LMFDB - Embedded newform 4664.2.a.k.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4664,2,Mod(1,4664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4664, 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("4664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4664 = 2^{3} \cdot 11 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4664.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:	$37.2422275027$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 7x^{9} + 53x^{8} + 13x^{7} - 189x^{6} - 16x^{5} + 260x^{4} + 32x^{3} - 118x^{2} - 8x + 16 $ x^11 - 5x^10 - 7x^9 + 53x^8 + 13x^7 - 189x^6 - 16x^5 + 260x^4 + 32x^3 - 118x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.15382$ of defining polynomial
Character	$\chi$	$=$	4664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15382 q^{3} +4.11573 q^{5} -1.30816 q^{7} -1.66869 q^{9} +O(q^{10})$	$q+1.15382 q^{3} +4.11573 q^{5} -1.30816 q^{7} -1.66869 q^{9} +1.00000 q^{11} -6.30041 q^{13} +4.74882 q^{15} -3.49128 q^{17} -7.02802 q^{19} -1.50938 q^{21} -4.30717 q^{23} +11.9392 q^{25} -5.38684 q^{27} +7.31878 q^{29} -4.67753 q^{31} +1.15382 q^{33} -5.38402 q^{35} -7.08950 q^{37} -7.26955 q^{39} -5.46003 q^{41} -0.276371 q^{43} -6.86789 q^{45} +2.61799 q^{47} -5.28873 q^{49} -4.02832 q^{51} +1.00000 q^{53} +4.11573 q^{55} -8.10910 q^{57} +6.72447 q^{59} +6.90519 q^{61} +2.18291 q^{63} -25.9308 q^{65} +6.91028 q^{67} -4.96971 q^{69} -11.0084 q^{71} +8.79341 q^{73} +13.7758 q^{75} -1.30816 q^{77} -5.94310 q^{79} -1.20939 q^{81} -11.4147 q^{83} -14.3692 q^{85} +8.44457 q^{87} +1.99497 q^{89} +8.24191 q^{91} -5.39705 q^{93} -28.9255 q^{95} -5.08777 q^{97} -1.66869 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9	$ 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9} + 11 q^{11} - 13 q^{13} - 8 q^{15} - 7 q^{17} - 21 q^{19} + 6 q^{21} + 11 q^{23} + 4 q^{25} - 6 q^{27} - 5 q^{31} - 6 q^{33} - 25 q^{35} - 4 q^{37} - 19 q^{39} - 11 q^{41} + 16 q^{45} - 17 q^{47} - 2 q^{49} - 18 q^{51} + 11 q^{53} + 3 q^{55} - 5 q^{57} - 19 q^{59} - 2 q^{61} - 36 q^{63} - 13 q^{65} + 25 q^{67} + 3 q^{69} - 30 q^{71} + 5 q^{73} - 5 q^{75} - 5 q^{77} - 23 q^{79} - 9 q^{81} - 19 q^{83} + 2 q^{85} - 7 q^{87} + 6 q^{89} - 20 q^{91} + 43 q^{93} - 50 q^{95} - 35 q^{97} + 7 q^{99}+O(q^{100}) $ 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9 + 11 * q^11 - 13 * q^13 - 8 * q^15 - 7 * q^17 - 21 * q^19 + 6 * q^21 + 11 * q^23 + 4 * q^25 - 6 * q^27 - 5 * q^31 - 6 * q^33 - 25 * q^35 - 4 * q^37 - 19 * q^39 - 11 * q^41 + 16 * q^45 - 17 * q^47 - 2 * q^49 - 18 * q^51 + 11 * q^53 + 3 * q^55 - 5 * q^57 - 19 * q^59 - 2 * q^61 - 36 * q^63 - 13 * q^65 + 25 * q^67 + 3 * q^69 - 30 * q^71 + 5 * q^73 - 5 * q^75 - 5 * q^77 - 23 * q^79 - 9 * q^81 - 19 * q^83 + 2 * q^85 - 7 * q^87 + 6 * q^89 - 20 * q^91 + 43 * q^93 - 50 * q^95 - 35 * q^97 + 7 * 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
$3$	1.15382	0.666160	0.333080	−	0.942899i	$-0.391912\pi$
$3$	1.15382	0.666160	0.333080	+	0.942899i	$0.391912\pi$
$4$	0	0
$5$	4.11573	1.84061	0.920305	−	0.391201i	$-0.127940\pi$
$5$	4.11573	1.84061	0.920305	+	0.391201i	$0.127940\pi$
$6$	0	0
$7$	−1.30816	−0.494437	−0.247218	−	0.968960i	$-0.579516\pi$
$7$	−1.30816	−0.494437	−0.247218	+	0.968960i	$0.579516\pi$
$8$	0	0
$9$	−1.66869	−0.556231
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.30041	−1.74742	−0.873709	−	0.486449i	$-0.838292\pi$
$13$	−6.30041	−1.74742	−0.873709	+	0.486449i	$0.838292\pi$
$14$	0	0
$15$	4.74882	1.22614
$16$	0	0
$17$	−3.49128	−0.846761	−0.423380	−	0.905952i	$-0.639157\pi$
$17$	−3.49128	−0.846761	−0.423380	+	0.905952i	$0.639157\pi$
$18$	0	0
$19$	−7.02802	−1.61234	−0.806170	−	0.591684i	$-0.798463\pi$
$19$	−7.02802	−1.61234	−0.806170	+	0.591684i	$0.798463\pi$
$20$	0	0
$21$	−1.50938	−0.329374
$22$	0	0
$23$	−4.30717	−0.898107	−0.449053	−	0.893505i	$-0.648239\pi$
$23$	−4.30717	−0.898107	−0.449053	+	0.893505i	$0.648239\pi$
$24$	0	0
$25$	11.9392	2.38785
$26$	0	0
$27$	−5.38684	−1.03670
$28$	0	0
$29$	7.31878	1.35906	0.679531	−	0.733646i	$-0.262183\pi$
$29$	7.31878	1.35906	0.679531	+	0.733646i	$0.262183\pi$
$30$	0	0
$31$	−4.67753	−0.840110	−0.420055	−	0.907499i	$-0.637989\pi$
$31$	−4.67753	−0.840110	−0.420055	+	0.907499i	$0.637989\pi$
$32$	0	0
$33$	1.15382	0.200855
$34$	0	0
$35$	−5.38402	−0.910065
$36$	0	0
$37$	−7.08950	−1.16551	−0.582753	−	0.812649i	$-0.698025\pi$
$37$	−7.08950	−1.16551	−0.582753	+	0.812649i	$0.698025\pi$
$38$	0	0
$39$	−7.26955	−1.16406
$40$	0	0
$41$	−5.46003	−0.852714	−0.426357	−	0.904555i	$-0.640203\pi$
$41$	−5.46003	−0.852714	−0.426357	+	0.904555i	$0.640203\pi$
$42$	0	0
$43$	−0.276371	−0.0421463	−0.0210731	−	0.999778i	$-0.506708\pi$
$43$	−0.276371	−0.0421463	−0.0210731	+	0.999778i	$0.506708\pi$
$44$	0	0
$45$	−6.86789	−1.02380
$46$	0	0
