LMFDB - Embedded newform 6544.2.a.i.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6544,2,Mod(1,6544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6544, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("6544.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6544 = 2^{4} \cdot 409 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6544.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.2541030827$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 126 x^{17} + 100 x^{16} - 1283 x^{15} + 247 x^{14} + 6767 x^{13} + \cdots - 4 $ x^20 - 5x^19 - 19x^18 + 126x^17 + 100x^16 - 1283x^15 + 247x^14 + 6767x^13 - 4554x^12 - 19689x^11 + 18771x^10 + 31011x^9 - 35515x^8 - 23548x^7 + 31466x^6 + 5354x^5 - 10552x^4 + 1129x^3 + 523x^2 - 54*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 409)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-1.38030$ of defining polynomial
Character	$\chi$	$=$	6544.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32806 q^{3} -2.04589 q^{5} +0.300238 q^{7} -1.23626 q^{9} +3.45738 q^{11} -3.89705 q^{13} -2.71707 q^{15} +5.59382 q^{17} +0.669118 q^{19} +0.398734 q^{21} -3.49818 q^{23} -0.814317 q^{25} -5.62600 q^{27} +8.88357 q^{29} -7.37344 q^{31} +4.59161 q^{33} -0.614254 q^{35} -3.54541 q^{37} -5.17552 q^{39} +7.92837 q^{41} -0.985097 q^{43} +2.52925 q^{45} +2.95435 q^{47} -6.90986 q^{49} +7.42893 q^{51} -6.43289 q^{53} -7.07343 q^{55} +0.888629 q^{57} -4.42090 q^{59} -0.236790 q^{61} -0.371170 q^{63} +7.97296 q^{65} -2.33250 q^{67} -4.64579 q^{69} -6.91848 q^{71} +4.76591 q^{73} -1.08146 q^{75} +1.03804 q^{77} -3.80018 q^{79} -3.76290 q^{81} +7.24637 q^{83} -11.4444 q^{85} +11.7979 q^{87} +14.2851 q^{89} -1.17004 q^{91} -9.79237 q^{93} -1.36894 q^{95} -10.8161 q^{97} -4.27421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{3} + 8 q^{5} - 5 q^{7} + 27 q^{9} - 30 q^{11} - 7 q^{13} - 26 q^{15} + 6 q^{17} - q^{19} - 3 q^{21} - 37 q^{23} + 14 q^{25} + 5 q^{27} + 12 q^{29} - 10 q^{31} - 22 q^{33} - 12 q^{35} - 4 q^{37}+ \cdots - 43 q^{99}+O(q^{100}) $ 20 * q - q^3 + 8 * q^5 - 5 * q^7 + 27 * q^9 - 30 * q^11 - 7 * q^13 - 26 * q^15 + 6 * q^17 - q^19 - 3 * q^21 - 37 * q^23 + 14 * q^25 + 5 * q^27 + 12 * q^29 - 10 * q^31 - 22 * q^33 - 12 * q^35 - 4 * q^37 - 31 * q^39 - 7 * q^41 + q^43 - 10 * q^45 - 51 * q^47 + 5 * q^49 - 6 * q^51 + 10 * q^53 + 5 * q^55 - 30 * q^57 - 43 * q^59 - 21 * q^61 - 8 * q^63 - 10 * q^65 - 4 * q^67 - 19 * q^69 - 97 * q^71 - 36 * q^73 + 19 * q^75 - 20 * q^77 - 16 * q^79 + 16 * q^81 - 28 * q^83 - 30 * q^85 + 14 * q^87 + 2 * q^89 + 25 * q^91 - 27 * q^93 - 61 * q^95 - 36 * q^97 - 43 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.32806	0.766756	0.383378	−	0.923592i	$-0.374761\pi$
$3$	1.32806	0.766756	0.383378	+	0.923592i	$0.374761\pi$
$4$	0	0
$5$	−2.04589	−0.914952	−0.457476	−	0.889222i	$-0.651246\pi$
$5$	−2.04589	−0.914952	−0.457476	+	0.889222i	$0.651246\pi$
$6$	0	0
$7$	0.300238	0.113479	0.0567396	−	0.998389i	$-0.481930\pi$
$7$	0.300238	0.113479	0.0567396	+	0.998389i	$0.481930\pi$
$8$	0	0
$9$	−1.23626	−0.412085
$10$	0	0
$11$	3.45738	1.04244	0.521220	−	0.853423i	$-0.325477\pi$
$11$	3.45738	1.04244	0.521220	+	0.853423i	$0.325477\pi$
$12$	0	0
$13$	−3.89705	−1.08085	−0.540424	−	0.841393i	$-0.681736\pi$
$13$	−3.89705	−1.08085	−0.540424	+	0.841393i	$0.681736\pi$
$14$	0	0
$15$	−2.71707	−0.701545
$16$	0	0
$17$	5.59382	1.35670	0.678350	−	0.734739i	$-0.262695\pi$
$17$	5.59382	1.35670	0.678350	+	0.734739i	$0.262695\pi$
$18$	0	0
$19$	0.669118	0.153506	0.0767531	−	0.997050i	$-0.475545\pi$
$19$	0.669118	0.153506	0.0767531	+	0.997050i	$0.475545\pi$
$20$	0	0
$21$	0.398734	0.0870108
$22$	0	0
$23$	−3.49818	−0.729420	−0.364710	−	0.931121i	$-0.618832\pi$
$23$	−3.49818	−0.729420	−0.364710	+	0.931121i	$0.618832\pi$
$24$	0	0
$25$	−0.814317	−0.162863
$26$	0	0
$27$	−5.62600	−1.08272
$28$	0	0
$29$	8.88357	1.64964	0.824819	−	0.565397i	$-0.191277\pi$
$29$	8.88357	1.64964	0.824819	+	0.565397i	$0.191277\pi$
$30$	0	0
$31$	−7.37344	−1.32431	−0.662154	−	0.749368i	$-0.730358\pi$
$31$	−7.37344	−1.32431	−0.662154	+	0.749368i	$0.730358\pi$
$32$	0	0
$33$	4.59161	0.799297
$34$	0	0
$35$	−0.614254	−0.103828
$36$	0	0
$37$	−3.54541	−0.582862	−0.291431	−	0.956592i	$-0.594131\pi$
$37$	−3.54541	−0.582862	−0.291431	+	0.956592i	$0.594131\pi$
$38$	0	0
$39$	−5.17552	−0.828747
$40$	0	0
$41$	7.92837	1.23820	0.619102	−	0.785311i	$-0.287497\pi$
$41$	7.92837	1.23820	0.619102	+	0.785311i	$0.287497\pi$
$42$	0	0
$43$	−0.985097	−0.150226	−0.0751130	−	0.997175i	$-0.523932\pi$
$43$	−0.985097	−0.150226	−0.0751130	+	0.997175i	$0.523932\pi$
$44$	0	0
$45$	2.52925	0.377038
$46$	0	0
$47$	2.95435	0.430936	0.215468	−	0.976511i	$-0.430872\pi$
$47$	2.95435	0.430936	0.215468	+	0.976511i	$0.430872\pi$
$48$	0	0
