LMFDB - Embedded newform 4851.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.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:	$38.7354300205$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} -2.23607 q^{8} +O(q^{10})$	$q-2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} -2.23607 q^{8} +4.47214 q^{10} +1.00000 q^{11} -3.23607 q^{13} -1.00000 q^{16} -3.23607 q^{17} -6.47214 q^{19} -6.00000 q^{20} -2.23607 q^{22} -2.47214 q^{23} -1.00000 q^{25} +7.23607 q^{26} -8.47214 q^{29} +2.76393 q^{31} +6.70820 q^{32} +7.23607 q^{34} -8.47214 q^{37} +14.4721 q^{38} +4.47214 q^{40} -11.2361 q^{41} +8.00000 q^{43} +3.00000 q^{44} +5.52786 q^{46} +2.76393 q^{47} +2.23607 q^{50} -9.70820 q^{52} +0.472136 q^{53} -2.00000 q^{55} +18.9443 q^{58} -1.23607 q^{59} +7.23607 q^{61} -6.18034 q^{62} -13.0000 q^{64} +6.47214 q^{65} +14.4721 q^{67} -9.70820 q^{68} +10.4721 q^{71} +0.763932 q^{73} +18.9443 q^{74} -19.4164 q^{76} -8.94427 q^{79} +2.00000 q^{80} +25.1246 q^{82} -11.4164 q^{83} +6.47214 q^{85} -17.8885 q^{86} -2.23607 q^{88} +2.00000 q^{89} -7.41641 q^{92} -6.18034 q^{94} +12.9443 q^{95} -17.4164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - 4 q^{5}+O(q^{10}) $ 2 * q + 6 * q^4 - 4 * q^5	$ 2 q + 6 q^{4} - 4 q^{5} + 2 q^{11} - 2 q^{13} - 2 q^{16} - 2 q^{17} - 4 q^{19} - 12 q^{20} + 4 q^{23} - 2 q^{25} + 10 q^{26} - 8 q^{29} + 10 q^{31} + 10 q^{34} - 8 q^{37} + 20 q^{38} - 18 q^{41} + 16 q^{43} + 6 q^{44} + 20 q^{46} + 10 q^{47} - 6 q^{52} - 8 q^{53} - 4 q^{55} + 20 q^{58} + 2 q^{59} + 10 q^{61} + 10 q^{62} - 26 q^{64} + 4 q^{65} + 20 q^{67} - 6 q^{68} + 12 q^{71} + 6 q^{73} + 20 q^{74} - 12 q^{76} + 4 q^{80} + 10 q^{82} + 4 q^{83} + 4 q^{85} + 4 q^{89} + 12 q^{92} + 10 q^{94} + 8 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 - 4 * q^5 + 2 * q^11 - 2 * q^13 - 2 * q^16 - 2 * q^17 - 4 * q^19 - 12 * q^20 + 4 * q^23 - 2 * q^25 + 10 * q^26 - 8 * q^29 + 10 * q^31 + 10 * q^34 - 8 * q^37 + 20 * q^38 - 18 * q^41 + 16 * q^43 + 6 * q^44 + 20 * q^46 + 10 * q^47 - 6 * q^52 - 8 * q^53 - 4 * q^55 + 20 * q^58 + 2 * q^59 + 10 * q^61 + 10 * q^62 - 26 * q^64 + 4 * q^65 + 20 * q^67 - 6 * q^68 + 12 * q^71 + 6 * q^73 + 20 * q^74 - 12 * q^76 + 4 * q^80 + 10 * q^82 + 4 * q^83 + 4 * q^85 + 4 * q^89 + 12 * q^92 + 10 * q^94 + 8 * q^95 - 8 * q^97

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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	0	0
$10$	4.47214	1.41421
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−3.23607	−0.784862	−0.392431	−	0.919781i	$-0.628366\pi$
$17$	−3.23607	−0.784862	−0.392431	+	0.919781i	$0.628366\pi$
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	−6.00000	−1.34164
$21$	0	0
$22$	−2.23607	−0.476731
$23$	−2.47214	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	7.23607	1.41911
$27$	0	0
$28$	0	0
$29$	−8.47214	−1.57324	−0.786618	−	0.617440i	$-0.788170\pi$
$29$	−8.47214	−1.57324	−0.786618	+	0.617440i	$0.788170\pi$
$30$	0	0
$31$	2.76393	0.496417	0.248208	−	0.968707i	$-0.420158\pi$
$31$	2.76393	0.496417	0.248208	+	0.968707i	$0.420158\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	7.23607	1.24098
$35$	0	0
$36$	0	0
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	14.4721	2.34769
$39$	0	0
$40$	4.47214	0.707107
$41$	−11.2361	−1.75478	−0.877390	−	0.479779i	$-0.840717\pi$
$41$	−11.2361	−1.75478	−0.877390	+	0.479779i	$0.840717\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	5.52786	0.815039
$47$	2.76393	0.403161	0.201580	−	0.979472i	$-0.435392\pi$
$47$	2.76393	0.403161	0.201580	+	0.979472i	$0.435392\pi$
$48$	0	0
$49$	0	0
$50$	2.23607	0.316228
$51$	0	0
$52$	−9.70820	−1.34629
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	18.9443	2.48750
$59$	−1.23607	−0.160922	−0.0804612	−	0.996758i	$-0.525639\pi$
$59$	−1.23607	−0.160922	−0.0804612	+	0.996758i	$0.525639\pi$
$60$	0	0
$61$	7.23607	0.926484	0.463242	−	0.886232i	$-0.346686\pi$
$61$	7.23607	0.926484	0.463242	+	0.886232i	$0.346686\pi$
$62$	−6.18034	−0.784904
$63$	0	0
$64$	−13.0000	−1.62500
$65$	6.47214	0.802770
$66$	0	0
$67$	14.4721	1.76805	0.884026	−	0.467437i	$-0.154823\pi$
$67$	14.4721	1.76805	0.884026	+	0.467437i	$0.154823\pi$
$68$	−9.70820	−1.17729
$69$	0	0
$70$	0	0
$71$	10.4721	1.24281	0.621407	−	0.783488i	$-0.286561\pi$
$71$	10.4721	1.24281	0.621407	+	0.783488i	$0.286561\pi$
$72$	0	0
$73$	0.763932	0.0894115	0.0447057	−	0.999000i	$-0.485765\pi$
$73$	0.763932	0.0894115	0.0447057	+	0.999000i	$0.485765\pi$
$74$	18.9443	2.20223
$75$	0	0
$76$	−19.4164	−2.22721
$77$	0	0
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	25.1246	2.77455
$83$	−11.4164	−1.25311	−0.626557	−	0.779376i	$-0.715536\pi$
$83$	−11.4164	−1.25311	−0.626557	+	0.779376i	$0.715536\pi$
