LMFDB - Embedded newform 1716.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1716,2,Mod(1,1716)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1716, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1716.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1716.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$13.7023289869$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1716.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.585786 q^{15} +2.24264 q^{17} -2.82843 q^{21} +0.828427 q^{23} -4.65685 q^{25} +1.00000 q^{27} -3.41421 q^{29} -2.58579 q^{31} -1.00000 q^{33} +1.65685 q^{35} -4.82843 q^{37} -1.00000 q^{39} -7.65685 q^{41} +8.24264 q^{43} -0.585786 q^{45} -8.00000 q^{47} +1.00000 q^{49} +2.24264 q^{51} -13.3137 q^{53} +0.585786 q^{55} -2.82843 q^{59} -8.82843 q^{61} -2.82843 q^{63} +0.585786 q^{65} -4.24264 q^{67} +0.828427 q^{69} -5.17157 q^{71} +5.65685 q^{73} -4.65685 q^{75} +2.82843 q^{77} +7.07107 q^{79} +1.00000 q^{81} -6.34315 q^{83} -1.31371 q^{85} -3.41421 q^{87} +7.89949 q^{89} +2.82843 q^{91} -2.58579 q^{93} -7.17157 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 4 q^{15} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 8 q^{31} - 2 q^{33} - 8 q^{35} - 4 q^{37} - 2 q^{39} - 4 q^{41} + 8 q^{43} - 4 q^{45} - 16 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 4 q^{55} - 12 q^{61} + 4 q^{65} - 4 q^{69} - 16 q^{71} + 2 q^{75} + 2 q^{81} - 24 q^{83} + 20 q^{85} - 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 - 2 * q^11 - 2 * q^13 - 4 * q^15 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 8 * q^31 - 2 * q^33 - 8 * q^35 - 4 * q^37 - 2 * q^39 - 4 * q^41 + 8 * q^43 - 4 * q^45 - 16 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 4 * q^55 - 12 * q^61 + 4 * q^65 - 4 * q^69 - 16 * q^71 + 2 * q^75 + 2 * q^81 - 24 * q^83 + 20 * q^85 - 4 * q^87 - 4 * q^89 - 8 * q^93 - 20 * q^97 - 2 * q^99

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	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.585786	−0.151249
$16$	0	0
$17$	2.24264	0.543920	0.271960	−	0.962309i	$-0.412328\pi$
$17$	2.24264	0.543920	0.271960	+	0.962309i	$0.412328\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	0.828427	0.172739	0.0863695	−	0.996263i	$-0.472473\pi$
$23$	0.828427	0.172739	0.0863695	+	0.996263i	$0.472473\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.41421	−0.634004	−0.317002	−	0.948425i	$-0.602676\pi$
$29$	−3.41421	−0.634004	−0.317002	+	0.948425i	$0.602676\pi$
$30$	0	0
$31$	−2.58579	−0.464421	−0.232210	−	0.972666i	$-0.574596\pi$
$31$	−2.58579	−0.464421	−0.232210	+	0.972666i	$0.574596\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	1.65685	0.280059
$36$	0	0
$37$	−4.82843	−0.793789	−0.396894	−	0.917864i	$-0.629912\pi$
$37$	−4.82843	−0.793789	−0.396894	+	0.917864i	$0.629912\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	8.24264	1.25699	0.628495	−	0.777813i	$-0.283671\pi$
$43$	8.24264	1.25699	0.628495	+	0.777813i	$0.283671\pi$
$44$	0	0
$45$	−0.585786	−0.0873239
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.24264	0.314033
$52$	0	0
$53$	−13.3137	−1.82878	−0.914389	−	0.404836i	$-0.867329\pi$
$53$	−13.3137	−1.82878	−0.914389	+	0.404836i	$0.867329\pi$
$54$	0	0
$55$	0.585786	0.0789874
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	0.585786	0.0726579
$66$	0	0
$67$	−4.24264	−0.518321	−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264	−0.518321	−0.259161	+	0.965834i	$0.583446\pi$
$68$	0	0
$69$	0.828427	0.0997309
$70$	0	0
$71$	−5.17157	−0.613753	−0.306876	−	0.951749i	$-0.599284\pi$
$71$	−5.17157	−0.613753	−0.306876	+	0.951749i	$0.599284\pi$
$72$	0	0
$73$	5.65685	0.662085	0.331042	−	0.943616i	$-0.392600\pi$
$73$	5.65685	0.662085	0.331042	+	0.943616i	$0.392600\pi$
$74$	0	0
$75$	−4.65685	−0.537727
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	7.07107	0.795557	0.397779	−	0.917481i	$-0.369781\pi$
$79$	7.07107	0.795557	0.397779	+	0.917481i	$0.369781\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.34315	−0.696251	−0.348125	−	0.937448i	$-0.613182\pi$
$83$	−6.34315	−0.696251	−0.348125	+	0.937448i	$0.613182\pi$
$84$	0	0
$85$	−1.31371	−0.142492
$86$	0	0
$87$	−3.41421	−0.366042
$88$	0	0
$89$	7.89949	0.837345	0.418672	−	0.908137i	$-0.362496\pi$
$89$	7.89949	0.837345	0.418672	+	0.908137i	$0.362496\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	−2.58579	−0.268134
