LMFDB - Embedded newform 4851.2.a.be.1.2

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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +O(q^{10})$	$q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +4.82843 q^{10} -1.00000 q^{11} -0.828427 q^{13} +3.00000 q^{16} +4.41421 q^{17} +7.24264 q^{19} +7.65685 q^{20} -2.41421 q^{22} +7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -3.24264 q^{29} +5.65685 q^{31} -1.58579 q^{32} +10.6569 q^{34} -9.48528 q^{37} +17.4853 q^{38} +8.82843 q^{40} +1.17157 q^{41} -2.75736 q^{43} -3.82843 q^{44} +16.8995 q^{46} -9.82843 q^{47} -2.41421 q^{50} -3.17157 q^{52} +7.17157 q^{53} -2.00000 q^{55} -7.82843 q^{58} +8.65685 q^{59} -4.00000 q^{61} +13.6569 q^{62} -9.82843 q^{64} -1.65685 q^{65} -3.17157 q^{67} +16.8995 q^{68} +4.17157 q^{71} +0.343146 q^{73} -22.8995 q^{74} +27.7279 q^{76} +13.3137 q^{79} +6.00000 q^{80} +2.82843 q^{82} +2.82843 q^{83} +8.82843 q^{85} -6.65685 q^{86} -4.41421 q^{88} +14.1421 q^{89} +26.7990 q^{92} -23.7279 q^{94} +14.4853 q^{95} -11.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} + 6 q^{16} + 6 q^{17} + 6 q^{19} + 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{29} - 6 q^{32} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 12 q^{40} + 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 2 q^{50} - 12 q^{52} + 20 q^{53} - 4 q^{55} - 10 q^{58} + 6 q^{59} - 8 q^{61} + 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} + 14 q^{68} + 14 q^{71} + 12 q^{73} - 26 q^{74} + 30 q^{76} + 4 q^{79} + 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} - 22 q^{94} + 12 q^{95} - 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^10 - 2 * q^11 + 4 * q^13 + 6 * q^16 + 6 * q^17 + 6 * q^19 + 4 * q^20 - 2 * q^22 + 14 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^29 - 6 * q^32 + 10 * q^34 - 2 * q^37 + 18 * q^38 + 12 * q^40 + 8 * q^41 - 14 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 2 * q^50 - 12 * q^52 + 20 * q^53 - 4 * q^55 - 10 * q^58 + 6 * q^59 - 8 * q^61 + 16 * q^62 - 14 * q^64 + 8 * q^65 - 12 * q^67 + 14 * q^68 + 14 * q^71 + 12 * q^73 - 26 * q^74 + 30 * q^76 + 4 * q^79 + 12 * q^80 + 12 * q^85 - 2 * q^86 - 6 * q^88 + 14 * q^92 - 22 * q^94 + 12 * q^95 - 6 * 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	4.41421	1.56066
$9$	0	0
$10$	4.82843	1.52688
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	4.41421	1.07060	0.535302	−	0.844661i	$-0.320198\pi$
$17$	4.41421	1.07060	0.535302	+	0.844661i	$0.320198\pi$
$18$	0	0
$19$	7.24264	1.66158	0.830788	−	0.556589i	$-0.187890\pi$
$19$	7.24264	1.66158	0.830788	+	0.556589i	$0.187890\pi$
$20$	7.65685	1.71212
$21$	0	0
$22$	−2.41421	−0.514712
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	−3.24264	−0.602143	−0.301072	−	0.953602i	$-0.597344\pi$
$29$	−3.24264	−0.602143	−0.301072	+	0.953602i	$0.597344\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	10.6569	1.82764
$35$	0	0
$36$	0	0
$37$	−9.48528	−1.55937	−0.779685	−	0.626172i	$-0.784621\pi$
$37$	−9.48528	−1.55937	−0.779685	+	0.626172i	$0.784621\pi$
$38$	17.4853	2.83649
$39$	0	0
$40$	8.82843	1.39590
$41$	1.17157	0.182969	0.0914845	−	0.995807i	$-0.470839\pi$
$41$	1.17157	0.182969	0.0914845	+	0.995807i	$0.470839\pi$
$42$	0	0
$43$	−2.75736	−0.420493	−0.210247	−	0.977648i	$-0.567427\pi$
$43$	−2.75736	−0.420493	−0.210247	+	0.977648i	$0.567427\pi$
$44$	−3.82843	−0.577157
$45$	0	0
$46$	16.8995	2.49169
$47$	−9.82843	−1.43362	−0.716812	−	0.697267i	$-0.754399\pi$
$47$	−9.82843	−1.43362	−0.716812	+	0.697267i	$0.754399\pi$
$48$	0	0
$49$	0	0
$50$	−2.41421	−0.341421
$51$	0	0
$52$	−3.17157	−0.439818
$53$	7.17157	0.985091	0.492546	−	0.870287i	$-0.336066\pi$
$53$	7.17157	0.985091	0.492546	+	0.870287i	$0.336066\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	−7.82843	−1.02792
$59$	8.65685	1.12703	0.563513	−	0.826107i	$-0.309449\pi$
$59$	8.65685	1.12703	0.563513	+	0.826107i	$0.309449\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	13.6569	1.73442
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−1.65685	−0.205507
$66$	0	0
$67$	−3.17157	−0.387469	−0.193735	−	0.981054i	$-0.562060\pi$
$67$	−3.17157	−0.387469	−0.193735	+	0.981054i	$0.562060\pi$
$68$	16.8995	2.04936
$69$	0	0
$70$	0	0
$71$	4.17157	0.495075	0.247537	−	0.968878i	$-0.420379\pi$
$71$	4.17157	0.495075	0.247537	+	0.968878i	$0.420379\pi$
$72$	0	0
$73$	0.343146	0.0401622	0.0200811	−	0.999798i	$-0.493608\pi$
$73$	0.343146	0.0401622	0.0200811	+	0.999798i	$0.493608\pi$
$74$	−22.8995	−2.66201
$75$	0	0
$76$	27.7279	3.18061
$77$	0	0
$78$	0	0
$79$	13.3137	1.49791	0.748955	−	0.662621i	$-0.230556\pi$
$79$	13.3137	1.49791	0.748955	+	0.662621i	$0.230556\pi$
$80$	6.00000	0.670820