$49$	−6.90986	−0.987122
$50$	0	0
$51$	7.42893	1.04026
$52$	0	0
$53$	−6.43289	−0.883625	−0.441813	−	0.897107i	$-0.645664\pi$
$53$	−6.43289	−0.883625	−0.441813	+	0.897107i	$0.645664\pi$
$54$	0	0
$55$	−7.07343	−0.953782
$56$	0	0
$57$	0.888629	0.117702
$58$	0	0
$59$	−4.42090	−0.575552	−0.287776	−	0.957698i	$-0.592916\pi$
$59$	−4.42090	−0.575552	−0.287776	+	0.957698i	$0.592916\pi$
$60$	0	0
$61$	−0.236790	−0.0303179	−0.0151590	−	0.999885i	$-0.504825\pi$
$61$	−0.236790	−0.0303179	−0.0151590	+	0.999885i	$0.504825\pi$
$62$	0	0
$63$	−0.371170	−0.0467631
$64$	0	0
$65$	7.97296	0.988924
$66$	0	0
$67$	−2.33250	−0.284960	−0.142480	−	0.989798i	$-0.545508\pi$
$67$	−2.33250	−0.284960	−0.142480	+	0.989798i	$0.545508\pi$
$68$	0	0
$69$	−4.64579	−0.559287
$70$	0	0
$71$	−6.91848	−0.821073	−0.410536	−	0.911844i	$-0.634659\pi$
$71$	−6.91848	−0.821073	−0.410536	+	0.911844i	$0.634659\pi$
$72$	0	0
$73$	4.76591	0.557807	0.278904	−	0.960319i	$-0.410029\pi$
$73$	4.76591	0.557807	0.278904	+	0.960319i	$0.410029\pi$
$74$	0	0
$75$	−1.08146	−0.124876
$76$	0	0
$77$	1.03804	0.118295
$78$	0	0
$79$	−3.80018	−0.427553	−0.213777	−	0.976883i	$-0.568577\pi$
$79$	−3.80018	−0.427553	−0.213777	+	0.976883i	$0.568577\pi$
$80$	0	0
$81$	−3.76290	−0.418100
$82$	0	0
$83$	7.24637	0.795392	0.397696	−	0.917517i	$-0.369810\pi$
$83$	7.24637	0.795392	0.397696	+	0.917517i	$0.369810\pi$
$84$	0	0
$85$	−11.4444	−1.24132
$86$	0	0
$87$	11.7979	1.26487
$88$	0	0
$89$	14.2851	1.51421	0.757107	−	0.653291i	$-0.226612\pi$
$89$	14.2851	1.51421	0.757107	+	0.653291i	$0.226612\pi$
$90$	0	0
$91$	−1.17004	−0.122654
$92$	0	0
$93$	−9.79237	−1.01542
$94$	0	0
$95$	−1.36894	−0.140451
$96$	0	0
$97$	−10.8161	−1.09820	−0.549102	−	0.835755i	$-0.685030\pi$
$97$	−10.8161	−1.09820	−0.549102	+	0.835755i	$0.685030\pi$
$98$	0	0
$99$	−4.27421	−0.429574
$100$	0	0
$101$	−14.6903	−1.46174	−0.730872	−	0.682515i	$-0.760886\pi$
$101$	−14.6903	−1.46174	−0.730872	+	0.682515i	$0.760886\pi$
$102$	0	0
$103$	13.1277	1.29351	0.646753	−	0.762699i	$-0.276126\pi$
$103$	13.1277	1.29351	0.646753	+	0.762699i	$0.276126\pi$
$104$	0	0
$105$	−0.815767	−0.0796107
$106$	0	0
$107$	−11.7666	−1.13752	−0.568760	−	0.822504i	$-0.692577\pi$
$107$	−11.7666	−1.13752	−0.568760	+	0.822504i	$0.692577\pi$
$108$	0	0
$109$	−2.17412	−0.208243	−0.104122	−	0.994565i	$-0.533203\pi$
$109$	−2.17412	−0.208243	−0.104122	+	0.994565i	$0.533203\pi$
$110$	0	0
$111$	−4.70852	−0.446913
$112$	0	0
$113$	−12.7111	−1.19576	−0.597881	−	0.801585i	$-0.703991\pi$
$113$	−12.7111	−1.19576	−0.597881	+	0.801585i	$0.703991\pi$
$114$	0	0
$115$	7.15690	0.667384
$116$	0	0
$117$	4.81776	0.445402
$118$	0	0
$119$	1.67947	0.153957
$120$	0	0
$121$	0.953476	0.0866796
$122$	0	0
$123$	10.5294	0.949400
$124$	0	0
$125$	11.8955	1.06396
$126$	0	0
$127$	−16.2137	−1.43873	−0.719365	−	0.694633i	$-0.755567\pi$
$127$	−16.2137	−1.43873	−0.719365	+	0.694633i	$0.755567\pi$
$128$	0	0
$129$	−1.30827	−0.115187
$130$	0	0
$131$	1.83753	0.160546	0.0802729	−	0.996773i	$-0.474421\pi$
$131$	1.83753	0.160546	0.0802729	+	0.996773i	$0.474421\pi$
$132$	0	0
$133$	0.200894	0.0174197
$134$	0	0
$135$	11.5102	0.990641
$136$	0	0
$137$	5.89871	0.503961	0.251980	−	0.967732i	$-0.418918\pi$
$137$	5.89871	0.503961	0.251980	+	0.967732i	$0.418918\pi$
$138$	0	0
$139$	−8.57012	−0.726908	−0.363454	−	0.931612i	$-0.618403\pi$
$139$	−8.57012	−0.726908	−0.363454	+	0.931612i	$0.618403\pi$
$140$	0	0
$141$	3.92356	0.330423
$142$	0	0
$143$	−13.4736	−1.12672
$144$	0	0
$145$	−18.1749	−1.50934
$146$	0	0
$147$	−9.17671	−0.756882
$148$	0	0
$149$	20.0774	1.64481	0.822403	−	0.568905i	$-0.192633\pi$
$149$	20.0774	1.64481	0.822403	+	0.568905i	$0.192633\pi$
$150$	0	0
$151$	−21.5706	−1.75539	−0.877696	−	0.479218i	$-0.840920\pi$
$151$	−21.5706	−1.75539	−0.877696	+	0.479218i	$0.840920\pi$
$152$	0	0
$153$	−6.91539	−0.559076
$154$	0	0
$155$	15.0853	1.21168
$156$	0	0
$157$	−4.99937	−0.398993	−0.199497	−	0.979899i	$-0.563931\pi$
$157$	−4.99937	−0.398993	−0.199497	+	0.979899i	$0.563931\pi$
$158$	0	0
$159$	−8.54327	−0.677525
$160$	0	0
$161$	−1.05028	−0.0827740
$162$	0	0
$163$	17.7843	1.39298	0.696488	−	0.717568i	$-0.254745\pi$
$163$	17.7843	1.39298	0.696488	+	0.717568i	$0.254745\pi$
$164$	0	0
$165$	−9.39395	−0.731318
$166$	0	0
$167$	−23.1226	−1.78928	−0.894640	−	0.446787i	$-0.852568\pi$
$167$	−23.1226	−1.78928	−0.894640	+	0.446787i	$0.852568\pi$
$168$	0	0
$169$	2.18703	0.168233
$170$	0	0
$171$	−0.827201	−0.0632576
$172$	0	0
$173$	−16.8117	−1.27817	−0.639087	−	0.769135i	$-0.720687\pi$
$173$	−16.8117	−1.27817	−0.639087	+	0.769135i	$0.720687\pi$
$174$	0	0
$175$	−0.244489	−0.0184816
$176$	0	0
$177$	−5.87122	−0.441308
$178$	0	0
$179$	−18.9790	−1.41856	−0.709280	−	0.704927i	$-0.750980\pi$
$179$	−18.9790	−1.41856	−0.709280	+	0.704927i	$0.750980\pi$