$84$	0	0
$85$	6.47214	0.702002
$86$	−17.8885	−1.92897
$87$	0	0
$88$	−2.23607	−0.238366
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	−7.41641	−0.773214
$93$	0	0
$94$	−6.18034	−0.637453
$95$	12.9443	1.32805
$96$	0	0
$97$	−17.4164	−1.76837	−0.884184	−	0.467139i	$-0.845285\pi$
$97$	−17.4164	−1.76837	−0.884184	+	0.467139i	$0.845285\pi$
$98$	0	0
$99$	0	0
$100$	−3.00000	−0.300000
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	0	0
$103$	−7.70820	−0.759512	−0.379756	−	0.925087i	$-0.623992\pi$
$103$	−7.70820	−0.759512	−0.379756	+	0.925087i	$0.623992\pi$
$104$	7.23607	0.709555
$105$	0	0
$106$	−1.05573	−0.102541
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−4.47214	−0.428353	−0.214176	−	0.976795i	$-0.568707\pi$
$109$	−4.47214	−0.428353	−0.214176	+	0.976795i	$0.568707\pi$
$110$	4.47214	0.426401
$111$	0	0
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	4.94427	0.461056
$116$	−25.4164	−2.35985
$117$	0	0
$118$	2.76393	0.254441
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−16.1803	−1.46490
$123$	0	0
$124$	8.29180	0.744625
$125$	12.0000	1.07331
$126$	0	0
$127$	3.05573	0.271152	0.135576	−	0.990767i	$-0.456712\pi$
$127$	3.05573	0.271152	0.135576	+	0.990767i	$0.456712\pi$
$128$	15.6525	1.38350
$129$	0	0
$130$	−14.4721	−1.26929
$131$	21.8885	1.91241	0.956205	−	0.292696i	$-0.0945525\pi$
$131$	21.8885	1.91241	0.956205	+	0.292696i	$0.0945525\pi$
$132$	0	0
$133$	0	0
$134$	−32.3607	−2.79554
$135$	0	0
$136$	7.23607	0.620488
$137$	−16.4721	−1.40731	−0.703655	−	0.710542i	$-0.748450\pi$
$137$	−16.4721	−1.40731	−0.703655	+	0.710542i	$0.748450\pi$
$138$	0	0
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	0	0
$141$	0	0
$142$	−23.4164	−1.96506
$143$	−3.23607	−0.270614
$144$	0	0
$145$	16.9443	1.40715
$146$	−1.70820	−0.141372
$147$	0	0
$148$	−25.4164	−2.08922
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	14.4721	1.17385
$153$	0	0
$154$	0	0
$155$	−5.52786	−0.444009
$156$	0	0
$157$	−10.9443	−0.873448	−0.436724	−	0.899596i	$-0.643861\pi$
$157$	−10.9443	−0.873448	−0.436724	+	0.899596i	$0.643861\pi$
$158$	20.0000	1.59111
$159$	0	0
$160$	−13.4164	−1.06066
$161$	0	0
$162$	0	0
$163$	−3.41641	−0.267594	−0.133797	−	0.991009i	$-0.542717\pi$
$163$	−3.41641	−0.267594	−0.133797	+	0.991009i	$0.542717\pi$
$164$	−33.7082	−2.63217
$165$	0	0
$166$	25.5279	1.98135
$167$	4.94427	0.382599	0.191300	−	0.981532i	$-0.438730\pi$
$167$	4.94427	0.382599	0.191300	+	0.981532i	$0.438730\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	−14.4721	−1.10996
$171$	0	0
$172$	24.0000	1.82998
$173$	−12.7639	−0.970424	−0.485212	−	0.874397i	$-0.661258\pi$
$173$	−12.7639	−0.970424	−0.485212	+	0.874397i	$0.661258\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−4.47214	−0.335201
$179$	8.94427	0.668526	0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427	0.668526	0.334263	+	0.942480i	$0.391513\pi$
$180$	0	0
$181$	25.4164	1.88919	0.944593	−	0.328243i	$-0.106456\pi$
$181$	25.4164	1.88919	0.944593	+	0.328243i	$0.106456\pi$
$182$	0	0
$183$	0	0
$184$	5.52786	0.407520
$185$	16.9443	1.24577
$186$	0	0
$187$	−3.23607	−0.236645
$188$	8.29180	0.604741
$189$	0	0
$190$	−28.9443	−2.09984
$191$	3.05573	0.221105	0.110552	−	0.993870i	$-0.464738\pi$
$191$	3.05573	0.221105	0.110552	+	0.993870i	$0.464738\pi$
$192$	0	0
$193$	11.8885	0.855756	0.427878	−	0.903836i	$-0.359261\pi$
$193$	11.8885	0.855756	0.427878	+	0.903836i	$0.359261\pi$
$194$	38.9443	2.79604
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	2.18034	0.154560	0.0772801	−	0.997009i	$-0.475376\pi$
$199$	2.18034	0.154560	0.0772801	+	0.997009i	$0.475376\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	10.6525	0.749506
$203$	0	0
$204$	0	0
$205$	22.4721	1.56952
$206$	17.2361	1.20089
$207$	0	0
$208$	3.23607	0.224381
$209$	−6.47214	−0.447687
$210$	0	0
$211$	−13.8885	−0.956127	−0.478063	−	0.878325i	$-0.658661\pi$
$211$	−13.8885	−0.956127	−0.478063	+	0.878325i	$0.658661\pi$
$212$	1.41641	0.0972793
$213$	0	0
$214$	−8.94427	−0.611418
$215$	−16.0000	−1.09119
$216$	0	0
$217$	0	0
$218$	10.0000	0.677285
$219$	0	0
$220$	−6.00000	−0.404520
$221$	10.4721	0.704432
$222$	0	0
$223$	10.1803	0.681726	0.340863	−	0.940113i	$-0.389281\pi$
$223$	10.1803	0.681726	0.340863	+	0.940113i	$0.389281\pi$
$224$	0	0
$225$	0	0
$226$	4.47214	0.297482
$227$	5.88854	0.390836	0.195418	−	0.980720i	$-0.437394\pi$
$227$	5.88854	0.390836	0.195418	+	0.980720i	$0.437394\pi$
$228$	0	0
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	−11.0557	−0.728993
$231$	0	0
$232$	18.9443	1.24375
$233$	−9.41641	−0.616889	−0.308445	−	0.951242i	$-0.599808\pi$
$233$	−9.41641	−0.616889	−0.308445	+	0.951242i	$0.599808\pi$
$234$	0	0
$235$	−5.52786	−0.360598