$81$	0	0
$82$	2.82843	0.312348
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	8.82843	0.957577
$86$	−6.65685	−0.717827
$87$	0	0
$88$	−4.41421	−0.470557
$89$	14.1421	1.49906	0.749532	−	0.661968i	$-0.230279\pi$
$89$	14.1421	1.49906	0.749532	+	0.661968i	$0.230279\pi$
$90$	0	0
$91$	0	0
$92$	26.7990	2.79399
$93$	0	0
$94$	−23.7279	−2.44735
$95$	14.4853	1.48616
$96$	0	0
$97$	−11.4853	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$97$	−11.4853	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$98$	0	0
$99$	0	0
$100$	−3.82843	−0.382843
$101$	−4.89949	−0.487518	−0.243759	−	0.969836i	$-0.578381\pi$
$101$	−4.89949	−0.487518	−0.243759	+	0.969836i	$0.578381\pi$
$102$	0	0
$103$	12.4853	1.23021	0.615106	−	0.788445i	$-0.289113\pi$
$103$	12.4853	1.23021	0.615106	+	0.788445i	$0.289113\pi$
$104$	−3.65685	−0.358584
$105$	0	0
$106$	17.3137	1.68166
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	0	0
$109$	−6.82843	−0.654045	−0.327022	−	0.945017i	$-0.606045\pi$
$109$	−6.82843	−0.654045	−0.327022	+	0.945017i	$0.606045\pi$
$110$	−4.82843	−0.460372
$111$	0	0
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	14.0000	1.30551
$116$	−12.4142	−1.15263
$117$	0	0
$118$	20.8995	1.92395
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−9.65685	−0.874291
$123$	0	0
$124$	21.6569	1.94484
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−16.0711	−1.42608	−0.713038	−	0.701125i	$-0.752681\pi$
$127$	−16.0711	−1.42608	−0.713038	+	0.701125i	$0.752681\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	−4.00000	−0.350823
$131$	5.17157	0.451842	0.225921	−	0.974146i	$-0.427461\pi$
$131$	5.17157	0.451842	0.225921	+	0.974146i	$0.427461\pi$
$132$	0	0
$133$	0	0
$134$	−7.65685	−0.661451
$135$	0	0
$136$	19.4853	1.67085
$137$	−0.485281	−0.0414604	−0.0207302	−	0.999785i	$-0.506599\pi$
$137$	−0.485281	−0.0414604	−0.0207302	+	0.999785i	$0.506599\pi$
$138$	0	0
$139$	8.41421	0.713684	0.356842	−	0.934165i	$-0.383853\pi$
$139$	8.41421	0.713684	0.356842	+	0.934165i	$0.383853\pi$
$140$	0	0
$141$	0	0
$142$	10.0711	0.845145
$143$	0.828427	0.0692766
$144$	0	0
$145$	−6.48528	−0.538573
$146$	0.828427	0.0685611
$147$	0	0
$148$	−36.3137	−2.98497
$149$	22.2132	1.81978	0.909888	−	0.414853i	$-0.136167\pi$
$149$	22.2132	1.81978	0.909888	+	0.414853i	$0.136167\pi$
$150$	0	0
$151$	−18.8995	−1.53802	−0.769010	−	0.639237i	$-0.779250\pi$
$151$	−18.8995	−1.53802	−0.769010	+	0.639237i	$0.779250\pi$
$152$	31.9706	2.59316
$153$	0	0
$154$	0	0
$155$	11.3137	0.908739
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	32.1421	2.55709
$159$	0	0
$160$	−3.17157	−0.250735
$161$	0	0
$162$	0	0
$163$	−20.9706	−1.64254	−0.821271	−	0.570539i	$-0.806734\pi$
$163$	−20.9706	−1.64254	−0.821271	+	0.570539i	$0.806734\pi$
$164$	4.48528	0.350242
$165$	0	0
$166$	6.82843	0.529989
$167$	−5.17157	−0.400188	−0.200094	−	0.979777i	$-0.564125\pi$
$167$	−5.17157	−0.400188	−0.200094	+	0.979777i	$0.564125\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	21.3137	1.63469
$171$	0	0
$172$	−10.5563	−0.804914
$173$	14.8284	1.12738	0.563692	−	0.825985i	$-0.309380\pi$
$173$	14.8284	1.12738	0.563692	+	0.825985i	$0.309380\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	34.1421	2.55906
$179$	−17.8284	−1.33256	−0.666280	−	0.745702i	$-0.732114\pi$
$179$	−17.8284	−1.33256	−0.666280	+	0.745702i	$0.732114\pi$
$180$	0	0
$181$	−11.6569	−0.866447	−0.433224	−	0.901286i	$-0.642624\pi$
$181$	−11.6569	−0.866447	−0.433224	+	0.901286i	$0.642624\pi$
$182$	0	0
$183$	0	0
$184$	30.8995	2.27794
$185$	−18.9706	−1.39474
$186$	0	0
$187$	−4.41421	−0.322799
$188$	−37.6274	−2.74426
$189$	0	0
$190$	34.9706	2.53703
$191$	21.6569	1.56703	0.783517	−	0.621370i	$-0.213423\pi$
$191$	21.6569	1.56703	0.783517	+	0.621370i	$0.213423\pi$
$192$	0	0
$193$	−12.1421	−0.874010	−0.437005	−	0.899459i	$-0.643961\pi$
$193$	−12.1421	−0.874010	−0.437005	+	0.899459i	$0.643961\pi$
$194$	−27.7279	−1.99075
$195$	0	0
$196$	0	0
$197$	16.4142	1.16946	0.584732	−	0.811226i	$-0.301200\pi$
$197$	16.4142	1.16946	0.584732	+	0.811226i	$0.301200\pi$
$198$	0	0
$199$	−19.7990	−1.40351	−0.701757	−	0.712417i	$-0.747601\pi$
$199$	−19.7990	−1.40351	−0.701757	+	0.712417i	$0.747601\pi$
$200$	−4.41421	−0.312132
$201$	0	0
$202$	−11.8284	−0.832245
$203$	0	0
$204$	0	0
$205$	2.34315	0.163652
$206$	30.1421	2.10010
$207$	0	0
$208$	−2.48528	−0.172323
$209$	−7.24264	−0.500984
$210$	0	0
$211$	14.9706	1.03062	0.515308	−	0.857005i	$-0.327678\pi$
$211$	14.9706	1.03062	0.515308	+	0.857005i	$0.327678\pi$
$212$	27.4558	1.88568
$213$	0	0
$214$	−23.3137	−1.59369
$215$	−5.51472	−0.376101
$216$	0	0
$217$	0	0
$218$	−16.4853	−1.11652
$219$	0	0
$220$	−7.65685	−0.516225