$180$	0	0
$181$	8.22419	0.611299	0.305649	−	0.952144i	$-0.401126\pi$
$181$	8.22419	0.611299	0.305649	+	0.952144i	$0.401126\pi$
$182$	0	0
$183$	−0.314472	−0.0232464
$184$	0	0
$185$	7.25354	0.533291
$186$	0	0
$187$	19.3400	1.41428
$188$	0	0
$189$	−1.68914	−0.122867
$190$	0	0
$191$	−3.31876	−0.240137	−0.120069	−	0.992766i	$-0.538311\pi$
$191$	−3.31876	−0.240137	−0.120069	+	0.992766i	$0.538311\pi$
$192$	0	0
$193$	22.2933	1.60471	0.802355	−	0.596848i	$-0.203580\pi$
$193$	22.2933	1.60471	0.802355	+	0.596848i	$0.203580\pi$
$194$	0	0
$195$	10.5886	0.758263
$196$	0	0
$197$	12.2417	0.872182	0.436091	−	0.899903i	$-0.356363\pi$
$197$	12.2417	0.872182	0.436091	+	0.899903i	$0.356363\pi$
$198$	0	0
$199$	−18.2365	−1.29275	−0.646377	−	0.763018i	$-0.723717\pi$
$199$	−18.2365	−1.29275	−0.646377	+	0.763018i	$0.723717\pi$
$200$	0	0
$201$	−3.09770	−0.218495
$202$	0	0
$203$	2.66718	0.187200
$204$	0	0
$205$	−16.2206	−1.13290
$206$	0	0
$207$	4.32464	0.300583
$208$	0	0
$209$	2.31339	0.160021
$210$	0	0
$211$	1.49736	0.103082	0.0515411	−	0.998671i	$-0.483587\pi$
$211$	1.49736	0.103082	0.0515411	+	0.998671i	$0.483587\pi$
$212$	0	0
$213$	−9.18816	−0.629563
$214$	0	0
$215$	2.01540	0.137449
$216$	0	0
$217$	−2.21378	−0.150281
$218$	0	0
$219$	6.32941	0.427702
$220$	0	0
$221$	−21.7994	−1.46639
$222$	0	0
$223$	−0.368166	−0.0246542	−0.0123271	−	0.999924i	$-0.503924\pi$
$223$	−0.368166	−0.0246542	−0.0123271	+	0.999924i	$0.503924\pi$
$224$	0	0
$225$	1.00670	0.0671136
$226$	0	0
$227$	−20.1086	−1.33465	−0.667326	−	0.744765i	$-0.732561\pi$
$227$	−20.1086	−1.33465	−0.667326	+	0.744765i	$0.732561\pi$
$228$	0	0
$229$	−19.5150	−1.28958	−0.644792	−	0.764358i	$-0.723056\pi$
$229$	−19.5150	−1.28958	−0.644792	+	0.764358i	$0.723056\pi$
$230$	0	0
$231$	1.37857	0.0907035
$232$	0	0
$233$	−10.3574	−0.678537	−0.339269	−	0.940690i	$-0.610180\pi$
$233$	−10.3574	−0.678537	−0.339269	+	0.940690i	$0.610180\pi$
$234$	0	0
$235$	−6.04429	−0.394286
$236$	0	0
$237$	−5.04687	−0.327829
$238$	0	0
$239$	9.67826	0.626034	0.313017	−	0.949747i	$-0.398660\pi$
$239$	9.67826	0.626034	0.313017	+	0.949747i	$0.398660\pi$
$240$	0	0
$241$	−23.4147	−1.50827	−0.754137	−	0.656718i	$-0.771944\pi$
$241$	−23.4147	−1.50827	−0.754137	+	0.656718i	$0.771944\pi$
$242$	0	0
$243$	11.8806	0.762144
$244$	0	0
$245$	14.1368	0.903169
$246$	0	0
$247$	−2.60759	−0.165917
$248$	0	0
$249$	9.62362	0.609872
$250$	0	0
$251$	−23.9836	−1.51383	−0.756917	−	0.653511i	$-0.773295\pi$
$251$	−23.9836	−1.51383	−0.756917	+	0.653511i	$0.773295\pi$
$252$	0	0
$253$	−12.0945	−0.760376
$254$	0	0
$255$	−15.1988	−0.951786
$256$	0	0
$257$	−19.8386	−1.23750	−0.618750	−	0.785588i	$-0.712361\pi$
$257$	−19.8386	−1.23750	−0.618750	+	0.785588i	$0.712361\pi$
$258$	0	0
$259$	−1.06447	−0.0661427
$260$	0	0
$261$	−10.9824	−0.679792
$262$	0	0
$263$	−5.57483	−0.343759	−0.171879	−	0.985118i	$-0.554984\pi$
$263$	−5.57483	−0.343759	−0.171879	+	0.985118i	$0.554984\pi$
$264$	0	0
$265$	13.1610	0.808475
$266$	0	0
$267$	18.9714	1.16103
$268$	0	0
$269$	−17.9676	−1.09550	−0.547752	−	0.836641i	$-0.684516\pi$
$269$	−17.9676	−1.09550	−0.547752	+	0.836641i	$0.684516\pi$
$270$	0	0
$271$	−4.64456	−0.282137	−0.141068	−	0.990000i	$-0.545054\pi$
$271$	−4.64456	−0.282137	−0.141068	+	0.990000i	$0.545054\pi$
$272$	0	0
$273$	−1.55389	−0.0940455
$274$	0	0
$275$	−2.81540	−0.169775
$276$	0	0
$277$	19.2078	1.15409	0.577043	−	0.816714i	$-0.304206\pi$
$277$	19.2078	1.15409	0.577043	+	0.816714i	$0.304206\pi$
$278$	0	0
$279$	9.11545	0.545728
$280$	0	0
$281$	17.7157	1.05683	0.528416	−	0.848986i	$-0.322786\pi$
$281$	17.7157	1.05683	0.528416	+	0.848986i	$0.322786\pi$
$282$	0	0
$283$	−17.7025	−1.05231	−0.526153	−	0.850390i	$-0.676366\pi$
$283$	−17.7025	−1.05231	−0.526153	+	0.850390i	$0.676366\pi$
$284$	0	0
$285$	−1.81804	−0.107691
$286$	0	0
$287$	2.38040	0.140510
$288$	0	0
$289$	14.2908	0.840636
$290$	0	0
$291$	−14.3644	−0.842054
$292$	0	0
$293$	−3.13261	−0.183009	−0.0915047	−	0.995805i	$-0.529168\pi$
$293$	−3.13261	−0.183009	−0.0915047	+	0.995805i	$0.529168\pi$
$294$	0	0
$295$	9.04469	0.526602
$296$	0	0
$297$	−19.4512	−1.12867
$298$	0	0
$299$	13.6326	0.788392
$300$	0	0
$301$	−0.295763	−0.0170475
$302$	0	0
$303$	−19.5097	−1.12080
$304$	0	0
$305$	0.484448	0.0277394
$306$	0	0
$307$	−0.217861	−0.0124340	−0.00621700	−	0.999981i	$-0.501979\pi$
$307$	−0.217861	−0.0124340	−0.00621700	+	0.999981i	$0.501979\pi$
$308$	0	0
$309$	17.4343	0.991804
$310$	0	0
$311$	17.2626	0.978872	0.489436	−	0.872039i	$-0.337203\pi$
$311$	17.2626	0.978872	0.489436	+	0.872039i	$0.337203\pi$
$312$	0	0
$313$	11.6304	0.657388	0.328694	−	0.944437i	$-0.393392\pi$
$313$	11.6304	0.657388	0.328694	+	0.944437i	$0.393392\pi$
$314$	0	0
$315$	0.759376	0.0427860
$316$	0	0
$317$	5.02237	0.282085	0.141042	−	0.990004i	$-0.454955\pi$