$47$	2.61799	0.381873	0.190936	−	0.981602i	$-0.438848\pi$
$47$	2.61799	0.381873	0.190936	+	0.981602i	$0.438848\pi$
$48$	0	0
$49$	−5.28873	−0.755532
$50$	0	0
$51$	−4.02832	−0.564078
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	4.11573	0.554965
$56$	0	0
$57$	−8.10910	−1.07408
$58$	0	0
$59$	6.72447	0.875452	0.437726	−	0.899108i	$-0.355784\pi$
$59$	6.72447	0.875452	0.437726	+	0.899108i	$0.355784\pi$
$60$	0	0
$61$	6.90519	0.884119	0.442059	−	0.896986i	$-0.354248\pi$
$61$	6.90519	0.884119	0.442059	+	0.896986i	$0.354248\pi$
$62$	0	0
$63$	2.18291	0.275021
$64$	0	0
$65$	−25.9308	−3.21632
$66$	0	0
$67$	6.91028	0.844225	0.422112	−	0.906544i	$-0.361289\pi$
$67$	6.91028	0.844225	0.422112	+	0.906544i	$0.361289\pi$
$68$	0	0
$69$	−4.96971	−0.598283
$70$	0	0
$71$	−11.0084	−1.30646	−0.653229	−	0.757160i	$-0.726586\pi$
$71$	−11.0084	−1.30646	−0.653229	+	0.757160i	$0.726586\pi$
$72$	0	0
$73$	8.79341	1.02919	0.514595	−	0.857433i	$-0.327942\pi$
$73$	8.79341	1.02919	0.514595	+	0.857433i	$0.327942\pi$
$74$	0	0
$75$	13.7758	1.59069
$76$	0	0
$77$	−1.30816	−0.149078
$78$	0	0
$79$	−5.94310	−0.668651	−0.334325	−	0.942458i	$-0.608508\pi$
$79$	−5.94310	−0.668651	−0.334325	+	0.942458i	$0.608508\pi$
$80$	0	0
$81$	−1.20939	−0.134376
$82$	0	0
$83$	−11.4147	−1.25292	−0.626462	−	0.779452i	$-0.715498\pi$
$83$	−11.4147	−1.25292	−0.626462	+	0.779452i	$0.715498\pi$
$84$	0	0
$85$	−14.3692	−1.55856
$86$	0	0
$87$	8.44457	0.905353
$88$	0	0
$89$	1.99497	0.211467	0.105733	−	0.994395i	$-0.466281\pi$
$89$	1.99497	0.211467	0.105733	+	0.994395i	$0.466281\pi$
$90$	0	0
$91$	8.24191	0.863987
$92$	0	0
$93$	−5.39705	−0.559648
$94$	0	0
$95$	−28.9255	−2.96769
$96$	0	0
$97$	−5.08777	−0.516585	−0.258292	−	0.966067i	$-0.583160\pi$
$97$	−5.08777	−0.516585	−0.258292	+	0.966067i	$0.583160\pi$
$98$	0	0
$99$	−1.66869	−0.167710
$100$	0	0
$101$	−8.51570	−0.847344	−0.423672	−	0.905816i	$-0.639259\pi$
$101$	−8.51570	−0.847344	−0.423672	+	0.905816i	$0.639259\pi$
$102$	0	0
$103$	1.12730	0.111076	0.0555382	−	0.998457i	$-0.482313\pi$
$103$	1.12730	0.111076	0.0555382	+	0.998457i	$0.482313\pi$
$104$	0	0
$105$	−6.21220	−0.606249
$106$	0	0
$107$	−10.9228	−1.05595	−0.527974	−	0.849261i	$-0.677048\pi$
$107$	−10.9228	−1.05595	−0.527974	+	0.849261i	$0.677048\pi$
$108$	0	0
$109$	−4.28427	−0.410359	−0.205179	−	0.978724i	$-0.565778\pi$
$109$	−4.28427	−0.410359	−0.205179	+	0.978724i	$0.565778\pi$
$110$	0	0
$111$	−8.18003	−0.776414
$112$	0	0
$113$	14.2746	1.34284	0.671422	−	0.741075i	$-0.265684\pi$
$113$	14.2746	1.34284	0.671422	+	0.741075i	$0.265684\pi$
$114$	0	0
$115$	−17.7271	−1.65306
$116$	0	0
$117$	10.5134	0.971968
$118$	0	0
$119$	4.56714	0.418669
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.29991	−0.568044
$124$	0	0
$125$	28.5600	2.55448
$126$	0	0
$127$	−5.84866	−0.518984	−0.259492	−	0.965745i	$-0.583555\pi$
$127$	−5.84866	−0.518984	−0.259492	+	0.965745i	$0.583555\pi$
$128$	0	0
$129$	−0.318884	−0.0280761
$130$	0	0
$131$	18.7066	1.63440	0.817200	−	0.576354i	$-0.195525\pi$
$131$	18.7066	1.63440	0.817200	+	0.576354i	$0.195525\pi$
$132$	0	0
$133$	9.19375	0.797200
$134$	0	0
$135$	−22.1708	−1.90816
$136$	0	0
$137$	−11.1836	−0.955481	−0.477741	−	0.878501i	$-0.658544\pi$
$137$	−11.1836	−0.955481	−0.477741	+	0.878501i	$0.658544\pi$
$138$	0	0
$139$	−5.28075	−0.447907	−0.223954	−	0.974600i	$-0.571896\pi$
$139$	−5.28075	−0.447907	−0.223954	+	0.974600i	$0.571896\pi$
$140$	0	0
$141$	3.02069	0.254388
$142$	0	0
$143$	−6.30041	−0.526866
$144$	0	0
$145$	30.1221	2.50150
$146$	0	0
$147$	−6.10226	−0.503306
$148$	0	0
$149$	−17.5235	−1.43558	−0.717790	−	0.696260i	$-0.754846\pi$
$149$	−17.5235	−1.43558	−0.717790	+	0.696260i	$0.754846\pi$
$150$	0	0
$151$	14.1247	1.14945	0.574727	−	0.818345i	$-0.305108\pi$
$151$	14.1247	1.14945	0.574727	+	0.818345i	$0.305108\pi$
$152$	0	0
$153$	5.82588	0.470994
$154$	0	0
$155$	−19.2515	−1.54631
$156$	0	0
$157$	23.5562	1.87999	0.939993	−	0.341193i	$-0.110831\pi$
$157$	23.5562	1.87999	0.939993	+	0.341193i	$0.110831\pi$
$158$	0	0
$159$	1.15382	0.0915041
$160$	0	0
$161$	5.63445	0.444057
$162$	0	0
$163$	5.75942	0.451113	0.225556	−	0.974230i	$-0.427580\pi$
$163$	5.75942	0.451113	0.225556	+	0.974230i	$0.427580\pi$
$164$	0	0
$165$	4.74882	0.369695
$166$	0	0
$167$	−3.68688	−0.285299	−0.142650	−	0.989773i	$-0.545562\pi$
$167$	−3.68688	−0.285299	−0.142650	+	0.989773i	$0.545562\pi$
$168$	0	0
$169$	26.6951	2.05347
$170$	0	0
$171$	11.7276	0.896833
$172$	0	0
$173$	9.10529	0.692263	0.346131	−	0.938186i	$-0.387495\pi$
$173$	9.10529	0.692263	0.346131	+	0.938186i	$0.387495\pi$
$174$	0	0
$175$	−15.6184	−1.18064
$176$	0	0
$177$	7.75885	0.583191
$178$	0	0
$179$	−7.52148	−0.562182	−0.281091	−	0.959681i	$-0.590696\pi$
$179$	−7.52148	−0.562182	−0.281091	+	0.959681i	$0.590696\pi$
$180$	0	0
$181$	4.93432	0.366765	0.183383	−	0.983042i	$-0.441295\pi$
