LMFDB - Embedded newform 4851.2.a.cc.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:	$1$
Dimension:	$6$
Coefficient field:	6.6.672323328.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 11x^{4} - 2x^{3} + 33x^{2} + 10x - 21 $ x^6 - 11x^4 - 2x^3 + 33x^2 + 10x - 21
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 693)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.42010$ 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)

$f(q)$	$=$	$q-2.42010 q^{2} +3.85689 q^{4} -2.85689 q^{5} -4.49387 q^{8} +O(q^{10})$	$q-2.42010 q^{2} +3.85689 q^{4} -2.85689 q^{5} -4.49387 q^{8} +6.91397 q^{10} -1.00000 q^{11} -1.62045 q^{13} +3.16184 q^{16} -8.11432 q^{17} -1.20561 q^{19} -11.0187 q^{20} +2.42010 q^{22} +6.24617 q^{23} +3.16184 q^{25} +3.92166 q^{26} +2.28468 q^{29} +6.14515 q^{31} +1.33577 q^{32} +19.6375 q^{34} +8.63544 q^{37} +2.91771 q^{38} +12.8385 q^{40} -8.09333 q^{41} +6.81225 q^{43} -3.85689 q^{44} -15.1164 q^{46} +7.76030 q^{47} -7.65197 q^{50} -6.24991 q^{52} -1.56864 q^{53} +2.85689 q^{55} -5.52916 q^{58} +0.246168 q^{59} +3.21975 q^{61} -14.8719 q^{62} -9.55638 q^{64} +4.62946 q^{65} -13.9063 q^{67} -31.2961 q^{68} +10.0713 q^{71} +10.4802 q^{73} -20.8987 q^{74} -4.64993 q^{76} -13.2392 q^{79} -9.03303 q^{80} +19.5867 q^{82} -3.59930 q^{83} +23.1817 q^{85} -16.4863 q^{86} +4.49387 q^{88} -4.38640 q^{89} +24.0908 q^{92} -18.7807 q^{94} +3.44431 q^{95} -1.69506 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 4 q^{5} + 6 q^{8}+O(q^{10}) $ 6 * q + 10 * q^4 - 4 * q^5 + 6 * q^8	$ 6 q + 10 q^{4} - 4 q^{5} + 6 q^{8} - 6 q^{10} - 6 q^{11} - 2 q^{13} + 2 q^{16} - 8 q^{17} + 6 q^{19} - 36 q^{20} + 2 q^{23} + 2 q^{25} - 8 q^{26} + 2 q^{29} + 4 q^{31} + 12 q^{32} + 14 q^{34} - 6 q^{37} - 10 q^{38} - 18 q^{40} - 30 q^{41} - 12 q^{43} - 10 q^{44} - 4 q^{46} - 14 q^{47} + 24 q^{50} - 22 q^{52} - 16 q^{53} + 4 q^{55} + 2 q^{58} - 34 q^{59} - 2 q^{61} - 26 q^{62} + 2 q^{64} + 20 q^{65} - 20 q^{67} - 18 q^{68} + 6 q^{71} + 26 q^{73} - 14 q^{74} - 4 q^{76} + 2 q^{79} - 44 q^{80} - 4 q^{82} - 8 q^{83} + 10 q^{85} - 78 q^{86} - 6 q^{88} - 20 q^{89} + 30 q^{92} - 10 q^{94} + 10 q^{95} - 14 q^{97}+O(q^{100}) $ 6 * q + 10 * q^4 - 4 * q^5 + 6 * q^8 - 6 * q^10 - 6 * q^11 - 2 * q^13 + 2 * q^16 - 8 * q^17 + 6 * q^19 - 36 * q^20 + 2 * q^23 + 2 * q^25 - 8 * q^26 + 2 * q^29 + 4 * q^31 + 12 * q^32 + 14 * q^34 - 6 * q^37 - 10 * q^38 - 18 * q^40 - 30 * q^41 - 12 * q^43 - 10 * q^44 - 4 * q^46 - 14 * q^47 + 24 * q^50 - 22 * q^52 - 16 * q^53 + 4 * q^55 + 2 * q^58 - 34 * q^59 - 2 * q^61 - 26 * q^62 + 2 * q^64 + 20 * q^65 - 20 * q^67 - 18 * q^68 + 6 * q^71 + 26 * q^73 - 14 * q^74 - 4 * q^76 + 2 * q^79 - 44 * q^80 - 4 * q^82 - 8 * q^83 + 10 * q^85 - 78 * q^86 - 6 * q^88 - 20 * q^89 + 30 * q^92 - 10 * q^94 + 10 * q^95 - 14 * 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 $

$2$	−2.42010	−1.71127	−0.855635	−	0.517579i	$-0.826833\pi$
$2$	−2.42010	−1.71127	−0.855635	+	0.517579i	$0.826833\pi$
$3$	0	0
$3$	0	0
$4$	3.85689	1.92845
$5$	−2.85689	−1.27764	−0.638821	−	0.769356i	$-0.720577\pi$
$5$	−2.85689	−1.27764	−0.638821	+	0.769356i	$0.720577\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−4.49387	−1.58882
$9$	0	0
$10$	6.91397	2.18639
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.62045	−0.449432	−0.224716	−	0.974424i	$-0.572145\pi$
$13$	−1.62045	−0.449432	−0.224716	+	0.974424i	$0.572145\pi$
$14$	0	0
$15$	0	0
$16$	3.16184	0.790459
$17$	−8.11432	−1.96801	−0.984006	−	0.178135i	$-0.942994\pi$
$17$	−8.11432	−1.96801	−0.984006	+	0.178135i	$0.942994\pi$
$18$	0	0
$19$	−1.20561	−0.276587	−0.138294	−	0.990391i	$-0.544162\pi$
$19$	−1.20561	−0.276587	−0.138294	+	0.990391i	$0.544162\pi$
$20$	−11.0187	−2.46386
$21$	0	0
$22$	2.42010	0.515967
$23$	6.24617	1.30242	0.651208	−	0.758899i	$-0.274263\pi$
$23$	6.24617	1.30242	0.651208	+	0.758899i	$0.274263\pi$
$24$	0	0
$25$	3.16184	0.632368
$26$	3.92166	0.769100
$27$	0	0
$28$	0	0
$29$	2.28468	0.424254	0.212127	−	0.977242i	$-0.431961\pi$
$29$	2.28468	0.424254	0.212127	+	0.977242i	$0.431961\pi$
$30$	0	0
$31$	6.14515	1.10370	0.551851	−	0.833943i	$-0.313922\pi$
$31$	6.14515	1.10370	0.551851	+	0.833943i	$0.313922\pi$
$32$	1.33577	0.236133
$33$	0	0
$34$	19.6375	3.36780
$35$	0	0
$36$	0	0
$37$	8.63544	1.41966	0.709829	−	0.704374i	$-0.248772\pi$
$37$	8.63544	1.41966	0.709829	+	0.704374i	$0.248772\pi$
$38$	2.91771	0.473315
$39$	0	0
$40$	12.8385	2.02995
$41$	−8.09333	−1.26397	−0.631983	−	0.774982i	$-0.717759\pi$
$41$	−8.09333	−1.26397	−0.631983	+	0.774982i	$0.717759\pi$
$42$	0	0
$43$	6.81225	1.03886	0.519429	−	0.854513i	$-0.326144\pi$
$43$	6.81225	1.03886	0.519429	+	0.854513i	$0.326144\pi$
$44$	−3.85689	−0.581448
$45$	0	0
$46$	−15.1164	−2.22879
$47$	7.76030	1.13196	0.565978	−	0.824420i	$-0.308499\pi$
$47$	7.76030	1.13196	0.565978	+	0.824420i	$0.308499\pi$
$48$	0	0
$49$	0	0
$50$	−7.65197	−1.08215
$51$	0	0
$52$	−6.24991	−0.866706
$53$	−1.56864	−0.215469	−0.107734	−	0.994180i	$-0.534360\pi$
$53$	−1.56864	−0.215469	−0.107734	+	0.994180i	$0.534360\pi$
