LMFDB - Embedded newform 4851.2.a.bn.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:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.91223$ 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)

$f(q)$	$=$	$q-0.656620 q^{2} -1.56885 q^{4} -3.56885 q^{5} +2.34338 q^{8} +O(q^{10})$	$q-0.656620 q^{2} -1.56885 q^{4} -3.56885 q^{5} +2.34338 q^{8} +2.34338 q^{10} +1.00000 q^{11} +5.91223 q^{13} +1.59899 q^{16} -1.65662 q^{17} -1.48108 q^{19} +5.59899 q^{20} -0.656620 q^{22} -3.34338 q^{23} +7.73669 q^{25} -3.88209 q^{26} -3.08007 q^{29} +7.08007 q^{31} -5.73669 q^{32} +1.08777 q^{34} -4.51122 q^{37} +0.972507 q^{38} -8.36317 q^{40} -1.28575 q^{41} +1.59899 q^{43} -1.56885 q^{44} +2.19533 q^{46} -1.65662 q^{47} -5.08007 q^{50} -9.27540 q^{52} -9.22547 q^{53} -3.56885 q^{55} +2.02243 q^{58} +8.85195 q^{59} +6.68676 q^{61} -4.64892 q^{62} +0.568850 q^{64} -21.0999 q^{65} -9.82446 q^{67} +2.59899 q^{68} +8.61878 q^{71} -4.56115 q^{73} +2.96216 q^{74} +2.32359 q^{76} -6.39331 q^{79} -5.70655 q^{80} +0.844248 q^{82} +0.167838 q^{83} +5.91223 q^{85} -1.04993 q^{86} +2.34338 q^{88} -2.56885 q^{89} +5.24526 q^{92} +1.08777 q^{94} +5.28575 q^{95} -9.73669 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{4} - 2 q^{5} + 9 q^{8}+O(q^{10}) $ 3 * q + 4 * q^4 - 2 * q^5 + 9 * q^8	$ 3 q + 4 q^{4} - 2 q^{5} + 9 q^{8} + 9 q^{10} + 3 q^{11} + 11 q^{13} + 2 q^{16} - 3 q^{17} + 11 q^{19} + 14 q^{20} - 12 q^{23} + 3 q^{25} + q^{26} + 9 q^{29} + 3 q^{31} + 3 q^{32} + 10 q^{34} - 4 q^{37} + 8 q^{38} + 3 q^{40} - 5 q^{41} + 2 q^{43} + 4 q^{44} - 10 q^{46} - 3 q^{47} + 3 q^{50} + 7 q^{52} - 17 q^{53} - 2 q^{55} - 13 q^{58} + 8 q^{59} + 24 q^{61} + 13 q^{62} - 7 q^{64} - 15 q^{65} - 16 q^{67} + 5 q^{68} - 7 q^{71} + 20 q^{73} - 22 q^{74} + 39 q^{76} + 3 q^{79} + 9 q^{80} - 41 q^{82} - 11 q^{83} + 11 q^{85} + 21 q^{86} + 9 q^{88} + q^{89} - 25 q^{92} + 10 q^{94} + 17 q^{95} - 9 q^{97}+O(q^{100}) $ 3 * q + 4 * q^4 - 2 * q^5 + 9 * q^8 + 9 * q^10 + 3 * q^11 + 11 * q^13 + 2 * q^16 - 3 * q^17 + 11 * q^19 + 14 * q^20 - 12 * q^23 + 3 * q^25 + q^26 + 9 * q^29 + 3 * q^31 + 3 * q^32 + 10 * q^34 - 4 * q^37 + 8 * q^38 + 3 * q^40 - 5 * q^41 + 2 * q^43 + 4 * q^44 - 10 * q^46 - 3 * q^47 + 3 * q^50 + 7 * q^52 - 17 * q^53 - 2 * q^55 - 13 * q^58 + 8 * q^59 + 24 * q^61 + 13 * q^62 - 7 * q^64 - 15 * q^65 - 16 * q^67 + 5 * q^68 - 7 * q^71 + 20 * q^73 - 22 * q^74 + 39 * q^76 + 3 * q^79 + 9 * q^80 - 41 * q^82 - 11 * q^83 + 11 * q^85 + 21 * q^86 + 9 * q^88 + q^89 - 25 * q^92 + 10 * q^94 + 17 * q^95 - 9 * 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 $

$2$	−0.656620	−0.464301	−0.232150	−	0.972680i	$-0.574576\pi$
$2$	−0.656620	−0.464301	−0.232150	+	0.972680i	$0.574576\pi$
$3$	0	0
$3$	0	0
$4$	−1.56885	−0.784425
$5$	−3.56885	−1.59604	−0.798019	−	0.602632i	$-0.794119\pi$
$5$	−3.56885	−1.59604	−0.798019	+	0.602632i	$0.794119\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.34338	0.828510
$9$	0	0
