68049	−0.0577422
$848$	−49.6489	−1.70495
$849$	0	0
$850$	−0.321010	−0.0110105
$851$	−20.8063	−0.713231
$852$	0	0
$853$	33.0733	1.13241	0.566203	−	0.824266i	$-0.308412\pi$
$853$	33.0733	1.13241	0.566203	+	0.824266i	$0.308412\pi$
$854$	−2.10258	−0.0719488
$855$	0	0
$856$	3.59069	0.122727
$857$	39.6295	1.35372	0.676859	−	0.736113i	$-0.263341\pi$
$857$	39.6295	1.35372	0.676859	+	0.736113i	$0.263341\pi$
$858$	0	0
$859$	−1.17288	−0.0400183	−0.0200091	−	0.999800i	$-0.506370\pi$
$859$	−1.17288	−0.0400183	−0.0200091	+	0.999800i	$0.506370\pi$
$860$	9.11152	0.310700
$861$	0	0
$862$	1.91049	0.0650715
$863$	54.3303	1.84943	0.924713	−	0.380665i	$-0.124305\pi$
$863$	54.3303	1.84943	0.924713	+	0.380665i	$0.124305\pi$
$864$	0	0
$865$	6.39771	0.217529
$866$	1.37202	0.0466232
$867$	0	0
$868$	37.5242	1.27365
$869$	−29.3376	−0.995211
$870$	0	0
$871$	13.5072	0.457673
$872$	−0.364598	−0.0123468
$873$	0	0
$874$	−1.13313	−0.0383286
$875$	2.46357	0.0832840
$876$	0	0
$877$	47.7735	1.61319	0.806597	−	0.591101i	$-0.201307\pi$
$877$	47.7735	1.61319	0.806597	+	0.591101i	$0.201307\pi$
$878$	−0.660265	−0.0222828
$879$	0	0
$880$	−12.7616	−0.430193
$881$	−28.0225	−0.944101	−0.472050	−	0.881572i	$-0.656486\pi$
$881$	−28.0225	−0.944101	−0.472050	+	0.881572i	$0.656486\pi$
$882$	0	0
$883$	−2.71952	−0.0915191	−0.0457595	−	0.998952i	$-0.514571\pi$
$883$	−2.71952	−0.0915191	−0.0457595	+	0.998952i	$0.514571\pi$
$884$	9.53198	0.320595
$885$	0	0
$886$	−1.32037	−0.0443586
$887$	4.76247	0.159908	0.0799541	−	0.996799i	$-0.474523\pi$
$887$	4.76247	0.159908	0.0799541	+	0.996799i	$0.474523\pi$
$888$	0	0
$889$	−14.7246	−0.493848
$890$	0.0649584	0.00217741
$891$	0	0
$892$	−15.1345	−0.506741
$893$	−24.7433	−0.828004
$894$	0	0
$895$	15.6316	0.522507
$896$	5.26797	0.175991
$897$	0	0
$898$	1.83941	0.0613820
$899$	−17.6040	−0.587126
$900$	0	0
$901$	−59.6946	−1.98871
$902$	0.546212	0.0181869
$903$	0	0
$904$	1.97538	0.0657003
$905$	11.1033	0.369085
$906$	0	0
$907$	−23.1462	−0.768556	−0.384278	−	0.923217i	$-0.625550\pi$
$907$	−23.1462	−0.768556	−0.384278	+	0.923217i	$0.625550\pi$
$908$	0.600715	0.0199354
$909$	0	0
$910$	0.165558	0.00548818
$911$	11.5589	0.382963	0.191481	−	0.981496i	$-0.438671\pi$
$911$	11.5589	0.382963	0.191481	+	0.981496i	$0.438671\pi$
$912$	0	0
$913$	33.7657	1.11748
$914$	1.64365	0.0543672
$915$	0	0
$916$	9.24541	0.305477
$917$	−44.0324	−1.45408
$918$	0	0
$919$	−45.9902	−1.51708	−0.758538	−	0.651629i	$-0.774086\pi$
$919$	−45.9902	−1.51708	−0.758538	+	0.651629i	$0.774086\pi$
$920$	−1.14646	−0.0377975
$921$	0	0
$922$	0.588740	0.0193891
$923$	7.79224	0.256485
$924$	0	0
$925$	4.87293	0.160221
$926$	−1.01497	−0.0333539
$927$	0	0
$928$	−1.85425	−0.0608689
$929$	−42.6792	−1.40026	−0.700130	−	0.714016i	$-0.746875\pi$
$929$	−42.6792	−1.40026	−0.700130	+	0.714016i	$0.746875\pi$
$930$	0	0
$931$	3.67579	0.120469