$236$	−3.70820	−0.241384
$237$	0	0
$238$	0	0
$239$	9.88854	0.639637	0.319818	−	0.947479i	$-0.396378\pi$
$239$	9.88854	0.639637	0.319818	+	0.947479i	$0.396378\pi$
$240$	0	0
$241$	13.1246	0.845431	0.422715	−	0.906263i	$-0.361077\pi$
$241$	13.1246	0.845431	0.422715	+	0.906263i	$0.361077\pi$
$242$	−2.23607	−0.143740
$243$	0	0
$244$	21.7082	1.38973
$245$	0	0
$246$	0	0
$247$	20.9443	1.33265
$248$	−6.18034	−0.392452
$249$	0	0
$250$	−26.8328	−1.69706
$251$	4.29180	0.270896	0.135448	−	0.990784i	$-0.456753\pi$
$251$	4.29180	0.270896	0.135448	+	0.990784i	$0.456753\pi$
$252$	0	0
$253$	−2.47214	−0.155422
$254$	−6.83282	−0.428729
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	19.4164	1.20415
$261$	0	0
$262$	−48.9443	−3.02379
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−0.944272	−0.0580062
$266$	0	0
$267$	0	0
$268$	43.4164	2.65208
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	10.4721	0.636137	0.318068	−	0.948068i	$-0.396966\pi$
$271$	10.4721	0.636137	0.318068	+	0.948068i	$0.396966\pi$
$272$	3.23607	0.196215
$273$	0	0
$274$	36.8328	2.22515
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	−3.41641	−0.204903
$279$	0	0
$280$	0	0
$281$	3.52786	0.210455	0.105227	−	0.994448i	$-0.466443\pi$
$281$	3.52786	0.210455	0.105227	+	0.994448i	$0.466443\pi$
$282$	0	0
$283$	−29.8885	−1.77669	−0.888345	−	0.459177i	$-0.848144\pi$
$283$	−29.8885	−1.77669	−0.888345	+	0.459177i	$0.848144\pi$
$284$	31.4164	1.86422
$285$	0	0
$286$	7.23607	0.427878
$287$	0	0
$288$	0	0
$289$	−6.52786	−0.383992
$290$	−37.8885	−2.22489
$291$	0	0
$292$	2.29180	0.134117
$293$	−25.1246	−1.46780	−0.733898	−	0.679260i	$-0.762301\pi$
$293$	−25.1246	−1.46780	−0.733898	+	0.679260i	$0.762301\pi$
$294$	0	0
$295$	2.47214	0.143933
$296$	18.9443	1.10111
$297$	0	0
$298$	31.3050	1.81345
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	−20.0000	−1.15087
$303$	0	0
$304$	6.47214	0.371202
$305$	−14.4721	−0.828672
$306$	0	0
$307$	8.94427	0.510477	0.255238	−	0.966878i	$-0.417846\pi$
$307$	8.94427	0.510477	0.255238	+	0.966878i	$0.417846\pi$
$308$	0	0
$309$	0	0
$310$	12.3607	0.702039
$311$	−8.29180	−0.470185	−0.235092	−	0.971973i	$-0.575539\pi$
$311$	−8.29180	−0.470185	−0.235092	+	0.971973i	$0.575539\pi$
$312$	0	0
$313$	−14.9443	−0.844700	−0.422350	−	0.906433i	$-0.638795\pi$
$313$	−14.9443	−0.844700	−0.422350	+	0.906433i	$0.638795\pi$
$314$	24.4721	1.38104
$315$	0	0
$316$	−26.8328	−1.50946
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−8.47214	−0.474349
$320$	26.0000	1.45344
$321$	0	0
$322$	0	0
$323$	20.9443	1.16537
$324$	0	0
$325$	3.23607	0.179505
$326$	7.63932	0.423103
$327$	0	0
$328$	25.1246	1.38727
$329$	0	0
$330$	0	0
$331$	−13.8885	−0.763383	−0.381692	−	0.924290i	$-0.624658\pi$
$331$	−13.8885	−0.763383	−0.381692	+	0.924290i	$0.624658\pi$
$332$	−34.2492	−1.87967
$333$	0	0
$334$	−11.0557	−0.604943
$335$	−28.9443	−1.58139
$336$	0	0
$337$	−11.5279	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$337$	−11.5279	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$338$	5.65248	0.307454
$339$	0	0
$340$	19.4164	1.05300
$341$	2.76393	0.149675
$342$	0	0
$343$	0	0
$344$	−17.8885	−0.964486
$345$	0	0
$346$	28.5410	1.53437
$347$	−20.9443	−1.12435	−0.562174	−	0.827019i	$-0.690035\pi$
$347$	−20.9443	−1.12435	−0.562174	+	0.827019i	$0.690035\pi$
$348$	0	0
$349$	7.23607	0.387338	0.193669	−	0.981067i	$-0.437961\pi$
$349$	7.23607	0.387338	0.193669	+	0.981067i	$0.437961\pi$
$350$	0	0
$351$	0	0
$352$	6.70820	0.357548
$353$	19.8885	1.05856	0.529280	−	0.848447i	$-0.322462\pi$
$353$	19.8885	1.05856	0.529280	+	0.848447i	$0.322462\pi$
$354$	0	0
$355$	−20.9443	−1.11161
$356$	6.00000	0.317999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	24.9443	1.31651	0.658254	−	0.752796i	$-0.271295\pi$
$359$	24.9443	1.31651	0.658254	+	0.752796i	$0.271295\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	−56.8328	−2.98707
$363$	0	0
$364$	0	0
$365$	−1.52786	−0.0799721
$366$	0	0
$367$	23.1246	1.20709	0.603547	−	0.797327i	$-0.293753\pi$
$367$	23.1246	1.20709	0.603547	+	0.797327i	$0.293753\pi$
$368$	2.47214	0.128869
$369$	0	0
$370$	−37.8885	−1.96973
$371$	0	0
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	7.23607	0.374168
$375$	0	0
$376$	−6.18034	−0.318727
$377$	27.4164	1.41202
$378$	0	0
$379$	37.3050	1.91623	0.958113	−	0.286389i	$-0.0924551\pi$
$379$	37.3050	1.91623	0.958113	+	0.286389i	$0.0924551\pi$
$380$	38.8328	1.99208
$381$	0	0
$382$	−6.83282	−0.349597
$383$	−4.65248	−0.237730	−0.118865	−	0.992910i	$-0.537926\pi$
$383$	−4.65248	−0.237730	−0.118865	+	0.992910i	$0.537926\pi$
$384$	0	0
$385$	0	0
$386$	−26.5836	−1.35307
$387$	0	0
$388$	−52.2492	−2.65255
$389$	−15.8885	−0.805581	−0.402791	−	0.915292i	$-0.631960\pi$