$10$	2.34338	0.741042
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.91223	1.63976	0.819879	−	0.572537i	$-0.194041\pi$
$13$	5.91223	1.63976	0.819879	+	0.572537i	$0.194041\pi$
$14$	0	0
$15$	0	0
$16$	1.59899	0.399747
$17$	−1.65662	−0.401789	−0.200895	−	0.979613i	$-0.564385\pi$
$17$	−1.65662	−0.401789	−0.200895	+	0.979613i	$0.564385\pi$
$18$	0	0
$19$	−1.48108	−0.339783	−0.169891	−	0.985463i	$-0.554342\pi$
$19$	−1.48108	−0.339783	−0.169891	+	0.985463i	$0.554342\pi$
$20$	5.59899	1.25197
$21$	0	0
$22$	−0.656620	−0.139992
$23$	−3.34338	−0.697143	−0.348571	−	0.937282i	$-0.613333\pi$
$23$	−3.34338	−0.697143	−0.348571	+	0.937282i	$0.613333\pi$
$24$	0	0
$25$	7.73669	1.54734
$26$	−3.88209	−0.761341
$27$	0	0
$28$	0	0
$29$	−3.08007	−0.571954	−0.285977	−	0.958236i	$-0.592318\pi$
$29$	−3.08007	−0.571954	−0.285977	+	0.958236i	$0.592318\pi$
$30$	0	0
$31$	7.08007	1.27162	0.635809	−	0.771847i	$-0.280667\pi$
$31$	7.08007	1.27162	0.635809	+	0.771847i	$0.280667\pi$
$32$	−5.73669	−1.01411
$33$	0	0
$34$	1.08777	0.186551
$35$	0	0
$36$	0	0
$37$	−4.51122	−0.741640	−0.370820	−	0.928705i	$-0.620923\pi$
$37$	−4.51122	−0.741640	−0.370820	+	0.928705i	$0.620923\pi$
$38$	0.972507	0.157761
$39$	0	0
$40$	−8.36317	−1.32233
$41$	−1.28575	−0.200800	−0.100400	−	0.994947i	$-0.532012\pi$
$41$	−1.28575	−0.200800	−0.100400	+	0.994947i	$0.532012\pi$
$42$	0	0
$43$	1.59899	0.243843	0.121922	−	0.992540i	$-0.461094\pi$
$43$	1.59899	0.243843	0.121922	+	0.992540i	$0.461094\pi$
$44$	−1.56885	−0.236513
$45$	0	0
$46$	2.19533	0.323684
$47$	−1.65662	−0.241643	−0.120821	−	0.992674i	$-0.538553\pi$
$47$	−1.65662	−0.241643	−0.120821	+	0.992674i	$0.538553\pi$
$48$	0	0
$49$	0	0
$50$	−5.08007	−0.718430
$51$	0	0
$52$	−9.27540	−1.28627
$53$	−9.22547	−1.26722	−0.633608	−	0.773654i	$-0.718427\pi$
$53$	−9.22547	−1.26722	−0.633608	+	0.773654i	$0.718427\pi$
$54$	0	0
$55$	−3.56885	−0.481224
$56$	0	0
$57$	0	0
$58$	2.02243	0.265559
$59$	8.85195	1.15243	0.576213	−	0.817300i	$-0.304530\pi$
$59$	8.85195	1.15243	0.576213	+	0.817300i	$0.304530\pi$
$60$	0	0
$61$	6.68676	0.856152	0.428076	−	0.903743i	$-0.359192\pi$
$61$	6.68676	0.856152	0.428076	+	0.903743i	$0.359192\pi$
$62$	−4.64892	−0.590413
$63$	0	0
$64$	0.568850	0.0711062
$65$	−21.0999	−2.61712
$66$	0	0
$67$	−9.82446	−1.20025	−0.600124	−	0.799907i	$-0.704882\pi$
$67$	−9.82446	−1.20025	−0.600124	+	0.799907i	$0.704882\pi$
$68$	2.59899	0.315174
$69$	0	0
$70$	0	0
$71$	8.61878	1.02286	0.511430	−	0.859325i	$-0.329116\pi$
$71$	8.61878	1.02286	0.511430	+	0.859325i	$0.329116\pi$
$72$	0	0
$73$	−4.56115	−0.533842	−0.266921	−	0.963718i	$-0.586006\pi$
$73$	−4.56115	−0.533842	−0.266921	+	0.963718i	$0.586006\pi$
$74$	2.96216	0.344344
$75$	0	0
$76$	2.32359	0.266534
$77$	0	0
$78$	0	0
$79$	−6.39331	−0.719303	−0.359652	−	0.933087i	$-0.617104\pi$
$79$	−6.39331	−0.719303	−0.359652	+	0.933087i	$0.617104\pi$
$80$	−5.70655	−0.638012
$81$	0	0
$82$	0.844248	0.0932316
$83$	0.167838	0.0184226	0.00921130	−	0.999958i	$-0.497068\pi$