$317$	5.02237	0.282085	0.141042	+	0.990004i	$0.454955\pi$
$318$	0	0
$319$	30.7139	1.71965
$320$	0	0
$321$	−15.6267	−0.872200
$322$	0	0
$323$	3.74292	0.208262
$324$	0	0
$325$	3.17344	0.176031
$326$	0	0
$327$	−2.88736	−0.159672
$328$	0	0
$329$	0.887007	0.0489023
$330$	0	0
$331$	−7.47113	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$331$	−7.47113	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$332$	0	0
$333$	4.38303	0.240189
$334$	0	0
$335$	4.77204	0.260724
$336$	0	0
$337$	14.1266	0.769525	0.384763	−	0.923016i	$-0.374283\pi$
$337$	14.1266	0.769525	0.384763	+	0.923016i	$0.374283\pi$
$338$	0	0
$339$	−16.8811	−0.916858
$340$	0	0
$341$	−25.4928	−1.38051
$342$	0	0
$343$	−4.17626	−0.225497
$344$	0	0
$345$	9.50479	0.511721
$346$	0	0
$347$	17.3816	0.933091	0.466546	−	0.884497i	$-0.345498\pi$
$347$	17.3816	0.933091	0.466546	+	0.884497i	$0.345498\pi$
$348$	0	0
$349$	−11.3743	−0.608850	−0.304425	−	0.952536i	$-0.598464\pi$
$349$	−11.3743	−0.608850	−0.304425	+	0.952536i	$0.598464\pi$
$350$	0	0
$351$	21.9248	1.17026
$352$	0	0
$353$	−27.8215	−1.48079	−0.740395	−	0.672172i	$-0.765362\pi$
$353$	−27.8215	−1.48079	−0.740395	+	0.672172i	$0.765362\pi$
$354$	0	0
$355$	14.1545	0.751242
$356$	0	0
$357$	2.23044	0.118048
$358$	0	0
$359$	−31.8150	−1.67913	−0.839567	−	0.543257i	$-0.817191\pi$
$359$	−31.8150	−1.67913	−0.839567	+	0.543257i	$0.817191\pi$
$360$	0	0
$361$	−18.5523	−0.976436
$362$	0	0
$363$	1.26627	0.0664621
$364$	0	0
$365$	−9.75054	−0.510367
$366$	0	0
$367$	3.39169	0.177045	0.0885224	−	0.996074i	$-0.471786\pi$
$367$	3.39169	0.177045	0.0885224	+	0.996074i	$0.471786\pi$
$368$	0	0
$369$	−9.80150	−0.510246
$370$	0	0
$371$	−1.93140	−0.100273
$372$	0	0
$373$	16.6667	0.862970	0.431485	−	0.902120i	$-0.357990\pi$
$373$	16.6667	0.862970	0.431485	+	0.902120i	$0.357990\pi$
$374$	0	0
$375$	15.7979	0.815801
$376$	0	0
$377$	−34.6198	−1.78301
$378$	0	0
$379$	−6.52444	−0.335138	−0.167569	−	0.985860i	$-0.553592\pi$
$379$	−6.52444	−0.335138	−0.167569	+	0.985860i	$0.553592\pi$
$380$	0	0
$381$	−21.5327	−1.10315
$382$	0	0
$383$	−8.66460	−0.442740	−0.221370	−	0.975190i	$-0.571053\pi$
$383$	−8.66460	−0.442740	−0.221370	+	0.975190i	$0.571053\pi$
$384$	0	0
$385$	−2.12371	−0.108234
$386$	0	0
$387$	1.21783	0.0619059
$388$	0	0
$389$	25.3450	1.28504	0.642522	−	0.766267i	$-0.277888\pi$
$389$	25.3450	1.28504	0.642522	+	0.766267i	$0.277888\pi$
$390$	0	0
$391$	−19.5682	−0.989604
$392$	0	0
$393$	2.44035	0.123099
$394$	0	0
$395$	7.77476	0.391191
$396$	0	0
$397$	17.7609	0.891394	0.445697	−	0.895184i	$-0.352956\pi$
$397$	17.7609	0.891394	0.445697	+	0.895184i	$0.352956\pi$
$398$	0	0
$399$	0.266800	0.0133567
$400$	0	0
$401$	6.25522	0.312371	0.156185	−	0.987728i	$-0.450080\pi$
$401$	6.25522	0.312371	0.156185	+	0.987728i	$0.450080\pi$
$402$	0	0
$403$	28.7347	1.43138
$404$	0	0
$405$	7.69850	0.382542
$406$	0	0
$407$	−12.2578	−0.607598
$408$	0	0
$409$	1.00000	0.0494468
$409$	1.00000	0.0494468
$410$	0	0
$411$	7.83384	0.386415
$412$	0	0
$413$	−1.32732	−0.0653132
$414$	0	0
$415$	−14.8253	−0.727746
$416$	0	0
$417$	−11.3816	−0.557361
$418$	0	0
$419$	17.4774	0.853825	0.426913	−	0.904293i	$-0.359601\pi$
$419$	17.4774	0.853825	0.426913	+	0.904293i	$0.359601\pi$
$420$	0	0
$421$	23.7864	1.15928	0.579640	−	0.814872i	$-0.303193\pi$
$421$	23.7864	1.15928	0.579640	+	0.814872i	$0.303193\pi$
$422$	0	0
$423$	−3.65233	−0.177583
$424$	0	0
$425$	−4.55514	−0.220957
$426$	0	0
$427$	−0.0710934	−0.00344045
$428$	0	0
$429$	−17.8937	−0.863918
$430$	0	0
$431$	−9.72228	−0.468306	−0.234153	−	0.972200i	$-0.575232\pi$
$431$	−9.72228	−0.468306	−0.234153	+	0.972200i	$0.575232\pi$
$432$	0	0
$433$	12.8718	0.618582	0.309291	−	0.950968i	$-0.399908\pi$
$433$	12.8718	0.618582	0.309291	+	0.950968i	$0.399908\pi$
$434$	0	0
$435$	−24.1373	−1.15729
$436$	0	0
$437$	−2.34069	−0.111970
$438$	0	0
$439$	−16.3107	−0.778466	−0.389233	−	0.921139i	$-0.627260\pi$
$439$	−16.3107	−0.778466	−0.389233	+	0.921139i	$0.627260\pi$
$440$	0	0
$441$	8.54235	0.406779
$442$	0	0
$443$	20.7520	0.985957	0.492978	−	0.870042i	$-0.335908\pi$
$443$	20.7520	0.985957	0.492978	+	0.870042i	$0.335908\pi$
$444$	0	0
$445$	−29.2257	−1.38543
$446$	0	0
$447$	26.6640	1.26116
$448$	0	0
$449$	−14.7710	−0.697086	−0.348543	−	0.937293i	$-0.613324\pi$
$449$	−14.7710	−0.697086	−0.348543	+	0.937293i	$0.613324\pi$
$450$	0	0
$451$	27.4114	1.29075
$452$	0	0
$453$	−28.6471	−1.34596
$454$	0	0
$455$	2.39378	0.112222
$456$	0	0
$457$	−15.0736	−0.705113	−0.352557	−	0.935790i	$-0.614688\pi$
$457$	−15.0736	−0.705113	−0.352557	+	0.935790i	$0.614688\pi$
$458$	0	0
$459$	−31.4708	−1.46893
$460$	0	0
$461$	−19.8978	−0.926733	−0.463366	−	0.886167i	$-0.653359\pi$
$461$	−19.8978	−0.926733	−0.463366	+	0.886167i	$0.653359\pi$
$462$	0	0
$463$	27.4958	1.27784	0.638919	−	0.769274i	$-0.279382\pi$