$221$	−3.65685	−0.245987
$222$	0	0
$223$	−22.9706	−1.53822	−0.769111	−	0.639115i	$-0.779301\pi$
$223$	−22.9706	−1.53822	−0.769111	+	0.639115i	$0.779301\pi$
$224$	0	0
$225$	0	0
$226$	18.4853	1.22962
$227$	−14.9706	−0.993631	−0.496816	−	0.867856i	$-0.665497\pi$
$227$	−14.9706	−0.993631	−0.496816	+	0.867856i	$0.665497\pi$
$228$	0	0
$229$	−15.6569	−1.03463	−0.517317	−	0.855794i	$-0.673069\pi$
$229$	−15.6569	−1.03463	−0.517317	+	0.855794i	$0.673069\pi$
$230$	33.7990	2.22864
$231$	0	0
$232$	−14.3137	−0.939741
$233$	−10.4142	−0.682258	−0.341129	−	0.940017i	$-0.610809\pi$
$233$	−10.4142	−0.682258	−0.341129	+	0.940017i	$0.610809\pi$
$234$	0	0
$235$	−19.6569	−1.28227
$236$	33.1421	2.15737
$237$	0	0
$238$	0	0
$239$	6.48528	0.419498	0.209749	−	0.977755i	$-0.432735\pi$
$239$	6.48528	0.419498	0.209749	+	0.977755i	$0.432735\pi$
$240$	0	0
$241$	7.31371	0.471117	0.235559	−	0.971860i	$-0.424308\pi$
$241$	7.31371	0.471117	0.235559	+	0.971860i	$0.424308\pi$
$242$	2.41421	0.155192
$243$	0	0
$244$	−15.3137	−0.980360
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	24.9706	1.58563
$249$	0	0
$250$	−28.9706	−1.83226
$251$	2.51472	0.158728	0.0793638	−	0.996846i	$-0.474711\pi$
$251$	2.51472	0.158728	0.0793638	+	0.996846i	$0.474711\pi$
$252$	0	0
$253$	−7.00000	−0.440086
$254$	−38.7990	−2.43447
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−21.4558	−1.33838	−0.669189	−	0.743092i	$-0.733359\pi$
$257$	−21.4558	−1.33838	−0.669189	+	0.743092i	$0.733359\pi$
$258$	0	0
$259$	0	0
$260$	−6.34315	−0.393385
$261$	0	0
$262$	12.4853	0.771343
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	14.3431	0.881092
$266$	0	0
$267$	0	0
$268$	−12.1421	−0.741699
$269$	27.7990	1.69493	0.847467	−	0.530848i	$-0.178126\pi$
$269$	27.7990	1.69493	0.847467	+	0.530848i	$0.178126\pi$
$270$	0	0
$271$	−22.9706	−1.39536	−0.697681	−	0.716408i	$-0.745785\pi$
$271$	−22.9706	−1.39536	−0.697681	+	0.716408i	$0.745785\pi$
$272$	13.2426	0.802953
$273$	0	0
$274$	−1.17157	−0.0707773
$275$	1.00000	0.0603023
$276$	0	0
$277$	−25.6569	−1.54157	−0.770785	−	0.637095i	$-0.780136\pi$
$277$	−25.6569	−1.54157	−0.770785	+	0.637095i	$0.780136\pi$
$278$	20.3137	1.21834
$279$	0	0
$280$	0	0
$281$	−14.4142	−0.859880	−0.429940	−	0.902857i	$-0.641465\pi$
$281$	−14.4142	−0.859880	−0.429940	+	0.902857i	$0.641465\pi$
$282$	0	0
$283$	20.3431	1.20927	0.604637	−	0.796501i	$-0.293318\pi$
$283$	20.3431	1.20927	0.604637	+	0.796501i	$0.293318\pi$
$284$	15.9706	0.947679
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	0	0
$289$	2.48528	0.146193
$290$	−15.6569	−0.919402
$291$	0	0
$292$	1.31371	0.0768790
$293$	−5.72792	−0.334629	−0.167314	−	0.985904i	$-0.553509\pi$
$293$	−5.72792	−0.334629	−0.167314	+	0.985904i	$0.553509\pi$
$294$	0	0
$295$	17.3137	1.00804
$296$	−41.8701	−2.43365
$297$	0	0
$298$	53.6274	3.10655
$299$	−5.79899	−0.335364
$300$	0	0
$301$	0	0
$302$	−45.6274	−2.62556
$303$	0	0
$304$	21.7279	1.24618
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−17.3137	−0.988146	−0.494073	−	0.869421i	$-0.664492\pi$
$307$	−17.3137	−0.988146	−0.494073	+	0.869421i	$0.664492\pi$
$308$	0	0
$309$	0	0
$310$	27.3137	1.55131
$311$	−29.6274	−1.68002	−0.840008	−	0.542573i	$-0.817450\pi$
$311$	−29.6274	−1.68002	−0.840008	+	0.542573i	$0.817450\pi$
$312$	0	0
$313$	4.79899	0.271255	0.135627	−	0.990760i	$-0.456695\pi$
$313$	4.79899	0.271255	0.135627	+	0.990760i	$0.456695\pi$
$314$	16.8995	0.953694
$315$	0	0
$316$	50.9706	2.86732
$317$	25.3137	1.42176	0.710880	−	0.703314i	$-0.248297\pi$
$317$	25.3137	1.42176	0.710880	+	0.703314i	$0.248297\pi$
$318$	0	0
$319$	3.24264	0.181553
$320$	−19.6569	−1.09885
$321$	0	0
$322$	0	0
$323$	31.9706	1.77889
$324$	0	0
$325$	0.828427	0.0459529
$326$	−50.6274	−2.80399
$327$	0	0
$328$	5.17157	0.285552
$329$	0	0
$330$	0	0
$331$	22.4853	1.23590	0.617951	−	0.786216i	$-0.287963\pi$
$331$	22.4853	1.23590	0.617951	+	0.786216i	$0.287963\pi$
$332$	10.8284	0.594287
$333$	0	0
$334$	−12.4853	−0.683164
$335$	−6.34315	−0.346563
$336$	0	0
$337$	3.85786	0.210151	0.105076	−	0.994464i	$-0.466492\pi$
$337$	3.85786	0.210151	0.105076	+	0.994464i	$0.466492\pi$
$338$	−29.7279	−1.61699
$339$	0	0
$340$	33.7990	1.83301
$341$	−5.65685	−0.306336
$342$	0	0
$343$	0	0
$344$	−12.1716	−0.656247
$345$	0	0
$346$	35.7990	1.92457
$347$	−14.9706	−0.803662	−0.401831	−	0.915714i	$-0.631626\pi$
$347$	−14.9706	−0.803662	−0.401831	+	0.915714i	$0.631626\pi$
$348$	0	0
$349$	10.9706	0.587241	0.293620	−	0.955922i	$-0.405140\pi$
$349$	10.9706	0.587241	0.293620	+	0.955922i	$0.405140\pi$
$350$	0	0
$351$	0	0
$352$	1.58579	0.0845227
$353$	−13.3137	−0.708617	−0.354309	−	0.935129i	$-0.615284\pi$
$353$	−13.3137	−0.708617	−0.354309	+	0.935129i	$0.615284\pi$