$83$	0.167838	0.0184226	0.00921130	+	0.999958i	$0.497068\pi$
$84$	0	0
$85$	5.91223	0.641271
$86$	−1.04993	−0.113217
$87$	0	0
$88$	2.34338	0.249805
$89$	−2.56885	−0.272298	−0.136149	−	0.990688i	$-0.543473\pi$
$89$	−2.56885	−0.272298	−0.136149	+	0.990688i	$0.543473\pi$
$90$	0	0
$91$	0	0
$92$	5.24526	0.546856
$93$	0	0
$94$	1.08777	0.112195
$95$	5.28575	0.542306
$96$	0	0
$97$	−9.73669	−0.988611	−0.494305	−	0.869288i	$-0.664578\pi$
$97$	−9.73669	−0.988611	−0.494305	+	0.869288i	$0.664578\pi$
$98$	0	0
$99$	0	0
$100$	−12.1377	−1.21377
$101$	1.85460	0.184539	0.0922697	−	0.995734i	$-0.470588\pi$
$101$	1.85460	0.184539	0.0922697	+	0.995734i	$0.470588\pi$
$102$	0	0
$103$	3.16784	0.312136	0.156068	−	0.987746i	$-0.450118\pi$
$103$	3.16784	0.312136	0.156068	+	0.987746i	$0.450118\pi$
$104$	13.8546	1.35856
$105$	0	0
$106$	6.05763	0.588369
$107$	4.76683	0.460826	0.230413	−	0.973093i	$-0.425992\pi$
$107$	4.76683	0.460826	0.230413	+	0.973093i	$0.425992\pi$
$108$	0	0
$109$	−14.8821	−1.42545	−0.712723	−	0.701446i	$-0.752538\pi$
$109$	−14.8821	−1.42545	−0.712723	+	0.701446i	$0.752538\pi$
$110$	2.34338	0.223432
$111$	0	0
$112$	0	0
$113$	−12.4432	−1.17056	−0.585281	−	0.810831i	$-0.699016\pi$
$113$	−12.4432	−1.17056	−0.585281	+	0.810831i	$0.699016\pi$
$114$	0	0
$115$	11.9320	1.11267
$116$	4.83216	0.448655
$117$	0	0
$118$	−5.81237	−0.535072
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.39066	−0.397512
$123$	0	0
$124$	−11.1076	−0.997488
$125$	−9.76683	−0.873571
$126$	0	0
$127$	−6.62142	−0.587556	−0.293778	−	0.955874i	$-0.594913\pi$
$127$	−6.62142	−0.587556	−0.293778	+	0.955874i	$0.594913\pi$
$128$	11.0999	0.981098
$129$	0	0
$130$	13.8546	1.21513
$131$	−6.05763	−0.529258	−0.264629	−	0.964350i	$-0.585249\pi$
$131$	−6.05763	−0.529258	−0.264629	+	0.964350i	$0.585249\pi$
$132$	0	0
$133$	0	0
$134$	6.45094	0.557276
$135$	0	0
$136$	−3.88209	−0.332887
$137$	7.42345	0.634228	0.317114	−	0.948387i	$-0.397286\pi$
$137$	7.42345	0.634228	0.317114	+	0.948387i	$0.397286\pi$
$138$	0	0
$139$	10.8245	0.918119	0.459059	−	0.888406i	$-0.348187\pi$
$139$	10.8245	0.918119	0.459059	+	0.888406i	$0.348187\pi$
$140$	0	0
$141$	0	0
$142$	−5.65927	−0.474915
$143$	5.91223	0.494405
$144$	0	0
$145$	10.9923	0.912861
$146$	2.99494	0.247863
$147$	0	0
$148$	7.07742	0.581760
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	16.4234	1.33652	0.668261	−	0.743927i	$-0.267039\pi$
$151$	16.4234	1.33652	0.668261	+	0.743927i	$0.267039\pi$
$152$	−3.47073	−0.281513
$153$	0	0
$154$	0	0
$155$	−25.2677	−2.02955
$156$	0	0
$157$	11.4509	0.913885	0.456942	−	0.889496i	$-0.348945\pi$
$157$	11.4509	0.913885	0.456942	+	0.889496i	$0.348945\pi$
$158$	4.19798	0.333973
$159$	0	0
$160$	20.4734	1.61856
$161$	0	0
$162$	0	0
$163$	−8.93972	−0.700213	−0.350107	−	0.936710i	$-0.613855\pi$
$163$	−8.93972	−0.700213	−0.350107	+	0.936710i	$0.613855\pi$
$164$	2.01714	0.157513
$165$	0	0
$166$	−0.110206	−0.00855363
$167$	−18.2178	−1.40973	−0.704867	−	0.709340i	$-0.748993\pi$