$181$	4.93432	0.366765	0.183383	+	0.983042i	$0.441295\pi$
$182$	0	0
$183$	7.96736	0.588964
$184$	0	0
$185$	−29.1785	−2.14524
$186$	0	0
$187$	−3.49128	−0.255308
$188$	0	0
$189$	7.04683	0.512582
$190$	0	0
$191$	7.18382	0.519803	0.259901	−	0.965635i	$-0.416310\pi$
$191$	7.18382	0.519803	0.259901	+	0.965635i	$0.416310\pi$
$192$	0	0
$193$	11.6700	0.840024	0.420012	−	0.907519i	$-0.362026\pi$
$193$	11.6700	0.840024	0.420012	+	0.907519i	$0.362026\pi$
$194$	0	0
$195$	−29.9195	−2.14258
$196$	0	0
$197$	23.8235	1.69735	0.848676	−	0.528914i	$-0.177400\pi$
$197$	23.8235	1.69735	0.848676	+	0.528914i	$0.177400\pi$
$198$	0	0
$199$	4.29240	0.304280	0.152140	−	0.988359i	$-0.451384\pi$
$199$	4.29240	0.304280	0.152140	+	0.988359i	$0.451384\pi$
$200$	0	0
$201$	7.97324	0.562389
$202$	0	0
$203$	−9.57410	−0.671970
$204$	0	0
$205$	−22.4720	−1.56951
$206$	0	0
$207$	7.18734	0.499555
$208$	0	0
$209$	−7.02802	−0.486139
$210$	0	0
$211$	−20.2946	−1.39714	−0.698568	−	0.715544i	$-0.746179\pi$
$211$	−20.2946	−1.39714	−0.698568	+	0.715544i	$0.746179\pi$
$212$	0	0
$213$	−12.7018	−0.870310
$214$	0	0
$215$	−1.13747	−0.0775748
$216$	0	0
$217$	6.11894	0.415381
$218$	0	0
$219$	10.1460	0.685606
$220$	0	0
$221$	21.9965	1.47965
$222$	0	0
$223$	9.57583	0.641245	0.320623	−	0.947207i	$-0.396108\pi$
$223$	9.57583	0.641245	0.320623	+	0.947207i	$0.396108\pi$
$224$	0	0
$225$	−19.9229	−1.32819
$226$	0	0
$227$	−16.3809	−1.08724	−0.543619	−	0.839332i	$-0.682946\pi$
$227$	−16.3809	−1.08724	−0.543619	+	0.839332i	$0.682946\pi$
$228$	0	0
$229$	13.4247	0.887128	0.443564	−	0.896243i	$-0.353714\pi$
$229$	13.4247	0.887128	0.443564	+	0.896243i	$0.353714\pi$
$230$	0	0
$231$	−1.50938	−0.0993100
$232$	0	0
$233$	25.0044	1.63809	0.819045	−	0.573729i	$-0.194504\pi$
$233$	25.0044	1.63809	0.819045	+	0.573729i	$0.194504\pi$
$234$	0	0
$235$	10.7749	0.702879
$236$	0	0
$237$	−6.85728	−0.445428
$238$	0	0
$239$	0.536658	0.0347135	0.0173568	−	0.999849i	$-0.494475\pi$
$239$	0.536658	0.0347135	0.0173568	+	0.999849i	$0.494475\pi$
$240$	0	0
$241$	−8.23891	−0.530715	−0.265357	−	0.964150i	$-0.585490\pi$
$241$	−8.23891	−0.530715	−0.265357	+	0.964150i	$0.585490\pi$
$242$	0	0
$243$	14.7651	0.947183
$244$	0	0
$245$	−21.7670	−1.39064
$246$	0	0
$247$	44.2794	2.81743
$248$	0	0
$249$	−13.1705	−0.834648
$250$	0	0
$251$	−20.2468	−1.27797	−0.638984	−	0.769220i	$-0.720645\pi$
$251$	−20.2468	−1.27797	−0.638984	+	0.769220i	$0.720645\pi$
$252$	0	0
$253$	−4.30717	−0.270789
$254$	0	0
$255$	−16.5795	−1.03825
$256$	0	0
$257$	21.8292	1.36167	0.680833	−	0.732439i	$-0.261618\pi$
$257$	21.8292	1.36167	0.680833	+	0.732439i	$0.261618\pi$
$258$	0	0
$259$	9.27417	0.576269
$260$	0	0
$261$	−12.2128	−0.755953
$262$	0	0
$263$	−26.6238	−1.64169	−0.820846	−	0.571150i	$-0.806498\pi$
$263$	−26.6238	−1.64169	−0.820846	+	0.571150i	$0.806498\pi$
$264$	0	0
$265$	4.11573	0.252827
$266$	0	0
$267$	2.30185	0.140871
$268$	0	0
$269$	−10.2414	−0.624431	−0.312215	−	0.950011i	$-0.601071\pi$
$269$	−10.2414	−0.624431	−0.312215	+	0.950011i	$0.601071\pi$
$270$	0	0
$271$	−20.7833	−1.26250	−0.631248	−	0.775581i	$-0.717457\pi$
$271$	−20.7833	−1.26250	−0.631248	+	0.775581i	$0.717457\pi$
$272$	0	0
$273$	9.50971	0.575554
$274$	0	0
$275$	11.9392	0.719963
$276$	0	0
$277$	−25.4749	−1.53064	−0.765320	−	0.643650i	$-0.777419\pi$
$277$	−25.4749	−1.53064	−0.765320	+	0.643650i	$0.777419\pi$
$278$	0	0
$279$	7.80536	0.467295
$280$	0	0
$281$	−24.9217	−1.48671	−0.743353	−	0.668900i	$-0.766766\pi$
$281$	−24.9217	−1.48671	−0.743353	+	0.668900i	$0.766766\pi$
$282$	0	0
$283$	−31.4413	−1.86899	−0.934496	−	0.355973i	$-0.884150\pi$
$283$	−31.4413	−1.86899	−0.934496	+	0.355973i	$0.884150\pi$
$284$	0	0
$285$	−33.3749	−1.97696
$286$	0	0
$287$	7.14258	0.421613
$288$	0	0
$289$	−4.81094	−0.282996
$290$	0	0
$291$	−5.87039	−0.344128
$292$	0	0
$293$	−11.7129	−0.684273	−0.342137	−	0.939650i	$-0.611151\pi$
$293$	−11.7129	−0.684273	−0.342137	+	0.939650i	$0.611151\pi$
$294$	0	0
$295$	27.6761	1.61137
$296$	0	0
$297$	−5.38684	−0.312576
$298$	0	0
$299$	27.1369	1.56937
$300$	0	0
$301$	0.361537	0.0208386
$302$	0	0
$303$	−9.82561	−0.564467
$304$	0	0
$305$	28.4199	1.62732
$306$	0	0
$307$	21.2771	1.21435	0.607174	−	0.794569i	$-0.292303\pi$
$307$	21.2771	1.21435	0.607174	+	0.794569i	$0.292303\pi$
$308$	0	0
$309$	1.30071	0.0739947
$310$	0	0
$311$	21.9674	1.24566	0.622828	−	0.782359i	$-0.285984\pi$
$311$	21.9674	1.24566	0.622828	+	0.782359i	$0.285984\pi$
$312$	0	0
$313$	10.4522	0.590795	0.295398	−	0.955374i	$-0.404548\pi$
$313$	10.4522	0.590795	0.295398	+	0.955374i	$0.404548\pi$
$314$	0	0
$315$	8.98427	0.506206
$316$	0	0
$317$	16.5526	0.929689	0.464845	−	0.885392i	$-0.346110\pi$
$317$	16.5526	0.929689	0.464845	+	0.885392i	$0.346110\pi$
$318$	0	0
$319$	7.31878	0.409773
$320$	0	0
$321$	−12.6030	−0.703430
$322$	0	0
$323$	24.5368	1.36527