$932$	11.2234	0.367634
$933$	0	0
$934$	−1.31369	−0.0429854
$935$	−15.3437	−0.501792
$936$	0	0
$937$	−47.1687	−1.54093	−0.770467	−	0.637480i	$-0.779977\pi$
$937$	−47.1687	−1.54093	−0.770467	+	0.637480i	$0.779977\pi$
$938$	2.23621	0.0730150
$939$	0	0
$940$	−12.5030	−0.407804
$941$	−40.1783	−1.30977	−0.654887	−	0.755727i	$-0.727284\pi$
$941$	−40.1783	−1.30977	−0.654887	+	0.755727i	$0.727284\pi$
$942$	0	0
$943$	−10.8041	−0.351829
$944$	−11.3236	−0.368551
$945$	0	0
$946$	−0.985646	−0.0320461
$947$	0.805932	0.0261893	0.0130946	−	0.999914i	$-0.495832\pi$
$947$	0.805932	0.0261893	0.0130946	+	0.999914i	$0.495832\pi$
$948$	0	0
$949$	−2.26796	−0.0736210
$950$	0.265384	0.00861018
$951$	0	0
$952$	3.15975	0.102408
$953$	8.33532	0.270007	0.135004	−	0.990845i	$-0.456895\pi$
$953$	8.33532	0.270007	0.135004	+	0.990845i	$0.456895\pi$
$954$	0	0
$955$	−15.5825	−0.504236
$956$	5.11214	0.165338
$957$	0	0
$958$	0.0878419	0.00283804
$959$	−25.8504	−0.834754
$960$	0	0
$961$	27.2634	0.879464
$962$	0.327472	0.0105581
$963$	0	0
$964$	29.8983	0.962960
$965$	−18.6960	−0.601846
$966$	0	0
$967$	−42.1925	−1.35682	−0.678410	−	0.734683i	$-0.737331\pi$
$967$	−42.1925	−1.35682	−0.678410	+	0.734683i	$0.737331\pi$
$968$	−0.183156	−0.00588687
$969$	0	0
$970$	1.26208	0.0405231
$971$	−25.3293	−0.812856	−0.406428	−	0.913683i	$-0.633226\pi$
$971$	−25.3293	−0.812856	−0.406428	+	0.913683i	$0.633226\pi$
$972$	0	0
$973$	−26.3771	−0.845611
$974$	−2.03981	−0.0653596
$975$	0	0
$976$	50.4561	1.61506
$977$	−36.0931	−1.15472	−0.577360	−	0.816489i	$-0.695917\pi$
$977$	−36.0931	−1.15472	−0.577360	+	0.816489i	$0.695917\pi$
$978$	0	0
$979$	3.10490	0.0992329
$980$	1.85741	0.0593329
$981$	0	0
$982$	0.562964	0.0179649
$983$	60.9563	1.94420	0.972102	−	0.234560i	$-0.0753650\pi$
$983$	60.9563	1.94420	0.972102	+	0.234560i	$0.0753650\pi$
$984$	0	0
$985$	10.2024	0.325077
$986$	−0.740341	−0.0235773
$987$	0	0
$988$	−7.88023	−0.250704
$989$	19.4961	0.619939
$990$	0	0
$991$	56.5677	1.79693	0.898466	−	0.439042i	$-0.144682\pi$
$991$	56.5677	1.79693	0.898466	+	0.439042i	$0.144682\pi$
$992$	6.13697	0.194849
$993$	0	0
$994$	1.29006	0.0409183
$995$	−16.4798	−0.522446
$996$	0	0
$997$	−2.37781	−0.0753060	−0.0376530	−	0.999291i	$-0.511988\pi$
$997$	−2.37781	−0.0753060	−0.0376530	+	0.999291i	$0.511988\pi$
$998$	−0.864237	−0.0273569
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.ba.1.6		8
3.2	odd	2		5265.2.a.bf.1.3		8
9.2	odd	6		585.2.i.e.391.6	yes	16
9.4	even	3		1755.2.i.f.586.3		16
9.5	odd	6		585.2.i.e.196.6	✓	16
9.7	even	3		1755.2.i.f.1171.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.e.196.6	✓	16	9.5	odd	6
585.2.i.e.391.6	yes	16	9.2	odd	6
1755.2.i.f.586.3		16	9.4	even	3
1755.2.i.f.1171.3		16	9.7	even	3
5265.2.a.ba.1.6		8	1.1	even	1	trivial