$54$	0	0
$55$	2.85689	0.385223
$56$	0	0
$57$	0	0
$58$	−5.52916	−0.726014
$59$	0.246168	0.0320483	0.0160242	−	0.999872i	$-0.494899\pi$
$59$	0.246168	0.0320483	0.0160242	+	0.999872i	$0.494899\pi$
$60$	0	0
$61$	3.21975	0.412247	0.206124	−	0.978526i	$-0.433915\pi$
$61$	3.21975	0.412247	0.206124	+	0.978526i	$0.433915\pi$
$62$	−14.8719	−1.88873
$63$	0	0
$64$	−9.55638	−1.19455
$65$	4.62946	0.574213
$66$	0	0
$67$	−13.9063	−1.69892	−0.849462	−	0.527650i	$-0.823073\pi$
$67$	−13.9063	−1.69892	−0.849462	+	0.527650i	$0.823073\pi$
$68$	−31.2961	−3.79521
$69$	0	0
$70$	0	0
$71$	10.0713	1.19525	0.597625	−	0.801776i	$-0.296111\pi$
$71$	10.0713	1.19525	0.597625	+	0.801776i	$0.296111\pi$
$72$	0	0
$73$	10.4802	1.22662	0.613309	−	0.789843i	$-0.289838\pi$
$73$	10.4802	1.22662	0.613309	+	0.789843i	$0.289838\pi$
$74$	−20.8987	−2.42942
$75$	0	0
$76$	−4.64993	−0.533383
$77$	0	0
$78$	0	0
$79$	−13.2392	−1.48953	−0.744763	−	0.667329i	$-0.767438\pi$
$79$	−13.2392	−1.48953	−0.744763	+	0.667329i	$0.767438\pi$
$80$	−9.03303	−1.00992
$81$	0	0
$82$	19.5867	2.16299
$83$	−3.59930	−0.395075	−0.197537	−	0.980295i	$-0.563294\pi$
$83$	−3.59930	−0.395075	−0.197537	+	0.980295i	$0.563294\pi$
$84$	0	0
$85$	23.1817	2.51441
$86$	−16.4863	−1.77777
$87$	0	0
$88$	4.49387	0.479048
$89$	−4.38640	−0.464957	−0.232479	−	0.972602i	$-0.574684\pi$
$89$	−4.38640	−0.464957	−0.232479	+	0.972602i	$0.574684\pi$
$90$	0	0
$91$	0	0
$92$	24.0908	2.51164
$93$	0	0
$94$	−18.7807	−1.93708
$95$	3.44431	0.353379
$96$	0	0
$97$	−1.69506	−0.172107	−0.0860534	−	0.996291i	$-0.527426\pi$
$97$	−1.69506	−0.172107	−0.0860534	+	0.996291i	$0.527426\pi$
$98$	0	0
$99$	0	0
$100$	12.1949	1.21949
$101$	−0.245661	−0.0244442	−0.0122221	−	0.999925i	$-0.503891\pi$
$101$	−0.245661	−0.0244442	−0.0122221	+	0.999925i	$0.503891\pi$
$102$	0	0
$103$	−8.08481	−0.796620	−0.398310	−	0.917251i	$-0.630403\pi$
$103$	−8.08481	−0.796620	−0.398310	+	0.917251i	$0.630403\pi$
$104$	7.28210	0.714068
$105$	0	0
$106$	3.79626	0.368726
$107$	−4.76068	−0.460232	−0.230116	−	0.973163i	$-0.573911\pi$
$107$	−4.76068	−0.460232	−0.230116	+	0.973163i	$0.573911\pi$
$108$	0	0
$109$	12.8508	1.23088	0.615440	−	0.788184i	$-0.288978\pi$
$109$	12.8508	1.23088	0.615440	+	0.788184i	$0.288978\pi$
$110$	−6.91397	−0.659221
$111$	0	0
$112$	0	0
$113$	8.94622	0.841589	0.420795	−	0.907156i	$-0.361751\pi$
$113$	8.94622	0.841589	0.420795	+	0.907156i	$0.361751\pi$
$114$	0	0
$115$	−17.8446	−1.66402
$116$	8.81177	0.818152
$117$	0	0
$118$	−0.595751	−0.0548434
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.79213	−0.705466
$123$	0	0
$124$	23.7012	2.12843
$125$	5.25143	0.469702
$126$	0	0
$127$	15.3288	1.36021	0.680106	−	0.733114i	$-0.261934\pi$
$127$	15.3288	1.36021	0.680106	+	0.733114i	$0.261934\pi$
$128$	20.4559	1.80806
$129$	0	0
$130$	−11.2038	−0.982634
$131$	3.54499	0.309727	0.154863	−	0.987936i	$-0.450506\pi$
$131$	3.54499	0.309727	0.154863	+	0.987936i	$0.450506\pi$
$132$	0	0
$133$	0	0
$134$	33.6546	2.90732
$135$	0	0
$136$	36.4647	3.12682
$137$	4.09541	0.349894	0.174947	−	0.984578i	$-0.444025\pi$
$137$	4.09541	0.349894	0.174947	+	0.984578i	$0.444025\pi$
$138$	0	0
$139$	−21.0198	−1.78287	−0.891437	−	0.453145i	$-0.850302\pi$
$139$	−21.0198	−1.78287	−0.891437	+	0.453145i	$0.850302\pi$
$140$	0	0
$141$	0	0
$142$	−24.3737	−2.04539
$143$	1.62045	0.135509
$144$	0	0
$145$	−6.52709	−0.542045
$146$	−25.3632	−2.09907
$147$	0	0
$148$	33.3060	2.73773
$149$	19.5070	1.59807	0.799036	−	0.601283i	$-0.205344\pi$
$149$	19.5070	1.59807	0.799036	+	0.601283i	$0.205344\pi$
$150$	0	0
$151$	−8.96883	−0.729873	−0.364936	−	0.931032i	$-0.618909\pi$
$151$	−8.96883	−0.729873	−0.364936	+	0.931032i	$0.618909\pi$
$152$	5.41788	0.439448
$153$	0	0
$154$	0	0
$155$	−17.5560	−1.41013
$156$	0	0
$157$	13.0586	1.04219	0.521094	−	0.853499i	$-0.325524\pi$
$157$	13.0586	1.04219	0.521094	+	0.853499i	$0.325524\pi$
$158$	32.0402	2.54898
$159$	0	0
$160$	−3.81616	−0.301694
$161$	0	0
$162$	0	0
$163$	−14.3175	−1.12143	−0.560717	−	0.828007i	$-0.689475\pi$
$163$	−14.3175	−1.12143	−0.560717	+	0.828007i	$0.689475\pi$
$164$	−31.2151	−2.43749
$165$	0	0
$166$	8.71068	0.676080
$167$	−2.14432	−0.165932	−0.0829661	−	0.996552i	$-0.526439\pi$
$167$	−2.14432	−0.165932	−0.0829661	+	0.996552i	$0.526439\pi$
$168$	0	0
$169$	−10.3741	−0.798011
$170$	−56.1022	−4.30284
$171$	0	0
$172$	26.2741	2.00338
$173$	20.0884	1.52729	0.763646	−	0.645636i	$-0.223407\pi$
$173$	20.0884	1.52729	0.763646	+	0.645636i	$0.223407\pi$
$174$	0	0
$175$	0	0
$176$	−3.16184	−0.238332
$177$	0	0
$178$	10.6155	0.795668
$179$	−13.0070	−0.972186	−0.486093	−	0.873907i	$-0.661578\pi$
$179$	−13.0070	−0.972186	−0.486093	+	0.873907i	$0.661578\pi$
$180$	0	0
$181$	−8.64048	−0.642241	−0.321121	−	0.947038i	$-0.604060\pi$
$181$	−8.64048	−0.642241	−0.321121	+	0.947038i	$0.604060\pi$
$182$	0	0
$183$	0	0
$184$	−28.0695	−2.06931
$185$	−24.6705	−1.81381
$186$	0	0
$187$	8.11432	0.593378
$188$	29.9307	2.18292
$189$	0	0
$190$	−8.33559	−0.604727