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.17157	−0.728163	−0.364081	−	0.931367i	$-0.618617\pi$
$97$	−7.17157	−0.728163	−0.364081	+	0.931367i	$0.618617\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	6.72792	0.669453	0.334727	−	0.942315i	$-0.391356\pi$
$101$	6.72792	0.669453	0.334727	+	0.942315i	$0.391356\pi$
$102$	0	0
$103$	2.82843	0.278693	0.139347	−	0.990244i	$-0.455500\pi$
$103$	2.82843	0.278693	0.139347	+	0.990244i	$0.455500\pi$
$104$	0	0
$105$	1.65685	0.161692
$106$	0	0
$107$	6.82843	0.660129	0.330064	−	0.943958i	$-0.392930\pi$
$107$	6.82843	0.660129	0.330064	+	0.943958i	$0.392930\pi$
$108$	0	0
$109$	7.31371	0.700526	0.350263	−	0.936651i	$-0.386092\pi$
$109$	7.31371	0.700526	0.350263	+	0.936651i	$0.386092\pi$
$110$	0	0
$111$	−4.82843	−0.458294
$112$	0	0
$113$	−14.9706	−1.40831	−0.704156	−	0.710045i	$-0.748674\pi$
$113$	−14.9706	−1.40831	−0.704156	+	0.710045i	$0.748674\pi$
$114$	0	0
$115$	−0.485281	−0.0452527
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−6.34315	−0.581475
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.65685	−0.690395
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	7.07107	0.627456	0.313728	−	0.949513i	$-0.398422\pi$
$127$	7.07107	0.627456	0.313728	+	0.949513i	$0.398422\pi$
$128$	0	0
$129$	8.24264	0.725724
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.585786	−0.0504165
$136$	0	0
$137$	−1.07107	−0.0915075	−0.0457537	−	0.998953i	$-0.514569\pi$
$137$	−1.07107	−0.0915075	−0.0457537	+	0.998953i	$0.514569\pi$
$138$	0	0
$139$	−0.242641	−0.0205805	−0.0102903	−	0.999947i	$-0.503276\pi$
$139$	−0.242641	−0.0205805	−0.0102903	+	0.999947i	$0.503276\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−14.4853	−1.18668	−0.593340	−	0.804952i	$-0.702191\pi$
$149$	−14.4853	−1.18668	−0.593340	+	0.804952i	$0.702191\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	2.24264	0.181307
$154$	0	0
$155$	1.51472	0.121665
$156$	0	0
$157$	17.6569	1.40917	0.704585	−	0.709619i	$-0.251133\pi$
$157$	17.6569	1.40917	0.704585	+	0.709619i	$0.251133\pi$
$158$	0	0
$159$	−13.3137	−1.05585
$160$	0	0
$161$	−2.34315	−0.184666
$162$	0	0
$163$	12.7279	0.996928	0.498464	−	0.866910i	$-0.333898\pi$
$163$	12.7279	0.996928	0.498464	+	0.866910i	$0.333898\pi$
$164$	0	0
$165$	0.585786	0.0456034
$166$	0	0
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.100505	0.00764126	0.00382063	−	0.999993i	$-0.498784\pi$
$173$	0.100505	0.00764126	0.00382063	+	0.999993i	$0.498784\pi$
$174$	0	0
$175$	13.1716	0.995677
$176$	0	0
$177$	−2.82843	−0.212598
$178$	0	0
$179$	−0.828427	−0.0619196	−0.0309598	−	0.999521i	$-0.509856\pi$
$179$	−0.828427	−0.0619196	−0.0309598	+	0.999521i	$0.509856\pi$
$180$	0	0
$181$	−8.97056	−0.666777	−0.333388	−	0.942790i	$-0.608192\pi$
$181$	−8.97056	−0.666777	−0.333388	+	0.942790i	$0.608192\pi$
$182$	0	0
$183$	−8.82843	−0.652616
$184$	0	0
$185$	2.82843	0.207950
$186$	0	0
$187$	−2.24264	−0.163998
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−13.6569	−0.988175	−0.494088	−	0.869412i	$-0.664498\pi$
$191$	−13.6569	−0.988175	−0.494088	+	0.869412i	$0.664498\pi$
$192$	0	0
$193$	5.65685	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$194$	0	0
$195$	0.585786	0.0419490
$196$	0	0
$197$	16.1421	1.15008	0.575040	−	0.818125i	$-0.304987\pi$
$197$	16.1421	1.15008	0.575040	+	0.818125i	$0.304987\pi$
$198$	0	0
$199$	12.4853	0.885058	0.442529	−	0.896754i	$-0.354081\pi$
$199$	12.4853	0.885058	0.442529	+	0.896754i	$0.354081\pi$
$200$	0	0
$201$	−4.24264	−0.299253
$202$	0	0
$203$	9.65685	0.677778
$204$	0	0
$205$	4.48528	0.313266
$206$	0	0
$207$	0.828427	0.0575797
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.4142	0.923473	0.461736	−	0.887017i	$-0.347227\pi$
$211$	13.4142	0.923473	0.461736	+	0.887017i	$0.347227\pi$
$212$	0	0
$213$	−5.17157	−0.354350
$214$	0	0
$215$	−4.82843	−0.329296
$216$	0	0
$217$	7.31371	0.496487
$218$	0	0
$219$	5.65685	0.382255
$220$	0	0
$221$	−2.24264	−0.150856
$222$	0	0
$223$	−2.10051	−0.140660	−0.0703301	−	0.997524i	$-0.522405\pi$
$223$	−2.10051	−0.140660	−0.0703301	+	0.997524i	$0.522405\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	0	0
$227$	−16.1421	−1.07139	−0.535696	−	0.844411i	$-0.679951\pi$
$227$	−16.1421	−1.07139	−0.535696	+	0.844411i	$0.679951\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	2.82843	0.186097
