LMFDB - Embedded newform 2816.2.a.q.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2816,2,Mod(1,2816)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2816.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2816, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,4,0,0,0,5,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.4858732092$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.380224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 5x + 2 $ x^5 - 8x^3 - 2x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.479072$ of defining polynomial
Character	$\chi$	$=$	2816.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33544 q^{3} +1.93119 q^{5} -1.83930 q^{7} -1.21660 q^{9} -1.00000 q^{11} +0.160700 q^{13} +2.57899 q^{15} +5.41442 q^{17} -2.79744 q^{19} -2.45628 q^{21} +4.44137 q^{23} -1.27050 q^{25} -5.63102 q^{27} +8.34948 q^{29} +9.05835 q^{31} -1.33544 q^{33} -3.55204 q^{35} -0.602073 q^{37} +0.214606 q^{39} +11.7639 q^{41} +4.47936 q^{43} -2.34948 q^{45} -6.29558 q^{47} -3.61698 q^{49} +7.23064 q^{51} +11.7211 q^{53} -1.93119 q^{55} -3.73582 q^{57} +9.63102 q^{59} +3.32912 q^{61} +2.23769 q^{63} +0.310343 q^{65} -5.76864 q^{67} +5.93119 q^{69} +9.62916 q^{71} +0.681940 q^{73} -1.69668 q^{75} +1.83930 q^{77} -5.77867 q^{79} -3.87010 q^{81} +2.92400 q^{83} +10.4563 q^{85} +11.1502 q^{87} -5.74214 q^{89} -0.295576 q^{91} +12.0969 q^{93} -5.40240 q^{95} -5.37829 q^{97} +1.21660 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{5} + 5 q^{9} - 5 q^{11} + 10 q^{13} - 4 q^{15} - 2 q^{17} + 12 q^{21} - 6 q^{23} + 3 q^{25} + 12 q^{27} + 10 q^{29} + 2 q^{31} + 16 q^{37} + 12 q^{39} - 2 q^{41} - 4 q^{43} + 20 q^{45} + 2 q^{47}+ \cdots - 5 q^{99}+O(q^{100}) $ 5 * q + 4 * q^5 + 5 * q^9 - 5 * q^11 + 10 * q^13 - 4 * q^15 - 2 * q^17 + 12 * q^21 - 6 * q^23 + 3 * q^25 + 12 * q^27 + 10 * q^29 + 2 * q^31 + 16 * q^37 + 12 * q^39 - 2 * q^41 - 4 * q^43 + 20 * q^45 + 2 * q^47 - 3 * q^49 + 8 * q^53 - 4 * q^55 - 8 * q^57 + 8 * q^59 + 30 * q^61 + 20 * q^63 + 8 * q^65 + 24 * q^69 - 6 * q^71 + 2 * q^73 - 36 * q^75 - 8 * q^79 - 3 * q^81 + 28 * q^85 + 16 * q^87 + 2 * q^89 + 32 * q^91 + 16 * q^93 - 12 * q^95 - 10 * q^97 - 5 * 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
$3$	1.33544	0.771017	0.385509	−	0.922704i	$-0.374026\pi$
$3$	1.33544	0.771017	0.385509	+	0.922704i	$0.374026\pi$
$4$	0	0
$5$	1.93119	0.863655	0.431827	−	0.901956i	$-0.357869\pi$
$5$	1.93119	0.863655	0.431827	+	0.901956i	$0.357869\pi$
$6$	0	0
$7$	−1.83930	−0.695190	−0.347595	−	0.937645i	$-0.613002\pi$
$7$	−1.83930	−0.695190	−0.347595	+	0.937645i	$0.613002\pi$
$8$	0	0
$9$	−1.21660	−0.405532
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.160700	0.0445702	0.0222851	−	0.999752i	$-0.492906\pi$
$13$	0.160700	0.0445702	0.0222851	+	0.999752i	$0.492906\pi$
$14$	0	0
$15$	2.57899	0.665893
$16$	0	0
$17$	5.41442	1.31319	0.656595	−	0.754243i	$-0.271996\pi$
$17$	5.41442	1.31319	0.656595	+	0.754243i	$0.271996\pi$
$18$	0	0
$19$	−2.79744	−0.641777	−0.320889	−	0.947117i	$-0.603982\pi$
$19$	−2.79744	−0.641777	−0.320889	+	0.947117i	$0.603982\pi$
$20$	0	0
$21$	−2.45628	−0.536003
$22$	0	0
$23$	4.44137	0.926090	0.463045	−	0.886335i	$-0.346757\pi$
$23$	4.44137	0.926090	0.463045	+	0.886335i	$0.346757\pi$
$24$	0	0
$25$	−1.27050	−0.254101
$26$	0	0
$27$	−5.63102	−1.08369
$28$	0	0
$29$	8.34948	1.55046	0.775230	−	0.631679i	$-0.217634\pi$
$29$	8.34948	1.55046	0.775230	+	0.631679i	$0.217634\pi$
$30$	0	0
$31$	9.05835	1.62693	0.813464	−	0.581616i	$-0.197579\pi$
$31$	9.05835	1.62693	0.813464	+	0.581616i	$0.197579\pi$
$32$	0	0
$33$	−1.33544	−0.232470
$34$	0	0
$35$	−3.55204	−0.600404
$36$	0	0
$37$	−0.602073	−0.0989801	−0.0494901	−	0.998775i	$-0.515760\pi$
$37$	−0.602073	−0.0989801	−0.0494901	+	0.998775i	$0.515760\pi$
$38$	0	0
$39$	0.214606	0.0343644
$40$	0	0
$41$	11.7639	1.83721	0.918606	−	0.395174i	$-0.129316\pi$
$41$	11.7639	1.83721	0.918606	+	0.395174i	$0.129316\pi$
$42$	0	0
$43$	4.47936	0.683096	0.341548	−	0.939864i	$-0.389049\pi$
$43$	4.47936	0.683096	0.341548	+	0.939864i	$0.389049\pi$
$44$	0	0
$45$	−2.34948	−0.350240
$46$	0	0
$47$	−6.29558	−0.918304	−0.459152	−	0.888358i	$-0.651847\pi$
$47$	−6.29558	−0.918304	−0.459152	+	0.888358i	$0.651847\pi$
$48$	0	0
$49$	−3.61698	−0.516711
$50$	0	0
$51$	7.23064	1.01249
$52$	0	0
$53$	11.7211	1.61001	0.805005	−	0.593268i	$-0.202162\pi$
$53$	11.7211	1.61001	0.805005	+	0.593268i	$0.202162\pi$
$54$	0	0
$55$	−1.93119	−0.260402
$56$	0	0
$57$	−3.73582	−0.494821
$58$	0	0
$59$	9.63102	1.25385	0.626926	−	0.779079i	$-0.284313\pi$
$59$	9.63102	1.25385	0.626926	+	0.779079i	$0.284313\pi$
$60$	0	0
$61$	3.32912	0.426250	0.213125	−	0.977025i	$-0.431636\pi$
$61$	3.32912	0.426250	0.213125	+	0.977025i	$0.431636\pi$
$62$	0	0
$63$	2.23769	0.281922
$64$	0	0
$65$	0.310343	0.0384933
$66$	0	0
$67$	−5.76864	−0.704751	−0.352376	−	0.935859i	$-0.614626\pi$
$67$	−5.76864	−0.704751	−0.352376	+	0.935859i	$0.614626\pi$
$68$	0	0