$324$	0	0
$325$	−75.2220	−4.17257
$326$	0	0
$327$	−4.94329	−0.273365
$328$	0	0
$329$	−3.42474	−0.188812
$330$	0	0
$331$	13.3432	0.733410	0.366705	−	0.930337i	$-0.380486\pi$
$331$	13.3432	0.733410	0.366705	+	0.930337i	$0.380486\pi$
$332$	0	0
$333$	11.8302	0.648291
$334$	0	0
$335$	28.4408	1.55389
$336$	0	0
$337$	18.8915	1.02909	0.514544	−	0.857464i	$-0.327961\pi$
$337$	18.8915	1.02909	0.514544	+	0.857464i	$0.327961\pi$
$338$	0	0
$339$	16.4704	0.894549
$340$	0	0
$341$	−4.67753	−0.253303
$342$	0	0
$343$	16.0756	0.867999
$344$	0	0
$345$	−20.4540	−1.10121
$346$	0	0
$347$	−23.8779	−1.28183	−0.640916	−	0.767611i	$-0.721445\pi$
$347$	−23.8779	−1.28183	−0.640916	+	0.767611i	$0.721445\pi$
$348$	0	0
$349$	1.42939	0.0765133	0.0382567	−	0.999268i	$-0.487820\pi$
$349$	1.42939	0.0765133	0.0382567	+	0.999268i	$0.487820\pi$
$350$	0	0
$351$	33.9393	1.81155
$352$	0	0
$353$	22.9672	1.22242	0.611209	−	0.791469i	$-0.290683\pi$
$353$	22.9672	1.22242	0.611209	+	0.791469i	$0.290683\pi$
$354$	0	0
$355$	−45.3077	−2.40468
$356$	0	0
$357$	5.26968	0.278901
$358$	0	0
$359$	−10.1940	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$359$	−10.1940	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$360$	0	0
$361$	30.3931	1.59964
$362$	0	0
$363$	1.15382	0.0605600
$364$	0	0
$365$	36.1913	1.89434
$366$	0	0
$367$	−17.6405	−0.920828	−0.460414	−	0.887704i	$-0.652299\pi$
$367$	−17.6405	−0.920828	−0.460414	+	0.887704i	$0.652299\pi$
$368$	0	0
$369$	9.11112	0.474306
$370$	0	0
$371$	−1.30816	−0.0679161
$372$	0	0
$373$	17.0507	0.882852	0.441426	−	0.897298i	$-0.354473\pi$
$373$	17.0507	0.882852	0.441426	+	0.897298i	$0.354473\pi$
$374$	0	0
$375$	32.9532	1.70170
$376$	0	0
$377$	−46.1113	−2.37485
$378$	0	0
$379$	−1.85690	−0.0953823	−0.0476912	−	0.998862i	$-0.515186\pi$
$379$	−1.85690	−0.0953823	−0.0476912	+	0.998862i	$0.515186\pi$
$380$	0	0
$381$	−6.74831	−0.345727
$382$	0	0
$383$	33.1569	1.69424	0.847118	−	0.531404i	$-0.178335\pi$
$383$	33.1569	1.69424	0.847118	+	0.531404i	$0.178335\pi$
$384$	0	0
$385$	−5.38402	−0.274395
$386$	0	0
$387$	0.461179	0.0234430
$388$	0	0
$389$	−33.3516	−1.69099	−0.845497	−	0.533981i	$-0.820696\pi$
$389$	−33.3516	−1.69099	−0.845497	+	0.533981i	$0.820696\pi$
$390$	0	0
$391$	15.0375	0.760481
$392$	0	0
$393$	21.5841	1.08877
$394$	0	0
$395$	−24.4602	−1.23073
$396$	0	0
$397$	−26.5823	−1.33413	−0.667063	−	0.745001i	$-0.732449\pi$
$397$	−26.5823	−1.33413	−0.667063	+	0.745001i	$0.732449\pi$
$398$	0	0
$399$	10.6080	0.531062
$400$	0	0
$401$	−27.4019	−1.36839	−0.684193	−	0.729301i	$-0.739845\pi$
$401$	−27.4019	−1.36839	−0.684193	+	0.729301i	$0.739845\pi$
$402$	0	0
$403$	29.4704	1.46802
$404$	0	0
$405$	−4.97752	−0.247335
$406$	0	0
$407$	−7.08950	−0.351413
$408$	0	0
$409$	15.6795	0.775303	0.387651	−	0.921806i	$-0.373286\pi$
$409$	15.6795	0.775303	0.387651	+	0.921806i	$0.373286\pi$
$410$	0	0
$411$	−12.9039	−0.636504
$412$	0	0
$413$	−8.79666	−0.432855
$414$	0	0
$415$	−46.9798	−2.30615
$416$	0	0
$417$	−6.09305	−0.298378
$418$	0	0
$419$	−10.3567	−0.505958	−0.252979	−	0.967472i	$-0.581410\pi$
$419$	−10.3567	−0.505958	−0.252979	+	0.967472i	$0.581410\pi$
$420$	0	0
$421$	−12.1411	−0.591721	−0.295861	−	0.955231i	$-0.595606\pi$
$421$	−12.1411	−0.591721	−0.295861	+	0.955231i	$0.595606\pi$
$422$	0	0
$423$	−4.36861	−0.212409
$424$	0	0
$425$	−41.6833	−2.02193
$426$	0	0
$427$	−9.03306	−0.437141
$428$	0	0
$429$	−7.26955	−0.350977
$430$	0	0
$431$	−7.26633	−0.350007	−0.175003	−	0.984568i	$-0.555994\pi$
$431$	−7.26633	−0.350007	−0.175003	+	0.984568i	$0.555994\pi$
$432$	0	0
$433$	8.58206	0.412428	0.206214	−	0.978507i	$-0.433886\pi$
$433$	8.58206	0.412428	0.206214	+	0.978507i	$0.433886\pi$
$434$	0	0
$435$	34.7556	1.66640
$436$	0	0
$437$	30.2709	1.44805
$438$	0	0
$439$	−19.9606	−0.952669	−0.476335	−	0.879264i	$-0.658035\pi$
$439$	−19.9606	−0.952669	−0.476335	+	0.879264i	$0.658035\pi$
$440$	0	0
$441$	8.82526	0.420250
$442$	0	0
$443$	1.27336	0.0604994	0.0302497	−	0.999542i	$-0.490370\pi$
$443$	1.27336	0.0604994	0.0302497	+	0.999542i	$0.490370\pi$
$444$	0	0
$445$	8.21077	0.389228
$446$	0	0
$447$	−20.2190	−0.956326
$448$	0	0
$449$	−4.50139	−0.212434	−0.106217	−	0.994343i	$-0.533874\pi$
$449$	−4.50139	−0.212434	−0.106217	+	0.994343i	$0.533874\pi$
$450$	0	0
$451$	−5.46003	−0.257103
$452$	0	0
$453$	16.2974	0.765720
$454$	0	0
$455$	33.9215	1.59026
$456$	0	0
$457$	35.9371	1.68106	0.840532	−	0.541761i	$-0.182242\pi$
$457$	35.9371	1.68106	0.840532	+	0.541761i	$0.182242\pi$
$458$	0	0
$459$	18.8070	0.877836
$460$	0	0
$461$	−37.9648	−1.76820	−0.884099	−	0.467300i	$-0.845227\pi$
$461$	−37.9648	−1.76820	−0.884099	+	0.467300i	$0.845227\pi$
$462$	0	0
$463$	13.4460	0.624889	0.312444	−	0.949936i	$-0.398852\pi$
$463$	13.4460	0.624889	0.312444	+	0.949936i	$0.398852\pi$
$464$	0	0
$465$	−22.2128	−1.03009
$466$	0	0