$232$	0	0
$233$	11.4142	0.747770	0.373885	−	0.927475i	$-0.378025\pi$
$233$	11.4142	0.747770	0.373885	+	0.927475i	$0.378025\pi$
$234$	0	0
$235$	4.68629	0.305700
$236$	0	0
$237$	7.07107	0.459315
$238$	0	0
$239$	−8.14214	−0.526671	−0.263335	−	0.964704i	$-0.584823\pi$
$239$	−8.14214	−0.526671	−0.263335	+	0.964704i	$0.584823\pi$
$240$	0	0
$241$	−11.6569	−0.750884	−0.375442	−	0.926846i	$-0.622509\pi$
$241$	−11.6569	−0.750884	−0.375442	+	0.926846i	$0.622509\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.585786	−0.0374245
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.34315	−0.401981
$250$	0	0
$251$	14.3431	0.905331	0.452666	−	0.891680i	$-0.350473\pi$
$251$	14.3431	0.905331	0.452666	+	0.891680i	$0.350473\pi$
$252$	0	0
$253$	−0.828427	−0.0520828
$254$	0	0
$255$	−1.31371	−0.0822676
$256$	0	0
$257$	4.82843	0.301189	0.150595	−	0.988596i	$-0.451881\pi$
$257$	4.82843	0.301189	0.150595	+	0.988596i	$0.451881\pi$
$258$	0	0
$259$	13.6569	0.848596
$260$	0	0
$261$	−3.41421	−0.211335
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	7.79899	0.479088
$266$	0	0
$267$	7.89949	0.483441
$268$	0	0
$269$	−3.17157	−0.193374	−0.0966871	−	0.995315i	$-0.530825\pi$
$269$	−3.17157	−0.193374	−0.0966871	+	0.995315i	$0.530825\pi$
$270$	0	0
$271$	−9.65685	−0.586612	−0.293306	−	0.956019i	$-0.594755\pi$
$271$	−9.65685	−0.586612	−0.293306	+	0.956019i	$0.594755\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	4.65685	0.280819
$276$	0	0
$277$	−18.9706	−1.13983	−0.569915	−	0.821703i	$-0.693024\pi$
$277$	−18.9706	−1.13983	−0.569915	+	0.821703i	$0.693024\pi$
$278$	0	0
$279$	−2.58579	−0.154807
$280$	0	0
$281$	−12.3431	−0.736330	−0.368165	−	0.929760i	$-0.620014\pi$
$281$	−12.3431	−0.736330	−0.368165	+	0.929760i	$0.620014\pi$
$282$	0	0
$283$	12.9289	0.768545	0.384273	−	0.923220i	$-0.374452\pi$
$283$	12.9289	0.768545	0.384273	+	0.923220i	$0.374452\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	21.6569	1.27836
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	−7.17157	−0.420405
$292$	0	0
$293$	−18.4853	−1.07992	−0.539961	−	0.841690i	$-0.681561\pi$
$293$	−18.4853	−1.07992	−0.539961	+	0.841690i	$0.681561\pi$
$294$	0	0
$295$	1.65685	0.0964658
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−0.828427	−0.0479092
$300$	0	0
$301$	−23.3137	−1.34378
$302$	0	0
$303$	6.72792	0.386509
$304$	0	0
$305$	5.17157	0.296123
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	2.82843	0.160904
$310$	0	0
$311$	−19.1716	−1.08712	−0.543560	−	0.839370i	$-0.682924\pi$
$311$	−19.1716	−1.08712	−0.543560	+	0.839370i	$0.682924\pi$
$312$	0	0
$313$	22.6274	1.27898	0.639489	−	0.768801i	$-0.279146\pi$
$313$	22.6274	1.27898	0.639489	+	0.768801i	$0.279146\pi$
$314$	0	0
$315$	1.65685	0.0933532
$316$	0	0
$317$	−21.0711	−1.18347	−0.591735	−	0.806133i	$-0.701557\pi$
$317$	−21.0711	−1.18347	−0.591735	+	0.806133i	$0.701557\pi$
$318$	0	0
$319$	3.41421	0.191159
$320$	0	0
$321$	6.82843	0.381126
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.65685	0.258316
$326$	0	0
$327$	7.31371	0.404449
$328$	0	0
$329$	22.6274	1.24749
$330$	0	0
$331$	0.443651	0.0243853	0.0121926	−	0.999926i	$-0.496119\pi$
$331$	0.443651	0.0243853	0.0121926	+	0.999926i	$0.496119\pi$
$332$	0	0
$333$	−4.82843	−0.264596
$334$	0	0
$335$	2.48528	0.135785
$336$	0	0
$337$	26.4853	1.44275	0.721373	−	0.692547i	$-0.243512\pi$
$337$	26.4853	1.44275	0.721373	+	0.692547i	$0.243512\pi$
$338$	0	0
$339$	−14.9706	−0.813089
$340$	0	0
$341$	2.58579	0.140028
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	−0.485281	−0.0261267
$346$	0	0
$347$	−14.3431	−0.769980	−0.384990	−	0.922921i	$-0.625795\pi$
$347$	−14.3431	−0.769980	−0.384990	+	0.922921i	$0.625795\pi$
$348$	0	0
$349$	31.9411	1.70977	0.854885	−	0.518818i	$-0.173628\pi$
$349$	31.9411	1.70977	0.854885	+	0.518818i	$0.173628\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−10.7279	−0.570990	−0.285495	−	0.958380i	$-0.592158\pi$
$353$	−10.7279	−0.570990	−0.285495	+	0.958380i	$0.592158\pi$
$354$	0	0
$355$	3.02944	0.160786
$356$	0	0
$357$	−6.34315	−0.335715
$358$	0	0
$359$	12.1421	0.640837	0.320419	−	0.947276i	$-0.396176\pi$
$359$	12.1421	0.640837	0.320419	+	0.947276i	$0.396176\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−3.31371	−0.173447
$366$	0	0
$367$	31.3137	1.63456	0.817281	−	0.576239i	$-0.195480\pi$