$69$	5.93119	0.714031
$70$	0	0
$71$	9.62916	1.14277	0.571386	−	0.820682i	$-0.306406\pi$
$71$	9.62916	1.14277	0.571386	+	0.820682i	$0.306406\pi$
$72$	0	0
$73$	0.681940	0.0798150	0.0399075	−	0.999203i	$-0.487294\pi$
$73$	0.681940	0.0798150	0.0399075	+	0.999203i	$0.487294\pi$
$74$	0	0
$75$	−1.69668	−0.195916
$76$	0	0
$77$	1.83930	0.209608
$78$	0	0
$79$	−5.77867	−0.650151	−0.325075	−	0.945688i	$-0.605390\pi$
$79$	−5.77867	−0.650151	−0.325075	+	0.945688i	$0.605390\pi$
$80$	0	0
$81$	−3.87010	−0.430011
$82$	0	0
$83$	2.92400	0.320951	0.160476	−	0.987040i	$-0.448697\pi$
$83$	2.92400	0.320951	0.160476	+	0.987040i	$0.448697\pi$
$84$	0	0
$85$	10.4563	1.13414
$86$	0	0
$87$	11.1502	1.19543
$88$	0	0
$89$	−5.74214	−0.608666	−0.304333	−	0.952566i	$-0.598434\pi$
$89$	−5.74214	−0.608666	−0.304333	+	0.952566i	$0.598434\pi$
$90$	0	0
$91$	−0.295576	−0.0309848
$92$	0	0
$93$	12.0969	1.25439
$94$	0	0
$95$	−5.40240	−0.554274
$96$	0	0
$97$	−5.37829	−0.546082	−0.273041	−	0.962002i	$-0.588030\pi$
$97$	−5.37829	−0.546082	−0.273041	+	0.962002i	$0.588030\pi$
$98$	0	0
$99$	1.21660	0.122273
$100$	0	0
$101$	9.31868	0.927244	0.463622	−	0.886033i	$-0.346550\pi$
$101$	9.31868	0.927244	0.463622	+	0.886033i	$0.346550\pi$
$102$	0	0
$103$	1.67860	0.165397	0.0826987	−	0.996575i	$-0.473646\pi$
$103$	1.67860	0.165397	0.0826987	+	0.996575i	$0.473646\pi$
$104$	0	0
$105$	−4.74354	−0.462922
$106$	0	0
$107$	−11.2768	−1.09017	−0.545085	−	0.838381i	$-0.683503\pi$
$107$	−11.2768	−1.09017	−0.545085	+	0.838381i	$0.683503\pi$
$108$	0	0
$109$	19.6373	1.88092	0.940458	−	0.339910i	$-0.110397\pi$
$109$	19.6373	1.88092	0.940458	+	0.339910i	$0.110397\pi$
$110$	0	0
$111$	−0.804032	−0.0763154
$112$	0	0
$113$	−12.9156	−1.21499	−0.607497	−	0.794322i	$-0.707826\pi$
$113$	−12.9156	−1.21499	−0.607497	+	0.794322i	$0.707826\pi$
$114$	0	0
$115$	8.57714	0.799822
$116$	0	0
$117$	−0.195508	−0.0180747
$118$	0	0
$119$	−9.95874	−0.912916
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.7100	1.41652
$124$	0	0
$125$	−12.1095	−1.08311
$126$	0	0
$127$	−1.12716	−0.100019	−0.0500096	−	0.998749i	$-0.515925\pi$
$127$	−1.12716	−0.100019	−0.0500096	+	0.998749i	$0.515925\pi$
$128$	0	0
$129$	5.98192	0.526679
$130$	0	0
$131$	10.9554	0.957178	0.478589	−	0.878039i	$-0.341148\pi$
$131$	10.9554	0.957178	0.478589	+	0.878039i	$0.341148\pi$
$132$	0	0
$133$	5.14534	0.446157
$134$	0	0
$135$	−10.8746	−0.935934
$136$	0	0
$137$	10.1753	0.869338	0.434669	−	0.900590i	$-0.356865\pi$
$137$	10.1753	0.869338	0.434669	+	0.900590i	$0.356865\pi$
$138$	0	0
$139$	−18.3347	−1.55513	−0.777565	−	0.628802i	$-0.783545\pi$
$139$	−18.3347	−1.55513	−0.777565	+	0.628802i	$0.783545\pi$
$140$	0	0
$141$	−8.40737	−0.708028
$142$	0	0
$143$	−0.160700	−0.0134384
$144$	0	0
$145$	16.1244	1.33906
$146$	0	0
$147$	−4.83026	−0.398393
$148$	0	0
$149$	−8.46892	−0.693801	−0.346901	−	0.937902i	$-0.612766\pi$
$149$	−8.46892	−0.693801	−0.346901	+	0.937902i	$0.612766\pi$
$150$	0	0
$151$	−10.0705	−0.819528	−0.409764	−	0.912191i	$-0.634389\pi$
$151$	−10.0705	−0.819528	−0.409764	+	0.912191i	$0.634389\pi$
$152$	0	0
$153$	−6.58717	−0.532541
$154$	0	0
$155$	17.4934	1.40510
$156$	0	0
$157$	8.73969	0.697503	0.348752	−	0.937215i	$-0.386606\pi$
$157$	8.73969	0.697503	0.348752	+	0.937215i	$0.386606\pi$
$158$	0	0
$159$	15.6528	1.24135
$160$	0	0
$161$	−8.16901	−0.643809
$162$	0	0
$163$	−24.4237	−1.91301	−0.956507	−	0.291710i	$-0.905776\pi$
$163$	−24.4237	−1.91301	−0.956507	+	0.291710i	$0.905776\pi$
$164$	0	0
$165$	−2.57899	−0.200774
$166$	0	0
$167$	−7.36485	−0.569909	−0.284954	−	0.958541i	$-0.591978\pi$
$167$	−7.36485	−0.569909	−0.284954	+	0.958541i	$0.591978\pi$
$168$	0	0
$169$	−12.9742	−0.998013
$170$	0	0
$171$	3.40336	0.260262
$172$	0	0
$173$	1.38302	0.105149	0.0525747	−	0.998617i	$-0.483257\pi$
$173$	1.38302	0.105149	0.0525747	+	0.998617i	$0.483257\pi$
$174$	0	0
$175$	2.33684	0.176648
$176$	0	0
$177$	12.8617	0.966741
$178$	0	0
$179$	−9.89679	−0.739720	−0.369860	−	0.929087i	$-0.620594\pi$
$179$	−9.89679	−0.739720	−0.369860	+	0.929087i	$0.620594\pi$
$180$	0	0
$181$	−11.6803	−0.868192	−0.434096	−	0.900867i	$-0.642932\pi$
$181$	−11.6803	−0.868192	−0.434096	+	0.900867i	$0.642932\pi$
$182$	0	0
$183$	4.44584	0.328646
$184$	0	0
$185$	−1.16272	−0.0854846
$186$	0	0
$187$	−5.41442	−0.395942
$188$	0	0
$189$	10.3571	0.753370
$190$	0	0
$191$	−12.0902	−0.874814	−0.437407	−	0.899264i	$-0.644103\pi$
$191$	−12.0902	−0.874814	−0.437407	+	0.899264i	$0.644103\pi$
$192$	0	0
$193$	−13.2620	−0.954622	−0.477311	−	0.878734i	$-0.658388\pi$
$193$	−13.2620	−0.954622	−0.477311	+	0.878734i	$0.658388\pi$
$194$	0	0
$195$	0.414445	0.0296790
$196$	0	0
$197$	11.6835	0.832415	0.416208	−	0.909270i	$-0.363359\pi$
$197$	11.6835	0.832415	0.416208	+	0.909270i	$0.363359\pi$
$198$	0	0
$199$	−24.9329	−1.76745	−0.883725	−	0.468007i	$-0.844972\pi$
$199$	−24.9329	−1.76745	−0.883725	+	0.468007i	$0.844972\pi$
$200$	0	0
$201$	−7.70367	−0.543375
$202$	0	0
$203$	−15.3572	−1.07786