$167$	−18.2178	−1.40973	−0.704867	+	0.709340i	$0.748993\pi$
$168$	0	0
$169$	21.9545	1.68880
$170$	−3.88209	−0.297743
$171$	0	0
$172$	−2.50857	−0.191277
$173$	19.5611	1.48721	0.743603	−	0.668621i	$-0.233115\pi$
$173$	19.5611	1.48721	0.743603	+	0.668621i	$0.233115\pi$
$174$	0	0
$175$	0	0
$176$	1.59899	0.120528
$177$	0	0
$178$	1.68676	0.126428
$179$	−3.24791	−0.242760	−0.121380	−	0.992606i	$-0.538732\pi$
$179$	−3.24791	−0.242760	−0.121380	+	0.992606i	$0.538732\pi$
$180$	0	0
$181$	10.3407	0.768621	0.384310	−	0.923204i	$-0.374439\pi$
$181$	10.3407	0.768621	0.384310	+	0.923204i	$0.374439\pi$
$182$	0	0
$183$	0	0
$184$	−7.83481	−0.577590
$185$	16.0999	1.18369
$186$	0	0
$187$	−1.65662	−0.121144
$188$	2.59899	0.189551
$189$	0	0
$190$	−3.47073	−0.251793
$191$	−10.0224	−0.725198	−0.362599	−	0.931945i	$-0.618111\pi$
$191$	−10.0224	−0.725198	−0.362599	+	0.931945i	$0.618111\pi$
$192$	0	0
$193$	25.3253	1.82296	0.911478	−	0.411348i	$-0.134942\pi$
$193$	25.3253	1.82296	0.911478	+	0.411348i	$0.134942\pi$
$194$	6.39331	0.459013
$195$	0	0
$196$	0	0
$197$	24.5809	1.75132	0.875660	−	0.482929i	$-0.160427\pi$
$197$	24.5809	1.75132	0.875660	+	0.482929i	$0.160427\pi$
$198$	0	0
$199$	−5.59128	−0.396356	−0.198178	−	0.980166i	$-0.563502\pi$
$199$	−5.59128	−0.396356	−0.198178	+	0.980166i	$0.563502\pi$
$200$	18.1300	1.28198
$201$	0	0
$202$	−1.21777	−0.0856817
$203$	0	0
$204$	0	0
$205$	4.58864	0.320484
$206$	−2.08007	−0.144925
$207$	0	0
$208$	9.45359	0.655488
$209$	−1.48108	−0.102448
$210$	0	0
$211$	12.0999	0.832988	0.416494	−	0.909138i	$-0.363259\pi$
$211$	12.0999	0.832988	0.416494	+	0.909138i	$0.363259\pi$
$212$	14.4734	0.994035
$213$	0	0
$214$	−3.13000	−0.213962
$215$	−5.70655	−0.389183
$216$	0	0
$217$	0	0
$218$	9.77188	0.661836
$219$	0	0
$220$	5.59899	0.377484
$221$	−9.79432	−0.658837
$222$	0	0
$223$	−15.6265	−1.04643	−0.523213	−	0.852202i	$-0.675267\pi$
$223$	−15.6265	−1.04643	−0.523213	+	0.852202i	$0.675267\pi$
$224$	0	0
$225$	0	0
$226$	8.17048	0.543492
$227$	−15.6687	−1.03997	−0.519984	−	0.854176i	$-0.674062\pi$
$227$	−15.6687	−1.03997	−0.519984	+	0.854176i	$0.674062\pi$
$228$	0	0
$229$	5.57149	0.368175	0.184087	−	0.982910i	$-0.441067\pi$
$229$	5.57149	0.368175	0.184087	+	0.982910i	$0.441067\pi$
$230$	−7.83481	−0.516612
$231$	0	0
$232$	−7.21777	−0.473870
$233$	19.2754	1.26277	0.631387	−	0.775468i	$-0.282486\pi$
$233$	19.2754	1.26277	0.631387	+	0.775468i	$0.282486\pi$
$234$	0	0
$235$	5.91223	0.385671
$236$	−13.8874	−0.903992
$237$	0	0
$238$	0	0
$239$	22.1575	1.43325	0.716624	−	0.697459i	$-0.245686\pi$
$239$	22.1575	1.43325	0.716624	+	0.697459i	$0.245686\pi$
$240$	0	0
$241$	19.8744	1.28022	0.640111	−	0.768283i	$-0.278888\pi$
$241$	19.8744	1.28022	0.640111	+	0.768283i	$0.278888\pi$
$242$	−0.656620	−0.0422092
$243$	0	0
$244$	−10.4905	−0.671587
$245$	0	0
$246$	0	0
$247$	−8.75648	−0.557161
$248$	16.5913	1.05355
$249$	0	0
$250$	6.41310	0.405600