$354$	0	0
$355$	8.34315	0.442808
$356$	54.1421	2.86953
$357$	0	0
$358$	−43.0416	−2.27482
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	33.4558	1.76083
$362$	−28.1421	−1.47912
$363$	0	0
$364$	0	0
$365$	0.686292	0.0359221
$366$	0	0
$367$	−4.14214	−0.216218	−0.108109	−	0.994139i	$-0.534480\pi$
$367$	−4.14214	−0.216218	−0.108109	+	0.994139i	$0.534480\pi$
$368$	21.0000	1.09470
$369$	0	0
$370$	−45.7990	−2.38098
$371$	0	0
$372$	0	0
$373$	−1.65685	−0.0857887	−0.0428943	−	0.999080i	$-0.513658\pi$
$373$	−1.65685	−0.0857887	−0.0428943	+	0.999080i	$0.513658\pi$
$374$	−10.6569	−0.551053
$375$	0	0
$376$	−43.3848	−2.23740
$377$	2.68629	0.138351
$378$	0	0
$379$	18.8284	0.967151	0.483576	−	0.875303i	$-0.339338\pi$
$379$	18.8284	0.967151	0.483576	+	0.875303i	$0.339338\pi$
$380$	55.4558	2.84482
$381$	0	0
$382$	52.2843	2.67510
$383$	−2.31371	−0.118225	−0.0591125	−	0.998251i	$-0.518827\pi$
$383$	−2.31371	−0.118225	−0.0591125	+	0.998251i	$0.518827\pi$
$384$	0	0
$385$	0	0
$386$	−29.3137	−1.49203
$387$	0	0
$388$	−43.9706	−2.23227
$389$	11.7990	0.598233	0.299116	−	0.954217i	$-0.403308\pi$
$389$	11.7990	0.598233	0.299116	+	0.954217i	$0.403308\pi$
$390$	0	0
$391$	30.8995	1.56265
$392$	0	0
$393$	0	0
$394$	39.6274	1.99640
$395$	26.6274	1.33977
$396$	0	0
$397$	17.4853	0.877561	0.438781	−	0.898594i	$-0.355411\pi$
$397$	17.4853	0.877561	0.438781	+	0.898594i	$0.355411\pi$
$398$	−47.7990	−2.39595
$399$	0	0
$400$	−3.00000	−0.150000
$401$	−7.51472	−0.375267	−0.187634	−	0.982239i	$-0.560082\pi$
$401$	−7.51472	−0.375267	−0.187634	+	0.982239i	$0.560082\pi$
$402$	0	0
$403$	−4.68629	−0.233441
$404$	−18.7574	−0.933214
$405$	0	0
$406$	0	0
$407$	9.48528	0.470168
$408$	0	0
$409$	−7.79899	−0.385635	−0.192818	−	0.981235i	$-0.561763\pi$
$409$	−7.79899	−0.385635	−0.192818	+	0.981235i	$0.561763\pi$
$410$	5.65685	0.279372
$411$	0	0
$412$	47.7990	2.35489
$413$	0	0
$414$	0	0
$415$	5.65685	0.277684
$416$	1.31371	0.0644099
$417$	0	0
$418$	−17.4853	−0.855233
$419$	−14.7990	−0.722978	−0.361489	−	0.932376i	$-0.617731\pi$
$419$	−14.7990	−0.722978	−0.361489	+	0.932376i	$0.617731\pi$
$420$	0	0
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	36.1421	1.75937
$423$	0	0
$424$	31.6569	1.53739
$425$	−4.41421	−0.214121
$426$	0	0
$427$	0	0
$428$	−36.9706	−1.78704
$429$	0	0
$430$	−13.3137	−0.642044
$431$	−6.97056	−0.335760	−0.167880	−	0.985807i	$-0.553692\pi$
$431$	−6.97056	−0.335760	−0.167880	+	0.985807i	$0.553692\pi$
$432$	0	0
$433$	0.857864	0.0412263	0.0206132	−	0.999788i	$-0.493438\pi$
$433$	0.857864	0.0412263	0.0206132	+	0.999788i	$0.493438\pi$
$434$	0	0
$435$	0	0
$436$	−26.1421	−1.25198
$437$	50.6985	2.42524
$438$	0	0
$439$	30.8995	1.47475	0.737376	−	0.675482i	$-0.236065\pi$
$439$	30.8995	1.47475	0.737376	+	0.675482i	$0.236065\pi$
$440$	−8.82843	−0.420879
$441$	0	0
$442$	−8.82843	−0.419925
$443$	−33.6274	−1.59769	−0.798843	−	0.601539i	$-0.794554\pi$
$443$	−33.6274	−1.59769	−0.798843	+	0.601539i	$0.794554\pi$
$444$	0	0
$445$	28.2843	1.34080
$446$	−55.4558	−2.62591
$447$	0	0
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	−1.17157	−0.0551672
$452$	29.3137	1.37880
$453$	0	0
$454$	−36.1421	−1.69623
$455$	0	0
$456$	0	0
$457$	18.8284	0.880757	0.440378	−	0.897812i	$-0.354844\pi$
$457$	18.8284	0.880757	0.440378	+	0.897812i	$0.354844\pi$
$458$	−37.7990	−1.76623
$459$	0	0
$460$	53.5980	2.49902
$461$	−6.75736	−0.314722	−0.157361	−	0.987541i	$-0.550299\pi$
$461$	−6.75736	−0.314722	−0.157361	+	0.987541i	$0.550299\pi$
$462$	0	0
$463$	18.6274	0.865689	0.432845	−	0.901468i	$-0.357510\pi$
$463$	18.6274	0.865689	0.432845	+	0.901468i	$0.357510\pi$
$464$	−9.72792	−0.451607
$465$	0	0
$466$	−25.1421	−1.16469
$467$	8.31371	0.384713	0.192356	−	0.981325i	$-0.438387\pi$
$467$	8.31371	0.384713	0.192356	+	0.981325i	$0.438387\pi$
$468$	0	0
$469$	0	0
$470$	−47.4558	−2.18897
$471$	0	0
$472$	38.2132	1.75891
$473$	2.75736	0.126784
$474$	0	0
$475$	−7.24264	−0.332315
$476$	0	0
$477$	0	0
$478$	15.6569	0.716128
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	0	0
$481$	7.85786	0.358288
$482$	17.6569	0.804248
$483$	0	0
$484$	3.82843	0.174019
$485$	−22.9706	−1.04304
$486$	0	0
$487$	−32.6274	−1.47849	−0.739245	−	0.673437i	$-0.764817\pi$
$487$	−32.6274	−1.47849	−0.739245	+	0.673437i	$0.764817\pi$
$488$	−17.6569	−0.799288
$489$	0	0
$490$	0	0
$491$	1.51472	0.0683583	0.0341791	−	0.999416i	$-0.489118\pi$
$491$	1.51472	0.0683583	0.0341791	+	0.999416i	$0.489118\pi$
$492$	0	0
$493$	−14.3137	−0.644657
$494$	−14.4853	−0.651724
$495$	0	0
$496$	16.9706	0.762001
$497$	0	0
$498$	0	0
$499$	−31.7990	−1.42352	−0.711759	−	0.702424i	$-0.752101\pi$
$499$	−31.7990	−1.42352	−0.711759	+	0.702424i	$0.752101\pi$