$367$	31.3137	1.63456	0.817281	+	0.576239i	$0.195480\pi$
$368$	0	0
$369$	−7.65685	−0.398600
$370$	0	0
$371$	37.6569	1.95505
$372$	0	0
$373$	−31.4558	−1.62872	−0.814361	−	0.580359i	$-0.802912\pi$
$373$	−31.4558	−1.62872	−0.814361	+	0.580359i	$0.802912\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	3.41421	0.175841
$378$	0	0
$379$	−29.6985	−1.52551	−0.762754	−	0.646688i	$-0.776153\pi$
$379$	−29.6985	−1.52551	−0.762754	+	0.646688i	$0.776153\pi$
$380$	0	0
$381$	7.07107	0.362262
$382$	0	0
$383$	19.7990	1.01168	0.505841	−	0.862627i	$-0.331182\pi$
$383$	19.7990	1.01168	0.505841	+	0.862627i	$0.331182\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	0	0
$387$	8.24264	0.418997
$388$	0	0
$389$	21.7990	1.10525	0.552626	−	0.833429i	$-0.313626\pi$
$389$	21.7990	1.10525	0.552626	+	0.833429i	$0.313626\pi$
$390$	0	0
$391$	1.85786	0.0939562
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	−4.14214	−0.208413
$396$	0	0
$397$	1.79899	0.0902887	0.0451444	−	0.998980i	$-0.485625\pi$
$397$	1.79899	0.0902887	0.0451444	+	0.998980i	$0.485625\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.2132	1.15921	0.579606	−	0.814897i	$-0.303206\pi$
$401$	23.2132	1.15921	0.579606	+	0.814897i	$0.303206\pi$
$402$	0	0
$403$	2.58579	0.128807
$404$	0	0
$405$	−0.585786	−0.0291080
$406$	0	0
$407$	4.82843	0.239336
$408$	0	0
$409$	13.6569	0.675288	0.337644	−	0.941274i	$-0.390370\pi$
$409$	13.6569	0.675288	0.337644	+	0.941274i	$0.390370\pi$
$410$	0	0
$411$	−1.07107	−0.0528319
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	3.71573	0.182398
$416$	0	0
$417$	−0.242641	−0.0118822
$418$	0	0
$419$	23.1716	1.13201	0.566003	−	0.824403i	$-0.308489\pi$
$419$	23.1716	1.13201	0.566003	+	0.824403i	$0.308489\pi$
$420$	0	0
$421$	−14.6863	−0.715766	−0.357883	−	0.933766i	$-0.616501\pi$
$421$	−14.6863	−0.715766	−0.357883	+	0.933766i	$0.616501\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−10.4437	−0.506591
$426$	0	0
$427$	24.9706	1.20841
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−2.34315	−0.112865	−0.0564327	−	0.998406i	$-0.517973\pi$
$431$	−2.34315	−0.112865	−0.0564327	+	0.998406i	$0.517973\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.58579	0.314322	0.157161	−	0.987573i	$-0.449766\pi$
$439$	6.58579	0.314322	0.157161	+	0.987573i	$0.449766\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−0.686292	−0.0326067	−0.0163033	−	0.999867i	$-0.505190\pi$
$443$	−0.686292	−0.0326067	−0.0163033	+	0.999867i	$0.505190\pi$
$444$	0	0
$445$	−4.62742	−0.219361
$446$	0	0
$447$	−14.4853	−0.685130
$448$	0	0
$449$	11.4142	0.538670	0.269335	−	0.963047i	$-0.413196\pi$
$449$	11.4142	0.538670	0.269335	+	0.963047i	$0.413196\pi$
$450$	0	0
$451$	7.65685	0.360547
$452$	0	0
$453$	12.0000	0.563809
$454$	0	0
$455$	−1.65685	−0.0776745
$456$	0	0
$457$	2.68629	0.125659	0.0628297	−	0.998024i	$-0.479988\pi$
$457$	2.68629	0.125659	0.0628297	+	0.998024i	$0.479988\pi$
$458$	0	0
$459$	2.24264	0.104678
$460$	0	0
$461$	−22.9706	−1.06985	−0.534923	−	0.844901i	$-0.679659\pi$
$461$	−22.9706	−1.06985	−0.534923	+	0.844901i	$0.679659\pi$
$462$	0	0
$463$	27.0711	1.25810	0.629050	−	0.777365i	$-0.283444\pi$
$463$	27.0711	1.25810	0.629050	+	0.777365i	$0.283444\pi$
$464$	0	0
$465$	1.51472	0.0702434
$466$	0	0
$467$	27.4558	1.27050	0.635252	−	0.772305i	$-0.280896\pi$
$467$	27.4558	1.27050	0.635252	+	0.772305i	$0.280896\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	17.6569	0.813585
$472$	0	0
$473$	−8.24264	−0.378997
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.3137	−0.609593
$478$	0	0
$479$	−4.14214	−0.189259	−0.0946295	−	0.995513i	$-0.530167\pi$
$479$	−4.14214	−0.189259	−0.0946295	+	0.995513i	$0.530167\pi$
$480$	0	0
$481$	4.82843	0.220157
$482$	0	0
$483$	−2.34315	−0.106617
$484$	0	0
$485$	4.20101	0.190758
$486$	0	0
$487$	19.5563	0.886183	0.443091	−	0.896476i	$-0.353882\pi$
$487$	19.5563	0.886183	0.443091	+	0.896476i	$0.353882\pi$
$488$	0	0
$489$	12.7279	0.575577
$490$	0	0
$491$	−12.9706	−0.585353	−0.292677	−	0.956211i	$-0.594546\pi$
$491$	−12.9706	−0.585353	−0.292677	+	0.956211i	$0.594546\pi$
$492$	0	0
$493$	−7.65685	−0.344847
$494$	0	0
$495$	0.585786	0.0263291
$496$	0	0
$497$	14.6274	0.656129
$498$	0	0
$499$	−9.89949	−0.443162	−0.221581	−	0.975142i	$-0.571122\pi$
$499$	−9.89949	−0.443162	−0.221581	+	0.975142i	$0.571122\pi$
$500$	0	0
$501$	−11.3137	−0.505459