$463$	27.4958	1.27784	0.638919	+	0.769274i	$0.279382\pi$
$464$	0	0
$465$	20.0342	0.929061
$466$	0	0
$467$	18.1186	0.838429	0.419215	−	0.907887i	$-0.362305\pi$
$467$	18.1186	0.838429	0.419215	+	0.907887i	$0.362305\pi$
$468$	0	0
$469$	−0.700303	−0.0323370
$470$	0	0
$471$	−6.63946	−0.305930
$472$	0	0
$473$	−3.40586	−0.156601
$474$	0	0
$475$	−0.544874	−0.0250005
$476$	0	0
$477$	7.95270	0.364129
$478$	0	0
$479$	−1.62305	−0.0741590	−0.0370795	−	0.999312i	$-0.511805\pi$
$479$	−1.62305	−0.0741590	−0.0370795	+	0.999312i	$0.511805\pi$
$480$	0	0
$481$	13.8167	0.629985
$482$	0	0
$483$	−1.39484	−0.0634674
$484$	0	0
$485$	22.1285	1.00480
$486$	0	0
$487$	−17.6722	−0.800805	−0.400403	−	0.916339i	$-0.631130\pi$
$487$	−17.6722	−0.800805	−0.400403	+	0.916339i	$0.631130\pi$
$488$	0	0
$489$	23.6187	1.06807
$490$	0	0
$491$	−40.4253	−1.82437	−0.912183	−	0.409782i	$-0.865605\pi$
$491$	−40.4253	−1.82437	−0.912183	+	0.409782i	$0.865605\pi$
$492$	0	0
$493$	49.6931	2.23806
$494$	0	0
$495$	8.74457	0.393039
$496$	0	0
$497$	−2.07719	−0.0931747
$498$	0	0
$499$	2.67593	0.119791	0.0598955	−	0.998205i	$-0.480923\pi$
$499$	2.67593	0.119791	0.0598955	+	0.998205i	$0.480923\pi$
$500$	0	0
$501$	−30.7082	−1.37194
$502$	0	0
$503$	−13.3355	−0.594599	−0.297300	−	0.954784i	$-0.596086\pi$
$503$	−13.3355	−0.594599	−0.297300	+	0.954784i	$0.596086\pi$
$504$	0	0
$505$	30.0549	1.33742
$506$	0	0
$507$	2.90450	0.128994
$508$	0	0
$509$	32.0232	1.41940	0.709702	−	0.704502i	$-0.248830\pi$
$509$	32.0232	1.41940	0.709702	+	0.704502i	$0.248830\pi$
$510$	0	0
$511$	1.43090	0.0632995
$512$	0	0
$513$	−3.76446	−0.166205
$514$	0	0
$515$	−26.8578	−1.18350
$516$	0	0
$517$	10.2143	0.449225
$518$	0	0
$519$	−22.3270	−0.980047
$520$	0	0
$521$	24.3133	1.06519	0.532594	−	0.846371i	$-0.321217\pi$
$521$	24.3133	1.06519	0.532594	+	0.846371i	$0.321217\pi$
$522$	0	0
$523$	3.22360	0.140958	0.0704791	−	0.997513i	$-0.477547\pi$
$523$	3.22360	0.140958	0.0704791	+	0.997513i	$0.477547\pi$
$524$	0	0
$525$	−0.324696	−0.0141709
$526$	0	0
$527$	−41.2457	−1.79669
$528$	0	0
$529$	−10.7628	−0.467946
$530$	0	0
$531$	5.46536	0.237177
$532$	0	0
$533$	−30.8973	−1.33831
$534$	0	0
$535$	24.0732	1.04078
$536$	0	0
$537$	−25.2053	−1.08769
$538$	0	0
$539$	−23.8900	−1.02902
$540$	0	0
$541$	−26.5420	−1.14113	−0.570564	−	0.821253i	$-0.693275\pi$
$541$	−26.5420	−1.14113	−0.570564	+	0.821253i	$0.693275\pi$
$542$	0	0
$543$	10.9222	0.468717
$544$	0	0
$545$	4.44802	0.190532
$546$	0	0
$547$	−20.5342	−0.877981	−0.438990	−	0.898492i	$-0.644664\pi$
$547$	−20.5342	−0.877981	−0.438990	+	0.898492i	$0.644664\pi$
$548$	0	0
$549$	0.292734	0.0124936
$550$	0	0
$551$	5.94416	0.253230
$552$	0	0
$553$	−1.14096	−0.0485184
$554$	0	0
$555$	9.63313	0.408904
$556$	0	0
$557$	33.5780	1.42275	0.711373	−	0.702814i	$-0.248073\pi$
$557$	33.5780	1.42275	0.711373	+	0.702814i	$0.248073\pi$
$558$	0	0
$559$	3.83898	0.162371
$560$	0	0
$561$	25.6846	1.08441
$562$	0	0
$563$	34.2480	1.44338	0.721692	−	0.692215i	$-0.243365\pi$
$563$	34.2480	1.44338	0.721692	+	0.692215i	$0.243365\pi$
$564$	0	0
$565$	26.0056	1.09406
$566$	0	0
$567$	−1.12977	−0.0474457
$568$	0	0
$569$	7.92621	0.332284	0.166142	−	0.986102i	$-0.446869\pi$
$569$	7.92621	0.332284	0.166142	+	0.986102i	$0.446869\pi$
$570$	0	0
$571$	−16.8742	−0.706165	−0.353082	−	0.935592i	$-0.614866\pi$
$571$	−16.8742	−0.706165	−0.353082	+	0.935592i	$0.614866\pi$
$572$	0	0
$573$	−4.40752	−0.184127
$574$	0	0
$575$	2.84862	0.118796
$576$	0	0
$577$	−4.20001	−0.174849	−0.0874244	−	0.996171i	$-0.527864\pi$
$577$	−4.20001	−0.174849	−0.0874244	+	0.996171i	$0.527864\pi$
$578$	0	0
$579$	29.6069	1.23042
$580$	0	0
$581$	2.17563	0.0902605
$582$	0	0
$583$	−22.2409	−0.921126
$584$	0	0
$585$	−9.85662	−0.407521
$586$	0	0
$587$	−26.0714	−1.07608	−0.538042	−	0.842918i	$-0.680836\pi$
$587$	−26.0714	−1.07608	−0.538042	+	0.842918i	$0.680836\pi$
$588$	0	0
$589$	−4.93370	−0.203289
$590$	0	0
$591$	16.2577	0.668751
$592$	0	0
$593$	−30.8085	−1.26515	−0.632577	−	0.774497i	$-0.718003\pi$
$593$	−30.8085	−1.26515	−0.632577	+	0.774497i	$0.718003\pi$
$594$	0	0
$595$	−3.43603	−0.140863
$596$	0	0
$597$	−24.2192	−0.991227
$598$	0	0
$599$	9.25916	0.378319	0.189159	−	0.981946i	$-0.439424\pi$
$599$	9.25916	0.378319	0.189159	+	0.981946i	$0.439424\pi$
$600$	0	0
$601$	6.56273	0.267699	0.133850	−	0.991002i	$-0.457266\pi$
$601$	6.56273	0.267699	0.133850	+	0.991002i	$0.457266\pi$
$602$	0	0
$603$	2.88356	0.117428
$604$	0	0
$605$	−1.95071	−0.0793077
$606$	0	0
$607$	14.5775	0.591682	0.295841	−	0.955237i	$-0.404400\pi$
$607$	14.5775	0.591682	0.295841	+	0.955237i	$0.404400\pi$
$608$	0	0
$609$	3.54218	0.143536
$610$	0	0
$611$	−11.5133	−0.465777
$612$	0	0
$613$	−6.37447	−0.257462	−0.128731	−	0.991680i	$-0.541090\pi$