$204$	0	0
$205$	22.7183	1.58672
$206$	0	0
$207$	−5.40336	−0.375560
$208$	0	0
$209$	2.79744	0.193503
$210$	0	0
$211$	5.96245	0.410472	0.205236	−	0.978713i	$-0.434204\pi$
$211$	5.96245	0.410472	0.205236	+	0.978713i	$0.434204\pi$
$212$	0	0
$213$	12.8592	0.881096
$214$	0	0
$215$	8.65049	0.589959
$216$	0	0
$217$	−16.6610	−1.13102
$218$	0	0
$219$	0.910690	0.0615387
$220$	0	0
$221$	0.870099	0.0585292
$222$	0	0
$223$	−1.03801	−0.0695103	−0.0347552	−	0.999396i	$-0.511065\pi$
$223$	−1.03801	−0.0695103	−0.0347552	+	0.999396i	$0.511065\pi$
$224$	0	0
$225$	1.54569	0.103046
$226$	0	0
$227$	6.65612	0.441782	0.220891	−	0.975299i	$-0.429104\pi$
$227$	6.65612	0.441782	0.220891	+	0.975299i	$0.429104\pi$
$228$	0	0
$229$	16.3885	1.08298	0.541491	−	0.840707i	$-0.317860\pi$
$229$	16.3885	1.08298	0.541491	+	0.840707i	$0.317860\pi$
$230$	0	0
$231$	2.45628	0.161611
$232$	0	0
$233$	16.0742	1.05306	0.526529	−	0.850157i	$-0.323493\pi$
$233$	16.0742	1.05306	0.526529	+	0.850157i	$0.323493\pi$
$234$	0	0
$235$	−12.1580	−0.793098
$236$	0	0
$237$	−7.71707	−0.501278
$238$	0	0
$239$	2.52062	0.163045	0.0815226	−	0.996671i	$-0.474022\pi$
$239$	2.52062	0.163045	0.0815226	+	0.996671i	$0.474022\pi$
$240$	0	0
$241$	6.11179	0.393695	0.196848	−	0.980434i	$-0.436930\pi$
$241$	6.11179	0.393695	0.196848	+	0.980434i	$0.436930\pi$
$242$	0	0
$243$	11.7248	0.752144
$244$	0	0
$245$	−6.98507	−0.446260
$246$	0	0
$247$	−0.449550	−0.0286042
$248$	0	0
$249$	3.90483	0.247459
$250$	0	0
$251$	−0.489390	−0.0308900	−0.0154450	−	0.999881i	$-0.504916\pi$
$251$	−0.489390	−0.0308900	−0.0154450	+	0.999881i	$0.504916\pi$
$252$	0	0
$253$	−4.44137	−0.279227
$254$	0	0
$255$	13.9637	0.874443
$256$	0	0
$257$	10.0886	0.629311	0.314655	−	0.949206i	$-0.398111\pi$
$257$	10.0886	0.629311	0.314655	+	0.949206i	$0.398111\pi$
$258$	0	0
$259$	1.10739	0.0688100
$260$	0	0
$261$	−10.1580	−0.628762
$262$	0	0
$263$	3.98466	0.245705	0.122852	−	0.992425i	$-0.460796\pi$
$263$	3.98466	0.245705	0.122852	+	0.992425i	$0.460796\pi$
$264$	0	0
$265$	22.6356	1.39049
$266$	0	0
$267$	−7.66829	−0.469292
$268$	0	0
$269$	2.70069	0.164664	0.0823320	−	0.996605i	$-0.473763\pi$
$269$	2.70069	0.164664	0.0823320	+	0.996605i	$0.473763\pi$
$270$	0	0
$271$	−22.3611	−1.35834	−0.679170	−	0.733981i	$-0.737660\pi$
$271$	−22.3611	−1.35834	−0.679170	+	0.733981i	$0.737660\pi$
$272$	0	0
$273$	−0.394724	−0.0238898
$274$	0	0
$275$	1.27050	0.0766142
$276$	0	0
$277$	31.4865	1.89184	0.945921	−	0.324396i	$-0.105161\pi$
$277$	31.4865	1.89184	0.945921	+	0.324396i	$0.105161\pi$
$278$	0	0
$279$	−11.0204	−0.659772
$280$	0	0
$281$	9.57079	0.570945	0.285473	−	0.958387i	$-0.407849\pi$
$281$	9.57079	0.570945	0.285473	+	0.958387i	$0.407849\pi$
$282$	0	0
$283$	16.3997	0.974860	0.487430	−	0.873162i	$-0.337934\pi$
$283$	16.3997	0.974860	0.487430	+	0.873162i	$0.337934\pi$
$284$	0	0
$285$	−7.21458	−0.427355
$286$	0	0
$287$	−21.6373	−1.27721
$288$	0	0
$289$	12.3159	0.724467
$290$	0	0
$291$	−7.18239	−0.421039
$292$	0	0
$293$	−10.0087	−0.584715	−0.292357	−	0.956309i	$-0.594440\pi$
$293$	−10.0087	−0.584715	−0.292357	+	0.956309i	$0.594440\pi$
$294$	0	0
$295$	18.5993	1.08289
$296$	0	0
$297$	5.63102	0.326745
$298$	0	0
$299$	0.713730	0.0412761
$300$	0	0
$301$	−8.23888	−0.474881
$302$	0	0
$303$	12.4446	0.714921
$304$	0	0
$305$	6.42916	0.368133
$306$	0	0
$307$	18.8436	1.07546	0.537731	−	0.843117i	$-0.319282\pi$
$307$	18.8436	1.07546	0.537731	+	0.843117i	$0.319282\pi$
$308$	0	0
$309$	2.24167	0.127524
$310$	0	0
$311$	−1.59661	−0.0905356	−0.0452678	−	0.998975i	$-0.514414\pi$
$311$	−1.59661	−0.0905356	−0.0452678	+	0.998975i	$0.514414\pi$
$312$	0	0
$313$	−3.94136	−0.222779	−0.111389	−	0.993777i	$-0.535530\pi$
$313$	−3.94136	−0.222779	−0.111389	+	0.993777i	$0.535530\pi$
$314$	0	0
$315$	4.32140	0.243483
$316$	0	0
$317$	1.94384	0.109177	0.0545884	−	0.998509i	$-0.482615\pi$
$317$	1.94384	0.109177	0.0545884	+	0.998509i	$0.482615\pi$
$318$	0	0
$319$	−8.34948	−0.467481
$320$	0	0
$321$	−15.0595	−0.840539
$322$	0	0
$323$	−15.1465	−0.842776
$324$	0	0
$325$	−0.204170	−0.0113253
$326$	0	0
$327$	26.2245	1.45022
$328$	0	0
$329$	11.5795	0.638396
$330$	0	0
$331$	7.52148	0.413418	0.206709	−	0.978402i	$-0.433725\pi$
$331$	7.52148	0.413418	0.206709	+	0.978402i	$0.433725\pi$
$332$	0	0
$333$	0.732480	0.0401397
$334$	0	0
$335$	−11.1403	−0.608662
$336$	0	0
$337$	−8.47233	−0.461517	−0.230759	−	0.973011i	$-0.574121\pi$
$337$	−8.47233	−0.461517	−0.230759	+	0.973011i	$0.574121\pi$
$338$	0	0
$339$	−17.2480	−0.936781
$340$	0	0
$341$	−9.05835	−0.490537
$342$	0	0
$343$	19.5278	1.05440
$344$	0	0
$345$	11.4543	0.616677
$346$	0	0
$347$	17.2487	0.925960	0.462980	−	0.886369i	$-0.346780\pi$
$347$	17.2487	0.925960	0.462980	+	0.886369i	$0.346780\pi$
$348$	0	0
$349$	−4.67588	−0.250294	−0.125147	−	0.992138i	$-0.539940\pi$
$349$	−4.67588	−0.250294	−0.125147	+	0.992138i	$0.539940\pi$
$350$	0	0
$351$	−0.904906	−0.0483003
$352$	0	0
$353$	−5.45055	−0.290104	−0.145052	−	0.989424i	$-0.546335\pi$