$191$	−19.8279	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$191$	−19.8279	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$192$	0	0
$193$	−7.27395	−0.523591	−0.261795	−	0.965123i	$-0.584315\pi$
$193$	−7.27395	−0.523591	−0.261795	+	0.965123i	$0.584315\pi$
$194$	4.10221	0.294521
$195$	0	0
$196$	0	0
$197$	−12.6879	−0.903974	−0.451987	−	0.892025i	$-0.649285\pi$
$197$	−12.6879	−0.903974	−0.451987	+	0.892025i	$0.649285\pi$
$198$	0	0
$199$	23.2098	1.64530	0.822651	−	0.568547i	$-0.192494\pi$
$199$	23.2098	1.64530	0.822651	+	0.568547i	$0.192494\pi$
$200$	−14.2089	−1.00472
$201$	0	0
$202$	0.594525	0.0418306
$203$	0	0
$204$	0	0
$205$	23.1218	1.61490
$206$	19.5661	1.36323
$207$	0	0
$208$	−5.12360	−0.355258
$209$	1.20561	0.0833941
$210$	0	0
$211$	−11.7680	−0.810142	−0.405071	−	0.914285i	$-0.632753\pi$
$211$	−11.7680	−0.810142	−0.405071	+	0.914285i	$0.632753\pi$
$212$	−6.05007	−0.415520
$213$	0	0
$214$	11.5213	0.787582
$215$	−19.4619	−1.32729
$216$	0	0
$217$	0	0
$218$	−31.1001	−2.10637
$219$	0	0
$220$	11.0187	0.742883
$221$	13.1489	0.884488
$222$	0	0
$223$	9.69505	0.649229	0.324614	−	0.945846i	$-0.394766\pi$
$223$	9.69505	0.649229	0.324614	+	0.945846i	$0.394766\pi$
$224$	0	0
$225$	0	0
$226$	−21.6508	−1.44019
$227$	−14.5569	−0.966178	−0.483089	−	0.875571i	$-0.660485\pi$
$227$	−14.5569	−0.966178	−0.483089	+	0.875571i	$0.660485\pi$
$228$	0	0
$229$	−14.5622	−0.962298	−0.481149	−	0.876639i	$-0.659780\pi$
$229$	−14.5622	−0.962298	−0.481149	+	0.876639i	$0.659780\pi$
$230$	43.1858	2.84759
$231$	0	0
$232$	−10.2671	−0.674065
$233$	7.51177	0.492112	0.246056	−	0.969256i	$-0.420865\pi$
$233$	7.51177	0.492112	0.246056	+	0.969256i	$0.420865\pi$
$234$	0	0
$235$	−22.1704	−1.44623
$236$	0.949443	0.0618035
$237$	0	0
$238$	0	0
$239$	5.23765	0.338795	0.169398	−	0.985548i	$-0.445818\pi$
$239$	5.23765	0.338795	0.169398	+	0.985548i	$0.445818\pi$
$240$	0	0
$241$	−24.3504	−1.56855	−0.784275	−	0.620414i	$-0.786965\pi$
$241$	−24.3504	−1.56855	−0.784275	+	0.620414i	$0.786965\pi$
$242$	−2.42010	−0.155570
$243$	0	0
$244$	12.4182	0.794996
$245$	0	0
$246$	0	0
$247$	1.95364	0.124307
$248$	−27.6155	−1.75359
$249$	0	0
$250$	−12.7090	−0.803788
$251$	13.3323	0.841528	0.420764	−	0.907170i	$-0.361762\pi$
$251$	13.3323	0.841528	0.420764	+	0.907170i	$0.361762\pi$
$252$	0	0
$253$	−6.24617	−0.392693
$254$	−37.0973	−2.32769
$255$	0	0
$256$	−30.3925	−1.89953
$257$	3.26713	0.203798	0.101899	−	0.994795i	$-0.467508\pi$
$257$	3.26713	0.203798	0.101899	+	0.994795i	$0.467508\pi$
$258$	0	0
$259$	0	0
$260$	17.8553	1.10734
$261$	0	0
$262$	−8.57923	−0.530026
$263$	19.0284	1.17334	0.586672	−	0.809825i	$-0.300438\pi$
$263$	19.0284	1.17334	0.586672	+	0.809825i	$0.300438\pi$
$264$	0	0
$265$	4.48143	0.275292
$266$	0	0
$267$	0	0
$268$	−53.6351	−3.27628
$269$	12.9823	0.791544	0.395772	−	0.918349i	$-0.370477\pi$
$269$	12.9823	0.791544	0.395772	+	0.918349i	$0.370477\pi$
$270$	0	0
$271$	4.73628	0.287708	0.143854	−	0.989599i	$-0.454050\pi$
$271$	4.73628	0.287708	0.143854	+	0.989599i	$0.454050\pi$
$272$	−25.6562	−1.55563
$273$	0	0
$274$	−9.91130	−0.598764
$275$	−3.16184	−0.190666
$276$	0	0
$277$	−11.5417	−0.693475	−0.346738	−	0.937962i	$-0.612711\pi$
$277$	−11.5417	−0.693475	−0.346738	+	0.937962i	$0.612711\pi$
$278$	50.8700	3.05098
$279$	0	0
$280$	0	0
$281$	−25.8579	−1.54255	−0.771277	−	0.636500i	$-0.780381\pi$
$281$	−25.8579	−1.54255	−0.771277	+	0.636500i	$0.780381\pi$
$282$	0	0
$283$	10.5871	0.629336	0.314668	−	0.949202i	$-0.398107\pi$
$283$	10.5871	0.629336	0.314668	+	0.949202i	$0.398107\pi$
$284$	38.8441	2.30497
$285$	0	0
$286$	−3.92166	−0.231892
$287$	0	0
$288$	0	0
$289$	48.8422	2.87307
$290$	15.7962	0.927585
$291$	0	0
$292$	40.4211	2.36547
$293$	−5.61873	−0.328250	−0.164125	−	0.986440i	$-0.552480\pi$
$293$	−5.61873	−0.328250	−0.164125	+	0.986440i	$0.552480\pi$
$294$	0	0
$295$	−0.703275	−0.0409463
$296$	−38.8066	−2.25559
$297$	0	0
$298$	−47.2088	−2.73473
$299$	−10.1216	−0.585348
$300$	0	0
$301$	0	0
$302$	21.7055	1.24901
$303$	0	0
$304$	−3.81196	−0.218631
$305$	−9.19849	−0.526704
$306$	0	0
$307$	−16.5271	−0.943251	−0.471625	−	0.881799i	$-0.656332\pi$
$307$	−16.5271	−0.943251	−0.471625	+	0.881799i	$0.656332\pi$
$308$	0	0
$309$	0	0
$310$	42.4874	2.41312
$311$	−17.5473	−0.995018	−0.497509	−	0.867459i	$-0.665752\pi$
$311$	−17.5473	−0.995018	−0.497509	+	0.867459i	$0.665752\pi$
$312$	0	0
$313$	12.4562	0.704065	0.352033	−	0.935988i	$-0.385491\pi$
$313$	12.4562	0.704065	0.352033	+	0.935988i	$0.385491\pi$
$314$	−31.6031	−1.78347
$315$	0	0
$316$	−51.0622	−2.87247
$317$	−7.18261	−0.403416	−0.201708	−	0.979446i	$-0.564649\pi$
$317$	−7.18261	−0.403416	−0.201708	+	0.979446i	$0.564649\pi$
$318$	0	0
$319$	−2.28468	−0.127918
$320$	27.3015	1.52620
$321$	0	0
$322$	0	0
$323$	9.78275	0.544327
$324$	0	0
$325$	−5.12360	−0.284206
$326$	34.6498	1.91908
$327$	0	0
$328$	36.3704	2.00822
$329$	0	0
$330$	0	0
$331$	30.6231	1.68320	0.841599	−	0.540103i	$-0.181615\pi$
$331$	30.6231	1.68320	0.841599	+	0.540103i	$0.181615\pi$
$332$	−13.8821	−0.761880
$333$	0	0