$613$	−6.37447	−0.257462	−0.128731	+	0.991680i	$0.541090\pi$
$614$	0	0
$615$	−21.5420	−0.868655
$616$	0	0
$617$	6.10386	0.245732	0.122866	−	0.992423i	$-0.460791\pi$
$617$	6.10386	0.245732	0.122866	+	0.992423i	$0.460791\pi$
$618$	0	0
$619$	20.0277	0.804981	0.402491	−	0.915424i	$-0.368145\pi$
$619$	20.0277	0.804981	0.402491	+	0.915424i	$0.368145\pi$
$620$	0	0
$621$	19.6807	0.789761
$622$	0	0
$623$	4.28891	0.171832
$624$	0	0
$625$	−20.2653	−0.810612
$626$	0	0
$627$	3.07233	0.122697
$628$	0	0
$629$	−19.8324	−0.790769
$630$	0	0
$631$	4.19378	0.166952	0.0834758	−	0.996510i	$-0.473398\pi$
$631$	4.19378	0.166952	0.0834758	+	0.996510i	$0.473398\pi$
$632$	0	0
$633$	1.98858	0.0790389
$634$	0	0
$635$	33.1714	1.31637
$636$	0	0
$637$	26.9281	1.06693
$638$	0	0
$639$	8.55302	0.338352
$640$	0	0
$641$	18.0977	0.714815	0.357407	−	0.933949i	$-0.383661\pi$
$641$	18.0977	0.714815	0.357407	+	0.933949i	$0.383661\pi$
$642$	0	0
$643$	34.3118	1.35313	0.676563	−	0.736384i	$-0.263468\pi$
$643$	34.3118	1.35313	0.676563	+	0.736384i	$0.263468\pi$
$644$	0	0
$645$	2.67658	0.105390
$646$	0	0
$647$	42.9999	1.69050	0.845250	−	0.534372i	$-0.179452\pi$
$647$	42.9999	1.69050	0.845250	+	0.534372i	$0.179452\pi$
$648$	0	0
$649$	−15.2847	−0.599978
$650$	0	0
$651$	−2.94004	−0.115229
$652$	0	0
$653$	−25.7091	−1.00607	−0.503037	−	0.864265i	$-0.667784\pi$
$653$	−25.7091	−1.00607	−0.503037	+	0.864265i	$0.667784\pi$
$654$	0	0
$655$	−3.75939	−0.146892
$656$	0	0
$657$	−5.89188	−0.229864
$658$	0	0
$659$	21.5003	0.837534	0.418767	−	0.908094i	$-0.362462\pi$
$659$	21.5003	0.837534	0.418767	+	0.908094i	$0.362462\pi$
$660$	0	0
$661$	43.9110	1.70794	0.853970	−	0.520322i	$-0.174188\pi$
$661$	43.9110	1.70794	0.853970	+	0.520322i	$0.174188\pi$
$662$	0	0
$663$	−28.9509	−1.12436
$664$	0	0
$665$	−0.411008	−0.0159382
$666$	0	0
$667$	−31.0763	−1.20328
$668$	0	0
$669$	−0.488946	−0.0189038
$670$	0	0
$671$	−0.818674	−0.0316046
$672$	0	0
$673$	49.1559	1.89482	0.947410	−	0.320022i	$-0.103690\pi$
$673$	49.1559	1.89482	0.947410	+	0.320022i	$0.103690\pi$
$674$	0	0
$675$	4.58135	0.176336
$676$	0	0
$677$	23.1357	0.889178	0.444589	−	0.895735i	$-0.353350\pi$
$677$	23.1357	0.889178	0.444589	+	0.895735i	$0.353350\pi$
$678$	0	0
$679$	−3.24739	−0.124623
$680$	0	0
$681$	−26.7054	−1.02335
$682$	0	0
$683$	44.0487	1.68548	0.842739	−	0.538323i	$-0.180942\pi$
$683$	44.0487	1.68548	0.842739	+	0.538323i	$0.180942\pi$
$684$	0	0
$685$	−12.0681	−0.461100
$686$	0	0
$687$	−25.9170	−0.988797
$688$	0	0
$689$	25.0693	0.955065
$690$	0	0
$691$	−16.0482	−0.610504	−0.305252	−	0.952272i	$-0.598741\pi$
$691$	−16.0482	−0.610504	−0.305252	+	0.952272i	$0.598741\pi$
$692$	0	0
$693$	−1.28328	−0.0487477
$694$	0	0
$695$	17.5336	0.665086
$696$	0	0
$697$	44.3499	1.67987
$698$	0	0
$699$	−13.7553	−0.520273
$700$	0	0
$701$	−50.2864	−1.89929	−0.949646	−	0.313324i	$-0.898557\pi$
$701$	−50.2864	−1.89929	−0.949646	+	0.313324i	$0.898557\pi$
$702$	0	0
$703$	−2.37230	−0.0894729
$704$	0	0
$705$	−8.02718	−0.302321
$706$	0	0
$707$	−4.41059	−0.165877
$708$	0	0
$709$	10.6030	0.398203	0.199102	−	0.979979i	$-0.436198\pi$
$709$	10.6030	0.398203	0.199102	+	0.979979i	$0.436198\pi$
$710$	0	0
$711$	4.69799	0.176188
$712$	0	0
$713$	25.7936	0.965977
$714$	0	0
$715$	27.5656	1.03089
$716$	0	0
$717$	12.8533	0.480016
$718$	0	0
$719$	−50.7851	−1.89397	−0.946983	−	0.321284i	$-0.895886\pi$
$719$	−50.7851	−1.89397	−0.946983	+	0.321284i	$0.895886\pi$
$720$	0	0
$721$	3.94142	0.146786
$722$	0	0
$723$	−31.0961	−1.15648
$724$	0	0
$725$	−7.23404	−0.268666
$726$	0	0
$727$	17.9301	0.664990	0.332495	−	0.943105i	$-0.392110\pi$
$727$	17.9301	0.664990	0.332495	+	0.943105i	$0.392110\pi$
$728$	0	0
$729$	27.0669	1.00248
$730$	0	0
$731$	−5.51046	−0.203812
$732$	0	0
$733$	26.8303	0.990999	0.495500	−	0.868608i	$-0.334985\pi$
$733$	26.8303	0.990999	0.495500	+	0.868608i	$0.334985\pi$
$734$	0	0
$735$	18.7746	0.692511
$736$	0	0
$737$	−8.06433	−0.297053
$738$	0	0
$739$	−25.1889	−0.926587	−0.463294	−	0.886205i	$-0.653332\pi$
$739$	−25.1889	−0.926587	−0.463294	+	0.886205i	$0.653332\pi$
$740$	0	0
$741$	−3.46303	−0.127218
$742$	0	0
$743$	−22.3360	−0.819430	−0.409715	−	0.912214i	$-0.634372\pi$
$743$	−22.3360	−0.819430	−0.409715	+	0.912214i	$0.634372\pi$
$744$	0	0
$745$	−41.0763	−1.50492
$746$	0	0
$747$	−8.95837	−0.327770
$748$	0	0
$749$	−3.53277	−0.129085
$750$	0	0
$751$	−23.4082	−0.854176	−0.427088	−	0.904210i	$-0.640461\pi$
$751$	−23.4082	−0.854176	−0.427088	+	0.904210i	$0.640461\pi$
$752$	0	0
$753$	−31.8517	−1.16074
$754$	0	0
$755$	44.1312	1.60610
$756$	0	0
$757$	15.8828	0.577271	0.288635	−	0.957439i	$-0.406798\pi$
$757$	15.8828	0.577271	0.288635	+	0.957439i	$0.406798\pi$
$758$	0	0
$759$	−16.0623	−0.583023
$760$	0	0
$761$	11.3967	0.413129	0.206564	−	0.978433i	$-0.433772\pi$