$502$	0	0
$503$	−7.51472	−0.335065	−0.167532	−	0.985867i	$-0.553580\pi$
$503$	−7.51472	−0.335065	−0.167532	+	0.985867i	$0.553580\pi$
$504$	0	0
$505$	−3.94113	−0.175378
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−21.5563	−0.955468	−0.477734	−	0.878504i	$-0.658542\pi$
$509$	−21.5563	−0.955468	−0.477734	+	0.878504i	$0.658542\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.65685	−0.0730097
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	0.100505	0.00441168
$520$	0	0
$521$	−18.9706	−0.831115	−0.415558	−	0.909567i	$-0.636414\pi$
$521$	−18.9706	−0.831115	−0.415558	+	0.909567i	$0.636414\pi$
$522$	0	0
$523$	−21.8995	−0.957598	−0.478799	−	0.877925i	$-0.658928\pi$
$523$	−21.8995	−0.957598	−0.478799	+	0.877925i	$0.658928\pi$
$524$	0	0
$525$	13.1716	0.574855
$526$	0	0
$527$	−5.79899	−0.252608
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	−2.82843	−0.122743
$532$	0	0
$533$	7.65685	0.331655
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−0.828427	−0.0357493
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	−8.97056	−0.384964
$544$	0	0
$545$	−4.28427	−0.183518
$546$	0	0
$547$	45.8995	1.96252	0.981260	−	0.192687i	$-0.0617201\pi$
$547$	45.8995	1.96252	0.981260	+	0.192687i	$0.0617201\pi$
$548$	0	0
$549$	−8.82843	−0.376788
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	2.82843	0.120060
$556$	0	0
$557$	9.51472	0.403152	0.201576	−	0.979473i	$-0.435394\pi$
$557$	9.51472	0.403152	0.201576	+	0.979473i	$0.435394\pi$
$558$	0	0
$559$	−8.24264	−0.348627
$560$	0	0
$561$	−2.24264	−0.0946844
$562$	0	0
$563$	9.65685	0.406988	0.203494	−	0.979076i	$-0.434770\pi$
$563$	9.65685	0.406988	0.203494	+	0.979076i	$0.434770\pi$
$564$	0	0
$565$	8.76955	0.368938
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−17.5563	−0.736000	−0.368000	−	0.929826i	$-0.619957\pi$
$569$	−17.5563	−0.736000	−0.368000	+	0.929826i	$0.619957\pi$
$570$	0	0
$571$	20.0416	0.838716	0.419358	−	0.907821i	$-0.362255\pi$
$571$	20.0416	0.838716	0.419358	+	0.907821i	$0.362255\pi$
$572$	0	0
$573$	−13.6569	−0.570523
$574$	0	0
$575$	−3.85786	−0.160884
$576$	0	0
$577$	−6.97056	−0.290188	−0.145094	−	0.989418i	$-0.546349\pi$
$577$	−6.97056	−0.290188	−0.145094	+	0.989418i	$0.546349\pi$
$578$	0	0
$579$	5.65685	0.235091
$580$	0	0
$581$	17.9411	0.744323
$582$	0	0
$583$	13.3137	0.551397
$584$	0	0
$585$	0.585786	0.0242193
$586$	0	0
$587$	7.79899	0.321899	0.160949	−	0.986963i	$-0.448544\pi$
$587$	7.79899	0.321899	0.160949	+	0.986963i	$0.448544\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	16.1421	0.663999
$592$	0	0
$593$	−24.8284	−1.01958	−0.509791	−	0.860298i	$-0.670277\pi$
$593$	−24.8284	−1.01958	−0.509791	+	0.860298i	$0.670277\pi$
$594$	0	0
$595$	3.71573	0.152330
$596$	0	0
$597$	12.4853	0.510989
$598$	0	0
$599$	−2.34315	−0.0957383	−0.0478692	−	0.998854i	$-0.515243\pi$
$599$	−2.34315	−0.0957383	−0.0478692	+	0.998854i	$0.515243\pi$
$600$	0	0
$601$	−47.2548	−1.92756	−0.963782	−	0.266690i	$-0.914070\pi$
$601$	−47.2548	−1.92756	−0.963782	+	0.266690i	$0.914070\pi$
$602$	0	0
$603$	−4.24264	−0.172774
$604$	0	0
$605$	−0.585786	−0.0238156
$606$	0	0
$607$	−29.2132	−1.18573	−0.592864	−	0.805303i	$-0.702003\pi$
$607$	−29.2132	−1.18573	−0.592864	+	0.805303i	$0.702003\pi$
$608$	0	0
$609$	9.65685	0.391315
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	29.9411	1.20931	0.604655	−	0.796487i	$-0.293311\pi$
$613$	29.9411	1.20931	0.604655	+	0.796487i	$0.293311\pi$
$614$	0	0
$615$	4.48528	0.180864
$616$	0	0
$617$	41.0711	1.65346	0.826729	−	0.562600i	$-0.190199\pi$
$617$	41.0711	1.65346	0.826729	+	0.562600i	$0.190199\pi$
$618$	0	0
$619$	6.58579	0.264705	0.132353	−	0.991203i	$-0.457747\pi$
$619$	6.58579	0.264705	0.132353	+	0.991203i	$0.457747\pi$
$620$	0	0
$621$	0.828427	0.0332436
$622$	0	0
$623$	−22.3431	−0.895159
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10.8284	−0.431758
$630$	0	0
$631$	−28.9289	−1.15164	−0.575821	−	0.817576i	$-0.695318\pi$
$631$	−28.9289	−1.15164	−0.575821	+	0.817576i	$0.695318\pi$
$632$	0	0
$633$	13.4142	0.533167
$634$	0	0
$635$	−4.14214	−0.164376
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−5.17157	−0.204584
$640$	0	0
$641$	−41.5980	−1.64302	−0.821511	−	0.570193i	$-0.806868\pi$
$641$	−41.5980	−1.64302	−0.821511	+	0.570193i	$0.806868\pi$
$642$	0	0
$643$	23.5563	0.928972	0.464486	−	0.885581i	$-0.346239\pi$