$467$	−28.1775	−1.30390	−0.651949	−	0.758263i	$-0.726048\pi$
$467$	−28.1775	−1.30390	−0.651949	+	0.758263i	$0.726048\pi$
$468$	0	0
$469$	−9.03972	−0.417416
$470$	0	0
$471$	27.1796	1.25237
$472$	0	0
$473$	−0.276371	−0.0127076
$474$	0	0
$475$	−83.9092	−3.85002
$476$	0	0
$477$	−1.66869	−0.0764042
$478$	0	0
$479$	−42.8308	−1.95699	−0.978495	−	0.206272i	$-0.933867\pi$
$479$	−42.8308	−1.95699	−0.978495	+	0.206272i	$0.933867\pi$
$480$	0	0
$481$	44.6667	2.03663
$482$	0	0
$483$	6.50116	0.295813
$484$	0	0
$485$	−20.9399	−0.950832
$486$	0	0
$487$	−28.9655	−1.31255	−0.656277	−	0.754520i	$-0.727870\pi$
$487$	−28.9655	−1.31255	−0.656277	+	0.754520i	$0.727870\pi$
$488$	0	0
$489$	6.64535	0.300513
$490$	0	0
$491$	−29.3683	−1.32537	−0.662686	−	0.748897i	$-0.730584\pi$
$491$	−29.3683	−1.32537	−0.662686	+	0.748897i	$0.730584\pi$
$492$	0	0
$493$	−25.5519	−1.15080
$494$	0	0
$495$	−6.86789	−0.308689
$496$	0	0
$497$	14.4007	0.645961
$498$	0	0
$499$	23.2052	1.03881	0.519405	−	0.854528i	$-0.326154\pi$
$499$	23.2052	1.03881	0.519405	+	0.854528i	$0.326154\pi$
$500$	0	0
$501$	−4.25400	−0.190055
$502$	0	0
$503$	13.2648	0.591449	0.295724	−	0.955273i	$-0.404439\pi$
$503$	13.2648	0.591449	0.295724	+	0.955273i	$0.404439\pi$
$504$	0	0
$505$	−35.0483	−1.55963
$506$	0	0
$507$	30.8014	1.36794
$508$	0	0
$509$	−0.817749	−0.0362461	−0.0181231	−	0.999836i	$-0.505769\pi$
$509$	−0.817749	−0.0362461	−0.0181231	+	0.999836i	$0.505769\pi$
$510$	0	0
$511$	−11.5032	−0.508870
$512$	0	0
$513$	37.8589	1.67151
$514$	0	0
$515$	4.63967	0.204448
$516$	0	0
$517$	2.61799	0.115139
$518$	0	0
$519$	10.5059	0.461158
$520$	0	0
$521$	−40.5941	−1.77846	−0.889229	−	0.457462i	$-0.848759\pi$
$521$	−40.5941	−1.77846	−0.889229	+	0.457462i	$0.848759\pi$
$522$	0	0
$523$	−39.0946	−1.70949	−0.854743	−	0.519052i	$-0.826285\pi$
$523$	−39.0946	−1.70949	−0.854743	+	0.519052i	$0.826285\pi$
$524$	0	0
$525$	−18.0208	−0.786494
$526$	0	0
$527$	16.3306	0.711372
$528$	0	0
$529$	−4.44830	−0.193404
$530$	0	0
$531$	−11.2211	−0.486953
$532$	0	0
$533$	34.4004	1.49005
$534$	0	0
$535$	−44.9553	−1.94359
$536$	0	0
$537$	−8.67846	−0.374503
$538$	0	0
$539$	−5.28873	−0.227802
$540$	0	0
$541$	18.8614	0.810913	0.405456	−	0.914114i	$-0.367113\pi$
$541$	18.8614	0.810913	0.405456	+	0.914114i	$0.367113\pi$
$542$	0	0
$543$	5.69333	0.244324
$544$	0	0
$545$	−17.6329	−0.755311
$546$	0	0
$547$	5.78932	0.247534	0.123767	−	0.992311i	$-0.460503\pi$
$547$	5.78932	0.247534	0.123767	+	0.992311i	$0.460503\pi$
$548$	0	0
$549$	−11.5226	−0.491774
$550$	0	0
$551$	−51.4365	−2.19127
$552$	0	0
$553$	7.77450	0.330605
$554$	0	0
$555$	−33.6668	−1.42908
$556$	0	0
$557$	28.7843	1.21963	0.609814	−	0.792544i	$-0.291244\pi$
$557$	28.7843	1.21963	0.609814	+	0.792544i	$0.291244\pi$
$558$	0	0
$559$	1.74125	0.0736471
$560$	0	0
$561$	−4.02832	−0.170076
$562$	0	0
$563$	14.3138	0.603254	0.301627	−	0.953426i	$-0.402470\pi$
$563$	14.3138	0.603254	0.301627	+	0.953426i	$0.402470\pi$
$564$	0	0
$565$	58.7505	2.47165
$566$	0	0
$567$	1.58207	0.0664407
$568$	0	0
$569$	−21.7658	−0.912470	−0.456235	−	0.889859i	$-0.650802\pi$
$569$	−21.7658	−0.912470	−0.456235	+	0.889859i	$0.650802\pi$
$570$	0	0
$571$	11.1385	0.466131	0.233066	−	0.972461i	$-0.425124\pi$
$571$	11.1385	0.466131	0.233066	+	0.972461i	$0.425124\pi$
$572$	0	0
$573$	8.28885	0.346272
$574$	0	0
$575$	−51.4243	−2.14454
$576$	0	0
$577$	16.2403	0.676094	0.338047	−	0.941129i	$-0.390234\pi$
$577$	16.2403	0.676094	0.338047	+	0.941129i	$0.390234\pi$
$578$	0	0
$579$	13.4651	0.559590
$580$	0	0
$581$	14.9322	0.619492
$582$	0	0
$583$	1.00000	0.0414158
$584$	0	0
$585$	43.2705	1.78901
$586$	0	0
$587$	−29.6853	−1.22524	−0.612621	−	0.790377i	$-0.709885\pi$
$587$	−29.6853	−1.22524	−0.612621	+	0.790377i	$0.709885\pi$
$588$	0	0
$589$	32.8738	1.35454
$590$	0	0
$591$	27.4881	1.13071
$592$	0	0
$593$	−9.87873	−0.405671	−0.202835	−	0.979213i	$-0.565016\pi$
$593$	−9.87873	−0.405671	−0.202835	+	0.979213i	$0.565016\pi$
$594$	0	0
$595$	18.7971	0.770607
$596$	0	0
$597$	4.95267	0.202699
$598$	0	0
$599$	−19.5550	−0.798998	−0.399499	−	0.916734i	$-0.630816\pi$
$599$	−19.5550	−0.798998	−0.399499	+	0.916734i	$0.630816\pi$
$600$	0	0
$601$	1.55000	0.0632258	0.0316129	−	0.999500i	$-0.489936\pi$
$601$	1.55000	0.0632258	0.0316129	+	0.999500i	$0.489936\pi$
$602$	0	0
$603$	−11.5311	−0.469584
$604$	0	0
$605$	4.11573	0.167328
$606$	0	0
$607$	−22.6802	−0.920561	−0.460280	−	0.887774i	$-0.652251\pi$
$607$	−22.6802	−0.920561	−0.460280	+	0.887774i	$0.652251\pi$
$608$	0	0
$609$	−11.0468	−0.447640
$610$	0	0
$611$	−16.4944	−0.667291
$612$	0	0
$613$	−21.6746	−0.875427	−0.437713	−	0.899115i	$-0.644212\pi$
$613$	−21.6746	−0.875427	−0.437713	+	0.899115i	$0.644212\pi$
$614$	0	0
$615$	−25.9287	−1.04555
$616$	0	0
$617$	−23.2906	−0.937645	−0.468822	−	0.883292i	$-0.655322\pi$