$389$	−15.8885	−0.805581	−0.402791	+	0.915292i	$0.631960\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	0	0
$394$	−4.47214	−0.225303
$395$	17.8885	0.900070
$396$	0	0
$397$	35.8885	1.80119	0.900597	−	0.434655i	$-0.143130\pi$
$397$	35.8885	1.80119	0.900597	+	0.434655i	$0.143130\pi$
$398$	−4.87539	−0.244381
$399$	0	0
$400$	1.00000	0.0500000
$401$	−22.9443	−1.14578	−0.572891	−	0.819631i	$-0.694178\pi$
$401$	−22.9443	−1.14578	−0.572891	+	0.819631i	$0.694178\pi$
$402$	0	0
$403$	−8.94427	−0.445546
$404$	−14.2918	−0.711043
$405$	0	0
$406$	0	0
$407$	−8.47214	−0.419948
$408$	0	0
$409$	−9.12461	−0.451183	−0.225592	−	0.974222i	$-0.572431\pi$
$409$	−9.12461	−0.451183	−0.225592	+	0.974222i	$0.572431\pi$
$410$	−50.2492	−2.48163
$411$	0	0
$412$	−23.1246	−1.13927
$413$	0	0
$414$	0	0
$415$	22.8328	1.12082
$416$	−21.7082	−1.06433
$417$	0	0
$418$	14.4721	0.707855
$419$	24.6525	1.20435	0.602176	−	0.798363i	$-0.294300\pi$
$419$	24.6525	1.20435	0.602176	+	0.798363i	$0.294300\pi$
$420$	0	0
$421$	22.3607	1.08979	0.544896	−	0.838503i	$-0.316569\pi$
$421$	22.3607	1.08979	0.544896	+	0.838503i	$0.316569\pi$
$422$	31.0557	1.51177
$423$	0	0
$424$	−1.05573	−0.0512707
$425$	3.23607	0.156972
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	35.7771	1.72532
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	8.47214	0.407145	0.203572	−	0.979060i	$-0.434745\pi$
$433$	8.47214	0.407145	0.203572	+	0.979060i	$0.434745\pi$
$434$	0	0
$435$	0	0
$436$	−13.4164	−0.642529
$437$	16.0000	0.765384
$438$	0	0
$439$	−10.4721	−0.499808	−0.249904	−	0.968271i	$-0.580399\pi$
$439$	−10.4721	−0.499808	−0.249904	+	0.968271i	$0.580399\pi$
$440$	4.47214	0.213201
$441$	0	0
$442$	−23.4164	−1.11380
$443$	24.9443	1.18514	0.592569	−	0.805520i	$-0.298114\pi$
$443$	24.9443	1.18514	0.592569	+	0.805520i	$0.298114\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	−22.7639	−1.07790
$447$	0	0
$448$	0	0
$449$	28.4721	1.34368	0.671842	−	0.740695i	$-0.265504\pi$
$449$	28.4721	1.34368	0.671842	+	0.740695i	$0.265504\pi$
$450$	0	0
$451$	−11.2361	−0.529086
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−13.1672	−0.617967
$455$	0	0
$456$	0	0
$457$	−28.8328	−1.34874	−0.674371	−	0.738393i	$-0.735585\pi$
$457$	−28.8328	−1.34874	−0.674371	+	0.738393i	$0.735585\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	14.8328	0.691584
$461$	−12.1803	−0.567295	−0.283647	−	0.958929i	$-0.591545\pi$
$461$	−12.1803	−0.567295	−0.283647	+	0.958929i	$0.591545\pi$
$462$	0	0
$463$	−5.52786	−0.256902	−0.128451	−	0.991716i	$-0.541000\pi$
$463$	−5.52786	−0.256902	−0.128451	+	0.991716i	$0.541000\pi$
$464$	8.47214	0.393309
$465$	0	0
$466$	21.0557	0.975388
$467$	24.0689	1.11378	0.556888	−	0.830588i	$-0.311995\pi$
$467$	24.0689	1.11378	0.556888	+	0.830588i	$0.311995\pi$
$468$	0	0
$469$	0	0
$470$	12.3607	0.570156
$471$	0	0
$472$	2.76393	0.127220
$473$	8.00000	0.367840
$474$	0	0
$475$	6.47214	0.296962
$476$	0	0
$477$	0	0
$478$	−22.1115	−1.01135
$479$	13.5279	0.618104	0.309052	−	0.951045i	$-0.399988\pi$
$479$	13.5279	0.618104	0.309052	+	0.951045i	$0.399988\pi$
$480$	0	0
$481$	27.4164	1.25008
$482$	−29.3475	−1.33674
$483$	0	0
$484$	3.00000	0.136364
$485$	34.8328	1.58168
$486$	0	0
$487$	−36.3607	−1.64766	−0.823830	−	0.566837i	$-0.808167\pi$
$487$	−36.3607	−1.64766	−0.823830	+	0.566837i	$0.808167\pi$
$488$	−16.1803	−0.732450
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	27.4164	1.23477
$494$	−46.8328	−2.10711
$495$	0	0
$496$	−2.76393	−0.124104
$497$	0	0
$498$	0	0
$499$	1.52786	0.0683966	0.0341983	−	0.999415i	$-0.489112\pi$
$499$	1.52786	0.0683966	0.0341983	+	0.999415i	$0.489112\pi$
$500$	36.0000	1.60997
$501$	0	0
$502$	−9.59675	−0.428324
$503$	23.4164	1.04409	0.522043	−	0.852919i	$-0.325170\pi$
$503$	23.4164	1.04409	0.522043	+	0.852919i	$0.325170\pi$
$504$	0	0
$505$	9.52786	0.423984
$506$	5.52786	0.245744
$507$	0	0
$508$	9.16718	0.406728
$509$	40.4721	1.79390	0.896948	−	0.442136i	$-0.145779\pi$
$509$	40.4721	1.79390	0.896948	+	0.442136i	$0.145779\pi$
$510$	0	0
$511$	0	0
$512$	−11.1803	−0.494106
$513$	0	0
$514$	13.4164	0.591772
$515$	15.4164	0.679328
$516$	0	0
$517$	2.76393	0.121558
$518$	0	0
$519$	0	0
$520$	−14.4721	−0.634645
$521$	30.3607	1.33013	0.665063	−	0.746787i	$-0.268405\pi$
$521$	30.3607	1.33013	0.665063	+	0.746787i	$0.268405\pi$
$522$	0	0
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	65.6656	2.86862
$525$	0	0
$526$	0	0
$527$	−8.94427	−0.389619
$528$	0	0
$529$	−16.8885	−0.734285
$530$	2.11146	0.0917158
$531$	0	0
$532$	0	0
$533$	36.3607	1.57496
$534$	0	0
$535$	−8.00000	−0.345870
$536$	−32.3607	−1.39777
$537$	0	0
$538$	−30.0000	−1.29339
$539$	0	0
$540$	0	0