$643$	23.5563	0.928972	0.464486	+	0.885581i	$0.346239\pi$
$644$	0	0
$645$	−4.82843	−0.190119
$646$	0	0
$647$	18.3431	0.721143	0.360572	−	0.932731i	$-0.382582\pi$
$647$	18.3431	0.721143	0.360572	+	0.932731i	$0.382582\pi$
$648$	0	0
$649$	2.82843	0.111025
$650$	0	0
$651$	7.31371	0.286647
$652$	0	0
$653$	−23.1716	−0.906774	−0.453387	−	0.891314i	$-0.649784\pi$
$653$	−23.1716	−0.906774	−0.453387	+	0.891314i	$0.649784\pi$
$654$	0	0
$655$	−4.68629	−0.183109
$656$	0	0
$657$	5.65685	0.220695
$658$	0	0
$659$	31.1127	1.21198	0.605989	−	0.795473i	$-0.292777\pi$
$659$	31.1127	1.21198	0.605989	+	0.795473i	$0.292777\pi$
$660$	0	0
$661$	3.17157	0.123360	0.0616799	−	0.998096i	$-0.480354\pi$
$661$	3.17157	0.123360	0.0616799	+	0.998096i	$0.480354\pi$
$662$	0	0
$663$	−2.24264	−0.0870969
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.82843	−0.109517
$668$	0	0
$669$	−2.10051	−0.0812102
$670$	0	0
$671$	8.82843	0.340818
$672$	0	0
$673$	−0.343146	−0.0132273	−0.00661365	−	0.999978i	$-0.502105\pi$
$673$	−0.343146	−0.0132273	−0.00661365	+	0.999978i	$0.502105\pi$
$674$	0	0
$675$	−4.65685	−0.179242
$676$	0	0
$677$	−22.2426	−0.854854	−0.427427	−	0.904050i	$-0.640580\pi$
$677$	−22.2426	−0.854854	−0.427427	+	0.904050i	$0.640580\pi$
$678$	0	0
$679$	20.2843	0.778439
$680$	0	0
$681$	−16.1421	−0.618568
$682$	0	0
$683$	1.17157	0.0448290	0.0224145	−	0.999749i	$-0.492865\pi$
$683$	1.17157	0.0448290	0.0224145	+	0.999749i	$0.492865\pi$
$684$	0	0
$685$	0.627417	0.0239724
$686$	0	0
$687$	−21.3137	−0.813169
$688$	0	0
$689$	13.3137	0.507212
$690$	0	0
$691$	−39.5563	−1.50479	−0.752397	−	0.658710i	$-0.771103\pi$
$691$	−39.5563	−1.50479	−0.752397	+	0.658710i	$0.771103\pi$
$692$	0	0
$693$	2.82843	0.107443
$694$	0	0
$695$	0.142136	0.00539151
$696$	0	0
$697$	−17.1716	−0.650420
$698$	0	0
$699$	11.4142	0.431725
$700$	0	0
$701$	21.7574	0.821764	0.410882	−	0.911689i	$-0.365221\pi$
$701$	21.7574	0.821764	0.410882	+	0.911689i	$0.365221\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4.68629	0.176496
$706$	0	0
$707$	−19.0294	−0.715676
$708$	0	0
$709$	−3.37258	−0.126660	−0.0633300	−	0.997993i	$-0.520172\pi$
$709$	−3.37258	−0.126660	−0.0633300	+	0.997993i	$0.520172\pi$
$710$	0	0
$711$	7.07107	0.265186
$712$	0	0
$713$	−2.14214	−0.0802236
$714$	0	0
$715$	−0.585786	−0.0219072
$716$	0	0
$717$	−8.14214	−0.304074
$718$	0	0
$719$	−12.6863	−0.473119	−0.236559	−	0.971617i	$-0.576020\pi$
$719$	−12.6863	−0.473119	−0.236559	+	0.971617i	$0.576020\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−11.6569	−0.433523
$724$	0	0
$725$	15.8995	0.590492
$726$	0	0
$727$	−22.8284	−0.846659	−0.423330	−	0.905976i	$-0.639139\pi$
$727$	−22.8284	−0.846659	−0.423330	+	0.905976i	$0.639139\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.4853	0.683703
$732$	0	0
$733$	−30.6274	−1.13125	−0.565625	−	0.824663i	$-0.691365\pi$
$733$	−30.6274	−1.13125	−0.565625	+	0.824663i	$0.691365\pi$
$734$	0	0
$735$	−0.585786	−0.0216071
$736$	0	0
$737$	4.24264	0.156280
$738$	0	0
$739$	14.8284	0.545473	0.272736	−	0.962089i	$-0.412071\pi$
$739$	14.8284	0.545473	0.272736	+	0.962089i	$0.412071\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.9411	−1.53867	−0.769335	−	0.638845i	$-0.779412\pi$
$743$	−41.9411	−1.53867	−0.769335	+	0.638845i	$0.779412\pi$
$744$	0	0
$745$	8.48528	0.310877
$746$	0	0
$747$	−6.34315	−0.232084
$748$	0	0
$749$	−19.3137	−0.705708
$750$	0	0
$751$	3.31371	0.120919	0.0604595	−	0.998171i	$-0.480743\pi$
$751$	3.31371	0.120919	0.0604595	+	0.998171i	$0.480743\pi$
$752$	0	0
$753$	14.3431	0.522693
$754$	0	0
$755$	−7.02944	−0.255827
$756$	0	0
$757$	32.2843	1.17339	0.586696	−	0.809807i	$-0.300428\pi$
$757$	32.2843	1.17339	0.586696	+	0.809807i	$0.300428\pi$
$758$	0	0
$759$	−0.828427	−0.0300700
$760$	0	0
$761$	−24.8284	−0.900030	−0.450015	−	0.893021i	$-0.648581\pi$
$761$	−24.8284	−0.900030	−0.450015	+	0.893021i	$0.648581\pi$
$762$	0	0
$763$	−20.6863	−0.748894
$764$	0	0
$765$	−1.31371	−0.0474972
$766$	0	0
$767$	2.82843	0.102129
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	4.82843	0.173892
$772$	0	0
$773$	−7.41421	−0.266671	−0.133335	−	0.991071i	$-0.542569\pi$
$773$	−7.41421	−0.266671	−0.133335	+	0.991071i	$0.542569\pi$
$774$	0	0
$775$	12.0416	0.432548
$776$	0	0
$777$	13.6569	0.489937
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.17157	0.185053
$782$	0	0