$353$	−5.45055	−0.290104	−0.145052	+	0.989424i	$0.546335\pi$
$354$	0	0
$355$	18.5957	0.986960
$356$	0	0
$357$	−13.2993	−0.703874
$358$	0	0
$359$	−28.0176	−1.47871	−0.739357	−	0.673313i	$-0.764871\pi$
$359$	−28.0176	−1.47871	−0.739357	+	0.673313i	$0.764871\pi$
$360$	0	0
$361$	−11.1743	−0.588122
$362$	0	0
$363$	1.33544	0.0700925
$364$	0	0
$365$	1.31696	0.0689326
$366$	0	0
$367$	28.8393	1.50540	0.752699	−	0.658365i	$-0.228752\pi$
$367$	28.8393	1.50540	0.752699	+	0.658365i	$0.228752\pi$
$368$	0	0
$369$	−14.3119	−0.745049
$370$	0	0
$371$	−21.5585	−1.11926
$372$	0	0
$373$	1.34080	0.0694239	0.0347119	−	0.999397i	$-0.488949\pi$
$373$	1.34080	0.0694239	0.0347119	+	0.999397i	$0.488949\pi$
$374$	0	0
$375$	−16.1716	−0.835096
$376$	0	0
$377$	1.34176	0.0691044
$378$	0	0
$379$	8.72245	0.448042	0.224021	−	0.974584i	$-0.428082\pi$
$379$	8.72245	0.448042	0.224021	+	0.974584i	$0.428082\pi$
$380$	0	0
$381$	−1.50525	−0.0771165
$382$	0	0
$383$	−7.76275	−0.396658	−0.198329	−	0.980136i	$-0.563551\pi$
$383$	−7.76275	−0.396658	−0.198329	+	0.980136i	$0.563551\pi$
$384$	0	0
$385$	3.55204	0.181029
$386$	0	0
$387$	−5.44957	−0.277017
$388$	0	0
$389$	−38.5938	−1.95678	−0.978392	−	0.206760i	$-0.933708\pi$
$389$	−38.5938	−1.95678	−0.978392	+	0.206760i	$0.933708\pi$
$390$	0	0
$391$	24.0475	1.21613
$392$	0	0
$393$	14.6303	0.738001
$394$	0	0
$395$	−11.1597	−0.561506
$396$	0	0
$397$	−20.4200	−1.02485	−0.512426	−	0.858732i	$-0.671253\pi$
$397$	−20.4200	−1.02485	−0.512426	+	0.858732i	$0.671253\pi$
$398$	0	0
$399$	6.87129	0.343995
$400$	0	0
$401$	4.96247	0.247814	0.123907	−	0.992294i	$-0.460458\pi$
$401$	4.96247	0.247814	0.123907	+	0.992294i	$0.460458\pi$
$402$	0	0
$403$	1.45568	0.0725125
$404$	0	0
$405$	−7.47390	−0.371381
$406$	0	0
$407$	0.602073	0.0298436
$408$	0	0
$409$	−15.3418	−0.758601	−0.379301	−	0.925273i	$-0.623835\pi$
$409$	−15.3418	−0.758601	−0.379301	+	0.925273i	$0.623835\pi$
$410$	0	0
$411$	13.5886	0.670274
$412$	0	0
$413$	−17.7143	−0.871665
$414$	0	0
$415$	5.64681	0.277191
$416$	0	0
$417$	−24.4849	−1.19903
$418$	0	0
$419$	12.0628	0.589306	0.294653	−	0.955604i	$-0.404796\pi$
$419$	12.0628	0.589306	0.294653	+	0.955604i	$0.404796\pi$
$420$	0	0
$421$	14.3042	0.697143	0.348572	−	0.937282i	$-0.386667\pi$
$421$	14.3042	0.697143	0.348572	+	0.937282i	$0.386667\pi$
$422$	0	0
$423$	7.65918	0.372402
$424$	0	0
$425$	−6.87904	−0.333682
$426$	0	0
$427$	−6.12325	−0.296325
$428$	0	0
$429$	−0.214606	−0.0103613
$430$	0	0
$431$	18.9520	0.912885	0.456443	−	0.889753i	$-0.349123\pi$
$431$	18.9520	0.912885	0.456443	+	0.889753i	$0.349123\pi$
$432$	0	0
$433$	−16.4130	−0.788760	−0.394380	−	0.918948i	$-0.629041\pi$
$433$	−16.4130	−0.788760	−0.394380	+	0.918948i	$0.629041\pi$
$434$	0	0
$435$	21.5332	1.03244
$436$	0	0
$437$	−12.4245	−0.594344
$438$	0	0
$439$	−24.5482	−1.17162	−0.585810	−	0.810448i	$-0.699224\pi$
$439$	−24.5482	−1.17162	−0.585810	+	0.810448i	$0.699224\pi$
$440$	0	0
$441$	4.40040	0.209543
$442$	0	0
$443$	11.0679	0.525854	0.262927	−	0.964816i	$-0.415312\pi$
$443$	11.0679	0.525854	0.262927	+	0.964816i	$0.415312\pi$
$444$	0	0
$445$	−11.0892	−0.525677
$446$	0	0
$447$	−11.3097	−0.534933
$448$	0	0
$449$	26.6647	1.25838	0.629192	−	0.777250i	$-0.283386\pi$
$449$	26.6647	1.25838	0.629192	+	0.777250i	$0.283386\pi$
$450$	0	0
$451$	−11.7639	−0.553940
$452$	0	0
$453$	−13.4486	−0.631871
$454$	0	0
$455$	−0.570814	−0.0267602
$456$	0	0
$457$	−3.28296	−0.153570	−0.0767851	−	0.997048i	$-0.524466\pi$
$457$	−3.28296	−0.153570	−0.0767851	+	0.997048i	$0.524466\pi$
$458$	0	0
$459$	−30.4887	−1.42309
$460$	0	0
$461$	−3.17514	−0.147881	−0.0739406	−	0.997263i	$-0.523558\pi$
$461$	−3.17514	−0.147881	−0.0739406	+	0.997263i	$0.523558\pi$
$462$	0	0
$463$	−21.8696	−1.01637	−0.508184	−	0.861248i	$-0.669683\pi$
$463$	−21.8696	−1.01637	−0.508184	+	0.861248i	$0.669683\pi$
$464$	0	0
$465$	23.3614	1.08336
$466$	0	0
$467$	4.95735	0.229399	0.114699	−	0.993400i	$-0.463410\pi$
$467$	4.95735	0.229399	0.114699	+	0.993400i	$0.463410\pi$
$468$	0	0
$469$	10.6103	0.489936
$470$	0	0
$471$	11.6713	0.537787
$472$	0	0
$473$	−4.47936	−0.205961
$474$	0	0
$475$	3.55416	0.163076
$476$	0	0
$477$	−14.2598	−0.652912
$478$	0	0
$479$	35.5423	1.62397	0.811984	−	0.583680i	$-0.198388\pi$
$479$	35.5423	1.62397	0.811984	+	0.583680i	$0.198388\pi$
$480$	0	0
$481$	−0.0967532	−0.00441157
$482$	0	0
$483$	−10.9092	−0.496387
$484$	0	0
$485$	−10.3865	−0.471627
$486$	0	0
$487$	10.1547	0.460153	0.230076	−	0.973173i	$-0.426102\pi$
$487$	10.1547	0.460153	0.230076	+	0.973173i	$0.426102\pi$
$488$	0	0
$489$	−32.6164	−1.47497
$490$	0	0
$491$	−0.367564	−0.0165879	−0.00829395	−	0.999966i	$-0.502640\pi$
$491$	−0.367564	−0.0165879	−0.00829395	+	0.999966i	$0.502640\pi$
$492$	0	0
$493$	45.2076	2.03605
$494$	0	0
$495$	2.34948	0.105601
$496$	0	0
$497$	−17.7109	−0.794443
$498$	0	0
$499$	31.1824	1.39591	0.697957	−	0.716140i	$-0.254093\pi$
$499$	31.1824	1.39591	0.697957	+	0.716140i	$0.254093\pi$