$617$	−23.2906	−0.937645	−0.468822	+	0.883292i	$0.655322\pi$
$618$	0	0
$619$	11.6202	0.467056	0.233528	−	0.972350i	$-0.424973\pi$
$619$	11.6202	0.467056	0.233528	+	0.972350i	$0.424973\pi$
$620$	0	0
$621$	23.2020	0.931066
$622$	0	0
$623$	−2.60974	−0.104557
$624$	0	0
$625$	57.8491	2.31396
$626$	0	0
$627$	−8.10910	−0.323846
$628$	0	0
$629$	24.7515	0.986905
$630$	0	0
$631$	6.16530	0.245437	0.122718	−	0.992442i	$-0.460839\pi$
$631$	6.16530	0.245437	0.122718	+	0.992442i	$0.460839\pi$
$632$	0	0
$633$	−23.4163	−0.930716
$634$	0	0
$635$	−24.0715	−0.955248
$636$	0	0
$637$	33.3211	1.32023
$638$	0	0
$639$	18.3697	0.726693
$640$	0	0
$641$	−33.9150	−1.33956	−0.669780	−	0.742559i	$-0.733612\pi$
$641$	−33.9150	−1.33956	−0.669780	+	0.742559i	$0.733612\pi$
$642$	0	0
$643$	−47.7894	−1.88463	−0.942315	−	0.334727i	$-0.891356\pi$
$643$	−47.7894	−1.88463	−0.942315	+	0.334727i	$0.891356\pi$
$644$	0	0
$645$	−1.31244	−0.0516772
$646$	0	0
$647$	27.2743	1.07226	0.536131	−	0.844135i	$-0.319885\pi$
$647$	27.2743	1.07226	0.536131	+	0.844135i	$0.319885\pi$
$648$	0	0
$649$	6.72447	0.263959
$650$	0	0
$651$	7.06018	0.276710
$652$	0	0
$653$	19.6868	0.770402	0.385201	−	0.922833i	$-0.374132\pi$
$653$	19.6868	0.770402	0.385201	+	0.922833i	$0.374132\pi$
$654$	0	0
$655$	76.9912	3.00829
$656$	0	0
$657$	−14.6735	−0.572468
$658$	0	0
$659$	42.6005	1.65948	0.829741	−	0.558149i	$-0.188488\pi$
$659$	42.6005	1.65948	0.829741	+	0.558149i	$0.188488\pi$
$660$	0	0
$661$	15.8979	0.618355	0.309178	−	0.951004i	$-0.399946\pi$
$661$	15.8979	0.618355	0.309178	+	0.951004i	$0.399946\pi$
$662$	0	0
$663$	25.3801	0.985680
$664$	0	0
$665$	37.8390	1.46733
$666$	0	0
$667$	−31.5232	−1.22058
$668$	0	0
$669$	11.0488	0.427172
$670$	0	0
$671$	6.90519	0.266572
$672$	0	0
$673$	−42.5917	−1.64179	−0.820894	−	0.571081i	$-0.806524\pi$
$673$	−42.5917	−1.64179	−0.820894	+	0.571081i	$0.806524\pi$
$674$	0	0
$675$	−64.3148	−2.47548
$676$	0	0
$677$	14.7915	0.568484	0.284242	−	0.958753i	$-0.408258\pi$
$677$	14.7915	0.568484	0.284242	+	0.958753i	$0.408258\pi$
$678$	0	0
$679$	6.65560	0.255418
$680$	0	0
$681$	−18.9007	−0.724275
$682$	0	0
$683$	−13.3364	−0.510303	−0.255152	−	0.966901i	$-0.582125\pi$
$683$	−13.3364	−0.510303	−0.255152	+	0.966901i	$0.582125\pi$
$684$	0	0
$685$	−46.0288	−1.75867
$686$	0	0
$687$	15.4897	0.590969
$688$	0	0
$689$	−6.30041	−0.240026
$690$	0	0
$691$	−34.6400	−1.31777	−0.658884	−	0.752244i	$-0.728971\pi$
$691$	−34.6400	−1.31777	−0.658884	+	0.752244i	$0.728971\pi$
$692$	0	0
$693$	2.18291	0.0829219
$694$	0	0
$695$	−21.7341	−0.824422
$696$	0	0
$697$	19.0625	0.722045
$698$	0	0
$699$	28.8506	1.09123
$700$	0	0
$701$	20.0963	0.759025	0.379513	−	0.925187i	$-0.376092\pi$
$701$	20.0963	0.759025	0.379513	+	0.925187i	$0.376092\pi$
$702$	0	0
$703$	49.8252	1.87919
$704$	0	0
$705$	12.4324	0.468230
$706$	0	0
$707$	11.1399	0.418958
$708$	0	0
$709$	−12.0196	−0.451405	−0.225703	−	0.974196i	$-0.572468\pi$
$709$	−12.0196	−0.451405	−0.225703	+	0.974196i	$0.572468\pi$
$710$	0	0
$711$	9.91720	0.371924
$712$	0	0
$713$	20.1469	0.754508
$714$	0	0
$715$	−25.9308	−0.969756
$716$	0	0
$717$	0.619208	0.0231247
$718$	0	0
$719$	25.2372	0.941190	0.470595	−	0.882349i	$-0.344039\pi$
$719$	25.2372	0.941190	0.470595	+	0.882349i	$0.344039\pi$
$720$	0	0
$721$	−1.47469	−0.0549203
$722$	0	0
$723$	−9.50624	−0.353541
$724$	0	0
$725$	87.3806	3.24523
$726$	0	0
$727$	20.8023	0.771515	0.385758	−	0.922600i	$-0.373940\pi$
$727$	20.8023	0.771515	0.385758	+	0.922600i	$0.373940\pi$
$728$	0	0
$729$	20.6645	0.765352
$730$	0	0
$731$	0.964891	0.0356878
$732$	0	0
$733$	−5.74459	−0.212181	−0.106091	−	0.994356i	$-0.533833\pi$
$733$	−5.74459	−0.212181	−0.106091	+	0.994356i	$0.533833\pi$
$734$	0	0
$735$	−25.1152	−0.926389
$736$	0	0
$737$	6.91028	0.254543
$738$	0	0
$739$	31.4689	1.15760	0.578802	−	0.815468i	$-0.303520\pi$
$739$	31.4689	1.15760	0.578802	+	0.815468i	$0.303520\pi$
$740$	0	0
$741$	51.0906	1.87686
$742$	0	0
$743$	47.3695	1.73782	0.868908	−	0.494973i	$-0.164822\pi$
$743$	47.3695	1.73782	0.868908	+	0.494973i	$0.164822\pi$
$744$	0	0
$745$	−72.1219	−2.64234
$746$	0	0
$747$	19.0476	0.696915
$748$	0	0
$749$	14.2887	0.522099
$750$	0	0
$751$	−37.2564	−1.35950	−0.679752	−	0.733442i	$-0.737913\pi$
$751$	−37.2564	−1.35950	−0.679752	+	0.733442i	$0.737913\pi$
$752$	0	0
$753$	−23.3613	−0.851332
$754$	0	0
$755$	58.1335	2.11570
$756$	0	0
$757$	−17.0078	−0.618158	−0.309079	−	0.951036i	$-0.600021\pi$
$757$	−17.0078	−0.618158	−0.309079	+	0.951036i	$0.600021\pi$
$758$	0	0
$759$	−4.96971	−0.180389
$760$	0	0
$761$	46.8598	1.69867	0.849334	−	0.527856i	$-0.177004\pi$
$761$	46.8598	1.69867	0.849334	+	0.527856i	$0.177004\pi$
$762$	0	0
$763$	5.60450	0.202896
$764$	0	0
$765$	23.9777	0.866917
$766$	0	0
$767$	−42.3669	−1.52978
$768$	0	0
$769$	−11.5847	−0.417755	−0.208878	−	0.977942i	$-0.566981\pi$