$783$	−3.41421	−0.122014
$784$	0	0
$785$	−10.3431	−0.369163
$786$	0	0
$787$	23.7990	0.848342	0.424171	−	0.905582i	$-0.360565\pi$
$787$	23.7990	0.848342	0.424171	+	0.905582i	$0.360565\pi$
$788$	0	0
$789$	−8.48528	−0.302084
$790$	0	0
$791$	42.3431	1.50555
$792$	0	0
$793$	8.82843	0.313507
$794$	0	0
$795$	7.79899	0.276602
$796$	0	0
$797$	22.4853	0.796470	0.398235	−	0.917284i	$-0.369623\pi$
$797$	22.4853	0.796470	0.398235	+	0.917284i	$0.369623\pi$
$798$	0	0
$799$	−17.9411	−0.634711
$800$	0	0
$801$	7.89949	0.279115
$802$	0	0
$803$	−5.65685	−0.199626
$804$	0	0
$805$	1.37258	0.0483772
$806$	0	0
$807$	−3.17157	−0.111645
$808$	0	0
$809$	16.3848	0.576058	0.288029	−	0.957622i	$-0.407000\pi$
$809$	16.3848	0.576058	0.288029	+	0.957622i	$0.407000\pi$
$810$	0	0
$811$	−40.2843	−1.41457	−0.707286	−	0.706927i	$-0.750081\pi$
$811$	−40.2843	−1.41457	−0.707286	+	0.706927i	$0.750081\pi$
$812$	0	0
$813$	−9.65685	−0.338681
$814$	0	0
$815$	−7.45584	−0.261167
$816$	0	0
$817$	0	0
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	4.62742	0.161498	0.0807490	−	0.996734i	$-0.474269\pi$
$821$	4.62742	0.161498	0.0807490	+	0.996734i	$0.474269\pi$
$822$	0	0
$823$	−49.4558	−1.72392	−0.861961	−	0.506974i	$-0.830764\pi$
$823$	−49.4558	−1.72392	−0.861961	+	0.506974i	$0.830764\pi$
$824$	0	0
$825$	4.65685	0.162131
$826$	0	0
$827$	−30.7696	−1.06996	−0.534981	−	0.844864i	$-0.679681\pi$
$827$	−30.7696	−1.06996	−0.534981	+	0.844864i	$0.679681\pi$
$828$	0	0
$829$	−18.6863	−0.649002	−0.324501	−	0.945885i	$-0.605196\pi$
$829$	−18.6863	−0.649002	−0.324501	+	0.945885i	$0.605196\pi$
$830$	0	0
$831$	−18.9706	−0.658082
$832$	0	0
$833$	2.24264	0.0777029
$834$	0	0
$835$	6.62742	0.229351
$836$	0	0
$837$	−2.58579	−0.0893779
$838$	0	0
$839$	−30.3431	−1.04756	−0.523781	−	0.851853i	$-0.675479\pi$
$839$	−30.3431	−1.04756	−0.523781	+	0.851853i	$0.675479\pi$
$840$	0	0
$841$	−17.3431	−0.598040
$842$	0	0
$843$	−12.3431	−0.425121
$844$	0	0
$845$	−0.585786	−0.0201517
$846$	0	0
$847$	−2.82843	−0.0971859
$848$	0	0
$849$	12.9289	0.443720
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	5.31371	0.181938	0.0909690	−	0.995854i	$-0.471004\pi$
$853$	5.31371	0.181938	0.0909690	+	0.995854i	$0.471004\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.9289	0.373325	0.186663	−	0.982424i	$-0.440233\pi$
$857$	10.9289	0.373325	0.186663	+	0.982424i	$0.440233\pi$
$858$	0	0
$859$	−17.1716	−0.585887	−0.292943	−	0.956130i	$-0.594635\pi$
$859$	−17.1716	−0.585887	−0.292943	+	0.956130i	$0.594635\pi$
$860$	0	0
$861$	21.6569	0.738064
$862$	0	0
$863$	−52.7696	−1.79630	−0.898148	−	0.439693i	$-0.855087\pi$
$863$	−52.7696	−1.79630	−0.898148	+	0.439693i	$0.855087\pi$
$864$	0	0
$865$	−0.0588745	−0.00200179
$866$	0	0
$867$	−11.9706	−0.406542
$868$	0	0
$869$	−7.07107	−0.239870
$870$	0	0
$871$	4.24264	0.143756
$872$	0	0
$873$	−7.17157	−0.242721
$874$	0	0
$875$	−16.0000	−0.540899
$876$	0	0
$877$	16.3431	0.551869	0.275934	−	0.961176i	$-0.411013\pi$
$877$	16.3431	0.551869	0.275934	+	0.961176i	$0.411013\pi$
$878$	0	0
$879$	−18.4853	−0.623493
$880$	0	0
$881$	46.9706	1.58248	0.791239	−	0.611507i	$-0.209436\pi$
$881$	46.9706	1.58248	0.791239	+	0.611507i	$0.209436\pi$
$882$	0	0
$883$	−29.1716	−0.981702	−0.490851	−	0.871244i	$-0.663314\pi$
$883$	−29.1716	−0.981702	−0.490851	+	0.871244i	$0.663314\pi$
$884$	0	0
$885$	1.65685	0.0556945
$886$	0	0
$887$	−8.48528	−0.284908	−0.142454	−	0.989801i	$-0.545499\pi$
$887$	−8.48528	−0.284908	−0.142454	+	0.989801i	$0.545499\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0.485281	0.0162212
$896$	0	0
$897$	−0.828427	−0.0276604
$898$	0	0
$899$	8.82843	0.294445
$900$	0	0
$901$	−29.8579	−0.994710
$902$	0	0
$903$	−23.3137	−0.775832
$904$	0	0
$905$	5.25483	0.174677
$906$	0	0
$907$	−11.3137	−0.375666	−0.187833	−	0.982201i	$-0.560146\pi$
$907$	−11.3137	−0.375666	−0.187833	+	0.982201i	$0.560146\pi$
$908$	0	0
$909$	6.72792	0.223151
$910$	0	0
$911$	−12.2843	−0.406996	−0.203498	−	0.979075i	$-0.565231\pi$
$911$	−12.2843	−0.406996	−0.203498	+	0.979075i	$0.565231\pi$
$912$	0	0
$913$	6.34315	0.209927
$914$	0	0
$915$	5.17157	0.170967
$916$	0	0
$917$	−22.6274	−0.747223
$918$	0	0
$919$	−38.5858	−1.27283	−0.636414	−	0.771348i	$-0.719583\pi$
$919$	−38.5858	−1.27283	−0.636414	+	0.771348i	$0.719583\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	5.17157	0.170224
$924$	0	0
$925$	22.4853	0.739311