$500$	−45.9411	−2.05455
$501$	0	0
$502$	6.07107	0.270965
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−9.79899	−0.436049
$506$	−16.8995	−0.751274
$507$	0	0
$508$	−61.5269	−2.72982
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	0	0
$512$	−31.2426	−1.38074
$513$	0	0
$514$	−51.7990	−2.28476
$515$	24.9706	1.10033
$516$	0	0
$517$	9.82843	0.432254
$518$	0	0
$519$	0	0
$520$	−7.31371	−0.320727
$521$	−31.1716	−1.36565	−0.682826	−	0.730581i	$-0.739249\pi$
$521$	−31.1716	−1.36565	−0.682826	+	0.730581i	$0.739249\pi$
$522$	0	0
$523$	−22.2843	−0.974423	−0.487212	−	0.873284i	$-0.661986\pi$
$523$	−22.2843	−0.974423	−0.487212	+	0.873284i	$0.661986\pi$
$524$	19.7990	0.864923
$525$	0	0
$526$	−43.4558	−1.89476
$527$	24.9706	1.08773
$528$	0	0
$529$	26.0000	1.13043
$530$	34.6274	1.50412
$531$	0	0
$532$	0	0
$533$	−0.970563	−0.0420397
$534$	0	0
$535$	−19.3137	−0.835004
$536$	−14.0000	−0.604708
$537$	0	0
$538$	67.1127	2.89343
$539$	0	0
$540$	0	0
$541$	19.6569	0.845114	0.422557	−	0.906336i	$-0.361133\pi$
$541$	19.6569	0.845114	0.422557	+	0.906336i	$0.361133\pi$
$542$	−55.4558	−2.38203
$543$	0	0
$544$	−7.00000	−0.300123
$545$	−13.6569	−0.584995
$546$	0	0
$547$	−24.2132	−1.03528	−0.517641	−	0.855598i	$-0.673190\pi$
$547$	−24.2132	−1.03528	−0.517641	+	0.855598i	$0.673190\pi$
$548$	−1.85786	−0.0793640
$549$	0	0
$550$	2.41421	0.102942
$551$	−23.4853	−1.00051
$552$	0	0
$553$	0	0
$554$	−61.9411	−2.63163
$555$	0	0
$556$	32.2132	1.36614
$557$	8.55635	0.362544	0.181272	−	0.983433i	$-0.441979\pi$
$557$	8.55635	0.362544	0.181272	+	0.983433i	$0.441979\pi$
$558$	0	0
$559$	2.28427	0.0966144
$560$	0	0
$561$	0	0
$562$	−34.7990	−1.46791
$563$	−24.8284	−1.04639	−0.523197	−	0.852212i	$-0.675261\pi$
$563$	−24.8284	−1.04639	−0.523197	+	0.852212i	$0.675261\pi$
$564$	0	0
$565$	15.3137	0.644253
$566$	49.1127	2.06436
$567$	0	0
$568$	18.4142	0.772643
$569$	−1.24264	−0.0520942	−0.0260471	−	0.999661i	$-0.508292\pi$
$569$	−1.24264	−0.0520942	−0.0260471	+	0.999661i	$0.508292\pi$
$570$	0	0
$571$	−3.92893	−0.164421	−0.0822103	−	0.996615i	$-0.526198\pi$
$571$	−3.92893	−0.164421	−0.0822103	+	0.996615i	$0.526198\pi$
$572$	3.17157	0.132610
$573$	0	0
$574$	0	0
$575$	−7.00000	−0.291920
$576$	0	0
$577$	7.65685	0.318759	0.159380	−	0.987217i	$-0.449051\pi$
$577$	7.65685	0.318759	0.159380	+	0.987217i	$0.449051\pi$
$578$	6.00000	0.249567
$579$	0	0
$580$	−24.8284	−1.03094
$581$	0	0
$582$	0	0
$583$	−7.17157	−0.297016
$584$	1.51472	0.0626795
$585$	0	0
$586$	−13.8284	−0.571247
$587$	−8.68629	−0.358522	−0.179261	−	0.983802i	$-0.557371\pi$
$587$	−8.68629	−0.358522	−0.179261	+	0.983802i	$0.557371\pi$
$588$	0	0
$589$	40.9706	1.68816
$590$	41.7990	1.72084
$591$	0	0
$592$	−28.4558	−1.16953
$593$	22.2721	0.914605	0.457302	−	0.889311i	$-0.348816\pi$
$593$	22.2721	0.914605	0.457302	+	0.889311i	$0.348816\pi$
$594$	0	0
$595$	0	0
$596$	85.0416	3.48344
$597$	0	0
$598$	−14.0000	−0.572503
$599$	1.37258	0.0560822	0.0280411	−	0.999607i	$-0.491073\pi$
$599$	1.37258	0.0560822	0.0280411	+	0.999607i	$0.491073\pi$
$600$	0	0
$601$	−14.4853	−0.590867	−0.295433	−	0.955363i	$-0.595464\pi$
$601$	−14.4853	−0.590867	−0.295433	+	0.955363i	$0.595464\pi$
$602$	0	0
$603$	0	0
$604$	−72.3553	−2.94410
$605$	2.00000	0.0813116
$606$	0	0
$607$	−18.6863	−0.758453	−0.379227	−	0.925304i	$-0.623810\pi$
$607$	−18.6863	−0.758453	−0.379227	+	0.925304i	$0.623810\pi$
$608$	−11.4853	−0.465790
$609$	0	0
$610$	−19.3137	−0.781989
$611$	8.14214	0.329396
$612$	0	0
$613$	25.1716	1.01667	0.508335	−	0.861159i	$-0.330261\pi$
$613$	25.1716	1.01667	0.508335	+	0.861159i	$0.330261\pi$
$614$	−41.7990	−1.68687
$615$	0	0
$616$	0	0
$617$	−13.1716	−0.530268	−0.265134	−	0.964212i	$-0.585416\pi$
$617$	−13.1716	−0.530268	−0.265134	+	0.964212i	$0.585416\pi$
$618$	0	0
$619$	34.6274	1.39179	0.695897	−	0.718142i	$-0.255007\pi$
$619$	34.6274	1.39179	0.695897	+	0.718142i	$0.255007\pi$
$620$	43.3137	1.73952
$621$	0	0
$622$	−71.5269	−2.86797
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	11.5858	0.463061
$627$	0	0
$628$	26.7990	1.06940
$629$	−41.8701	−1.66947
$630$	0	0
$631$	26.2843	1.04636	0.523180	−	0.852222i	$-0.324745\pi$
$631$	26.2843	1.04636	0.523180	+	0.852222i	$0.324745\pi$
$632$	58.7696	2.33773
$633$	0	0
$634$	61.1127	2.42710
$635$	−32.1421	−1.27552
$636$	0	0
$637$	0	0
$638$	7.82843	0.309930
$639$	0	0
$640$	−41.1127	−1.62512
$641$	12.4853	0.493139	0.246569	−	0.969125i	$-0.420697\pi$
$641$	12.4853	0.493139	0.246569	+	0.969125i	$0.420697\pi$
$642$	0	0
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	0	0
$645$	0	0
$646$	77.1838	3.03675
$647$	19.3137	0.759300	0.379650	−	0.925130i	$-0.376044\pi$
$647$	19.3137	0.759300	0.379650	+	0.925130i	$0.376044\pi$