$769$	−11.5847	−0.417755	−0.208878	+	0.977942i	$0.566981\pi$
$770$	0	0
$771$	25.1870	0.907087
$772$	0	0
$773$	−30.4521	−1.09529	−0.547643	−	0.836712i	$-0.684475\pi$
$773$	−30.4521	−1.09529	−0.547643	+	0.836712i	$0.684475\pi$
$774$	0	0
$775$	−55.8462	−2.00605
$776$	0	0
$777$	10.7008	0.383887
$778$	0	0
$779$	38.3732	1.37486
$780$	0	0
$781$	−11.0084	−0.393912
$782$	0	0
$783$	−39.4251	−1.40894
$784$	0	0
$785$	96.9508	3.46032
$786$	0	0
$787$	29.1677	1.03972	0.519858	−	0.854253i	$-0.325985\pi$
$787$	29.1677	1.03972	0.519858	+	0.854253i	$0.325985\pi$
$788$	0	0
$789$	−30.7191	−1.09363
$790$	0	0
$791$	−18.6734	−0.663951
$792$	0	0
$793$	−43.5055	−1.54492
$794$	0	0
$795$	4.74882	0.168423
$796$	0	0
$797$	−19.2009	−0.680130	−0.340065	−	0.940402i	$-0.610449\pi$
$797$	−19.2009	−0.680130	−0.340065	+	0.940402i	$0.610449\pi$
$798$	0	0
$799$	−9.14013	−0.323355
$800$	0	0
$801$	−3.32900	−0.117624
$802$	0	0
$803$	8.79341	0.310313
$804$	0	0
$805$	23.1899	0.817335
$806$	0	0
$807$	−11.8168	−0.415971
$808$	0	0
$809$	−2.63813	−0.0927519	−0.0463759	−	0.998924i	$-0.514767\pi$
$809$	−2.63813	−0.0927519	−0.0463759	+	0.998924i	$0.514767\pi$
$810$	0	0
$811$	−34.0960	−1.19727	−0.598636	−	0.801021i	$-0.704291\pi$
$811$	−34.0960	−1.19727	−0.598636	+	0.801021i	$0.704291\pi$
$812$	0	0
$813$	−23.9802	−0.841024
$814$	0	0
$815$	23.7042	0.830323
$816$	0	0
$817$	1.94235	0.0679541
$818$	0	0
$819$	−13.7532	−0.480576
$820$	0	0
$821$	11.1491	0.389106	0.194553	−	0.980892i	$-0.437674\pi$
$821$	11.1491	0.389106	0.194553	+	0.980892i	$0.437674\pi$
$822$	0	0
$823$	25.0186	0.872095	0.436047	−	0.899924i	$-0.356378\pi$
$823$	25.0186	0.872095	0.436047	+	0.899924i	$0.356378\pi$
$824$	0	0
$825$	13.7758	0.479610
$826$	0	0
$827$	−45.0779	−1.56751	−0.783756	−	0.621069i	$-0.786699\pi$
$827$	−45.0779	−1.56751	−0.783756	+	0.621069i	$0.786699\pi$
$828$	0	0
$829$	−31.1578	−1.08216	−0.541078	−	0.840973i	$-0.681984\pi$
$829$	−31.1578	−1.08216	−0.541078	+	0.840973i	$0.681984\pi$
$830$	0	0
$831$	−29.3936	−1.01965
$832$	0	0
$833$	18.4644	0.639755
$834$	0	0
$835$	−15.1742	−0.525125
$836$	0	0
$837$	25.1971	0.870941
$838$	0	0
$839$	50.2054	1.73328	0.866642	−	0.498930i	$-0.166274\pi$
$839$	50.2054	1.73328	0.866642	+	0.498930i	$0.166274\pi$
$840$	0	0
$841$	24.5645	0.847051
$842$	0	0
$843$	−28.7553	−0.990384
$844$	0	0
$845$	109.870	3.77964
$846$	0	0
$847$	−1.30816	−0.0449488
$848$	0	0
$849$	−36.2777	−1.24505
$850$	0	0
$851$	30.5357	1.04675
$852$	0	0
$853$	−5.00136	−0.171243	−0.0856217	−	0.996328i	$-0.527288\pi$
$853$	−5.00136	−0.171243	−0.0856217	+	0.996328i	$0.527288\pi$
$854$	0	0
$855$	48.2677	1.65072
$856$	0	0
$857$	−23.7088	−0.809878	−0.404939	−	0.914344i	$-0.632707\pi$
$857$	−23.7088	−0.809878	−0.404939	+	0.914344i	$0.632707\pi$
$858$	0	0
$859$	−53.3384	−1.81988	−0.909942	−	0.414736i	$-0.863874\pi$
$859$	−53.3384	−1.81988	−0.909942	+	0.414736i	$0.863874\pi$
$860$	0	0
$861$	8.24127	0.280862
$862$	0	0
$863$	49.3478	1.67982	0.839909	−	0.542728i	$-0.182608\pi$
$863$	49.3478	1.67982	0.839909	+	0.542728i	$0.182608\pi$
$864$	0	0
$865$	37.4749	1.27419
$866$	0	0
$867$	−5.55097	−0.188521
$868$	0	0
$869$	−5.94310	−0.201606
$870$	0	0
$871$	−43.5375	−1.47521
$872$	0	0
$873$	8.48993	0.287340
$874$	0	0
$875$	−37.3609	−1.26303
$876$	0	0
$877$	−26.9603	−0.910385	−0.455192	−	0.890393i	$-0.650430\pi$
$877$	−26.9603	−0.910385	−0.455192	+	0.890393i	$0.650430\pi$
$878$	0	0
$879$	−13.5146	−0.455835
$880$	0	0
$881$	−31.0528	−1.04620	−0.523098	−	0.852273i	$-0.675224\pi$
$881$	−31.0528	−1.04620	−0.523098	+	0.852273i	$0.675224\pi$
$882$	0	0
$883$	37.8767	1.27465	0.637327	−	0.770594i	$-0.280040\pi$
$883$	37.8767	1.27465	0.637327	+	0.770594i	$0.280040\pi$
$884$	0	0
$885$	31.9333	1.07343
$886$	0	0
$887$	−12.5382	−0.420991	−0.210495	−	0.977595i	$-0.567508\pi$
$887$	−12.5382	−0.420991	−0.210495	+	0.977595i	$0.567508\pi$
$888$	0	0
$889$	7.65095	0.256605
$890$	0	0
$891$	−1.20939	−0.0405160
$892$	0	0
$893$	−18.3993	−0.615708
$894$	0	0
$895$	−30.9564	−1.03476
$896$	0	0
$897$	31.3112	1.04545
$898$	0	0
$899$	−34.2338	−1.14176
$900$	0	0
$901$	−3.49128	−0.116312
$902$	0	0
$903$	0.417150	0.0138819
$904$	0	0
$905$	20.3083	0.675072
$906$	0	0
$907$	−32.5278	−1.08007	−0.540033	−	0.841644i	$-0.681588\pi$
$907$	−32.5278	−1.08007	−0.540033	+	0.841644i	$0.681588\pi$
$908$	0	0
$909$	14.2101	0.471319
$910$	0	0
$911$	−24.2515	−0.803488	−0.401744	−	0.915752i	$-0.631596\pi$
$911$	−24.2515	−0.803488	−0.401744	+	0.915752i	$0.631596\pi$
$912$	0	0
$913$	−11.4147	−0.377771
$914$	0	0
$915$	32.7915	1.08405
$916$	0	0
$917$	−24.4711	−0.808107
$918$	0	0
$919$	27.9119	0.920728	0.460364	−	0.887730i	$-0.347719\pi$
$919$	27.9119	0.920728	0.460364	+	0.887730i	$0.347719\pi$
$920$	0	0
$921$	24.5500	0.808950
$922$	0	0
$923$	69.3575	2.28293
$924$	0	0
$925$	−84.6432	−2.78305
$926$	0	0
$927$	−1.88112	−0.0617842
$928$	0	0