$334$	5.18947	0.283955
$335$	39.7288	2.17061
$336$	0	0
$337$	8.16699	0.444884	0.222442	−	0.974946i	$-0.428597\pi$
$337$	8.16699	0.444884	0.222442	+	0.974946i	$0.428597\pi$
$338$	25.1065	1.36561
$339$	0	0
$340$	89.4095	4.84891
$341$	−6.14515	−0.332778
$342$	0	0
$343$	0	0
$344$	−30.6134	−1.65056
$345$	0	0
$346$	−48.6159	−2.61361
$347$	20.2393	1.08650	0.543252	−	0.839570i	$-0.317193\pi$
$347$	20.2393	1.08650	0.543252	+	0.839570i	$0.317193\pi$
$348$	0	0
$349$	−29.7317	−1.59150	−0.795749	−	0.605626i	$-0.792923\pi$
$349$	−29.7317	−1.59150	−0.795749	+	0.605626i	$0.792923\pi$
$350$	0	0
$351$	0	0
$352$	−1.33577	−0.0711969
$353$	21.7877	1.15964	0.579820	−	0.814744i	$-0.303123\pi$
$353$	21.7877	1.15964	0.579820	+	0.814744i	$0.303123\pi$
$354$	0	0
$355$	−28.7728	−1.52710
$356$	−16.9179	−0.896645
$357$	0	0
$358$	31.4782	1.66367
$359$	−4.27399	−0.225572	−0.112786	−	0.993619i	$-0.535978\pi$
$359$	−4.27399	−0.225572	−0.112786	+	0.993619i	$0.535978\pi$
$360$	0	0
$361$	−17.5465	−0.923500
$362$	20.9108	1.09905
$363$	0	0
$364$	0	0
$365$	−29.9409	−1.56718
$366$	0	0
$367$	−3.00541	−0.156881	−0.0784405	−	0.996919i	$-0.524994\pi$
$367$	−3.00541	−0.156881	−0.0784405	+	0.996919i	$0.524994\pi$
$368$	19.7494	1.02951
$369$	0	0
$370$	59.7052	3.10393
$371$	0	0
$372$	0	0
$373$	0.254419	0.0131733	0.00658665	−	0.999978i	$-0.497903\pi$
$373$	0.254419	0.0131733	0.00658665	+	0.999978i	$0.497903\pi$
$374$	−19.6375	−1.01543
$375$	0	0
$376$	−34.8738	−1.79848
$377$	−3.70221	−0.190674
$378$	0	0
$379$	−2.87695	−0.147779	−0.0738895	−	0.997266i	$-0.523541\pi$
$379$	−2.87695	−0.147779	−0.0738895	+	0.997266i	$0.523541\pi$
$380$	13.2843	0.681473
$381$	0	0
$382$	47.9856	2.45516
$383$	−35.0077	−1.78881	−0.894404	−	0.447259i	$-0.852400\pi$
$383$	−35.0077	−1.78881	−0.894404	+	0.447259i	$0.852400\pi$
$384$	0	0
$385$	0	0
$386$	17.6037	0.896005
$387$	0	0
$388$	−6.53765	−0.331899
$389$	−37.9476	−1.92402	−0.962009	−	0.273017i	$-0.911979\pi$
$389$	−37.9476	−1.92402	−0.962009	+	0.273017i	$0.911979\pi$
$390$	0	0
$391$	−50.6834	−2.56317
$392$	0	0
$393$	0	0
$394$	30.7060	1.54694
$395$	37.8230	1.90308
$396$	0	0
$397$	1.26608	0.0635428	0.0317714	−	0.999495i	$-0.489885\pi$
$397$	1.26608	0.0635428	0.0317714	+	0.999495i	$0.489885\pi$
$398$	−56.1702	−2.81556
$399$	0	0
$400$	9.99722	0.499861
$401$	23.8832	1.19267	0.596334	−	0.802736i	$-0.296623\pi$
$401$	23.8832	1.19267	0.596334	+	0.802736i	$0.296623\pi$
$402$	0	0
$403$	−9.95791	−0.496039
$404$	−0.947488	−0.0471393
$405$	0	0
$406$	0	0
$407$	−8.63544	−0.428043
$408$	0	0
$409$	−19.3958	−0.959059	−0.479530	−	0.877526i	$-0.659193\pi$
$409$	−19.3958	−0.959059	−0.479530	+	0.877526i	$0.659193\pi$
$410$	−55.9571	−2.76352
$411$	0	0
$412$	−31.1823	−1.53624
$413$	0	0
$414$	0	0
$415$	10.2828	0.504764
$416$	−2.16455	−0.106126
$417$	0	0
$418$	−2.91771	−0.142710
$419$	7.86765	0.384360	0.192180	−	0.981360i	$-0.438444\pi$
$419$	7.86765	0.384360	0.192180	+	0.981360i	$0.438444\pi$
$420$	0	0
$421$	23.7578	1.15788	0.578942	−	0.815369i	$-0.303466\pi$
$421$	23.7578	1.15788	0.578942	+	0.815369i	$0.303466\pi$
$422$	28.4797	1.38637
$423$	0	0
$424$	7.04925	0.342342
$425$	−25.6562	−1.24451
$426$	0	0
$427$	0	0
$428$	−18.3614	−0.887534
$429$	0	0
$430$	47.0997	2.27135
$431$	−1.92721	−0.0928304	−0.0464152	−	0.998922i	$-0.514780\pi$
$431$	−1.92721	−0.0928304	−0.0464152	+	0.998922i	$0.514780\pi$
$432$	0	0
$433$	8.35461	0.401497	0.200748	−	0.979643i	$-0.435663\pi$
$433$	8.35461	0.401497	0.200748	+	0.979643i	$0.435663\pi$
$434$	0	0
$435$	0	0
$436$	49.5640	2.37369
$437$	−7.53047	−0.360231
$438$	0	0
$439$	−14.3857	−0.686590	−0.343295	−	0.939228i	$-0.611543\pi$
$439$	−14.3857	−0.686590	−0.343295	+	0.939228i	$0.611543\pi$
$440$	−12.8385	−0.612052
$441$	0	0
$442$	−31.8216	−1.51360
$443$	−8.58037	−0.407666	−0.203833	−	0.979006i	$-0.565340\pi$
$443$	−8.58037	−0.407666	−0.203833	+	0.979006i	$0.565340\pi$
$444$	0	0
$445$	12.5315	0.594049
$446$	−23.4630	−1.11101
$447$	0	0
$448$	0	0
$449$	18.2061	0.859200	0.429600	−	0.903019i	$-0.358655\pi$
$449$	18.2061	0.859200	0.429600	+	0.903019i	$0.358655\pi$
$450$	0	0
$451$	8.09333	0.381100
$452$	34.5046	1.62296
$453$	0	0
$454$	35.2293	1.65339
$455$	0	0
$456$	0	0
$457$	−40.4573	−1.89251	−0.946255	−	0.323420i	$-0.895167\pi$
$457$	−40.4573	−1.89251	−0.946255	+	0.323420i	$0.895167\pi$
$458$	35.2420	1.64675
$459$	0	0
$460$	−68.8248	−3.20897
$461$	−40.7664	−1.89868	−0.949341	−	0.314247i	$-0.898248\pi$
$461$	−40.7664	−1.89868	−0.949341	+	0.314247i	$0.898248\pi$
$462$	0	0
$463$	−24.5461	−1.14076	−0.570378	−	0.821382i	$-0.693203\pi$
$463$	−24.5461	−1.14076	−0.570378	+	0.821382i	$0.693203\pi$
$464$	7.22379	0.335356
$465$	0	0
$466$	−18.1792	−0.842137
$467$	13.4785	0.623712	0.311856	−	0.950129i	$-0.399049\pi$
$467$	13.4785	0.623712	0.311856	+	0.950129i	$0.399049\pi$
$468$	0	0
$469$	0	0
$470$	53.6545	2.47490
$471$	0	0
$472$	−1.10625	−0.0509191
$473$	−6.81225	−0.313228
$474$	0	0
$475$	−3.81196	−0.174905
$476$	0	0
$477$	0	0
$478$	−12.6756	−0.579770