$500$	0	0
$501$	−9.83532	−0.439410
$502$	0	0
$503$	20.2004	0.900692	0.450346	−	0.892854i	$-0.351301\pi$
$503$	20.2004	0.900692	0.450346	+	0.892854i	$0.351301\pi$
$504$	0	0
$505$	17.9962	0.800818
$506$	0	0
$507$	−17.3262	−0.769486
$508$	0	0
$509$	−22.0209	−0.976058	−0.488029	−	0.872827i	$-0.662284\pi$
$509$	−22.0209	−0.976058	−0.488029	+	0.872827i	$0.662284\pi$
$510$	0	0
$511$	−1.25429	−0.0554866
$512$	0	0
$513$	15.7524	0.695488
$514$	0	0
$515$	3.24170	0.142846
$516$	0	0
$517$	6.29558	0.276879
$518$	0	0
$519$	1.84695	0.0810719
$520$	0	0
$521$	−36.1120	−1.58209	−0.791047	−	0.611755i	$-0.790464\pi$
$521$	−36.1120	−1.58209	−0.791047	+	0.611755i	$0.790464\pi$
$522$	0	0
$523$	−34.5388	−1.51028	−0.755139	−	0.655565i	$-0.772431\pi$
$523$	−34.5388	−1.51028	−0.755139	+	0.655565i	$0.772431\pi$
$524$	0	0
$525$	3.12071	0.136199
$526$	0	0
$527$	49.0457	2.13646
$528$	0	0
$529$	−3.27421	−0.142357
$530$	0	0
$531$	−11.7171	−0.508478
$532$	0	0
$533$	1.89046	0.0818850
$534$	0	0
$535$	−21.7777	−0.941530
$536$	0	0
$537$	−13.2166	−0.570337
$538$	0	0
$539$	3.61698	0.155794
$540$	0	0
$541$	6.14527	0.264206	0.132103	−	0.991236i	$-0.457827\pi$
$541$	6.14527	0.264206	0.132103	+	0.991236i	$0.457827\pi$
$542$	0	0
$543$	−15.5984	−0.669391
$544$	0	0
$545$	37.9234	1.62446
$546$	0	0
$547$	−17.3331	−0.741110	−0.370555	−	0.928811i	$-0.620832\pi$
$547$	−17.3331	−0.741110	−0.370555	+	0.928811i	$0.620832\pi$
$548$	0	0
$549$	−4.05020	−0.172858
$550$	0	0
$551$	−23.3572	−0.995050
$552$	0	0
$553$	10.6287	0.451978
$554$	0	0
$555$	−1.55274	−0.0659101
$556$	0	0
$557$	−38.1630	−1.61702	−0.808508	−	0.588485i	$-0.799725\pi$
$557$	−38.1630	−1.61702	−0.808508	+	0.588485i	$0.799725\pi$
$558$	0	0
$559$	0.719834	0.0304457
$560$	0	0
$561$	−7.23064	−0.305278
$562$	0	0
$563$	33.2131	1.39976	0.699882	−	0.714258i	$-0.253236\pi$
$563$	33.2131	1.39976	0.699882	+	0.714258i	$0.253236\pi$
$564$	0	0
$565$	−24.9424	−1.04934
$566$	0	0
$567$	7.11827	0.298939
$568$	0	0
$569$	−8.23504	−0.345231	−0.172615	−	0.984989i	$-0.555222\pi$
$569$	−8.23504	−0.345231	−0.172615	+	0.984989i	$0.555222\pi$
$570$	0	0
$571$	32.7618	1.37104	0.685518	−	0.728055i	$-0.259576\pi$
$571$	32.7618	1.37104	0.685518	+	0.728055i	$0.259576\pi$
$572$	0	0
$573$	−16.1457	−0.674496
$574$	0	0
$575$	−5.64278	−0.235320
$576$	0	0
$577$	−6.50814	−0.270937	−0.135469	−	0.990782i	$-0.543254\pi$
$577$	−6.50814	−0.270937	−0.135469	+	0.990782i	$0.543254\pi$
$578$	0	0
$579$	−17.7107	−0.736030
$580$	0	0
$581$	−5.37812	−0.223122
$582$	0	0
$583$	−11.7211	−0.485436
$584$	0	0
$585$	−0.377562	−0.0156103
$586$	0	0
$587$	9.95325	0.410815	0.205407	−	0.978677i	$-0.434148\pi$
$587$	9.95325	0.410815	0.205407	+	0.978677i	$0.434148\pi$
$588$	0	0
$589$	−25.3402	−1.04413
$590$	0	0
$591$	15.6026	0.641806
$592$	0	0
$593$	8.80877	0.361733	0.180867	−	0.983508i	$-0.442110\pi$
$593$	8.80877	0.361733	0.180867	+	0.983508i	$0.442110\pi$
$594$	0	0
$595$	−19.2322	−0.788444
$596$	0	0
$597$	−33.2965	−1.36273
$598$	0	0
$599$	31.9145	1.30399	0.651996	−	0.758223i	$-0.273932\pi$
$599$	31.9145	1.30399	0.651996	+	0.758223i	$0.273932\pi$
$600$	0	0
$601$	−11.0314	−0.449981	−0.224991	−	0.974361i	$-0.572235\pi$
$601$	−11.0314	−0.449981	−0.224991	+	0.974361i	$0.572235\pi$
$602$	0	0
$603$	7.01811	0.285799
$604$	0	0
$605$	1.93119	0.0785141
$606$	0	0
$607$	−12.2159	−0.495828	−0.247914	−	0.968782i	$-0.579745\pi$
$607$	−12.2159	−0.495828	−0.247914	+	0.968782i	$0.579745\pi$
$608$	0	0
$609$	−20.5086	−0.831052
$610$	0	0
$611$	−1.01170	−0.0409290
$612$	0	0
$613$	3.57482	0.144386	0.0721928	−	0.997391i	$-0.477000\pi$
$613$	3.57482	0.144386	0.0721928	+	0.997391i	$0.477000\pi$
$614$	0	0
$615$	30.3390	1.22339
$616$	0	0
$617$	−30.5057	−1.22811	−0.614057	−	0.789262i	$-0.710463\pi$
$617$	−30.5057	−1.22811	−0.614057	+	0.789262i	$0.710463\pi$
$618$	0	0
$619$	−10.6118	−0.426526	−0.213263	−	0.976995i	$-0.568409\pi$
$619$	−10.6118	−0.426526	−0.213263	+	0.976995i	$0.568409\pi$
$620$	0	0
$621$	−25.0094	−1.00359
$622$	0	0
$623$	10.5615	0.423138
$624$	0	0
$625$	−17.0333	−0.681332
$626$	0	0
$627$	3.73582	0.149194
$628$	0	0
$629$	−3.25987	−0.129980
$630$	0	0
$631$	4.29235	0.170876	0.0854379	−	0.996343i	$-0.472771\pi$
$631$	4.29235	0.170876	0.0854379	+	0.996343i	$0.472771\pi$
$632$	0	0
$633$	7.96250	0.316481
$634$	0	0
$635$	−2.17676	−0.0863820
$636$	0	0
$637$	−0.581249	−0.0230299
$638$	0	0
$639$	−11.7148	−0.463431
$640$	0	0
$641$	−26.5606	−1.04908	−0.524540	−	0.851386i	$-0.675763\pi$
$641$	−26.5606	−1.04908	−0.524540	+	0.851386i	$0.675763\pi$
$642$	0	0
$643$	−28.0614	−1.10663	−0.553317	−	0.832971i	$-0.686638\pi$
$643$	−28.0614	−1.10663	−0.553317	+	0.832971i	$0.686638\pi$
$644$	0	0
$645$	11.5522	0.454868
$646$	0	0
$647$	−32.2983	−1.26978	−0.634888	−	0.772604i	$-0.718954\pi$
$647$	−32.2983	−1.26978	−0.634888	+	0.772604i	$0.718954\pi$
$648$	0	0
$649$	−9.63102	−0.378050
$650$	0	0
$651$	−22.2498	−0.872039
$652$	0	0
$653$	−32.9200	−1.28826	−0.644130	−	0.764916i	$-0.722780\pi$