$251$	22.1076	1.39542	0.697708	−	0.716382i	$-0.254203\pi$
$251$	22.1076	1.39542	0.697708	+	0.716382i	$0.254203\pi$
$252$	0	0
$253$	−3.34338	−0.210196
$254$	4.34776	0.272803
$255$	0	0
$256$	−8.42609	−0.526631
$257$	29.1196	1.81643	0.908217	−	0.418500i	$-0.137444\pi$
$257$	29.1196	1.81643	0.908217	+	0.418500i	$0.137444\pi$
$258$	0	0
$259$	0	0
$260$	33.1025	2.05293
$261$	0	0
$262$	3.97757	0.245735
$263$	15.5035	0.955988	0.477994	−	0.878363i	$-0.341364\pi$
$263$	15.5035	0.955988	0.477994	+	0.878363i	$0.341364\pi$
$264$	0	0
$265$	32.9243	2.02252
$266$	0	0
$267$	0	0
$268$	15.4131	0.941505
$269$	1.70655	0.104050	0.0520251	−	0.998646i	$-0.483432\pi$
$269$	1.70655	0.104050	0.0520251	+	0.998646i	$0.483432\pi$
$270$	0	0
$271$	20.5284	1.24701	0.623505	−	0.781820i	$-0.285708\pi$
$271$	20.5284	1.24701	0.623505	+	0.781820i	$0.285708\pi$
$272$	−2.64892	−0.160614
$273$	0	0
$274$	−4.87439	−0.294472
$275$	7.73669	0.466540
$276$	0	0
$277$	−26.6610	−1.60190	−0.800952	−	0.598728i	$-0.795673\pi$
$277$	−26.6610	−1.60190	−0.800952	+	0.598728i	$0.795673\pi$
$278$	−7.10756	−0.426283
$279$	0	0
$280$	0	0
$281$	−15.7444	−0.939232	−0.469616	−	0.882871i	$-0.655608\pi$
$281$	−15.7444	−0.939232	−0.469616	+	0.882871i	$0.655608\pi$
$282$	0	0
$283$	16.0697	0.955246	0.477623	−	0.878565i	$-0.341499\pi$
$283$	16.0697	0.955246	0.477623	+	0.878565i	$0.341499\pi$
$284$	−13.5216	−0.802357
$285$	0	0
$286$	−3.88209	−0.229553
$287$	0	0
$288$	0	0
$289$	−14.2556	−0.838565
$290$	−7.21777	−0.423842
$291$	0	0
$292$	7.15575	0.418759
$293$	15.3357	0.895920	0.447960	−	0.894054i	$-0.352151\pi$
$293$	15.3357	0.895920	0.447960	+	0.894054i	$0.352151\pi$
$294$	0	0
$295$	−31.5913	−1.83932
$296$	−10.5715	−0.614456
$297$	0	0
$298$	−0.656620	−0.0380370
$299$	−19.7668	−1.14315
$300$	0	0
$301$	0	0
$302$	−10.7840	−0.620548
$303$	0	0
$304$	−2.36823	−0.135827
$305$	−23.8640	−1.36645
$306$	0	0
$307$	16.4707	0.940034	0.470017	−	0.882657i	$-0.344248\pi$
$307$	16.4707	0.940034	0.470017	+	0.882657i	$0.344248\pi$
$308$	0	0
$309$	0	0
$310$	16.5913	0.942322
$311$	21.6291	1.22648	0.613238	−	0.789898i	$-0.289867\pi$
$311$	21.6291	1.22648	0.613238	+	0.789898i	$0.289867\pi$
$312$	0	0
$313$	−16.3882	−0.926319	−0.463159	−	0.886275i	$-0.653284\pi$
$313$	−16.3882	−0.926319	−0.463159	+	0.886275i	$0.653284\pi$
$314$	−7.51892	−0.424317
$315$	0	0
$316$	10.0301	0.564239
$317$	−4.46129	−0.250571	−0.125285	−	0.992121i	$-0.539985\pi$
$317$	−4.46129	−0.250571	−0.125285	+	0.992121i	$0.539985\pi$
$318$	0	0
$319$	−3.08007	−0.172451
$320$	−2.03014	−0.113488
$321$	0	0
$322$	0	0
$323$	2.45359	0.136521
$324$	0	0
$325$	45.7411	2.53726
$326$	5.87000	0.325109
$327$	0	0
$328$	−3.01299	−0.166365
$329$	0	0
$330$	0	0
$331$	19.0396	1.04651	0.523255	−	0.852176i	$-0.324718\pi$
$331$	19.0396	1.04651	0.523255	+	0.852176i	$0.324718\pi$
$332$	−0.263312	−0.0144511
$333$	0	0
$334$	11.9622	0.654540
$335$	35.0620	1.91564
$336$	0	0
$337$	27.0147	1.47159	0.735793	−	0.677206i	$-0.236810\pi$