$479$	−20.2749	−0.926385	−0.463193	−	0.886258i	$-0.653296\pi$
$479$	−20.2749	−0.926385	−0.463193	+	0.886258i	$0.653296\pi$
$480$	0	0
$481$	−13.9933	−0.638040
$482$	58.9305	2.68421
$483$	0	0
$484$	3.85689	0.175313
$485$	4.84259	0.219891
$486$	0	0
$487$	−0.995835	−0.0451256	−0.0225628	−	0.999745i	$-0.507183\pi$
$487$	−0.995835	−0.0451256	−0.0225628	+	0.999745i	$0.507183\pi$
$488$	−14.4691	−0.654988
$489$	0	0
$490$	0	0
$491$	−22.1296	−0.998693	−0.499346	−	0.866402i	$-0.666426\pi$
$491$	−22.1296	−0.998693	−0.499346	+	0.866402i	$0.666426\pi$
$492$	0	0
$493$	−18.5386	−0.834938
$494$	−4.72801	−0.212723
$495$	0	0
$496$	19.4300	0.872431
$497$	0	0
$498$	0	0
$499$	−8.70354	−0.389624	−0.194812	−	0.980841i	$-0.562410\pi$
$499$	−8.70354	−0.389624	−0.194812	+	0.980841i	$0.562410\pi$
$500$	20.2542	0.905796
$501$	0	0
$502$	−32.2655	−1.44008
$503$	25.3699	1.13119	0.565595	−	0.824683i	$-0.308647\pi$
$503$	25.3699	1.13119	0.565595	+	0.824683i	$0.308647\pi$
$504$	0	0
$505$	0.701827	0.0312309
$506$	15.1164	0.672004
$507$	0	0
$508$	59.1216	2.62310
$509$	−35.5513	−1.57578	−0.787891	−	0.615815i	$-0.788827\pi$
$509$	−35.5513	−1.57578	−0.787891	+	0.615815i	$0.788827\pi$
$510$	0	0
$511$	0	0
$512$	32.6413	1.44255
$513$	0	0
$514$	−7.90679	−0.348754
$515$	23.0974	1.01780
$516$	0	0
$517$	−7.76030	−0.341298
$518$	0	0
$519$	0	0
$520$	−20.8042	−0.912323
$521$	1.92775	0.0844564	0.0422282	−	0.999108i	$-0.486554\pi$
$521$	1.92775	0.0844564	0.0422282	+	0.999108i	$0.486554\pi$
$522$	0	0
$523$	−34.9541	−1.52844	−0.764218	−	0.644957i	$-0.776875\pi$
$523$	−34.9541	−1.52844	−0.764218	+	0.644957i	$0.776875\pi$
$524$	13.6726	0.597292
$525$	0	0
$526$	−46.0507	−2.00791
$527$	−49.8637	−2.17210
$528$	0	0
$529$	16.0146	0.696288
$530$	−10.8455	−0.471099
$531$	0	0
$532$	0	0
$533$	13.1149	0.568067
$534$	0	0
$535$	13.6008	0.588012
$536$	62.4930	2.69929
$537$	0	0
$538$	−31.4184	−1.35455
$539$	0	0
$540$	0	0
$541$	−7.94735	−0.341683	−0.170842	−	0.985299i	$-0.554649\pi$
$541$	−7.94735	−0.341683	−0.170842	+	0.985299i	$0.554649\pi$
$542$	−11.4623	−0.492347
$543$	0	0
$544$	−10.8389	−0.464713
$545$	−36.7132	−1.57262
$546$	0	0
$547$	−3.68742	−0.157663	−0.0788313	−	0.996888i	$-0.525119\pi$
$547$	−3.68742	−0.157663	−0.0788313	+	0.996888i	$0.525119\pi$
$548$	15.7955	0.674752
$549$	0	0
$550$	7.65197	0.326281
$551$	−2.75444	−0.117343
$552$	0	0
$553$	0	0
$554$	27.9322	1.18672
$555$	0	0
$556$	−81.0710	−3.43818
$557$	−13.5013	−0.572069	−0.286035	−	0.958219i	$-0.592337\pi$
$557$	−13.5013	−0.572069	−0.286035	+	0.958219i	$0.592337\pi$
$558$	0	0
$559$	−11.0389	−0.466897
$560$	0	0
$561$	0	0
$562$	62.5788	2.63973
$563$	27.9652	1.17859	0.589296	−	0.807917i	$-0.299405\pi$
$563$	27.9652	1.17859	0.589296	+	0.807917i	$0.299405\pi$
$564$	0	0
$565$	−25.5584	−1.07525
$566$	−25.6218	−1.07696
$567$	0	0
$568$	−45.2593	−1.89904
$569$	12.6463	0.530162	0.265081	−	0.964226i	$-0.414601\pi$
$569$	12.6463	0.530162	0.265081	+	0.964226i	$0.414601\pi$
$570$	0	0
$571$	−19.0816	−0.798541	−0.399270	−	0.916833i	$-0.630737\pi$
$571$	−19.0816	−0.798541	−0.399270	+	0.916833i	$0.630737\pi$
$572$	6.24991	0.261322
$573$	0	0
$574$	0	0
$575$	19.7494	0.823606
$576$	0	0
$577$	−39.5908	−1.64819	−0.824093	−	0.566454i	$-0.808315\pi$
$577$	−39.5908	−1.64819	−0.824093	+	0.566454i	$0.808315\pi$
$578$	−118.203	−4.91660
$579$	0	0
$580$	−25.1743	−1.04530
$581$	0	0
$582$	0	0
$583$	1.56864	0.0649663
$584$	−47.0968	−1.94888
$585$	0	0
$586$	13.5979	0.561724
$587$	−30.9761	−1.27852	−0.639261	−	0.768990i	$-0.720760\pi$
$587$	−30.9761	−1.27852	−0.639261	+	0.768990i	$0.720760\pi$
$588$	0	0
$589$	−7.40868	−0.305269
$590$	1.70200	0.0700702
$591$	0	0
$592$	27.3039	1.12218
$593$	36.5697	1.50174	0.750868	−	0.660452i	$-0.229635\pi$
$593$	36.5697	1.50174	0.750868	+	0.660452i	$0.229635\pi$
$594$	0	0
$595$	0	0
$596$	75.2362	3.08180
$597$	0	0
$598$	24.4953	1.00169
$599$	15.8382	0.647130	0.323565	−	0.946206i	$-0.395119\pi$
$599$	15.8382	0.647130	0.323565	+	0.946206i	$0.395119\pi$
$600$	0	0
$601$	−3.42710	−0.139794	−0.0698971	−	0.997554i	$-0.522267\pi$
$601$	−3.42710	−0.139794	−0.0698971	+	0.997554i	$0.522267\pi$
$602$	0	0
$603$	0	0
$604$	−34.5918	−1.40752
$605$	−2.85689	−0.116149
$606$	0	0
$607$	−9.28141	−0.376721	−0.188360	−	0.982100i	$-0.560317\pi$
$607$	−9.28141	−0.376721	−0.188360	+	0.982100i	$0.560317\pi$
$608$	−1.61043	−0.0653114
$609$	0	0
$610$	22.2613	0.901333
$611$	−12.5752	−0.508738
$612$	0	0
$613$	15.7933	0.637887	0.318943	−	0.947774i	$-0.396672\pi$
$613$	15.7933	0.637887	0.318943	+	0.947774i	$0.396672\pi$
$614$	39.9972	1.61416
$615$	0	0
$616$	0	0
$617$	−46.6541	−1.87823	−0.939113	−	0.343610i	$-0.888350\pi$
$617$	−46.6541	−1.87823	−0.939113	+	0.343610i	$0.888350\pi$
$618$	0	0
$619$	−38.4167	−1.54410	−0.772048	−	0.635564i	$-0.780768\pi$
$619$	−38.4167	−1.54410	−0.772048	+	0.635564i	$0.780768\pi$
$620$	−67.7117	−2.71937
$621$	0	0
$622$	42.4663	1.70274
$623$	0	0
$624$	0	0
$625$	−30.8120	−1.23248
$626$	−30.1452	−1.20485
$627$	0	0
$628$	50.3656	2.00980
$629$	−70.0708	−2.79390
$630$	0	0