$653$	−32.9200	−1.28826	−0.644130	+	0.764916i	$0.722780\pi$
$654$	0	0
$655$	21.1570	0.826671
$656$	0	0
$657$	−0.829646	−0.0323676
$658$	0	0
$659$	−3.99602	−0.155663	−0.0778314	−	0.996967i	$-0.524800\pi$
$659$	−3.99602	−0.155663	−0.0778314	+	0.996967i	$0.524800\pi$
$660$	0	0
$661$	−19.2377	−0.748258	−0.374129	−	0.927377i	$-0.622058\pi$
$661$	−19.2377	−0.748258	−0.374129	+	0.927377i	$0.622058\pi$
$662$	0	0
$663$	1.16197	0.0451270
$664$	0	0
$665$	9.93663	0.385326
$666$	0	0
$667$	37.0832	1.43587
$668$	0	0
$669$	−1.38620	−0.0535937
$670$	0	0
$671$	−3.32912	−0.128519
$672$	0	0
$673$	5.41202	0.208618	0.104309	−	0.994545i	$-0.466737\pi$
$673$	5.41202	0.208618	0.104309	+	0.994545i	$0.466737\pi$
$674$	0	0
$675$	7.15422	0.275366
$676$	0	0
$677$	−41.1062	−1.57984	−0.789921	−	0.613209i	$-0.789878\pi$
$677$	−41.1062	−1.57984	−0.789921	+	0.613209i	$0.789878\pi$
$678$	0	0
$679$	9.89229	0.379631
$680$	0	0
$681$	8.88885	0.340621
$682$	0	0
$683$	−35.8592	−1.37211	−0.686057	−	0.727548i	$-0.740660\pi$
$683$	−35.8592	−1.37211	−0.686057	+	0.727548i	$0.740660\pi$
$684$	0	0
$685$	19.6505	0.750808
$686$	0	0
$687$	21.8859	0.834997
$688$	0	0
$689$	1.88358	0.0717585
$690$	0	0
$691$	45.1462	1.71744	0.858721	−	0.512444i	$-0.171260\pi$
$691$	45.1462	1.71744	0.858721	+	0.512444i	$0.171260\pi$
$692$	0	0
$693$	−2.23769	−0.0850027
$694$	0	0
$695$	−35.4078	−1.34310
$696$	0	0
$697$	63.6947	2.41261
$698$	0	0
$699$	21.4662	0.811926
$700$	0	0
$701$	18.5867	0.702009	0.351005	−	0.936374i	$-0.385840\pi$
$701$	18.5867	0.702009	0.351005	+	0.936374i	$0.385840\pi$
$702$	0	0
$703$	1.68426	0.0635232
$704$	0	0
$705$	−16.2362	−0.611492
$706$	0	0
$707$	−17.1399	−0.644610
$708$	0	0
$709$	27.2703	1.02416	0.512079	−	0.858938i	$-0.328876\pi$
$709$	27.2703	1.02416	0.512079	+	0.858938i	$0.328876\pi$
$710$	0	0
$711$	7.03031	0.263657
$712$	0	0
$713$	40.2315	1.50668
$714$	0	0
$715$	−0.310343	−0.0116062
$716$	0	0
$717$	3.36614	0.125711
$718$	0	0
$719$	−34.1800	−1.27470	−0.637349	−	0.770575i	$-0.719969\pi$
$719$	−34.1800	−1.27470	−0.637349	+	0.770575i	$0.719969\pi$
$720$	0	0
$721$	−3.08745	−0.114983
$722$	0	0
$723$	8.16194	0.303546
$724$	0	0
$725$	−10.6080	−0.393973
$726$	0	0
$727$	−25.0380	−0.928608	−0.464304	−	0.885676i	$-0.653696\pi$
$727$	−25.0380	−0.928608	−0.464304	+	0.885676i	$0.653696\pi$
$728$	0	0
$729$	27.2680	1.00993
$730$	0	0
$731$	24.2531	0.897034
$732$	0	0
$733$	17.2695	0.637862	0.318931	−	0.947778i	$-0.396676\pi$
$733$	17.2695	0.637862	0.318931	+	0.947778i	$0.396676\pi$
$734$	0	0
$735$	−9.32815	−0.344074
$736$	0	0
$737$	5.76864	0.212490
$738$	0	0
$739$	−17.2687	−0.635240	−0.317620	−	0.948218i	$-0.602884\pi$
$739$	−17.2687	−0.635240	−0.317620	+	0.948218i	$0.602884\pi$
$740$	0	0
$741$	−0.600347	−0.0220543
$742$	0	0
$743$	34.6374	1.27072	0.635361	−	0.772215i	$-0.280851\pi$
$743$	34.6374	1.27072	0.635361	+	0.772215i	$0.280851\pi$
$744$	0	0
$745$	−16.3551	−0.599205
$746$	0	0
$747$	−3.55734	−0.130156
$748$	0	0
$749$	20.7414	0.757875
$750$	0	0
$751$	−51.5526	−1.88118	−0.940591	−	0.339543i	$-0.889728\pi$
$751$	−51.5526	−1.88118	−0.940591	+	0.339543i	$0.889728\pi$
$752$	0	0
$753$	−0.653551	−0.0238167
$754$	0	0
$755$	−19.4481	−0.707790
$756$	0	0
$757$	−32.6049	−1.18504	−0.592522	−	0.805554i	$-0.701868\pi$
$757$	−32.6049	−1.18504	−0.592522	+	0.805554i	$0.701868\pi$
$758$	0	0
$759$	−5.93119	−0.215289
$760$	0	0
$761$	2.79094	0.101172	0.0505858	−	0.998720i	$-0.483891\pi$
$761$	2.79094	0.101172	0.0505858	+	0.998720i	$0.483891\pi$
$762$	0	0
$763$	−36.1190	−1.30759
$764$	0	0
$765$	−12.7211	−0.459932
$766$	0	0
$767$	1.54771	0.0558845
$768$	0	0
$769$	23.1604	0.835187	0.417593	−	0.908634i	$-0.362874\pi$
$769$	23.1604	0.835187	0.417593	+	0.908634i	$0.362874\pi$
$770$	0	0
$771$	13.4728	0.485209
$772$	0	0
$773$	20.7913	0.747811	0.373905	−	0.927467i	$-0.378018\pi$
$773$	20.7913	0.747811	0.373905	+	0.927467i	$0.378018\pi$
$774$	0	0
$775$	−11.5087	−0.413403
$776$	0	0
$777$	1.47886	0.0530537
$778$	0	0
$779$	−32.9088	−1.17908
$780$	0	0
$781$	−9.62916	−0.344559
$782$	0	0
$783$	−47.0161	−1.68022
$784$	0	0
$785$	16.8780	0.602402
$786$	0	0
$787$	−40.3874	−1.43965	−0.719827	−	0.694153i	$-0.755779\pi$
$787$	−40.3874	−1.43965	−0.719827	+	0.694153i	$0.755779\pi$
$788$	0	0
$789$	5.32128	0.189443
$790$	0	0
$791$	23.7556	0.844651
$792$	0	0
$793$	0.534990	0.0189981
$794$	0	0
$795$	30.2285	1.07209
$796$	0	0
$797$	−31.1190	−1.10229	−0.551145	−	0.834409i	$-0.685809\pi$
$797$	−31.1190	−1.10229	−0.551145	+	0.834409i	$0.685809\pi$
$798$	0	0
$799$	−34.0869	−1.20591
$800$	0	0
$801$	6.98588	0.246834
$802$	0	0
$803$	−0.681940	−0.0240651
$804$	0	0
$805$	−15.7759	−0.556028
$806$	0	0
$807$	3.60661	0.126959
$808$	0	0
$809$	−23.6252	−0.830619	−0.415309	−	0.909680i	$-0.636327\pi$
$809$	−23.6252	−0.830619	−0.415309	+	0.909680i	$0.636327\pi$
$810$	0	0
$811$	26.2978	0.923442	0.461721	−	0.887025i	$-0.347232\pi$
$811$	26.2978	0.923442	0.461721	+	0.887025i	$0.347232\pi$
$812$	0	0