$541$	20.8328	0.895673	0.447836	−	0.894116i	$-0.352195\pi$
$541$	20.8328	0.895673	0.447836	+	0.894116i	$0.352195\pi$
$542$	−23.4164	−1.00582
$543$	0	0
$544$	−21.7082	−0.930732
$545$	8.94427	0.383131
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−49.4164	−2.11096
$549$	0	0
$550$	2.23607	0.0953463
$551$	54.8328	2.33596
$552$	0	0
$553$	0	0
$554$	44.4721	1.88944
$555$	0	0
$556$	4.58359	0.194388
$557$	38.9443	1.65012	0.825061	−	0.565044i	$-0.191141\pi$
$557$	38.9443	1.65012	0.825061	+	0.565044i	$0.191141\pi$
$558$	0	0
$559$	−25.8885	−1.09497
$560$	0	0
$561$	0	0
$562$	−7.88854	−0.332758
$563$	12.5836	0.530335	0.265168	−	0.964202i	$-0.414573\pi$
$563$	12.5836	0.530335	0.265168	+	0.964202i	$0.414573\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	66.8328	2.80919
$567$	0	0
$568$	−23.4164	−0.982531
$569$	−7.52786	−0.315584	−0.157792	−	0.987472i	$-0.550438\pi$
$569$	−7.52786	−0.315584	−0.157792	+	0.987472i	$0.550438\pi$
$570$	0	0
$571$	−15.0557	−0.630063	−0.315031	−	0.949081i	$-0.602015\pi$
$571$	−15.0557	−0.630063	−0.315031	+	0.949081i	$0.602015\pi$
$572$	−9.70820	−0.405920
$573$	0	0
$574$	0	0
$575$	2.47214	0.103095
$576$	0	0
$577$	19.5279	0.812956	0.406478	−	0.913661i	$-0.366757\pi$
$577$	19.5279	0.812956	0.406478	+	0.913661i	$0.366757\pi$
$578$	14.5967	0.607145
$579$	0	0
$580$	50.8328	2.11072
$581$	0	0
$582$	0	0
$583$	0.472136	0.0195539
$584$	−1.70820	−0.0706860
$585$	0	0
$586$	56.1803	2.32079
$587$	27.1246	1.11955	0.559776	−	0.828644i	$-0.310887\pi$
$587$	27.1246	1.11955	0.559776	+	0.828644i	$0.310887\pi$
$588$	0	0
$589$	−17.8885	−0.737085
$590$	−5.52786	−0.227579
$591$	0	0
$592$	8.47214	0.348203
$593$	−45.7082	−1.87701	−0.938505	−	0.345264i	$-0.887789\pi$
$593$	−45.7082	−1.87701	−0.938505	+	0.345264i	$0.887789\pi$
$594$	0	0
$595$	0	0
$596$	−42.0000	−1.72039
$597$	0	0
$598$	−17.8885	−0.731517
$599$	23.4164	0.956768	0.478384	−	0.878151i	$-0.341223\pi$
$599$	23.4164	0.956768	0.478384	+	0.878151i	$0.341223\pi$
$600$	0	0
$601$	37.1246	1.51434	0.757172	−	0.653215i	$-0.226580\pi$
$601$	37.1246	1.51434	0.757172	+	0.653215i	$0.226580\pi$
$602$	0	0
$603$	0	0
$604$	26.8328	1.09181
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−12.9443	−0.525392	−0.262696	−	0.964879i	$-0.584612\pi$
$607$	−12.9443	−0.525392	−0.262696	+	0.964879i	$0.584612\pi$
$608$	−43.4164	−1.76077
$609$	0	0
$610$	32.3607	1.31025
$611$	−8.94427	−0.361847
$612$	0	0
$613$	15.3050	0.618161	0.309081	−	0.951036i	$-0.399979\pi$
$613$	15.3050	0.618161	0.309081	+	0.951036i	$0.399979\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	−6.58359	−0.265045	−0.132523	−	0.991180i	$-0.542308\pi$
$617$	−6.58359	−0.265045	−0.132523	+	0.991180i	$0.542308\pi$
$618$	0	0
$619$	−11.1246	−0.447136	−0.223568	−	0.974688i	$-0.571770\pi$
$619$	−11.1246	−0.447136	−0.223568	+	0.974688i	$0.571770\pi$
$620$	−16.5836	−0.666013
$621$	0	0
$622$	18.5410	0.743427
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	33.4164	1.33559
$627$	0	0
$628$	−32.8328	−1.31017
$629$	27.4164	1.09316
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	20.0000	0.795557
$633$	0	0
$634$	31.3050	1.24328
$635$	−6.11146	−0.242526
$636$	0	0
$637$	0	0
$638$	18.9443	0.750011
$639$	0	0
$640$	−31.3050	−1.23744
$641$	15.5279	0.613314	0.306657	−	0.951820i	$-0.400790\pi$
$641$	15.5279	0.613314	0.306657	+	0.951820i	$0.400790\pi$
$642$	0	0
$643$	−11.1246	−0.438712	−0.219356	−	0.975645i	$-0.570396\pi$
$643$	−11.1246	−0.438712	−0.219356	+	0.975645i	$0.570396\pi$
$644$	0	0
$645$	0	0
$646$	−46.8328	−1.84261
$647$	36.0689	1.41801	0.709007	−	0.705201i	$-0.249143\pi$
$647$	36.0689	1.41801	0.709007	+	0.705201i	$0.249143\pi$
$648$	0	0
$649$	−1.23607	−0.0485199
$650$	−7.23607	−0.283822
$651$	0	0
$652$	−10.2492	−0.401391
$653$	−25.0557	−0.980506	−0.490253	−	0.871580i	$-0.663096\pi$
$653$	−25.0557	−0.980506	−0.490253	+	0.871580i	$0.663096\pi$
$654$	0	0
$655$	−43.7771	−1.71051
$656$	11.2361	0.438695
$657$	0	0
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	40.8328	1.58821	0.794106	−	0.607779i	$-0.207939\pi$
$661$	40.8328	1.58821	0.794106	+	0.607779i	$0.207939\pi$
$662$	31.0557	1.20702
$663$	0	0
$664$	25.5279	0.990673
$665$	0	0
$666$	0	0
$667$	20.9443	0.810965
$668$	14.8328	0.573899
$669$	0	0
$670$	64.7214	2.50040
$671$	7.23607	0.279345
$672$	0	0
$673$	−21.4164	−0.825542	−0.412771	−	0.910835i	$-0.635439\pi$
$673$	−21.4164	−0.825542	−0.412771	+	0.910835i	$0.635439\pi$
$674$	25.7771	0.992896
$675$	0	0
$676$	−7.58359	−0.291677
$677$	−9.70820	−0.373117	−0.186558	−	0.982444i	$-0.559733\pi$
$677$	−9.70820	−0.373117	−0.186558	+	0.982444i	$0.559733\pi$
$678$	0	0
$679$	0	0
$680$	−14.4721	−0.554981
$681$	0	0
$682$	−6.18034	−0.236657
$683$	5.88854	0.225319	0.112659	−	0.993634i	$-0.464063\pi$