$761$	11.3967	0.413129	0.206564	+	0.978433i	$0.433772\pi$
$762$	0	0
$763$	−0.652753	−0.0236312
$764$	0	0
$765$	14.1482	0.511528
$766$	0	0
$767$	17.2285	0.622085
$768$	0	0
$769$	28.4670	1.02655	0.513274	−	0.858225i	$-0.328432\pi$
$769$	28.4670	1.02655	0.513274	+	0.858225i	$0.328432\pi$
$770$	0	0
$771$	−26.3469	−0.948861
$772$	0	0
$773$	22.8660	0.822432	0.411216	−	0.911538i	$-0.365104\pi$
$773$	22.8660	0.822432	0.411216	+	0.911538i	$0.365104\pi$
$774$	0	0
$775$	6.00431	0.215681
$776$	0	0
$777$	−1.41367	−0.0507153
$778$	0	0
$779$	5.30501	0.190072
$780$	0	0
$781$	−23.9198	−0.855919
$782$	0	0
$783$	−49.9790	−1.78610
$784$	0	0
$785$	10.2282	0.365059
$786$	0	0
$787$	0.582587	0.0207670	0.0103835	−	0.999946i	$-0.496695\pi$
$787$	0.582587	0.0207670	0.0103835	+	0.999946i	$0.496695\pi$
$788$	0	0
$789$	−7.40371	−0.263579
$790$	0	0
$791$	−3.81636	−0.135694
$792$	0	0
$793$	0.922785	0.0327691
$794$	0	0
$795$	17.4786	0.619903
$796$	0	0
$797$	−4.59138	−0.162635	−0.0813176	−	0.996688i	$-0.525913\pi$
$797$	−4.59138	−0.162635	−0.0813176	+	0.996688i	$0.525913\pi$
$798$	0	0
$799$	16.5261	0.584651
$800$	0	0
$801$	−17.6600	−0.623985
$802$	0	0
$803$	16.4776	0.581480
$804$	0	0
$805$	2.14877	0.0757342
$806$	0	0
$807$	−23.8621	−0.839985
$808$	0	0
$809$	−18.1700	−0.638824	−0.319412	−	0.947616i	$-0.603485\pi$
$809$	−18.1700	−0.638824	−0.319412	+	0.947616i	$0.603485\pi$
$810$	0	0
$811$	53.0220	1.86185	0.930926	−	0.365207i	$-0.119002\pi$
$811$	53.0220	1.86185	0.930926	+	0.365207i	$0.119002\pi$
$812$	0	0
$813$	−6.16825	−0.216330
$814$	0	0
$815$	−36.3849	−1.27451
$816$	0	0
$817$	−0.659146	−0.0230606
$818$	0	0
$819$	1.44647	0.0505438
$820$	0	0
$821$	36.7124	1.28127	0.640635	−	0.767845i	$-0.278671\pi$
$821$	36.7124	1.28127	0.640635	+	0.767845i	$0.278671\pi$
$822$	0	0
$823$	49.3411	1.71992	0.859961	−	0.510360i	$-0.170488\pi$
$823$	49.3411	1.71992	0.859961	+	0.510360i	$0.170488\pi$
$824$	0	0
$825$	−3.73902	−0.130176
$826$	0	0
$827$	34.0759	1.18494	0.592468	−	0.805594i	$-0.298154\pi$
$827$	34.0759	1.18494	0.592468	+	0.805594i	$0.298154\pi$
$828$	0	0
$829$	−16.5307	−0.574134	−0.287067	−	0.957911i	$-0.592680\pi$
$829$	−16.5307	−0.574134	−0.287067	+	0.957911i	$0.592680\pi$
$830$	0	0
$831$	25.5092	0.884903
$832$	0	0
$833$	−38.6525	−1.33923
$834$	0	0
$835$	47.3064	1.63711
$836$	0	0
$837$	41.4830	1.43386
$838$	0	0
$839$	13.6583	0.471538	0.235769	−	0.971809i	$-0.424239\pi$
$839$	13.6583	0.471538	0.235769	+	0.971809i	$0.424239\pi$
$840$	0	0
$841$	49.9179	1.72131
$842$	0	0
$843$	23.5276	0.810332
$844$	0	0
$845$	−4.47443	−0.153925
$846$	0	0
$847$	0.286269	0.00983633
$848$	0	0
$849$	−23.5100	−0.806862
$850$	0	0
$851$	12.4025	0.425151
$852$	0	0
$853$	−28.4297	−0.973414	−0.486707	−	0.873565i	$-0.661802\pi$
$853$	−28.4297	−0.973414	−0.486707	+	0.873565i	$0.661802\pi$
$854$	0	0
$855$	1.69236	0.0578777
$856$	0	0
$857$	27.5759	0.941975	0.470988	−	0.882140i	$-0.343898\pi$
$857$	27.5759	0.941975	0.470988	+	0.882140i	$0.343898\pi$
$858$	0	0
$859$	28.7472	0.980843	0.490422	−	0.871485i	$-0.336843\pi$
$859$	28.7472	0.980843	0.490422	+	0.871485i	$0.336843\pi$
$860$	0	0
$861$	3.16131	0.107737
$862$	0	0
$863$	−28.4792	−0.969444	−0.484722	−	0.874668i	$-0.661079\pi$
$863$	−28.4792	−0.969444	−0.484722	+	0.874668i	$0.661079\pi$
$864$	0	0
$865$	34.3950	1.16947
$866$	0	0
$867$	18.9791	0.644562
$868$	0	0
$869$	−13.1387	−0.445699
$870$	0	0
$871$	9.08986	0.307998
$872$	0	0
$873$	13.3714	0.452554
$874$	0	0
$875$	3.57147	0.120738
$876$	0	0
$877$	−38.9562	−1.31546	−0.657729	−	0.753255i	$-0.728483\pi$
$877$	−38.9562	−1.31546	−0.657729	+	0.753255i	$0.728483\pi$
$878$	0	0
$879$	−4.16030	−0.140323
$880$	0	0
$881$	−47.3697	−1.59593	−0.797963	−	0.602707i	$-0.794089\pi$
$881$	−47.3697	−1.59593	−0.797963	+	0.602707i	$0.794089\pi$
$882$	0	0
$883$	−30.7696	−1.03548	−0.517739	−	0.855539i	$-0.673226\pi$
$883$	−30.7696	−1.03548	−0.517739	+	0.855539i	$0.673226\pi$
$884$	0	0
$885$	12.0119	0.403776
$886$	0	0
$887$	10.3300	0.346846	0.173423	−	0.984847i	$-0.444517\pi$
$887$	10.3300	0.346846	0.173423	+	0.984847i	$0.444517\pi$
$888$	0	0
$889$	−4.86795	−0.163266
$890$	0	0
$891$	−13.0098	−0.435844
$892$	0	0
$893$	1.97681	0.0661514
$894$	0	0
$895$	38.8291	1.29791
$896$	0	0
$897$	18.1049	0.604505
$898$	0	0
$899$	−65.5025	−2.18463
$900$	0	0
$901$	−35.9844	−1.19881
$902$	0	0
$903$	−0.392791	−0.0130713
$904$	0	0
$905$	−16.8258	−0.559309
$906$	0	0
$907$	48.0569	1.59570	0.797851	−	0.602854i	$-0.205970\pi$
$907$	48.0569	1.59570	0.797851	+	0.602854i	$0.205970\pi$
$908$	0	0
$909$	18.1610	0.602363
$910$	0	0
$911$	−32.9823	−1.09275	−0.546376	−	0.837540i	$-0.683993\pi$
$911$	−32.9823	−1.09275	−0.546376	+	0.837540i	$0.683993\pi$
$912$	0	0
$913$	25.0535	0.829148
$914$	0	0
$915$	0.643376	0.0212694
$916$	0	0
$917$	0.551696	0.0182186
$918$	0	0