$631$	29.9006	1.19032	0.595162	−	0.803606i	$-0.297088\pi$
$631$	29.9006	1.19032	0.595162	+	0.803606i	$0.297088\pi$
$632$	59.4953	2.36659
$633$	0	0
$634$	17.3827	0.690354
$635$	−43.7928	−1.73786
$636$	0	0
$637$	0	0
$638$	5.52916	0.218901
$639$	0	0
$640$	−58.4402	−2.31005
$641$	−6.61600	−0.261316	−0.130658	−	0.991427i	$-0.541709\pi$
$641$	−6.61600	−0.261316	−0.130658	+	0.991427i	$0.541709\pi$
$642$	0	0
$643$	12.5670	0.495594	0.247797	−	0.968812i	$-0.420293\pi$
$643$	12.5670	0.495594	0.247797	+	0.968812i	$0.420293\pi$
$644$	0	0
$645$	0	0
$646$	−23.6752	−0.931490
$647$	−17.2752	−0.679158	−0.339579	−	0.940578i	$-0.610285\pi$
$647$	−17.2752	−0.679158	−0.339579	+	0.940578i	$0.610285\pi$
$648$	0	0
$649$	−0.246168	−0.00966294
$650$	12.3996	0.486354
$651$	0	0
$652$	−55.2211	−2.16263
$653$	4.33620	0.169689	0.0848444	−	0.996394i	$-0.472961\pi$
$653$	4.33620	0.169689	0.0848444	+	0.996394i	$0.472961\pi$
$654$	0	0
$655$	−10.1276	−0.395720
$656$	−25.5898	−0.999114
$657$	0	0
$658$	0	0
$659$	−36.6318	−1.42697	−0.713486	−	0.700669i	$-0.752885\pi$
$659$	−36.6318	−1.42697	−0.713486	+	0.700669i	$0.752885\pi$
$660$	0	0
$661$	26.8767	1.04538	0.522691	−	0.852522i	$-0.324928\pi$
$661$	26.8767	1.04538	0.522691	+	0.852522i	$0.324928\pi$
$662$	−74.1110	−2.88041
$663$	0	0
$664$	16.1748	0.627704
$665$	0	0
$666$	0	0
$667$	14.2705	0.552556
$668$	−8.27040	−0.319992
$669$	0	0
$670$	−96.1477	−3.71451
$671$	−3.21975	−0.124297
$672$	0	0
$673$	0.0568409	0.00219105	0.00109553	−	0.999999i	$-0.499651\pi$
$673$	0.0568409	0.00219105	0.00109553	+	0.999999i	$0.499651\pi$
$674$	−19.7649	−0.761317
$675$	0	0
$676$	−40.0119	−1.53892
$677$	−33.9724	−1.30567	−0.652833	−	0.757502i	$-0.726420\pi$
$677$	−33.9724	−1.30567	−0.652833	+	0.757502i	$0.726420\pi$
$678$	0	0
$679$	0	0
$680$	−104.176	−3.99496
$681$	0	0
$682$	14.8719	0.569474
$683$	−7.33397	−0.280626	−0.140313	−	0.990107i	$-0.544811\pi$
$683$	−7.33397	−0.280626	−0.140313	+	0.990107i	$0.544811\pi$
$684$	0	0
$685$	−11.7001	−0.447039
$686$	0	0
$687$	0	0
$688$	21.5392	0.821176
$689$	2.54190	0.0968387
$690$	0	0
$691$	0.233419	0.00887969	0.00443985	−	0.999990i	$-0.498587\pi$
$691$	0.233419	0.00887969	0.00443985	+	0.999990i	$0.498587\pi$
$692$	77.4787	2.94530
$693$	0	0
$694$	−48.9812	−1.85930
$695$	60.0512	2.27787
$696$	0	0
$697$	65.6719	2.48750
$698$	71.9536	2.72349
$699$	0	0
$700$	0	0
$701$	34.8430	1.31600	0.658001	−	0.753017i	$-0.271402\pi$
$701$	34.8430	1.31600	0.658001	+	0.753017i	$0.271402\pi$
$702$	0	0
$703$	−10.4110	−0.392659
$704$	9.55638	0.360170
$705$	0	0
$706$	−52.7284	−1.98446
$707$	0	0
$708$	0	0
$709$	−1.92092	−0.0721417	−0.0360708	−	0.999349i	$-0.511484\pi$
$709$	−1.92092	−0.0721417	−0.0360708	+	0.999349i	$0.511484\pi$
$710$	69.6330	2.61328
$711$	0	0
$712$	19.7119	0.738735
$713$	38.3836	1.43748
$714$	0	0
$715$	−4.62946	−0.173132
$716$	−50.1665	−1.87481
$717$	0	0
$718$	10.3435	0.386015
$719$	−19.9041	−0.742299	−0.371149	−	0.928573i	$-0.621036\pi$
$719$	−19.9041	−0.742299	−0.371149	+	0.928573i	$0.621036\pi$
$720$	0	0
$721$	0	0
$722$	42.4643	1.58036
$723$	0	0
$724$	−33.3254	−1.23853
$725$	7.22379	0.268285
$726$	0	0
$727$	45.7396	1.69639	0.848195	−	0.529685i	$-0.177690\pi$
$727$	45.7396	1.69639	0.848195	+	0.529685i	$0.177690\pi$
$728$	0	0
$729$	0	0
$730$	72.4600	2.68186
$731$	−55.2768	−2.04449
$732$	0	0
$733$	47.6258	1.75910	0.879550	−	0.475807i	$-0.157844\pi$
$733$	47.6258	1.75910	0.879550	+	0.475807i	$0.157844\pi$
$734$	7.27339	0.268466
$735$	0	0
$736$	8.34345	0.307544
$737$	13.9063	0.512245
$738$	0	0
$739$	−14.8536	−0.546397	−0.273199	−	0.961958i	$-0.588082\pi$
$739$	−14.8536	−0.546397	−0.273199	+	0.961958i	$0.588082\pi$
$740$	−95.1516	−3.49784
$741$	0	0
$742$	0	0
$743$	5.27520	0.193528	0.0967641	−	0.995307i	$-0.469151\pi$
$743$	5.27520	0.193528	0.0967641	+	0.995307i	$0.469151\pi$
$744$	0	0
$745$	−55.7293	−2.04176
$746$	−0.615719	−0.0225431
$747$	0	0
$748$	31.2961	1.14430
$749$	0	0
$750$	0	0
$751$	18.8822	0.689020	0.344510	−	0.938783i	$-0.388045\pi$
$751$	18.8822	0.689020	0.344510	+	0.938783i	$0.388045\pi$
$752$	24.5368	0.894766
$753$	0	0
$754$	8.95973	0.326294
$755$	25.6230	0.932516
$756$	0	0
$757$	43.0350	1.56413	0.782067	−	0.623194i	$-0.214165\pi$
$757$	43.0350	1.56413	0.782067	+	0.623194i	$0.214165\pi$
$758$	6.96251	0.252890
$759$	0	0
$760$	−15.4783	−0.561457
$761$	−18.8023	−0.681585	−0.340792	−	0.940139i	$-0.610695\pi$
$761$	−18.8023	−0.681585	−0.340792	+	0.940139i	$0.610695\pi$
$762$	0	0
$763$	0	0
$764$	−76.4743	−2.76674
$765$	0	0
$766$	84.7221	3.06114
$767$	−0.398903	−0.0144036
$768$	0	0
$769$	−28.8942	−1.04195	−0.520975	−	0.853572i	$-0.674432\pi$
$769$	−28.8942	−1.04195	−0.520975	+	0.853572i	$0.674432\pi$
$770$	0	0
$771$	0	0
$772$	−28.0549	−1.00972
$773$	−27.9267	−1.00445	−0.502227	−	0.864736i	$-0.667486\pi$
$773$	−27.9267	−1.00445	−0.502227	+	0.864736i	$0.667486\pi$
$774$	0	0
$775$	19.4300	0.697945
$776$	7.61736	0.273447
$777$	0	0
$778$	91.8370	3.29252
$779$	9.75744	0.349597
$780$	0	0
$781$	−10.0713	−0.360381
$782$	122.659	4.38628