$648$	0	0
$649$	−8.65685	−0.339811
$650$	2.00000	0.0784465
$651$	0	0
$652$	−80.2843	−3.14417
$653$	−31.1127	−1.21753	−0.608767	−	0.793349i	$-0.708336\pi$
$653$	−31.1127	−1.21753	−0.608767	+	0.793349i	$0.708336\pi$
$654$	0	0
$655$	10.3431	0.404140
$656$	3.51472	0.137227
$657$	0	0
$658$	0	0
$659$	−34.4853	−1.34336	−0.671678	−	0.740843i	$-0.734426\pi$
$659$	−34.4853	−1.34336	−0.671678	+	0.740843i	$0.734426\pi$
$660$	0	0
$661$	11.9706	0.465601	0.232800	−	0.972525i	$-0.425211\pi$
$661$	11.9706	0.465601	0.232800	+	0.972525i	$0.425211\pi$
$662$	54.2843	2.10982
$663$	0	0
$664$	12.4853	0.484523
$665$	0	0
$666$	0	0
$667$	−22.6985	−0.878889
$668$	−19.7990	−0.766046
$669$	0	0
$670$	−15.3137	−0.591620
$671$	4.00000	0.154418
$672$	0	0
$673$	27.1716	1.04739	0.523694	−	0.851907i	$-0.324554\pi$
$673$	27.1716	1.04739	0.523694	+	0.851907i	$0.324554\pi$
$674$	9.31371	0.358751
$675$	0	0
$676$	−47.1421	−1.81316
$677$	32.0122	1.23033	0.615164	−	0.788399i	$-0.289090\pi$
$677$	32.0122	1.23033	0.615164	+	0.788399i	$0.289090\pi$
$678$	0	0
$679$	0	0
$680$	38.9706	1.49445
$681$	0	0
$682$	−13.6569	−0.522948
$683$	−3.20101	−0.122483	−0.0612416	−	0.998123i	$-0.519506\pi$
$683$	−3.20101	−0.122483	−0.0612416	+	0.998123i	$0.519506\pi$
$684$	0	0
$685$	−0.970563	−0.0370833
$686$	0	0
$687$	0	0
$688$	−8.27208	−0.315370
$689$	−5.94113	−0.226339
$690$	0	0
$691$	−11.4558	−0.435801	−0.217900	−	0.975971i	$-0.569921\pi$
$691$	−11.4558	−0.435801	−0.217900	+	0.975971i	$0.569921\pi$
$692$	56.7696	2.15805
$693$	0	0
$694$	−36.1421	−1.37194
$695$	16.8284	0.638339
$696$	0	0
$697$	5.17157	0.195887
$698$	26.4853	1.00248
$699$	0	0
$700$	0	0
$701$	2.89949	0.109512	0.0547562	−	0.998500i	$-0.482562\pi$
$701$	2.89949	0.109512	0.0547562	+	0.998500i	$0.482562\pi$
$702$	0	0
$703$	−68.6985	−2.59101
$704$	9.82843	0.370423
$705$	0	0
$706$	−32.1421	−1.20969
$707$	0	0
$708$	0	0
$709$	25.7696	0.967796	0.483898	−	0.875124i	$-0.339221\pi$
$709$	25.7696	0.967796	0.483898	+	0.875124i	$0.339221\pi$
$710$	20.1421	0.755921
$711$	0	0
$712$	62.4264	2.33953
$713$	39.5980	1.48296
$714$	0	0
$715$	1.65685	0.0619628
$716$	−68.2548	−2.55080
$717$	0	0
$718$	19.3137	0.720781
$719$	−11.2010	−0.417727	−0.208864	−	0.977945i	$-0.566976\pi$
$719$	−11.2010	−0.417727	−0.208864	+	0.977945i	$0.566976\pi$
$720$	0	0
$721$	0	0
$722$	80.7696	3.00593
$723$	0	0
$724$	−44.6274	−1.65856
$725$	3.24264	0.120429
$726$	0	0
$727$	46.0833	1.70913	0.854567	−	0.519342i	$-0.173823\pi$
$727$	46.0833	1.70913	0.854567	+	0.519342i	$0.173823\pi$
$728$	0	0
$729$	0	0
$730$	1.65685	0.0613229
$731$	−12.1716	−0.450182
$732$	0	0
$733$	9.65685	0.356684	0.178342	−	0.983969i	$-0.442927\pi$
$733$	9.65685	0.356684	0.178342	+	0.983969i	$0.442927\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−11.1005	−0.409170
$737$	3.17157	0.116826
$738$	0	0
$739$	−39.6569	−1.45880	−0.729400	−	0.684087i	$-0.760201\pi$
$739$	−39.6569	−1.45880	−0.729400	+	0.684087i	$0.760201\pi$
$740$	−72.6274	−2.66984
$741$	0	0
$742$	0	0
$743$	−33.7990	−1.23996	−0.619982	−	0.784616i	$-0.712860\pi$
$743$	−33.7990	−1.23996	−0.619982	+	0.784616i	$0.712860\pi$
$744$	0	0
$745$	44.4264	1.62766
$746$	−4.00000	−0.146450
$747$	0	0
$748$	−16.8995	−0.617907
$749$	0	0
$750$	0	0
$751$	−29.3137	−1.06967	−0.534836	−	0.844956i	$-0.679627\pi$
$751$	−29.3137	−1.06967	−0.534836	+	0.844956i	$0.679627\pi$
$752$	−29.4853	−1.07522
$753$	0	0
$754$	6.48528	0.236180
$755$	−37.7990	−1.37565
$756$	0	0
$757$	−7.68629	−0.279363	−0.139682	−	0.990196i	$-0.544608\pi$
$757$	−7.68629	−0.279363	−0.139682	+	0.990196i	$0.544608\pi$
$758$	45.4558	1.65103
$759$	0	0
$760$	63.9411	2.31939
$761$	0.201010	0.00728661	0.00364331	−	0.999993i	$-0.498840\pi$
$761$	0.201010	0.00728661	0.00364331	+	0.999993i	$0.498840\pi$
$762$	0	0
$763$	0	0
$764$	82.9117	2.99964
$765$	0	0
$766$	−5.58579	−0.201823
$767$	−7.17157	−0.258950
$768$	0	0
$769$	33.7990	1.21882	0.609411	−	0.792854i	$-0.291406\pi$
$769$	33.7990	1.21882	0.609411	+	0.792854i	$0.291406\pi$
$770$	0	0
$771$	0	0
$772$	−46.4853	−1.67304
$773$	51.6569	1.85797	0.928984	−	0.370120i	$-0.120683\pi$
$773$	51.6569	1.85797	0.928984	+	0.370120i	$0.120683\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	−50.6985	−1.81997
$777$	0	0
$778$	28.4853	1.02125
$779$	8.48528	0.304017
$780$	0	0
$781$	−4.17157	−0.149271
$782$	74.5980	2.66762
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	29.5269	1.05252	0.526260	−	0.850323i	$-0.323594\pi$
$787$	29.5269	1.05252	0.526260	+	0.850323i	$0.323594\pi$
$788$	62.8406	2.23860
$789$	0	0
$790$	64.2843	2.28713
$791$	0	0
$792$	0	0
$793$	3.31371	0.117673
$794$	42.2132	1.49809
$795$	0	0
$796$	−75.7990	−2.68662