$683$	5.88854	0.225319	0.112659	+	0.993634i	$0.464063\pi$
$684$	0	0
$685$	32.9443	1.25874
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−1.52786	−0.0582070
$690$	0	0
$691$	−18.5410	−0.705334	−0.352667	−	0.935749i	$-0.614725\pi$
$691$	−18.5410	−0.705334	−0.352667	+	0.935749i	$0.614725\pi$
$692$	−38.2918	−1.45564
$693$	0	0
$694$	46.8328	1.77775
$695$	−3.05573	−0.115910
$696$	0	0
$697$	36.3607	1.37726
$698$	−16.1803	−0.612435
$699$	0	0
$700$	0	0
$701$	15.5279	0.586479	0.293240	−	0.956039i	$-0.405267\pi$
$701$	15.5279	0.586479	0.293240	+	0.956039i	$0.405267\pi$
$702$	0	0
$703$	54.8328	2.06806
$704$	−13.0000	−0.489956
$705$	0	0
$706$	−44.4721	−1.67373
$707$	0	0
$708$	0	0
$709$	−14.9443	−0.561244	−0.280622	−	0.959818i	$-0.590541\pi$
$709$	−14.9443	−0.561244	−0.280622	+	0.959818i	$0.590541\pi$
$710$	46.8328	1.75760
$711$	0	0
$712$	−4.47214	−0.167600
$713$	−6.83282	−0.255891
$714$	0	0
$715$	6.47214	0.242044
$716$	26.8328	1.00279
$717$	0	0
$718$	−55.7771	−2.08158
$719$	−51.4853	−1.92008	−0.960039	−	0.279867i	$-0.909710\pi$
$719$	−51.4853	−1.92008	−0.960039	+	0.279867i	$0.909710\pi$
$720$	0	0
$721$	0	0
$722$	−51.1803	−1.90474
$723$	0	0
$724$	76.2492	2.83378
$725$	8.47214	0.314647
$726$	0	0
$727$	−25.0132	−0.927687	−0.463843	−	0.885917i	$-0.653530\pi$
$727$	−25.0132	−0.927687	−0.463843	+	0.885917i	$0.653530\pi$
$728$	0	0
$729$	0	0
$730$	3.41641	0.126447
$731$	−25.8885	−0.957522
$732$	0	0
$733$	−8.76393	−0.323703	−0.161852	−	0.986815i	$-0.551747\pi$
$733$	−8.76393	−0.323703	−0.161852	+	0.986815i	$0.551747\pi$
$734$	−51.7082	−1.90858
$735$	0	0
$736$	−16.5836	−0.611279
$737$	14.4721	0.533088
$738$	0	0
$739$	24.9443	0.917590	0.458795	−	0.888542i	$-0.348281\pi$
$739$	24.9443	0.917590	0.458795	+	0.888542i	$0.348281\pi$
$740$	50.8328	1.86865
$741$	0	0
$742$	0	0
$743$	−1.88854	−0.0692840	−0.0346420	−	0.999400i	$-0.511029\pi$
$743$	−1.88854	−0.0692840	−0.0346420	+	0.999400i	$0.511029\pi$
$744$	0	0
$745$	28.0000	1.02584
$746$	−13.4164	−0.491210
$747$	0	0
$748$	−9.70820	−0.354967
$749$	0	0
$750$	0	0
$751$	29.5279	1.07749	0.538744	−	0.842470i	$-0.318899\pi$
$751$	29.5279	1.07749	0.538744	+	0.842470i	$0.318899\pi$
$752$	−2.76393	−0.100790
$753$	0	0
$754$	−61.3050	−2.23259
$755$	−17.8885	−0.651031
$756$	0	0
$757$	15.8885	0.577479	0.288739	−	0.957408i	$-0.406764\pi$
$757$	15.8885	0.577479	0.288739	+	0.957408i	$0.406764\pi$
$758$	−83.4164	−3.02982
$759$	0	0
$760$	−28.9443	−1.04992
$761$	−31.5967	−1.14538	−0.572691	−	0.819772i	$-0.694100\pi$
$761$	−31.5967	−1.14538	−0.572691	+	0.819772i	$0.694100\pi$
$762$	0	0
$763$	0	0
$764$	9.16718	0.331657
$765$	0	0
$766$	10.4033	0.375885
$767$	4.00000	0.144432
$768$	0	0
$769$	−18.2918	−0.659619	−0.329810	−	0.944047i	$-0.606985\pi$
$769$	−18.2918	−0.659619	−0.329810	+	0.944047i	$0.606985\pi$
$770$	0	0
$771$	0	0
$772$	35.6656	1.28363
$773$	−38.3607	−1.37974	−0.689869	−	0.723934i	$-0.742332\pi$
$773$	−38.3607	−1.37974	−0.689869	+	0.723934i	$0.742332\pi$
$774$	0	0
$775$	−2.76393	−0.0992834
$776$	38.9443	1.39802
$777$	0	0
$778$	35.5279	1.27374
$779$	72.7214	2.60551
$780$	0	0
$781$	10.4721	0.374722
$782$	−17.8885	−0.639693
$783$	0	0
$784$	0	0
$785$	21.8885	0.781236
$786$	0	0
$787$	43.4164	1.54763	0.773814	−	0.633413i	$-0.218347\pi$
$787$	43.4164	1.54763	0.773814	+	0.633413i	$0.218347\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−40.0000	−1.42314
$791$	0	0
$792$	0	0
$793$	−23.4164	−0.831541
$794$	−80.2492	−2.84794
$795$	0	0
$796$	6.54102	0.231840
$797$	−14.9443	−0.529353	−0.264677	−	0.964337i	$-0.585265\pi$
$797$	−14.9443	−0.529353	−0.264677	+	0.964337i	$0.585265\pi$
$798$	0	0
$799$	−8.94427	−0.316426
$800$	−6.70820	−0.237171
$801$	0	0
$802$	51.3050	1.81164
$803$	0.763932	0.0269586
$804$	0	0
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	10.6525	0.374753
$809$	−21.0557	−0.740280	−0.370140	−	0.928976i	$-0.620690\pi$
$809$	−21.0557	−0.740280	−0.370140	+	0.928976i	$0.620690\pi$
$810$	0	0
$811$	34.8328	1.22315	0.611573	−	0.791188i	$-0.290537\pi$
$811$	34.8328	1.22315	0.611573	+	0.791188i	$0.290537\pi$
$812$	0	0
$813$	0	0
$814$	18.9443	0.663996
$815$	6.83282	0.239343
$816$	0	0
$817$	−51.7771	−1.81145
$818$	20.4033	0.713383
$819$	0	0
$820$	67.4164	2.35428
$821$	−44.8328	−1.56468	−0.782338	−	0.622854i	$-0.785973\pi$
$821$	−44.8328	−1.56468	−0.782338	+	0.622854i	$0.785973\pi$
$822$	0	0
$823$	−14.1115	−0.491894	−0.245947	−	0.969283i	$-0.579099\pi$
$823$	−14.1115	−0.491894	−0.245947	+	0.969283i	$0.579099\pi$
$824$	17.2361	0.600447
$825$	0	0
$826$	0	0
$827$	12.9443	0.450116	0.225058	−	0.974345i	$-0.427743\pi$
$827$	12.9443	0.450116	0.225058	+	0.974345i	$0.427743\pi$
$828$	0	0