$926$	0	0
$927$	2.82843	0.0928977
$928$	0	0
$929$	−33.0711	−1.08503	−0.542513	−	0.840047i	$-0.682527\pi$
$929$	−33.0711	−1.08503	−0.542513	+	0.840047i	$0.682527\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−19.1716	−0.627649
$934$	0	0
$935$	1.31371	0.0429629
$936$	0	0
$937$	−34.2843	−1.12002	−0.560009	−	0.828486i	$-0.689202\pi$
$937$	−34.2843	−1.12002	−0.560009	+	0.828486i	$0.689202\pi$
$938$	0	0
$939$	22.6274	0.738418
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−6.34315	−0.206561
$944$	0	0
$945$	1.65685	0.0538975
$946$	0	0
$947$	34.6274	1.12524	0.562620	−	0.826716i	$-0.309793\pi$
$947$	34.6274	1.12524	0.562620	+	0.826716i	$0.309793\pi$
$948$	0	0
$949$	−5.65685	−0.183629
$950$	0	0
$951$	−21.0711	−0.683276
$952$	0	0
$953$	32.5858	1.05556	0.527779	−	0.849382i	$-0.323025\pi$
$953$	32.5858	1.05556	0.527779	+	0.849382i	$0.323025\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	3.41421	0.110366
$958$	0	0
$959$	3.02944	0.0978256
$960$	0	0
$961$	−24.3137	−0.784313
$962$	0	0
$963$	6.82843	0.220043
$964$	0	0
$965$	−3.31371	−0.106672
$966$	0	0
$967$	−38.4264	−1.23571	−0.617855	−	0.786292i	$-0.711998\pi$
$967$	−38.4264	−1.23571	−0.617855	+	0.786292i	$0.711998\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.8284	−1.18188	−0.590940	−	0.806715i	$-0.701243\pi$
$971$	−36.8284	−1.18188	−0.590940	+	0.806715i	$0.701243\pi$
$972$	0	0
$973$	0.686292	0.0220015
$974$	0	0
$975$	4.65685	0.149139
$976$	0	0
$977$	−17.5563	−0.561677	−0.280839	−	0.959755i	$-0.590613\pi$
$977$	−17.5563	−0.561677	−0.280839	+	0.959755i	$0.590613\pi$
$978$	0	0
$979$	−7.89949	−0.252469
$980$	0	0
$981$	7.31371	0.233509
$982$	0	0
$983$	−51.5980	−1.64572	−0.822860	−	0.568244i	$-0.807623\pi$
$983$	−51.5980	−1.64572	−0.822860	+	0.568244i	$0.807623\pi$
$984$	0	0
$985$	−9.45584	−0.301288
$986$	0	0
$987$	22.6274	0.720239
$988$	0	0
$989$	6.82843	0.217131
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	0	0
$993$	0.443651	0.0140788
$994$	0	0
$995$	−7.31371	−0.231860
$996$	0	0
$997$	−32.1421	−1.01795	−0.508976	−	0.860781i	$-0.669976\pi$
$997$	−32.1421	−1.01795	−0.508976	+	0.860781i	$0.669976\pi$
$998$	0	0
$999$	−4.82843	−0.152765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1716.2.a.e.1.2	✓	2
3.2	odd	2		5148.2.a.j.1.1		2
4.3	odd	2		6864.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.e.1.2	✓	2	1.1	even	1	trivial
5148.2.a.j.1.1		2	3.2	odd	2
6864.2.a.bb.1.2		2	4.3	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:	\( N \)	\(=\)	\( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1716.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.585786 q^{5} -2.82843 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.585786 q^{15} +2.24264 q^{17} -2.82843 q^{21} +0.828427 q^{23} -4.65685 q^{25} +1.00000 q^{27} -3.41421 q^{29} -2.58579 q^{31} -1.00000 q^{33} +1.65685 q^{35} -4.82843 q^{37} -1.00000 q^{39} -7.65685 q^{41} +8.24264 q^{43} -0.585786 q^{45} -8.00000 q^{47} +1.00000 q^{49} +2.24264 q^{51} -13.3137 q^{53} +0.585786 q^{55} -2.82843 q^{59} -8.82843 q^{61} -2.82843 q^{63} +0.585786 q^{65} -4.24264 q^{67} +0.828427 q^{69} -5.17157 q^{71} +5.65685 q^{73} -4.65685 q^{75} +2.82843 q^{77} +7.07107 q^{79} +1.00000 q^{81} -6.34315 q^{83} -1.31371 q^{85} -3.41421 q^{87} +7.89949 q^{89} +2.82843 q^{91} -2.58579 q^{93} -7.17157 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 4 q^{15} - 4 q^{17} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 8 q^{31} - 2 q^{33} - 8 q^{35} - 4 q^{37} - 2 q^{39} - 4 q^{41} + 8 q^{43} - 4 q^{45} - 16 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 4 q^{55} - 12 q^{61} + 4 q^{65} - 4 q^{69} - 16 q^{71} + 2 q^{75} + 2 q^{81} - 24 q^{83} + 20 q^{85} - 4 q^{87} - 4 q^{89} - 8 q^{93} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 - 2 * q^11 - 2 * q^13 - 4 * q^15 - 4 * q^17 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 8 * q^31 - 2 * q^33 - 8 * q^35 - 4 * q^37 - 2 * q^39 - 4 * q^41 + 8 * q^43 - 4 * q^45 - 16 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 + 4 * q^55 - 12 * q^61 + 4 * q^65 - 4 * q^69 - 16 * q^71 + 2 * q^75 + 2 * q^81 - 24 * q^83 + 20 * q^85 - 4 * q^87 - 4 * q^89 - 8 * q^93 - 20 * q^97 - 2 * q^99

Introduction

L-functions

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Database

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