$783$	0	0
$784$	0	0
$785$	−37.3070	−1.33154
$786$	0	0
$787$	−15.0882	−0.537838	−0.268919	−	0.963163i	$-0.586666\pi$
$787$	−15.0882	−0.537838	−0.268919	+	0.963163i	$0.586666\pi$
$788$	−48.9358	−1.74327
$789$	0	0
$790$	−91.5355	−3.25669
$791$	0	0
$792$	0	0
$793$	−5.21745	−0.185277
$794$	−3.06405	−0.108739
$795$	0	0
$796$	89.5178	3.17288
$797$	22.7824	0.806993	0.403496	−	0.914981i	$-0.367795\pi$
$797$	22.7824	0.806993	0.403496	+	0.914981i	$0.367795\pi$
$798$	0	0
$799$	−62.9696	−2.22770
$800$	4.22349	0.149323
$801$	0	0
$802$	−57.7997	−2.04098
$803$	−10.4802	−0.369839
$804$	0	0
$805$	0	0
$806$	24.0992	0.848857
$807$	0	0
$808$	1.10397	0.0388375
$809$	56.3710	1.98190	0.990949	−	0.134238i	$-0.0428587\pi$
$809$	56.3710	1.98190	0.990949	+	0.134238i	$0.0428587\pi$
$810$	0	0
$811$	38.7478	1.36062	0.680310	−	0.732924i	$-0.261845\pi$
$811$	38.7478	1.36062	0.680310	+	0.732924i	$0.261845\pi$
$812$	0	0
$813$	0	0
$814$	20.8987	0.732497
$815$	40.9036	1.43279
$816$	0	0
$817$	−8.21295	−0.287335
$818$	46.9398	1.64121
$819$	0	0
$820$	89.1783	3.11424
$821$	−36.0895	−1.25953	−0.629766	−	0.776785i	$-0.716849\pi$
$821$	−36.0895	−1.25953	−0.629766	+	0.776785i	$0.716849\pi$
$822$	0	0
$823$	−1.85754	−0.0647497	−0.0323748	−	0.999476i	$-0.510307\pi$
$823$	−1.85754	−0.0647497	−0.0323748	+	0.999476i	$0.510307\pi$
$824$	36.3321	1.26569
$825$	0	0
$826$	0	0
$827$	3.54179	0.123160	0.0615801	−	0.998102i	$-0.480386\pi$
$827$	3.54179	0.123160	0.0615801	+	0.998102i	$0.480386\pi$
$828$	0	0
$829$	37.8869	1.31587	0.657934	−	0.753076i	$-0.271431\pi$
$829$	37.8869	1.31587	0.657934	+	0.753076i	$0.271431\pi$
$830$	−24.8855	−0.863787
$831$	0	0
$832$	15.4856	0.536868
$833$	0	0
$834$	0	0
$835$	6.12609	0.212002
$836$	4.64993	0.160821
$837$	0	0
$838$	−19.0405	−0.657744
$839$	−15.0661	−0.520140	−0.260070	−	0.965590i	$-0.583746\pi$
$839$	−15.0661	−0.520140	−0.260070	+	0.965590i	$0.583746\pi$
$840$	0	0
$841$	−23.7802	−0.820008
$842$	−57.4963	−1.98145
$843$	0	0
$844$	−45.3879	−1.56231
$845$	29.6378	1.01957
$846$	0	0
$847$	0	0
$848$	−4.95978	−0.170319
$849$	0	0
$850$	62.0905	2.12969
$851$	53.9384	1.84899
$852$	0	0
$853$	−38.8605	−1.33056	−0.665278	−	0.746595i	$-0.731687\pi$
$853$	−38.8605	−1.33056	−0.665278	+	0.746595i	$0.731687\pi$
$854$	0	0
$855$	0	0
$856$	21.3939	0.731228
$857$	−23.2119	−0.792903	−0.396452	−	0.918056i	$-0.629759\pi$
$857$	−23.2119	−0.792903	−0.396452	+	0.918056i	$0.629759\pi$
$858$	0	0
$859$	−27.6313	−0.942768	−0.471384	−	0.881928i	$-0.656245\pi$
$859$	−27.6313	−0.942768	−0.471384	+	0.881928i	$0.656245\pi$
$860$	−75.0624	−2.55961
$861$	0	0
$862$	4.66404	0.158858
$863$	6.31852	0.215085	0.107543	−	0.994200i	$-0.465702\pi$
$863$	6.31852	0.215085	0.107543	+	0.994200i	$0.465702\pi$
$864$	0	0
$865$	−57.3904	−1.95133
$866$	−20.2190	−0.687070
$867$	0	0
$868$	0	0
$869$	13.2392	0.449109
$870$	0	0
$871$	22.5345	0.763551
$872$	−57.7496	−1.95565
$873$	0	0
$874$	18.2245	0.616453
$875$	0	0
$876$	0	0
$877$	−10.8076	−0.364947	−0.182473	−	0.983211i	$-0.558410\pi$
$877$	−10.8076	−0.364947	−0.182473	+	0.983211i	$0.558410\pi$
$878$	34.8147	1.17494
$879$	0	0
$880$	9.03303	0.304503
$881$	32.9475	1.11003	0.555015	−	0.831841i	$-0.312713\pi$
$881$	32.9475	1.11003	0.555015	+	0.831841i	$0.312713\pi$
$882$	0	0
$883$	−13.9550	−0.469622	−0.234811	−	0.972041i	$-0.575447\pi$
$883$	−13.9550	−0.469622	−0.234811	+	0.972041i	$0.575447\pi$
$884$	50.7138	1.70569
$885$	0	0
$886$	20.7654	0.697627
$887$	−8.75154	−0.293848	−0.146924	−	0.989148i	$-0.546937\pi$
$887$	−8.75154	−0.293848	−0.146924	+	0.989148i	$0.546937\pi$
$888$	0	0
$889$	0	0
$890$	−30.3274	−1.01658
$891$	0	0
$892$	37.3928	1.25200
$893$	−9.35594	−0.313084
$894$	0	0
$895$	37.1595	1.24210
$896$	0	0
$897$	0	0
$898$	−44.0607	−1.47032
$899$	14.0397	0.468250
$900$	0	0
$901$	12.7284	0.424046
$902$	−19.5867	−0.652166
$903$	0	0
$904$	−40.2031	−1.33714
$905$	24.6849	0.820554
$906$	0	0
$907$	0.528441	0.0175466	0.00877330	−	0.999962i	$-0.497207\pi$
$907$	0.528441	0.0175466	0.00877330	+	0.999962i	$0.497207\pi$
$908$	−56.1446	−1.86322
$909$	0	0
$910$	0	0
$911$	36.6894	1.21557	0.607787	−	0.794100i	$-0.292057\pi$
$911$	36.6894	1.21557	0.607787	+	0.794100i	$0.292057\pi$
$912$	0	0
$913$	3.59930	0.119119
$914$	97.9107	3.23860
$915$	0	0
$916$	−56.1649	−1.85574
$917$	0	0
$918$	0	0
$919$	−22.7580	−0.750717	−0.375358	−	0.926880i	$-0.622480\pi$
$919$	−22.7580	−0.750717	−0.375358	+	0.926880i	$0.622480\pi$
$920$	80.1915	2.64383
$921$	0	0
$922$	98.6590	3.24916
$923$	−16.3201	−0.537184
$924$	0	0
$925$	27.3039	0.897746
$926$	59.4042	1.95214
$927$	0	0
$928$	3.05181	0.100181
$929$	12.5099	0.410438	0.205219	−	0.978716i	$-0.434209\pi$
$929$	12.5099	0.410438	0.205219	+	0.978716i	$0.434209\pi$
$930$	0	0
$931$	0	0
$932$	28.9721	0.949012
$933$	0	0
$934$	−32.6194	−1.06734
$935$	−23.1817	−0.758124
$936$	0	0
$937$	45.8451	1.49770	0.748848	−	0.662742i	$-0.230607\pi$
$937$	45.8451	1.49770	0.748848	+	0.662742i	$0.230607\pi$
$938$	0	0
$939$	0	0
$940$	−85.5087	−2.78899