$337$	27.0147	1.47159	0.735793	+	0.677206i	$0.236810\pi$
$338$	−14.4157	−0.784113
$339$	0	0
$340$	−9.27540	−0.503029
$341$	7.08007	0.383407
$342$	0	0
$343$	0	0
$344$	3.74704	0.202027
$345$	0	0
$346$	−12.8442	−0.690511
$347$	20.2178	1.08535	0.542673	−	0.839944i	$-0.317412\pi$
$347$	20.2178	1.08535	0.542673	+	0.839944i	$0.317412\pi$
$348$	0	0
$349$	12.0224	0.643546	0.321773	−	0.946817i	$-0.395721\pi$
$349$	12.0224	0.643546	0.321773	+	0.946817i	$0.395721\pi$
$350$	0	0
$351$	0	0
$352$	−5.73669	−0.305766
$353$	−10.7591	−0.572650	−0.286325	−	0.958133i	$-0.592434\pi$
$353$	−10.7591	−0.572650	−0.286325	+	0.958133i	$0.592434\pi$
$354$	0	0
$355$	−30.7591	−1.63252
$356$	4.03014	0.213597
$357$	0	0
$358$	2.13264	0.112714
$359$	−24.2901	−1.28198	−0.640992	−	0.767548i	$-0.721477\pi$
$359$	−24.2901	−1.28198	−0.640992	+	0.767548i	$0.721477\pi$
$360$	0	0
$361$	−16.8064	−0.884548
$362$	−6.78994	−0.356871
$363$	0	0
$364$	0	0
$365$	16.2780	0.852032
$366$	0	0
$367$	−10.8442	−0.566065	−0.283033	−	0.959110i	$-0.591340\pi$
$367$	−10.8442	−0.566065	−0.283033	+	0.959110i	$0.591340\pi$
$368$	−5.34602	−0.278681
$369$	0	0
$370$	−10.5715	−0.549586
$371$	0	0
$372$	0	0
$373$	−33.0242	−1.70993	−0.854963	−	0.518688i	$-0.826421\pi$
$373$	−33.0242	−1.70993	−0.854963	+	0.518688i	$0.826421\pi$
$374$	1.08777	0.0562473
$375$	0	0
$376$	−3.88209	−0.200204
$377$	−18.2101	−0.937866
$378$	0	0
$379$	−21.9320	−1.12657	−0.563286	−	0.826262i	$-0.690463\pi$
$379$	−21.9320	−1.12657	−0.563286	+	0.826262i	$0.690463\pi$
$380$	−8.29254	−0.425398
$381$	0	0
$382$	6.58094	0.336710
$383$	36.4630	1.86317	0.931587	−	0.363519i	$-0.118425\pi$
$383$	36.4630	1.86317	0.931587	+	0.363519i	$0.118425\pi$
$384$	0	0
$385$	0	0
$386$	−16.6291	−0.846400
$387$	0	0
$388$	15.2754	0.775491
$389$	−19.4760	−0.987473	−0.493737	−	0.869611i	$-0.664369\pi$
$389$	−19.4760	−0.987473	−0.493737	+	0.869611i	$0.664369\pi$
$390$	0	0
$391$	5.53871	0.280105
$392$	0	0
$393$	0	0
$394$	−16.1403	−0.813139
$395$	22.8168	1.14804
$396$	0	0
$397$	7.83987	0.393472	0.196736	−	0.980457i	$-0.436966\pi$
$397$	7.83987	0.393472	0.196736	+	0.980457i	$0.436966\pi$
$398$	3.67135	0.184028
$399$	0	0
$400$	12.3709	0.618544
$401$	24.4459	1.22077	0.610385	−	0.792105i	$-0.291015\pi$
$401$	24.4459	1.22077	0.610385	+	0.792105i	$0.291015\pi$
$402$	0	0
$403$	41.8590	2.08514
$404$	−2.90958	−0.144757
$405$	0	0
$406$	0	0
$407$	−4.51122	−0.223613
$408$	0	0
$409$	2.35373	0.116384	0.0581922	−	0.998305i	$-0.481466\pi$
$409$	2.35373	0.116384	0.0581922	+	0.998305i	$0.481466\pi$
$410$	−3.01299	−0.148801
$411$	0	0
$412$	−4.96986	−0.244847
$413$	0	0
$414$	0	0
$415$	−0.598988	−0.0294032
$416$	−33.9166	−1.66290
$417$	0	0
$418$	0.972507	0.0475669
$419$	9.29081	0.453886	0.226943	−	0.973908i	$-0.427127\pi$
$419$	9.29081	0.453886	0.226943	+	0.973908i	$0.427127\pi$
$420$	0	0
$421$	−39.0319	−1.90230	−0.951149	−	0.308733i	$-0.900095\pi$
$421$	−39.0319	−1.90230	−0.951149	+	0.308733i	$0.900095\pi$
$422$	−7.94501	−0.386757
$423$	0	0