$929$	−18.8400	−0.618121	−0.309060	−	0.951042i	$-0.600014\pi$
$929$	−18.8400	−0.618121	−0.309060	+	0.951042i	$0.600014\pi$
$930$	0	0
$931$	37.1693	1.21817
$932$	0	0
$933$	25.3465	0.829806
$934$	0	0
$935$	−14.3692	−0.469923
$936$	0	0
$937$	3.94281	0.128806	0.0644030	−	0.997924i	$-0.479486\pi$
$937$	3.94281	0.128806	0.0644030	+	0.997924i	$0.479486\pi$
$938$	0	0
$939$	12.0600	0.393564
$940$	0	0
$941$	−6.86945	−0.223938	−0.111969	−	0.993712i	$-0.535716\pi$
$941$	−6.86945	−0.223938	−0.111969	+	0.993712i	$0.535716\pi$
$942$	0	0
$943$	23.5173	0.765828
$944$	0	0
$945$	29.0029	0.943463
$946$	0	0
$947$	−7.24536	−0.235442	−0.117721	−	0.993047i	$-0.537559\pi$
$947$	−7.24536	−0.235442	−0.117721	+	0.993047i	$0.537559\pi$
$948$	0	0
$949$	−55.4020	−1.79843
$950$	0	0
$951$	19.0988	0.619322
$952$	0	0
$953$	36.1357	1.17055	0.585275	−	0.810835i	$-0.300987\pi$
$953$	36.1357	1.17055	0.585275	+	0.810835i	$0.300987\pi$
$954$	0	0
$955$	29.5667	0.956754
$956$	0	0
$957$	8.44457	0.272974
$958$	0	0
$959$	14.6299	0.472425
$960$	0	0
$961$	−9.12068	−0.294216
$962$	0	0
$963$	18.2268	0.587350
$964$	0	0
$965$	48.0305	1.54616
$966$	0	0
$967$	23.9947	0.771617	0.385809	−	0.922579i	$-0.373923\pi$
$967$	23.9947	0.771617	0.385809	+	0.922579i	$0.373923\pi$
$968$	0	0
$969$	28.3112	0.909486
$970$	0	0
$971$	22.7932	0.731469	0.365734	−	0.930719i	$-0.380818\pi$
$971$	22.7932	0.731469	0.365734	+	0.930719i	$0.380818\pi$
$972$	0	0
$973$	6.90804	0.221462
$974$	0	0
$975$	−86.7929	−2.77960
$976$	0	0
$977$	−28.8090	−0.921682	−0.460841	−	0.887483i	$-0.652452\pi$
$977$	−28.8090	−0.921682	−0.460841	+	0.887483i	$0.652452\pi$
$978$	0	0
$979$	1.99497	0.0637596
$980$	0	0
$981$	7.14913	0.228254
$982$	0	0
$983$	−33.9789	−1.08376	−0.541880	−	0.840456i	$-0.682287\pi$
$983$	−33.9789	−1.08376	−0.541880	+	0.840456i	$0.682287\pi$
$984$	0	0
$985$	98.0509	3.12416
$986$	0	0
$987$	−3.95154	−0.125779
$988$	0	0
$989$	1.19038	0.0378518
$990$	0	0
$991$	−15.4655	−0.491276	−0.245638	−	0.969362i	$-0.578997\pi$
$991$	−15.4655	−0.491276	−0.245638	+	0.969362i	$0.578997\pi$
$992$	0	0
$993$	15.3957	0.488568
$994$	0	0
$995$	17.6664	0.560061
$996$	0	0
$997$	7.37612	0.233604	0.116802	−	0.993155i	$-0.462736\pi$
$997$	7.37612	0.233604	0.116802	+	0.993155i	$0.462736\pi$
$998$	0	0
$999$	38.1900	1.20828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4664.2.a.k.1.9	✓	11
4.3	odd	2		9328.2.a.bm.1.3		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4664.2.a.k.1.9	✓	11	1.1	even	1	trivial
9328.2.a.bm.1.3		11	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:	\( N \)	\(=\)	\( 4664 = 2^{3} \cdot 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4664.a (trivial)

\(f(q)\)	\(=\)	\(q+1.15382 q^{3} +4.11573 q^{5} -1.30816 q^{7} -1.66869 q^{9} +O(q^{10})\)	\(q+1.15382 q^{3} +4.11573 q^{5} -1.30816 q^{7} -1.66869 q^{9} +1.00000 q^{11} -6.30041 q^{13} +4.74882 q^{15} -3.49128 q^{17} -7.02802 q^{19} -1.50938 q^{21} -4.30717 q^{23} +11.9392 q^{25} -5.38684 q^{27} +7.31878 q^{29} -4.67753 q^{31} +1.15382 q^{33} -5.38402 q^{35} -7.08950 q^{37} -7.26955 q^{39} -5.46003 q^{41} -0.276371 q^{43} -6.86789 q^{45} +2.61799 q^{47} -5.28873 q^{49} -4.02832 q^{51} +1.00000 q^{53} +4.11573 q^{55} -8.10910 q^{57} +6.72447 q^{59} +6.90519 q^{61} +2.18291 q^{63} -25.9308 q^{65} +6.91028 q^{67} -4.96971 q^{69} -11.0084 q^{71} +8.79341 q^{73} +13.7758 q^{75} -1.30816 q^{77} -5.94310 q^{79} -1.20939 q^{81} -11.4147 q^{83} -14.3692 q^{85} +8.44457 q^{87} +1.99497 q^{89} +8.24191 q^{91} -5.39705 q^{93} -28.9255 q^{95} -5.08777 q^{97} -1.66869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9	\( 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9} + 11 q^{11} - 13 q^{13} - 8 q^{15} - 7 q^{17} - 21 q^{19} + 6 q^{21} + 11 q^{23} + 4 q^{25} - 6 q^{27} - 5 q^{31} - 6 q^{33} - 25 q^{35} - 4 q^{37} - 19 q^{39} - 11 q^{41} + 16 q^{45} - 17 q^{47} - 2 q^{49} - 18 q^{51} + 11 q^{53} + 3 q^{55} - 5 q^{57} - 19 q^{59} - 2 q^{61} - 36 q^{63} - 13 q^{65} + 25 q^{67} + 3 q^{69} - 30 q^{71} + 5 q^{73} - 5 q^{75} - 5 q^{77} - 23 q^{79} - 9 q^{81} - 19 q^{83} + 2 q^{85} - 7 q^{87} + 6 q^{89} - 20 q^{91} + 43 q^{93} - 50 q^{95} - 35 q^{97} + 7 q^{99}+O(q^{100}) \) 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9 + 11 * q^11 - 13 * q^13 - 8 * q^15 - 7 * q^17 - 21 * q^19 + 6 * q^21 + 11 * q^23 + 4 * q^25 - 6 * q^27 - 5 * q^31 - 6 * q^33 - 25 * q^35 - 4 * q^37 - 19 * q^39 - 11 * q^41 + 16 * q^45 - 17 * q^47 - 2 * q^49 - 18 * q^51 + 11 * q^53 + 3 * q^55 - 5 * q^57 - 19 * q^59 - 2 * q^61 - 36 * q^63 - 13 * q^65 + 25 * q^67 + 3 * q^69 - 30 * q^71 + 5 * q^73 - 5 * q^75 - 5 * q^77 - 23 * q^79 - 9 * q^81 - 19 * q^83 + 2 * q^85 - 7 * q^87 + 6 * q^89 - 20 * q^91 + 43 * q^93 - 50 * q^95 - 35 * q^97 + 7 * q^99

Introduction

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