$941$	3.27780	0.106853	0.0534266	−	0.998572i	$-0.482986\pi$
$941$	3.27780	0.106853	0.0534266	+	0.998572i	$0.482986\pi$
$942$	0	0
$943$	−50.5523	−1.64621
$944$	0.778343	0.0253329
$945$	0	0
$946$	16.4863	0.536017
$947$	8.92671	0.290079	0.145040	−	0.989426i	$-0.453669\pi$
$947$	8.92671	0.290079	0.145040	+	0.989426i	$0.453669\pi$
$948$	0	0
$949$	−16.9827	−0.551281
$950$	9.22533	0.299309
$951$	0	0
$952$	0	0
$953$	9.92412	0.321474	0.160737	−	0.986997i	$-0.448613\pi$
$953$	9.92412	0.321474	0.160737	+	0.986997i	$0.448613\pi$
$954$	0	0
$955$	56.6463	1.83303
$956$	20.2010	0.653348
$957$	0	0
$958$	49.0674	1.58530
$959$	0	0
$960$	0	0
$961$	6.76285	0.218156
$962$	33.8652	1.09186
$963$	0	0
$964$	−93.9170	−3.02486
$965$	20.7809	0.668961
$966$	0	0
$967$	−17.8164	−0.572936	−0.286468	−	0.958090i	$-0.592481\pi$
$967$	−17.8164	−0.572936	−0.286468	+	0.958090i	$0.592481\pi$
$968$	−4.49387	−0.144438
$969$	0	0
$970$	−11.7196	−0.376293
$971$	−46.2719	−1.48494	−0.742468	−	0.669881i	$-0.766345\pi$
$971$	−46.2719	−1.48494	−0.742468	+	0.669881i	$0.766345\pi$
$972$	0	0
$973$	0	0
$974$	2.41002	0.0772221
$975$	0	0
$976$	10.1803	0.325865
$977$	−31.4368	−1.00575	−0.502876	−	0.864359i	$-0.667725\pi$
$977$	−31.4368	−1.00575	−0.502876	+	0.864359i	$0.667725\pi$
$978$	0	0
$979$	4.38640	0.140190
$980$	0	0
$981$	0	0
$982$	53.5558	1.70903
$983$	13.5501	0.432182	0.216091	−	0.976373i	$-0.430669\pi$
$983$	13.5501	0.432182	0.216091	+	0.976373i	$0.430669\pi$
$984$	0	0
$985$	36.2479	1.15495
$986$	44.8654	1.42880
$987$	0	0
$988$	7.53498	0.239720
$989$	42.5505	1.35303
$990$	0	0
$991$	−30.0405	−0.954267	−0.477134	−	0.878831i	$-0.658324\pi$
$991$	−30.0405	−0.954267	−0.477134	+	0.878831i	$0.658324\pi$
$992$	8.20851	0.260621
$993$	0	0
$994$	0	0
$995$	−66.3080	−2.10211
$996$	0	0
$997$	19.4933	0.617359	0.308680	−	0.951166i	$-0.400113\pi$
$997$	19.4933	0.617359	0.308680	+	0.951166i	$0.400113\pi$
$998$	21.0635	0.666752
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.cc.1.1		6
3.2	odd	2		4851.2.a.cd.1.6		6
7.2	even	3		693.2.i.l.298.6	yes	12
7.4	even	3		693.2.i.l.100.6	yes	12
7.6	odd	2		4851.2.a.ce.1.1		6
21.2	odd	6		693.2.i.k.298.1	yes	12
21.11	odd	6		693.2.i.k.100.1	✓	12
21.20	even	2		4851.2.a.cb.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.i.k.100.1	✓	12	21.11	odd	6
693.2.i.k.298.1	yes	12	21.2	odd	6
693.2.i.l.100.6	yes	12	7.4	even	3
693.2.i.l.298.6	yes	12	7.2	even	3
4851.2.a.cb.1.6		6	21.20	even	2
4851.2.a.cc.1.1		6	1.1	even	1	trivial
4851.2.a.cd.1.6		6	3.2	odd	2
4851.2.a.ce.1.1		6	7.6	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:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42010 q^{2} +3.85689 q^{4} -2.85689 q^{5} -4.49387 q^{8} +O(q^{10})\)	\(q-2.42010 q^{2} +3.85689 q^{4} -2.85689 q^{5} -4.49387 q^{8} +6.91397 q^{10} -1.00000 q^{11} -1.62045 q^{13} +3.16184 q^{16} -8.11432 q^{17} -1.20561 q^{19} -11.0187 q^{20} +2.42010 q^{22} +6.24617 q^{23} +3.16184 q^{25} +3.92166 q^{26} +2.28468 q^{29} +6.14515 q^{31} +1.33577 q^{32} +19.6375 q^{34} +8.63544 q^{37} +2.91771 q^{38} +12.8385 q^{40} -8.09333 q^{41} +6.81225 q^{43} -3.85689 q^{44} -15.1164 q^{46} +7.76030 q^{47} -7.65197 q^{50} -6.24991 q^{52} -1.56864 q^{53} +2.85689 q^{55} -5.52916 q^{58} +0.246168 q^{59} +3.21975 q^{61} -14.8719 q^{62} -9.55638 q^{64} +4.62946 q^{65} -13.9063 q^{67} -31.2961 q^{68} +10.0713 q^{71} +10.4802 q^{73} -20.8987 q^{74} -4.64993 q^{76} -13.2392 q^{79} -9.03303 q^{80} +19.5867 q^{82} -3.59930 q^{83} +23.1817 q^{85} -16.4863 q^{86} +4.49387 q^{88} -4.38640 q^{89} +24.0908 q^{92} -18.7807 q^{94} +3.44431 q^{95} -1.69506 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 4 q^{5} + 6 q^{8}+O(q^{10}) \) 6 * q + 10 * q^4 - 4 * q^5 + 6 * q^8	\( 6 q + 10 q^{4} - 4 q^{5} + 6 q^{8} - 6 q^{10} - 6 q^{11} - 2 q^{13} + 2 q^{16} - 8 q^{17} + 6 q^{19} - 36 q^{20} + 2 q^{23} + 2 q^{25} - 8 q^{26} + 2 q^{29} + 4 q^{31} + 12 q^{32} + 14 q^{34} - 6 q^{37} - 10 q^{38} - 18 q^{40} - 30 q^{41} - 12 q^{43} - 10 q^{44} - 4 q^{46} - 14 q^{47} + 24 q^{50} - 22 q^{52} - 16 q^{53} + 4 q^{55} + 2 q^{58} - 34 q^{59} - 2 q^{61} - 26 q^{62} + 2 q^{64} + 20 q^{65} - 20 q^{67} - 18 q^{68} + 6 q^{71} + 26 q^{73} - 14 q^{74} - 4 q^{76} + 2 q^{79} - 44 q^{80} - 4 q^{82} - 8 q^{83} + 10 q^{85} - 78 q^{86} - 6 q^{88} - 20 q^{89} + 30 q^{92} - 10 q^{94} + 10 q^{95} - 14 q^{97}+O(q^{100}) \) 6 * q + 10 * q^4 - 4 * q^5 + 6 * q^8 - 6 * q^10 - 6 * q^11 - 2 * q^13 + 2 * q^16 - 8 * q^17 + 6 * q^19 - 36 * q^20 + 2 * q^23 + 2 * q^25 - 8 * q^26 + 2 * q^29 + 4 * q^31 + 12 * q^32 + 14 * q^34 - 6 * q^37 - 10 * q^38 - 18 * q^40 - 30 * q^41 - 12 * q^43 - 10 * q^44 - 4 * q^46 - 14 * q^47 + 24 * q^50 - 22 * q^52 - 16 * q^53 + 4 * q^55 + 2 * q^58 - 34 * q^59 - 2 * q^61 - 26 * q^62 + 2 * q^64 + 20 * q^65 - 20 * q^67 - 18 * q^68 + 6 * q^71 + 26 * q^73 - 14 * q^74 - 4 * q^76 + 2 * q^79 - 44 * q^80 - 4 * q^82 - 8 * q^83 + 10 * q^85 - 78 * q^86 - 6 * q^88 - 20 * q^89 + 30 * q^92 - 10 * q^94 + 10 * q^95 - 14 * q^97

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