$829$	−36.8328	−1.27926	−0.639628	−	0.768684i	$-0.720912\pi$
$829$	−36.8328	−1.27926	−0.639628	+	0.768684i	$0.720912\pi$
$830$	−51.0557	−1.77217
$831$	0	0
$832$	42.0689	1.45848
$833$	0	0
$834$	0	0
$835$	−9.88854	−0.342207
$836$	−19.4164	−0.671531
$837$	0	0
$838$	−55.1246	−1.90425
$839$	44.0689	1.52143	0.760713	−	0.649088i	$-0.224849\pi$
$839$	44.0689	1.52143	0.760713	+	0.649088i	$0.224849\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	−50.0000	−1.72311
$843$	0	0
$844$	−41.6656	−1.43419
$845$	5.05573	0.173922
$846$	0	0
$847$	0	0
$848$	−0.472136	−0.0162132
$849$	0	0
$850$	−7.23607	−0.248195
$851$	20.9443	0.717960
$852$	0	0
$853$	30.6525	1.04952	0.524760	−	0.851250i	$-0.324155\pi$
$853$	30.6525	1.04952	0.524760	+	0.851250i	$0.324155\pi$
$854$	0	0
$855$	0	0
$856$	−8.94427	−0.305709
$857$	15.2361	0.520454	0.260227	−	0.965547i	$-0.416203\pi$
$857$	15.2361	0.520454	0.260227	+	0.965547i	$0.416203\pi$
$858$	0	0
$859$	26.5410	0.905568	0.452784	−	0.891620i	$-0.350431\pi$
$859$	26.5410	0.905568	0.452784	+	0.891620i	$0.350431\pi$
$860$	−48.0000	−1.63679
$861$	0	0
$862$	26.8328	0.913929
$863$	3.05573	0.104018	0.0520091	−	0.998647i	$-0.483438\pi$
$863$	3.05573	0.104018	0.0520091	+	0.998647i	$0.483438\pi$
$864$	0	0
$865$	25.5279	0.867973
$866$	−18.9443	−0.643753
$867$	0	0
$868$	0	0
$869$	−8.94427	−0.303414
$870$	0	0
$871$	−46.8328	−1.58687
$872$	10.0000	0.338643
$873$	0	0
$874$	−35.7771	−1.21018
$875$	0	0
$876$	0	0
$877$	14.5836	0.492453	0.246226	−	0.969212i	$-0.420809\pi$
$877$	14.5836	0.492453	0.246226	+	0.969212i	$0.420809\pi$
$878$	23.4164	0.790265
$879$	0	0
$880$	2.00000	0.0674200
$881$	2.58359	0.0870434	0.0435217	−	0.999052i	$-0.486142\pi$
$881$	2.58359	0.0870434	0.0435217	+	0.999052i	$0.486142\pi$
$882$	0	0
$883$	−8.94427	−0.300999	−0.150499	−	0.988610i	$-0.548088\pi$
$883$	−8.94427	−0.300999	−0.150499	+	0.988610i	$0.548088\pi$
$884$	31.4164	1.05665
$885$	0	0
$886$	−55.7771	−1.87387
$887$	−4.36068	−0.146417	−0.0732086	−	0.997317i	$-0.523324\pi$
$887$	−4.36068	−0.146417	−0.0732086	+	0.997317i	$0.523324\pi$
$888$	0	0
$889$	0	0
$890$	8.94427	0.299813
$891$	0	0
$892$	30.5410	1.02259
$893$	−17.8885	−0.598617
$894$	0	0
$895$	−17.8885	−0.597948
$896$	0	0
$897$	0	0
$898$	−63.6656	−2.12455
$899$	−23.4164	−0.780981
$900$	0	0
$901$	−1.52786	−0.0509005
$902$	25.1246	0.836558
$903$	0	0
$904$	4.47214	0.148741
$905$	−50.8328	−1.68974
$906$	0	0
$907$	22.4721	0.746175	0.373088	−	0.927796i	$-0.378299\pi$
$907$	22.4721	0.746175	0.373088	+	0.927796i	$0.378299\pi$
$908$	17.6656	0.586255
$909$	0	0
$910$	0	0
$911$	−42.4721	−1.40716	−0.703582	−	0.710614i	$-0.748417\pi$
$911$	−42.4721	−1.40716	−0.703582	+	0.710614i	$0.748417\pi$
$912$	0	0
$913$	−11.4164	−0.377828
$914$	64.4721	2.13255
$915$	0	0
$916$	13.4164	0.443291
$917$	0	0
$918$	0	0
$919$	41.8885	1.38178	0.690888	−	0.722962i	$-0.257220\pi$
$919$	41.8885	1.38178	0.690888	+	0.722962i	$0.257220\pi$
$920$	−11.0557	−0.364497
$921$	0	0
$922$	27.2361	0.896972
$923$	−33.8885	−1.11546
$924$	0	0
$925$	8.47214	0.278562
$926$	12.3607	0.406197
$927$	0	0
$928$	−56.8328	−1.86563
$929$	−52.2492	−1.71424	−0.857121	−	0.515116i	$-0.827749\pi$
$929$	−52.2492	−1.71424	−0.857121	+	0.515116i	$0.827749\pi$
$930$	0	0
$931$	0	0
$932$	−28.2492	−0.925334
$933$	0	0
$934$	−53.8197	−1.76103
$935$	6.47214	0.211661
$936$	0	0
$937$	10.6525	0.348001	0.174001	−	0.984746i	$-0.444331\pi$
$937$	10.6525	0.348001	0.174001	+	0.984746i	$0.444331\pi$
$938$	0	0
$939$	0	0
$940$	−16.5836	−0.540897
$941$	7.59675	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$941$	7.59675	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$942$	0	0
$943$	27.7771	0.904546
$944$	1.23607	0.0402306
$945$	0	0
$946$	−17.8885	−0.581607
$947$	5.16718	0.167911	0.0839555	−	0.996470i	$-0.473245\pi$
$947$	5.16718	0.167911	0.0839555	+	0.996470i	$0.473245\pi$
$948$	0	0
$949$	−2.47214	−0.0802489
$950$	−14.4721	−0.469538
$951$	0	0
$952$	0	0
$953$	−22.9443	−0.743238	−0.371619	−	0.928385i	$-0.621197\pi$
$953$	−22.9443	−0.743238	−0.371619	+	0.928385i	$0.621197\pi$
$954$	0	0
$955$	−6.11146	−0.197762
$956$	29.6656	0.959455
$957$	0	0
$958$	−30.2492	−0.977308
$959$	0	0
$960$	0	0
$961$	−23.3607	−0.753570
$962$	−61.3050	−1.97655
$963$	0	0
$964$	39.3738	1.26815
$965$	−23.7771	−0.765412
$966$	0	0
$967$	−13.8885	−0.446625	−0.223313	−	0.974747i	$-0.571687\pi$
$967$	−13.8885	−0.446625	−0.223313	+	0.974747i	$0.571687\pi$
$968$	−2.23607	−0.0718699
$969$	0	0
$970$	−77.8885	−2.50085
$971$	11.1246	0.357006	0.178503	−	0.983939i	$-0.442875\pi$
$971$	11.1246	0.357006	0.178503	+	0.983939i	$0.442875\pi$
$972$	0	0
$973$	0	0
$974$	81.3050	2.60518
$975$	0	0
$976$	−7.23607	−0.231621
$977$	−22.9443	−0.734052	−0.367026	−	0.930211i	$-0.619624\pi$