$424$	−21.6188	−1.04990
$425$	−12.8168	−0.621704
$426$	0	0
$427$	0	0
$428$	−7.47843	−0.361484
$429$	0	0
$430$	3.74704	0.180698
$431$	3.80202	0.183137	0.0915685	−	0.995799i	$-0.470812\pi$
$431$	3.80202	0.183137	0.0915685	+	0.995799i	$0.470812\pi$
$432$	0	0
$433$	−8.22041	−0.395048	−0.197524	−	0.980298i	$-0.563290\pi$
$433$	−8.22041	−0.395048	−0.197524	+	0.980298i	$0.563290\pi$
$434$	0	0
$435$	0	0
$436$	23.3478	1.11815
$437$	4.95181	0.236877
$438$	0	0
$439$	−4.54136	−0.216747	−0.108374	−	0.994110i	$-0.534564\pi$
$439$	−4.54136	−0.216747	−0.108374	+	0.994110i	$0.534564\pi$
$440$	−8.36317	−0.398698
$441$	0	0
$442$	6.43115	0.305899
$443$	−13.7444	−0.653016	−0.326508	−	0.945194i	$-0.605872\pi$
$443$	−13.7444	−0.653016	−0.326508	+	0.945194i	$0.605872\pi$
$444$	0	0
$445$	9.16784	0.434597
$446$	10.2607	0.485857
$447$	0	0
$448$	0	0
$449$	−21.5662	−1.01777	−0.508886	−	0.860834i	$-0.669943\pi$
$449$	−21.5662	−1.01777	−0.508886	+	0.860834i	$0.669943\pi$
$450$	0	0
$451$	−1.28575	−0.0605435
$452$	19.5216	0.918217
$453$	0	0
$454$	10.2884	0.482858
$455$	0	0
$456$	0	0
$457$	−3.30554	−0.154627	−0.0773133	−	0.997007i	$-0.524634\pi$
$457$	−3.30554	−0.154627	−0.0773133	+	0.997007i	$0.524634\pi$
$458$	−3.65836	−0.170944
$459$	0	0
$460$	−18.7195	−0.872803
$461$	−32.1524	−1.49749	−0.748744	−	0.662859i	$-0.769343\pi$
$461$	−32.1524	−1.49749	−0.748744	+	0.662859i	$0.769343\pi$
$462$	0	0
$463$	5.82181	0.270563	0.135281	−	0.990807i	$-0.456806\pi$
$463$	5.82181	0.270563	0.135281	+	0.990807i	$0.456806\pi$
$464$	−4.92499	−0.228637
$465$	0	0
$466$	−12.6566	−0.586307
$467$	−6.01473	−0.278329	−0.139164	−	0.990269i	$-0.544442\pi$
$467$	−6.01473	−0.278329	−0.139164	+	0.990269i	$0.544442\pi$
$468$	0	0
$469$	0	0
$470$	−3.88209	−0.179067
$471$	0	0
$472$	20.7435	0.954796
$473$	1.59899	0.0735216
$474$	0	0
$475$	−11.4586	−0.525759
$476$	0	0
$477$	0	0
$478$	−14.5491	−0.665459
$479$	17.1351	0.782921	0.391460	−	0.920195i	$-0.371970\pi$
$479$	17.1351	0.782921	0.391460	+	0.920195i	$0.371970\pi$
$480$	0	0
$481$	−26.6714	−1.21611
$482$	−13.0499	−0.594408
$483$	0	0
$484$	−1.56885	−0.0713113
$485$	34.7488	1.57786
$486$	0	0
$487$	5.71425	0.258937	0.129469	−	0.991584i	$-0.458673\pi$
$487$	5.71425	0.258937	0.129469	+	0.991584i	$0.458673\pi$
$488$	15.6696	0.709330
$489$	0	0
$490$	0	0
$491$	−24.0673	−1.08614	−0.543071	−	0.839687i	$-0.682739\pi$
$491$	−24.0673	−1.08614	−0.543071	+	0.839687i	$0.682739\pi$
$492$	0	0
$493$	5.10250	0.229805
$494$	5.74968	0.258690
$495$	0	0
$496$	11.3209	0.508325
$497$	0	0
$498$	0	0
$499$	10.7893	0.482994	0.241497	−	0.970402i	$-0.422362\pi$
$499$	10.7893	0.482994	0.241497	+	0.970402i	$0.422362\pi$
$500$	15.3227	0.685251
$501$	0	0
$502$	−14.5163	−0.647893
$503$	28.0121	1.24900	0.624499	−	0.781026i	$-0.285303\pi$
$503$	28.0121	1.24900	0.624499	+	0.781026i	$0.285303\pi$
$504$	0	0
$505$	−6.61878	−0.294532
$506$	2.19533	0.0975944
$507$	0	0
$508$	10.3880	0.460894
$509$	−1.91487	−0.0848753	−0.0424377	−	0.999099i	$-0.513512\pi$