$797$	−11.1716	−0.395717	−0.197859	−	0.980231i	$-0.563399\pi$
$797$	−11.1716	−0.395717	−0.197859	+	0.980231i	$0.563399\pi$
$798$	0	0
$799$	−43.3848	−1.53484
$800$	1.58579	0.0560660
$801$	0	0
$802$	−18.1421	−0.640621
$803$	−0.343146	−0.0121094
$804$	0	0
$805$	0	0
$806$	−11.3137	−0.398508
$807$	0	0
$808$	−21.6274	−0.760850
$809$	−10.8284	−0.380707	−0.190354	−	0.981716i	$-0.560963\pi$
$809$	−10.8284	−0.380707	−0.190354	+	0.981716i	$0.560963\pi$
$810$	0	0
$811$	14.9706	0.525688	0.262844	−	0.964838i	$-0.415340\pi$
$811$	14.9706	0.525688	0.262844	+	0.964838i	$0.415340\pi$
$812$	0	0
$813$	0	0
$814$	22.8995	0.802627
$815$	−41.9411	−1.46913
$816$	0	0
$817$	−19.9706	−0.698682
$818$	−18.8284	−0.658321
$819$	0	0
$820$	8.97056	0.313266
$821$	−38.8284	−1.35512	−0.677561	−	0.735467i	$-0.736963\pi$
$821$	−38.8284	−1.35512	−0.677561	+	0.735467i	$0.736963\pi$
$822$	0	0
$823$	−33.1716	−1.15629	−0.578144	−	0.815935i	$-0.696223\pi$
$823$	−33.1716	−1.15629	−0.578144	+	0.815935i	$0.696223\pi$
$824$	55.1127	1.91994
$825$	0	0
$826$	0	0
$827$	37.3137	1.29752	0.648762	−	0.760991i	$-0.275287\pi$
$827$	37.3137	1.29752	0.648762	+	0.760991i	$0.275287\pi$
$828$	0	0
$829$	−6.17157	−0.214348	−0.107174	−	0.994240i	$-0.534180\pi$
$829$	−6.17157	−0.214348	−0.107174	+	0.994240i	$0.534180\pi$
$830$	13.6569	0.474036
$831$	0	0
$832$	8.14214	0.282278
$833$	0	0
$834$	0	0
$835$	−10.3431	−0.357939
$836$	−27.7279	−0.958990
$837$	0	0
$838$	−35.7279	−1.23420
$839$	−8.97056	−0.309698	−0.154849	−	0.987938i	$-0.549489\pi$
$839$	−8.97056	−0.309698	−0.154849	+	0.987938i	$0.549489\pi$
$840$	0	0
$841$	−18.4853	−0.637423
$842$	−16.8995	−0.582395
$843$	0	0
$844$	57.3137	1.97282
$845$	−24.6274	−0.847209
$846$	0	0
$847$	0	0
$848$	21.5147	0.738818
$849$	0	0
$850$	−10.6569	−0.365527
$851$	−66.3970	−2.27606
$852$	0	0
$853$	−12.4853	−0.427488	−0.213744	−	0.976890i	$-0.568566\pi$
$853$	−12.4853	−0.427488	−0.213744	+	0.976890i	$0.568566\pi$
$854$	0	0
$855$	0	0
$856$	−42.6274	−1.45698
$857$	44.3553	1.51515	0.757575	−	0.652748i	$-0.226384\pi$
$857$	44.3553	1.51515	0.757575	+	0.652748i	$0.226384\pi$
$858$	0	0
$859$	−24.4853	−0.835427	−0.417714	−	0.908579i	$-0.637168\pi$
$859$	−24.4853	−0.835427	−0.417714	+	0.908579i	$0.637168\pi$
$860$	−21.1127	−0.719937
$861$	0	0
$862$	−16.8284	−0.573179
$863$	−2.62742	−0.0894383	−0.0447192	−	0.999000i	$-0.514239\pi$
$863$	−2.62742	−0.0894383	−0.0447192	+	0.999000i	$0.514239\pi$
$864$	0	0
$865$	29.6569	1.00836
$866$	2.07107	0.0703777
$867$	0	0
$868$	0	0
$869$	−13.3137	−0.451637
$870$	0	0
$871$	2.62742	0.0890266
$872$	−30.1421	−1.02074
$873$	0	0
$874$	122.397	4.14014
$875$	0	0
$876$	0	0
$877$	2.48528	0.0839220	0.0419610	−	0.999119i	$-0.486639\pi$
$877$	2.48528	0.0839220	0.0419610	+	0.999119i	$0.486639\pi$
$878$	74.5980	2.51756
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−16.6863	−0.562175	−0.281088	−	0.959682i	$-0.590695\pi$
$881$	−16.6863	−0.562175	−0.281088	+	0.959682i	$0.590695\pi$
$882$	0	0
$883$	39.6569	1.33456	0.667280	−	0.744807i	$-0.267459\pi$
$883$	39.6569	1.33456	0.667280	+	0.744807i	$0.267459\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	−81.1838	−2.72742
$887$	13.3137	0.447031	0.223515	−	0.974700i	$-0.428247\pi$
$887$	13.3137	0.447031	0.223515	+	0.974700i	$0.428247\pi$
$888$	0	0
$889$	0	0
$890$	68.2843	2.28889
$891$	0	0
$892$	−87.9411	−2.94449
$893$	−71.1838	−2.38207
$894$	0	0
$895$	−35.6569	−1.19188
$896$	0	0
$897$	0	0
$898$	9.65685	0.322253
$899$	−18.3431	−0.611778
$900$	0	0
$901$	31.6569	1.05464
$902$	−2.82843	−0.0941763
$903$	0	0
$904$	33.7990	1.12414
$905$	−23.3137	−0.774974
$906$	0	0
$907$	−17.5147	−0.581567	−0.290783	−	0.956789i	$-0.593916\pi$
$907$	−17.5147	−0.581567	−0.290783	+	0.956789i	$0.593916\pi$
$908$	−57.3137	−1.90202
$909$	0	0
$910$	0	0
$911$	4.51472	0.149579	0.0747897	−	0.997199i	$-0.476171\pi$
$911$	4.51472	0.149579	0.0747897	+	0.997199i	$0.476171\pi$
$912$	0	0
$913$	−2.82843	−0.0936073
$914$	45.4558	1.50355
$915$	0	0
$916$	−59.9411	−1.98051
$917$	0	0
$918$	0	0
$919$	56.3553	1.85899	0.929496	−	0.368833i	$-0.120243\pi$
$919$	56.3553	1.85899	0.929496	+	0.368833i	$0.120243\pi$
$920$	61.7990	2.03745
$921$	0	0
$922$	−16.3137	−0.537263
$923$	−3.45584	−0.113750
$924$	0	0
$925$	9.48528	0.311874
$926$	44.9706	1.47782
$927$	0	0
$928$	5.14214	0.168799
$929$	22.8284	0.748976	0.374488	−	0.927232i	$-0.377818\pi$
$929$	22.8284	0.748976	0.374488	+	0.927232i	$0.377818\pi$
$930$	0	0
$931$	0	0
$932$	−39.8701	−1.30599
$933$	0	0
$934$	20.0711	0.656745
$935$	−8.82843	−0.288720
$936$	0	0
$937$	5.85786	0.191368	0.0956840	−	0.995412i	$-0.469496\pi$
$937$	5.85786	0.191368	0.0956840	+	0.995412i	$0.469496\pi$
$938$	0	0
$939$	0	0