5265.2.a.bf.1.3		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0672022 q^{2} -1.99548 q^{4} +1.00000 q^{5} +2.46357 q^{7} +0.268505 q^{8} +O(q^{10})\)	\(q-0.0672022 q^{2} -1.99548 q^{4} +1.00000 q^{5} +2.46357 q^{7} +0.268505 q^{8} -0.0672022 q^{10} -3.21214 q^{11} -1.00000 q^{13} -0.165558 q^{14} +3.97292 q^{16} +4.77678 q^{17} -3.94903 q^{19} -1.99548 q^{20} +0.215863 q^{22} -4.26977 q^{23} +1.00000 q^{25} +0.0672022 q^{26} -4.91602 q^{28} +2.30629 q^{29} -7.63305 q^{31} -0.804000 q^{32} -0.321010 q^{34} +2.46357 q^{35} +4.87293 q^{37} +0.265384 q^{38} +0.268505 q^{40} +2.53036 q^{41} -4.56607 q^{43} +6.40978 q^{44} +0.286938 q^{46} +6.26567 q^{47} -0.930808 q^{49} -0.0672022 q^{50} +1.99548 q^{52} -12.4968 q^{53} -3.21214 q^{55} +0.661482 q^{56} -0.154987 q^{58} -2.85019 q^{59} +12.7000 q^{61} +0.512957 q^{62} -7.89182 q^{64} -1.00000 q^{65} -13.5072 q^{67} -9.53198 q^{68} -0.165558 q^{70} -7.79224 q^{71} +2.26796 q^{73} -0.327472 q^{74} +7.88023 q^{76} -7.91335 q^{77} +9.13335 q^{79} +3.97292 q^{80} -0.170046 q^{82} -10.5119 q^{83} +4.77678 q^{85} +0.306850 q^{86} -0.862477 q^{88} -0.966612 q^{89} -2.46357 q^{91} +8.52026 q^{92} -0.421067 q^{94} -3.94903 q^{95} -18.7804 q^{97} +0.0625523 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8	\( 8 q - 3 q^{2} + 9 q^{4} + 8 q^{5} - 11 q^{7} + 6 q^{8} - 3 q^{10} - 6 q^{11} - 8 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} - 10 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 8 q^{25} + 3 q^{26} - 34 q^{28} - 14 q^{29} - 31 q^{31} - q^{32} - 7 q^{34} - 11 q^{35} + q^{37} - 9 q^{38} + 6 q^{40} + 12 q^{41} + 15 q^{43} - 16 q^{44} - 32 q^{46} + 18 q^{47} + 17 q^{49} - 3 q^{50} - 9 q^{52} - 2 q^{53} - 6 q^{55} - 16 q^{56} - 42 q^{58} - 24 q^{59} - 9 q^{61} + 20 q^{62} - 30 q^{64} - 8 q^{65} - 18 q^{67} + 14 q^{68} - 10 q^{70} - 10 q^{71} + 6 q^{73} + 37 q^{74} - 53 q^{76} + 34 q^{77} - 3 q^{79} + 11 q^{80} - 34 q^{82} + 10 q^{83} + 2 q^{85} - 60 q^{86} - 14 q^{88} + 13 q^{89} + 11 q^{91} - 5 q^{92} + 17 q^{94} - 10 q^{95} - 34 q^{97} + 30 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 9 * q^4 + 8 * q^5 - 11 * q^7 + 6 * q^8 - 3 * q^10 - 6 * q^11 - 8 * q^13 - 10 * q^14 + 11 * q^16 + 2 * q^17 - 10 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 8 * q^25 + 3 * q^26 - 34 * q^28 - 14 * q^29 - 31 * q^31 - q^32 - 7 * q^34 - 11 * q^35 + q^37 - 9 * q^38 + 6 * q^40 + 12 * q^41 + 15 * q^43 - 16 * q^44 - 32 * q^46 + 18 * q^47 + 17 * q^49 - 3 * q^50 - 9 * q^52 - 2 * q^53 - 6 * q^55 - 16 * q^56 - 42 * q^58 - 24 * q^59 - 9 * q^61 + 20 * q^62 - 30 * q^64 - 8 * q^65 - 18 * q^67 + 14 * q^68 - 10 * q^70 - 10 * q^71 + 6 * q^73 + 37 * q^74 - 53 * q^76 + 34 * q^77 - 3 * q^79 + 11 * q^80 - 34 * q^82 + 10 * q^83 + 2 * q^85 - 60 * q^86 - 14 * q^88 + 13 * q^89 + 11 * q^91 - 5 * q^92 + 17 * q^94 - 10 * q^95 - 34 * q^97 + 30 * q^98

Introduction

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