$919$	22.9775	0.757959	0.378979	−	0.925405i	$-0.376275\pi$
$919$	22.9775	0.757959	0.378979	+	0.925405i	$0.376275\pi$
$920$	0	0
$921$	−0.289333	−0.00953384
$922$	0	0
$923$	26.9617	0.887455
$924$	0	0
$925$	2.88709	0.0949269
$926$	0	0
$927$	−16.2291	−0.533035
$928$	0	0
$929$	33.4643	1.09793	0.548964	−	0.835846i	$-0.315022\pi$
$929$	33.4643	1.09793	0.548964	+	0.835846i	$0.315022\pi$
$930$	0	0
$931$	−4.62351	−0.151529
$932$	0	0
$933$	22.9258	0.750556
$934$	0	0
$935$	−39.5675	−1.29400
$936$	0	0
$937$	−12.3697	−0.404101	−0.202050	−	0.979375i	$-0.564760\pi$
$937$	−12.3697	−0.404101	−0.202050	+	0.979375i	$0.564760\pi$
$938$	0	0
$939$	15.4458	0.504056
$940$	0	0
$941$	−14.7118	−0.479590	−0.239795	−	0.970824i	$-0.577080\pi$
$941$	−14.7118	−0.479590	−0.239795	+	0.970824i	$0.577080\pi$
$942$	0	0
$943$	−27.7348	−0.903171
$944$	0	0
$945$	3.45580	0.112417
$946$	0	0
$947$	−49.1993	−1.59876	−0.799381	−	0.600824i	$-0.794839\pi$
$947$	−49.1993	−1.59876	−0.799381	+	0.600824i	$0.794839\pi$
$948$	0	0
$949$	−18.5730	−0.602905
$950$	0	0
$951$	6.67001	0.216290
$952$	0	0
$953$	2.42440	0.0785342	0.0392671	−	0.999229i	$-0.487498\pi$
$953$	2.42440	0.0785342	0.0392671	+	0.999229i	$0.487498\pi$
$954$	0	0
$955$	6.78984	0.219714
$956$	0	0
$957$	40.7899	1.31855
$958$	0	0
$959$	1.77101	0.0571890
$960$	0	0
$961$	23.3676	0.753792
$962$	0	0
$963$	14.5465	0.468755
$964$	0	0
$965$	−45.6098	−1.46823
$966$	0	0
$967$	−37.1967	−1.19617	−0.598083	−	0.801434i	$-0.704071\pi$
$967$	−37.1967	−1.19617	−0.598083	+	0.801434i	$0.704071\pi$
$968$	0	0
$969$	4.97083	0.159686
$970$	0	0
$971$	−42.2843	−1.35697	−0.678484	−	0.734615i	$-0.737363\pi$
$971$	−42.2843	−1.35697	−0.678484	+	0.734615i	$0.737363\pi$
$972$	0	0
$973$	−2.57307	−0.0824889
$974$	0	0
$975$	4.21451	0.134973
$976$	0	0
$977$	−47.6762	−1.52530	−0.762648	−	0.646813i	$-0.776101\pi$
$977$	−47.6762	−1.52530	−0.762648	+	0.646813i	$0.776101\pi$
$978$	0	0
$979$	49.3889	1.57848
$980$	0	0
$981$	2.68777	0.0858139
$982$	0	0
$983$	−26.4277	−0.842912	−0.421456	−	0.906849i	$-0.638481\pi$
$983$	−26.4277	−0.842912	−0.421456	+	0.906849i	$0.638481\pi$
$984$	0	0
$985$	−25.0451	−0.798004
$986$	0	0
$987$	1.17800	0.0374961
$988$	0	0
$989$	3.44604	0.109578
$990$	0	0
$991$	20.0705	0.637561	0.318780	−	0.947829i	$-0.396727\pi$
$991$	20.0705	0.637561	0.318780	+	0.947829i	$0.396727\pi$
$992$	0	0
$993$	−9.92211	−0.314869
$994$	0	0
$995$	37.3100	1.18281
$996$	0	0
$997$	−42.8122	−1.35587	−0.677937	−	0.735120i	$-0.737126\pi$
$997$	−42.8122	−1.35587	−0.677937	+	0.735120i	$0.737126\pi$
$998$	0	0
$999$	19.9465	0.631079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6544.2.a.i.1.14		20
4.3	odd	2		409.2.a.b.1.5	✓	20
12.11	even	2		3681.2.a.i.1.16		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
409.2.a.b.1.5	✓	20	4.3	odd	2
3681.2.a.i.1.16		20	12.11	even	2
6544.2.a.i.1.14		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6544 = 2^{4} \cdot 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6544.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32806 q^{3} -2.04589 q^{5} +0.300238 q^{7} -1.23626 q^{9} +3.45738 q^{11} -3.89705 q^{13} -2.71707 q^{15} +5.59382 q^{17} +0.669118 q^{19} +0.398734 q^{21} -3.49818 q^{23} -0.814317 q^{25} -5.62600 q^{27} +8.88357 q^{29} -7.37344 q^{31} +4.59161 q^{33} -0.614254 q^{35} -3.54541 q^{37} -5.17552 q^{39} +7.92837 q^{41} -0.985097 q^{43} +2.52925 q^{45} +2.95435 q^{47} -6.90986 q^{49} +7.42893 q^{51} -6.43289 q^{53} -7.07343 q^{55} +0.888629 q^{57} -4.42090 q^{59} -0.236790 q^{61} -0.371170 q^{63} +7.97296 q^{65} -2.33250 q^{67} -4.64579 q^{69} -6.91848 q^{71} +4.76591 q^{73} -1.08146 q^{75} +1.03804 q^{77} -3.80018 q^{79} -3.76290 q^{81} +7.24637 q^{83} -11.4444 q^{85} +11.7979 q^{87} +14.2851 q^{89} -1.17004 q^{91} -9.79237 q^{93} -1.36894 q^{95} -10.8161 q^{97} -4.27421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{3} + 8 q^{5} - 5 q^{7} + 27 q^{9} - 30 q^{11} - 7 q^{13} - 26 q^{15} + 6 q^{17} - q^{19} - 3 q^{21} - 37 q^{23} + 14 q^{25} + 5 q^{27} + 12 q^{29} - 10 q^{31} - 22 q^{33} - 12 q^{35} - 4 q^{37}+ \cdots - 43 q^{99}+O(q^{100}) \) 20 * q - q^3 + 8 * q^5 - 5 * q^7 + 27 * q^9 - 30 * q^11 - 7 * q^13 - 26 * q^15 + 6 * q^17 - q^19 - 3 * q^21 - 37 * q^23 + 14 * q^25 + 5 * q^27 + 12 * q^29 - 10 * q^31 - 22 * q^33 - 12 * q^35 - 4 * q^37 - 31 * q^39 - 7 * q^41 + q^43 - 10 * q^45 - 51 * q^47 + 5 * q^49 - 6 * q^51 + 10 * q^53 + 5 * q^55 - 30 * q^57 - 43 * q^59 - 21 * q^61 - 8 * q^63 - 10 * q^65 - 4 * q^67 - 19 * q^69 - 97 * q^71 - 36 * q^73 + 19 * q^75 - 20 * q^77 - 16 * q^79 + 16 * q^81 - 28 * q^83 - 30 * q^85 + 14 * q^87 + 2 * q^89 + 25 * q^91 - 27 * q^93 - 61 * q^95 - 36 * q^97 - 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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