$977$	−22.9443	−0.734052	−0.367026	+	0.930211i	$0.619624\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.8197	0.695939	0.347970	−	0.937506i	$-0.386871\pi$
$983$	21.8197	0.695939	0.347970	+	0.937506i	$0.386871\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	−61.3050	−1.95235
$987$	0	0
$988$	62.8328	1.99898
$989$	−19.7771	−0.628875
$990$	0	0
$991$	54.2492	1.72328	0.861642	−	0.507517i	$-0.169437\pi$
$991$	54.2492	1.72328	0.861642	+	0.507517i	$0.169437\pi$
$992$	18.5410	0.588678
$993$	0	0
$994$	0	0
$995$	−4.36068	−0.138243
$996$	0	0
$997$	−1.34752	−0.0426765	−0.0213383	−	0.999772i	$-0.506793\pi$
$997$	−1.34752	−0.0426765	−0.0213383	+	0.999772i	$0.506793\pi$
$998$	−3.41641	−0.108145
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.y.1.1		2
3.2	odd	2		539.2.a.f.1.2		2
7.6	odd	2		693.2.a.h.1.1		2
12.11	even	2		8624.2.a.ce.1.1		2
21.2	odd	6		539.2.e.j.67.1		4
21.5	even	6		539.2.e.i.67.1		4
21.11	odd	6		539.2.e.j.177.1		4
21.17	even	6		539.2.e.i.177.1		4
21.20	even	2		77.2.a.d.1.2	✓	2
33.32	even	2		5929.2.a.m.1.1		2
77.76	even	2		7623.2.a.bl.1.2		2
84.83	odd	2		1232.2.a.m.1.2		2
105.62	odd	4		1925.2.b.h.1849.3		4
105.83	odd	4		1925.2.b.h.1849.2		4
105.104	even	2		1925.2.a.r.1.1		2
168.83	odd	2		4928.2.a.bv.1.1		2
168.125	even	2		4928.2.a.bm.1.2		2
231.20	even	10		847.2.f.a.323.1		4
231.41	odd	10		847.2.f.b.372.1		4
231.62	odd	10		847.2.f.b.148.1		4
231.83	odd	10		847.2.f.m.729.1		4
231.104	even	10		847.2.f.a.729.1		4
231.125	even	10		847.2.f.n.148.1		4
231.146	even	10		847.2.f.n.372.1		4
231.167	odd	10		847.2.f.m.323.1		4
231.230	odd	2		847.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.d.1.2	✓	2	21.20	even	2
539.2.a.f.1.2		2	3.2	odd	2
539.2.e.i.67.1		4	21.5	even	6
539.2.e.i.177.1		4	21.17	even	6
539.2.e.j.67.1		4	21.2	odd	6
539.2.e.j.177.1		4	21.11	odd	6
693.2.a.h.1.1		2	7.6	odd	2
847.2.a.f.1.1		2	231.230	odd	2
847.2.f.a.323.1		4	231.20	even	10
847.2.f.a.729.1		4	231.104	even	10
847.2.f.b.148.1		4	231.62	odd	10
847.2.f.b.372.1		4	231.41	odd	10
847.2.f.m.323.1		4	231.167	odd	10
847.2.f.m.729.1		4	231.83	odd	10
847.2.f.n.148.1		4	231.125	even	10
847.2.f.n.372.1		4	231.146	even	10
1232.2.a.m.1.2		2	84.83	odd	2
1925.2.a.r.1.1		2	105.104	even	2
1925.2.b.h.1849.2		4	105.83	odd	4
1925.2.b.h.1849.3		4	105.62	odd	4
4851.2.a.y.1.1		2	1.1	even	1	trivial
4928.2.a.bm.1.2		2	168.125	even	2
4928.2.a.bv.1.1		2	168.83	odd	2
5929.2.a.m.1.1		2	33.32	even	2
7623.2.a.bl.1.2		2	77.76	even	2
8624.2.a.ce.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} -2.23607 q^{8} +O(q^{10})\)	\(q-2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} -2.23607 q^{8} +4.47214 q^{10} +1.00000 q^{11} -3.23607 q^{13} -1.00000 q^{16} -3.23607 q^{17} -6.47214 q^{19} -6.00000 q^{20} -2.23607 q^{22} -2.47214 q^{23} -1.00000 q^{25} +7.23607 q^{26} -8.47214 q^{29} +2.76393 q^{31} +6.70820 q^{32} +7.23607 q^{34} -8.47214 q^{37} +14.4721 q^{38} +4.47214 q^{40} -11.2361 q^{41} +8.00000 q^{43} +3.00000 q^{44} +5.52786 q^{46} +2.76393 q^{47} +2.23607 q^{50} -9.70820 q^{52} +0.472136 q^{53} -2.00000 q^{55} +18.9443 q^{58} -1.23607 q^{59} +7.23607 q^{61} -6.18034 q^{62} -13.0000 q^{64} +6.47214 q^{65} +14.4721 q^{67} -9.70820 q^{68} +10.4721 q^{71} +0.763932 q^{73} +18.9443 q^{74} -19.4164 q^{76} -8.94427 q^{79} +2.00000 q^{80} +25.1246 q^{82} -11.4164 q^{83} +6.47214 q^{85} -17.8885 q^{86} -2.23607 q^{88} +2.00000 q^{89} -7.41641 q^{92} -6.18034 q^{94} +12.9443 q^{95} -17.4164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - 4 q^{5}+O(q^{10}) \) 2 * q + 6 * q^4 - 4 * q^5	\( 2 q + 6 q^{4} - 4 q^{5} + 2 q^{11} - 2 q^{13} - 2 q^{16} - 2 q^{17} - 4 q^{19} - 12 q^{20} + 4 q^{23} - 2 q^{25} + 10 q^{26} - 8 q^{29} + 10 q^{31} + 10 q^{34} - 8 q^{37} + 20 q^{38} - 18 q^{41} + 16 q^{43} + 6 q^{44} + 20 q^{46} + 10 q^{47} - 6 q^{52} - 8 q^{53} - 4 q^{55} + 20 q^{58} + 2 q^{59} + 10 q^{61} + 10 q^{62} - 26 q^{64} + 4 q^{65} + 20 q^{67} - 6 q^{68} + 12 q^{71} + 6 q^{73} + 20 q^{74} - 12 q^{76} + 4 q^{80} + 10 q^{82} + 4 q^{83} + 4 q^{85} + 4 q^{89} + 12 q^{92} + 10 q^{94} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 - 4 * q^5 + 2 * q^11 - 2 * q^13 - 2 * q^16 - 2 * q^17 - 4 * q^19 - 12 * q^20 + 4 * q^23 - 2 * q^25 + 10 * q^26 - 8 * q^29 + 10 * q^31 + 10 * q^34 - 8 * q^37 + 20 * q^38 - 18 * q^41 + 16 * q^43 + 6 * q^44 + 20 * q^46 + 10 * q^47 - 6 * q^52 - 8 * q^53 - 4 * q^55 + 20 * q^58 + 2 * q^59 + 10 * q^61 + 10 * q^62 - 26 * q^64 + 4 * q^65 + 20 * q^67 - 6 * q^68 + 12 * q^71 + 6 * q^73 + 20 * q^74 - 12 * q^76 + 4 * q^80 + 10 * q^82 + 4 * q^83 + 4 * q^85 + 4 * q^89 + 12 * q^92 + 10 * q^94 + 8 * q^95 - 8 * q^97

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