$940$	−75.2548	−2.45454
$941$	12.0711	0.393506	0.196753	−	0.980453i	$-0.436960\pi$
$941$	12.0711	0.393506	0.196753	+	0.980453i	$0.436960\pi$
$942$	0	0
$943$	8.20101	0.267062
$944$	25.9706	0.845270
$945$	0	0
$946$	6.65685	0.216433
$947$	−37.4853	−1.21811	−0.609054	−	0.793129i	$-0.708451\pi$
$947$	−37.4853	−1.21811	−0.609054	+	0.793129i	$0.708451\pi$
$948$	0	0
$949$	−0.284271	−0.00922784
$950$	−17.4853	−0.567297
$951$	0	0
$952$	0	0
$953$	14.1421	0.458109	0.229054	−	0.973414i	$-0.426437\pi$
$953$	14.1421	0.458109	0.229054	+	0.973414i	$0.426437\pi$
$954$	0	0
$955$	43.3137	1.40160
$956$	24.8284	0.803009
$957$	0	0
$958$	−33.7990	−1.09200
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	18.9706	0.611635
$963$	0	0
$964$	28.0000	0.901819
$965$	−24.2843	−0.781738
$966$	0	0
$967$	−21.0416	−0.676653	−0.338327	−	0.941029i	$-0.609861\pi$
$967$	−21.0416	−0.676653	−0.338327	+	0.941029i	$0.609861\pi$
$968$	4.41421	0.141878
$969$	0	0
$970$	−55.4558	−1.78058
$971$	−4.97056	−0.159513	−0.0797565	−	0.996814i	$-0.525414\pi$
$971$	−4.97056	−0.159513	−0.0797565	+	0.996814i	$0.525414\pi$
$972$	0	0
$973$	0	0
$974$	−78.7696	−2.52394
$975$	0	0
$976$	−12.0000	−0.384111
$977$	12.8284	0.410418	0.205209	−	0.978718i	$-0.434213\pi$
$977$	12.8284	0.410418	0.205209	+	0.978718i	$0.434213\pi$
$978$	0	0
$979$	−14.1421	−0.451985
$980$	0	0
$981$	0	0
$982$	3.65685	0.116695
$983$	2.02944	0.0647290	0.0323645	−	0.999476i	$-0.489696\pi$
$983$	2.02944	0.0647290	0.0323645	+	0.999476i	$0.489696\pi$
$984$	0	0
$985$	32.8284	1.04600
$986$	−34.5563	−1.10050
$987$	0	0
$988$	−22.9706	−0.730791
$989$	−19.3015	−0.613752
$990$	0	0
$991$	−24.6863	−0.784186	−0.392093	−	0.919926i	$-0.628249\pi$
$991$	−24.6863	−0.784186	−0.392093	+	0.919926i	$0.628249\pi$
$992$	−8.97056	−0.284816
$993$	0	0
$994$	0	0
$995$	−39.5980	−1.25534
$996$	0	0
$997$	9.85786	0.312202	0.156101	−	0.987741i	$-0.450108\pi$
$997$	9.85786	0.312202	0.156101	+	0.987741i	$0.450108\pi$
$998$	−76.7696	−2.43010
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.be.1.2		2
3.2	odd	2		1617.2.a.n.1.1		2
7.2	even	3		693.2.i.f.298.1		4
7.4	even	3		693.2.i.f.100.1		4
7.6	odd	2		4851.2.a.bd.1.2		2
21.2	odd	6		231.2.i.d.67.2	✓	4
21.11	odd	6		231.2.i.d.100.2	yes	4
21.20	even	2		1617.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.i.d.67.2	✓	4	21.2	odd	6
231.2.i.d.100.2	yes	4	21.11	odd	6
693.2.i.f.100.1		4	7.4	even	3
693.2.i.f.298.1		4	7.2	even	3
1617.2.a.m.1.1		2	21.20	even	2
1617.2.a.n.1.1		2	3.2	odd	2
4851.2.a.bd.1.2		2	7.6	odd	2
4851.2.a.be.1.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{5} +4.41421 q^{8} +4.82843 q^{10} -1.00000 q^{11} -0.828427 q^{13} +3.00000 q^{16} +4.41421 q^{17} +7.24264 q^{19} +7.65685 q^{20} -2.41421 q^{22} +7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} -3.24264 q^{29} +5.65685 q^{31} -1.58579 q^{32} +10.6569 q^{34} -9.48528 q^{37} +17.4853 q^{38} +8.82843 q^{40} +1.17157 q^{41} -2.75736 q^{43} -3.82843 q^{44} +16.8995 q^{46} -9.82843 q^{47} -2.41421 q^{50} -3.17157 q^{52} +7.17157 q^{53} -2.00000 q^{55} -7.82843 q^{58} +8.65685 q^{59} -4.00000 q^{61} +13.6569 q^{62} -9.82843 q^{64} -1.65685 q^{65} -3.17157 q^{67} +16.8995 q^{68} +4.17157 q^{71} +0.343146 q^{73} -22.8995 q^{74} +27.7279 q^{76} +13.3137 q^{79} +6.00000 q^{80} +2.82843 q^{82} +2.82843 q^{83} +8.82843 q^{85} -6.65685 q^{86} -4.41421 q^{88} +14.1421 q^{89} +26.7990 q^{92} -23.7279 q^{94} +14.4853 q^{95} -11.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} + 6 q^{16} + 6 q^{17} + 6 q^{19} + 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{29} - 6 q^{32} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 12 q^{40} + 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 2 q^{50} - 12 q^{52} + 20 q^{53} - 4 q^{55} - 10 q^{58} + 6 q^{59} - 8 q^{61} + 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} + 14 q^{68} + 14 q^{71} + 12 q^{73} - 26 q^{74} + 30 q^{76} + 4 q^{79} + 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} - 22 q^{94} + 12 q^{95} - 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^10 - 2 * q^11 + 4 * q^13 + 6 * q^16 + 6 * q^17 + 6 * q^19 + 4 * q^20 - 2 * q^22 + 14 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^29 - 6 * q^32 + 10 * q^34 - 2 * q^37 + 18 * q^38 + 12 * q^40 + 8 * q^41 - 14 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 2 * q^50 - 12 * q^52 + 20 * q^53 - 4 * q^55 - 10 * q^58 + 6 * q^59 - 8 * q^61 + 16 * q^62 - 14 * q^64 + 8 * q^65 - 12 * q^67 + 14 * q^68 + 14 * q^71 + 12 * q^73 - 26 * q^74 + 30 * q^76 + 4 * q^79 + 12 * q^80 + 12 * q^85 - 2 * q^86 - 6 * q^88 + 14 * q^92 - 22 * q^94 + 12 * q^95 - 6 * q^97

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