$813$	−29.8619	−1.04730
$814$	0	0
$815$	−47.1669	−1.65218
$816$	0	0
$817$	−12.5307	−0.438395
$818$	0	0
$819$	0.359597	0.0125653
$820$	0	0
$821$	55.6264	1.94137	0.970687	−	0.240348i	$-0.0772615\pi$
$821$	55.6264	1.94137	0.970687	+	0.240348i	$0.0772615\pi$
$822$	0	0
$823$	28.0971	0.979402	0.489701	−	0.871890i	$-0.337106\pi$
$823$	28.0971	0.979402	0.489701	+	0.871890i	$0.337106\pi$
$824$	0	0
$825$	1.69668	0.0590709
$826$	0	0
$827$	−38.5555	−1.34070	−0.670352	−	0.742043i	$-0.733857\pi$
$827$	−38.5555	−1.34070	−0.670352	+	0.742043i	$0.733857\pi$
$828$	0	0
$829$	44.4581	1.54409	0.772047	−	0.635566i	$-0.219233\pi$
$829$	44.4581	1.54409	0.772047	+	0.635566i	$0.219233\pi$
$830$	0	0
$831$	42.0484	1.45864
$832$	0	0
$833$	−19.5838	−0.678539
$834$	0	0
$835$	−14.2229	−0.492204
$836$	0	0
$837$	−51.0077	−1.76308
$838$	0	0
$839$	−39.7880	−1.37364	−0.686818	−	0.726830i	$-0.740993\pi$
$839$	−39.7880	−1.37364	−0.686818	+	0.726830i	$0.740993\pi$
$840$	0	0
$841$	40.7138	1.40393
$842$	0	0
$843$	12.7812	0.440209
$844$	0	0
$845$	−25.0556	−0.861939
$846$	0	0
$847$	−1.83930	−0.0631991
$848$	0	0
$849$	21.9008	0.751634
$850$	0	0
$851$	−2.67403	−0.0916645
$852$	0	0
$853$	4.19523	0.143642	0.0718210	−	0.997418i	$-0.477119\pi$
$853$	4.19523	0.143642	0.0718210	+	0.997418i	$0.477119\pi$
$854$	0	0
$855$	6.57254	0.224776
$856$	0	0
$857$	−43.4106	−1.48288	−0.741438	−	0.671021i	$-0.765856\pi$
$857$	−43.4106	−1.48288	−0.741438	+	0.671021i	$0.765856\pi$
$858$	0	0
$859$	31.7672	1.08388	0.541941	−	0.840416i	$-0.317690\pi$
$859$	31.7672	1.08388	0.541941	+	0.840416i	$0.317690\pi$
$860$	0	0
$861$	−28.8954	−0.984752
$862$	0	0
$863$	51.0969	1.73936	0.869679	−	0.493618i	$-0.164326\pi$
$863$	51.0969	1.73936	0.869679	+	0.493618i	$0.164326\pi$
$864$	0	0
$865$	2.67088	0.0908127
$866$	0	0
$867$	16.4472	0.558577
$868$	0	0
$869$	5.77867	0.196028
$870$	0	0
$871$	−0.927021	−0.0314109
$872$	0	0
$873$	6.54321	0.221454
$874$	0	0
$875$	22.2731	0.752967
$876$	0	0
$877$	−45.8912	−1.54963	−0.774817	−	0.632185i	$-0.782158\pi$
$877$	−45.8912	−1.54963	−0.774817	+	0.632185i	$0.782158\pi$
$878$	0	0
$879$	−13.3660	−0.450825
$880$	0	0
$881$	−19.3333	−0.651356	−0.325678	−	0.945481i	$-0.605593\pi$
$881$	−19.3333	−0.651356	−0.325678	+	0.945481i	$0.605593\pi$
$882$	0	0
$883$	22.5862	0.760088	0.380044	−	0.924968i	$-0.375909\pi$
$883$	22.5862	0.760088	0.380044	+	0.924968i	$0.375909\pi$
$884$	0	0
$885$	24.8383	0.834931
$886$	0	0
$887$	−34.2005	−1.14834	−0.574170	−	0.818736i	$-0.694675\pi$
$887$	−34.2005	−1.14834	−0.574170	+	0.818736i	$0.694675\pi$
$888$	0	0
$889$	2.07318	0.0695323
$890$	0	0
$891$	3.87010	0.129653
$892$	0	0
$893$	17.6115	0.589347
$894$	0	0
$895$	−19.1126	−0.638863
$896$	0	0
$897$	0.953144	0.0318246
$898$	0	0
$899$	75.6325	2.52249
$900$	0	0
$901$	63.4627	2.11425
$902$	0	0
$903$	−11.0025	−0.366142
$904$	0	0
$905$	−22.5569	−0.749818
$906$	0	0
$907$	17.2305	0.572129	0.286065	−	0.958210i	$-0.407653\pi$
$907$	17.2305	0.572129	0.286065	+	0.958210i	$0.407653\pi$
$908$	0	0
$909$	−11.3371	−0.376027
$910$	0	0
$911$	28.1534	0.932763	0.466382	−	0.884584i	$-0.345557\pi$
$911$	28.1534	0.932763	0.466382	+	0.884584i	$0.345557\pi$
$912$	0	0
$913$	−2.92400	−0.0967704
$914$	0	0
$915$	8.58577	0.283837
$916$	0	0
$917$	−20.1503	−0.665420
$918$	0	0
$919$	1.87464	0.0618387	0.0309193	−	0.999522i	$-0.490157\pi$
$919$	1.87464	0.0618387	0.0309193	+	0.999522i	$0.490157\pi$
$920$	0	0
$921$	25.1645	0.829199
$922$	0	0
$923$	1.54741	0.0509336
$924$	0	0
$925$	0.764935	0.0251509
$926$	0	0
$927$	−2.04218	−0.0670740
$928$	0	0
$929$	23.2555	0.762989	0.381495	−	0.924371i	$-0.375409\pi$
$929$	23.2555	0.762989	0.381495	+	0.924371i	$0.375409\pi$
$930$	0	0
$931$	10.1183	0.331613
$932$	0	0
$933$	−2.13218	−0.0698045
$934$	0	0
$935$	−10.4563	−0.341957
$936$	0	0
$937$	11.1266	0.363490	0.181745	−	0.983346i	$-0.441826\pi$
$937$	11.1266	0.363490	0.181745	+	0.983346i	$0.441826\pi$
$938$	0	0
$939$	−5.26345	−0.171766
$940$	0	0
$941$	23.5923	0.769088	0.384544	−	0.923107i	$-0.374359\pi$
$941$	23.5923	0.769088	0.384544	+	0.923107i	$0.374359\pi$
$942$	0	0
$943$	52.2479	1.70142
$944$	0	0
$945$	20.0016	0.650652
$946$	0	0
$947$	23.1038	0.750772	0.375386	−	0.926869i	$-0.377510\pi$
$947$	23.1038	0.750772	0.375386	+	0.926869i	$0.377510\pi$
$948$	0	0
$949$	0.109588	0.00355737
$950$	0	0
$951$	2.59588	0.0841771
$952$	0	0
$953$	−2.75964	−0.0893934	−0.0446967	−	0.999001i	$-0.514232\pi$
$953$	−2.75964	−0.0893934	−0.0446967	+	0.999001i	$0.514232\pi$
$954$	0	0
$955$	−23.3484	−0.755537
$956$	0	0
$957$	−11.1502	−0.360436
$958$	0	0
$959$	−18.7155	−0.604355
$960$	0	0
$961$	51.0537	1.64689
$962$	0	0
$963$	13.7193	0.442099
$964$	0	0
$965$	−25.6115	−0.824464
$966$	0	0
$967$	50.7176	1.63097	0.815484	−	0.578780i	$-0.196471\pi$
$967$	50.7176	1.63097	0.815484	+	0.578780i	$0.196471\pi$
$968$	0	0
$969$	−20.2273	−0.649794
$970$	0	0
$971$	−38.8913	−1.24808	−0.624041	−	0.781392i	$-0.714510\pi$
$971$	−38.8913	−1.24808	−0.624041	+	0.781392i	$0.714510\pi$
$972$	0	0
$973$	33.7230	1.08111
$974$	0	0
$975$	−0.272657	−0.00873202
$976$	0	0
$977$	−17.1186	−0.547673	−0.273837	−	0.961776i	$-0.588293\pi$
$977$	−17.1186	−0.547673	−0.273837	+	0.961776i	$0.588293\pi$
$978$	0	0
$979$	5.74214	0.183520
$980$	0	0
$981$	−23.8907	−0.762773
$982$	0	0
$983$	−5.78625	−0.184553	−0.0922763	−	0.995733i	$-0.529414\pi$
$983$	−5.78625	−0.184553	−0.0922763	+	0.995733i	$0.529414\pi$
$984$	0	0
$985$	22.5631	0.718919
$986$	0	0
$987$	15.4637	0.492214
$988$	0	0
$989$	19.8945	0.632608
$990$	0	0
$991$	4.97362	0.157992	0.0789961	−	0.996875i	$-0.474829\pi$
$991$	4.97362	0.157992	0.0789961	+	0.996875i	$0.474829\pi$
$992$	0	0
$993$	10.0445	0.318752
$994$	0	0
$995$	−48.1503	−1.52647
$996$	0	0
$997$	−22.3402	−0.707521	−0.353760	−	0.935336i	$-0.615097\pi$
$997$	−22.3402	−0.707521	−0.353760	+	0.935336i	$0.615097\pi$
$998$	0	0
$999$	3.39028	0.107264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.a.q.1.4		5
4.3	odd	2		2816.2.a.r.1.2		5
8.3	odd	2		2816.2.a.o.1.4		5
8.5	even	2		2816.2.a.p.1.2		5
16.3	odd	4		88.2.c.a.45.3	✓	10
16.5	even	4		352.2.c.a.177.4		10
16.11	odd	4		88.2.c.a.45.4	yes	10
16.13	even	4		352.2.c.a.177.7		10
48.5	odd	4		3168.2.f.g.1585.3		10
48.11	even	4		792.2.f.g.397.7		10
48.29	odd	4		3168.2.f.g.1585.8		10
48.35	even	4		792.2.f.g.397.8		10
176.3	odd	20		968.2.o.g.493.1		40
176.19	even	20		968.2.o.h.493.10		40
176.21	odd	4		3872.2.c.f.1937.4		10
176.27	odd	20		968.2.o.g.245.5		40
176.35	even	20		968.2.o.h.565.6		40
176.43	even	4		968.2.c.d.485.7		10
176.51	even	20		968.2.o.h.269.3		40
176.59	odd	20		968.2.o.g.269.1		40
176.75	odd	20		968.2.o.g.565.9		40
176.83	even	20		968.2.o.h.245.2		40
176.91	odd	20		968.2.o.g.493.8		40
176.107	even	20		968.2.o.h.493.3		40
176.109	odd	4		3872.2.c.f.1937.7		10
176.115	odd	20		968.2.o.g.245.9		40
176.123	even	20		968.2.o.h.565.2		40
176.131	even	4		968.2.c.d.485.8		10
176.139	even	20		968.2.o.h.269.10		40
176.147	odd	20		968.2.o.g.269.8		40
176.163	odd	20		968.2.o.g.565.5		40
176.171	even	20		968.2.o.h.245.6		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.c.a.45.3	✓	10	16.3	odd	4
88.2.c.a.45.4	yes	10	16.11	odd	4
352.2.c.a.177.4		10	16.5	even	4
352.2.c.a.177.7		10	16.13	even	4
792.2.f.g.397.7		10	48.11	even	4
792.2.f.g.397.8		10	48.35	even	4
968.2.c.d.485.7		10	176.43	even	4
968.2.c.d.485.8		10	176.131	even	4
968.2.o.g.245.5		40	176.27	odd	20
968.2.o.g.245.9		40	176.115	odd	20
968.2.o.g.269.1		40	176.59	odd	20
968.2.o.g.269.8		40	176.147	odd	20
968.2.o.g.493.1		40	176.3	odd	20
968.2.o.g.493.8		40	176.91	odd	20
968.2.o.g.565.5		40	176.163	odd	20
968.2.o.g.565.9		40	176.75	odd	20
968.2.o.h.245.2		40	176.83	even	20
968.2.o.h.245.6		40	176.171	even	20
968.2.o.h.269.3		40	176.51	even	20
968.2.o.h.269.10		40	176.139	even	20
968.2.o.h.493.3		40	176.107	even	20
968.2.o.h.493.10		40	176.19	even	20
968.2.o.h.565.2		40	176.123	even	20
968.2.o.h.565.6		40	176.35	even	20
2816.2.a.o.1.4		5	8.3	odd	2
2816.2.a.p.1.2		5	8.5	even	2
2816.2.a.q.1.4		5	1.1	even	1	trivial
2816.2.a.r.1.2		5	4.3	odd	2
3168.2.f.g.1585.3		10	48.5	odd	4
3168.2.f.g.1585.8		10	48.29	odd	4
3872.2.c.f.1937.4		10	176.21	odd	4
3872.2.c.f.1937.7		10	176.109	odd	4

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:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33544 q^{3} +1.93119 q^{5} -1.83930 q^{7} -1.21660 q^{9} -1.00000 q^{11} +0.160700 q^{13} +2.57899 q^{15} +5.41442 q^{17} -2.79744 q^{19} -2.45628 q^{21} +4.44137 q^{23} -1.27050 q^{25} -5.63102 q^{27} +8.34948 q^{29} +9.05835 q^{31} -1.33544 q^{33} -3.55204 q^{35} -0.602073 q^{37} +0.214606 q^{39} +11.7639 q^{41} +4.47936 q^{43} -2.34948 q^{45} -6.29558 q^{47} -3.61698 q^{49} +7.23064 q^{51} +11.7211 q^{53} -1.93119 q^{55} -3.73582 q^{57} +9.63102 q^{59} +3.32912 q^{61} +2.23769 q^{63} +0.310343 q^{65} -5.76864 q^{67} +5.93119 q^{69} +9.62916 q^{71} +0.681940 q^{73} -1.69668 q^{75} +1.83930 q^{77} -5.77867 q^{79} -3.87010 q^{81} +2.92400 q^{83} +10.4563 q^{85} +11.1502 q^{87} -5.74214 q^{89} -0.295576 q^{91} +12.0969 q^{93} -5.40240 q^{95} -5.37829 q^{97} +1.21660 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{5} + 5 q^{9} - 5 q^{11} + 10 q^{13} - 4 q^{15} - 2 q^{17} + 12 q^{21} - 6 q^{23} + 3 q^{25} + 12 q^{27} + 10 q^{29} + 2 q^{31} + 16 q^{37} + 12 q^{39} - 2 q^{41} - 4 q^{43} + 20 q^{45} + 2 q^{47}+ \cdots - 5 q^{99}+O(q^{100}) \) 5 * q + 4 * q^5 + 5 * q^9 - 5 * q^11 + 10 * q^13 - 4 * q^15 - 2 * q^17 + 12 * q^21 - 6 * q^23 + 3 * q^25 + 12 * q^27 + 10 * q^29 + 2 * q^31 + 16 * q^37 + 12 * q^39 - 2 * q^41 - 4 * q^43 + 20 * q^45 + 2 * q^47 - 3 * q^49 + 8 * q^53 - 4 * q^55 - 8 * q^57 + 8 * q^59 + 30 * q^61 + 20 * q^63 + 8 * q^65 + 24 * q^69 - 6 * q^71 + 2 * q^73 - 36 * q^75 - 8 * q^79 - 3 * q^81 + 28 * q^85 + 16 * q^87 + 2 * q^89 + 32 * q^91 + 16 * q^93 - 12 * q^95 - 10 * q^97 - 5 * q^99

Introduction

L-functions

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