$509$	−1.91487	−0.0848753	−0.0424377	+	0.999099i	$0.513512\pi$
$510$	0	0
$511$	0	0
$512$	−16.6670	−0.736583
$513$	0	0
$514$	−19.1206	−0.843372
$515$	−11.3055	−0.498181
$516$	0	0
$517$	−1.65662	−0.0728581
$518$	0	0
$519$	0	0
$520$	−49.4450	−2.16831
$521$	1.57920	0.0691859	0.0345930	−	0.999401i	$-0.488987\pi$
$521$	1.57920	0.0691859	0.0345930	+	0.999401i	$0.488987\pi$
$522$	0	0
$523$	8.96986	0.392225	0.196112	−	0.980581i	$-0.437168\pi$
$523$	8.96986	0.392225	0.196112	+	0.980581i	$0.437168\pi$
$524$	9.50351	0.415163
$525$	0	0
$526$	−10.1799	−0.443866
$527$	−11.7290	−0.510923
$528$	0	0
$529$	−11.8218	−0.513992
$530$	−21.6188	−0.939060
$531$	0	0
$532$	0	0
$533$	−7.60163	−0.329263
$534$	0	0
$535$	−17.0121	−0.735497
$536$	−23.0224	−0.994418
$537$	0	0
$538$	−1.12055	−0.0483105
$539$	0	0
$540$	0	0
$541$	18.1025	0.778287	0.389144	−	0.921177i	$-0.372771\pi$
$541$	18.1025	0.778287	0.389144	+	0.921177i	$0.372771\pi$
$542$	−13.4793	−0.578987
$543$	0	0
$544$	9.50351	0.407460
$545$	53.1119	2.27507
$546$	0	0
$547$	22.6885	0.970090	0.485045	−	0.874489i	$-0.338803\pi$
$547$	22.6885	0.970090	0.485045	+	0.874489i	$0.338803\pi$
$548$	−11.6463	−0.497504
$549$	0	0
$550$	−5.08007	−0.216615
$551$	4.56182	0.194340
$552$	0	0
$553$	0	0
$554$	17.5062	0.743765
$555$	0	0
$556$	−16.9819	−0.720195
$557$	−38.3555	−1.62517	−0.812587	−	0.582840i	$-0.801941\pi$
$557$	−38.3555	−1.62517	−0.812587	+	0.582840i	$0.801941\pi$
$558$	0	0
$559$	9.45359	0.399844
$560$	0	0
$561$	0	0
$562$	10.3381	0.436086
$563$	−41.7739	−1.76056	−0.880279	−	0.474456i	$-0.842645\pi$
$563$	−41.7739	−1.76056	−0.880279	+	0.474456i	$0.842645\pi$
$564$	0	0
$565$	44.4080	1.86826
$566$	−10.5517	−0.443521
$567$	0	0
$568$	20.1971	0.847450
$569$	11.8700	0.497616	0.248808	−	0.968553i	$-0.419961\pi$
$569$	11.8700	0.497616	0.248808	+	0.968553i	$0.419961\pi$
$570$	0	0
$571$	19.8013	0.828661	0.414330	−	0.910127i	$-0.364016\pi$
$571$	19.8013	0.828661	0.414330	+	0.910127i	$0.364016\pi$
$572$	−9.27540	−0.387824
$573$	0	0
$574$	0	0
$575$	−25.8667	−1.07872
$576$	0	0
$577$	−28.6791	−1.19392	−0.596962	−	0.802269i	$-0.703626\pi$
$577$	−28.6791	−1.19392	−0.596962	+	0.802269i	$0.703626\pi$
$578$	9.36052	0.389346
$579$	0	0
$580$	−17.2453	−0.716070
$581$	0	0
$582$	0	0
$583$	−9.22547	−0.382080
$584$	−10.6885	−0.442293
$585$	0	0
$586$	−10.0697	−0.415976
$587$	1.01209	0.0417733	0.0208866	−	0.999782i	$-0.493351\pi$
$587$	1.01209	0.0417733	0.0208866	+	0.999782i	$0.493351\pi$
$588$	0	0
$589$	−10.4861	−0.432074
$590$	20.7435	0.853996
$591$	0	0
$592$	−7.21338	−0.296468
$593$	14.2332	0.584486	0.292243	−	0.956344i	$-0.405598\pi$
$593$	14.2332	0.584486	0.292243	+	0.956344i	$0.405598\pi$
$594$	0	0
$595$	0	0
$596$	−1.56885	−0.0642626
$597$	0	0
$598$	12.9793	0.530763
$599$	26.5457	1.08463	0.542315	−	0.840175i	$-0.317548\pi$
$599$	26.5457	1.08463	0.542315	+	0.840175i	$0.317548\pi$
$600$	0	0
$601$	12.1558	0.495843	0.247922	−	0.968780i	$-0.420252\pi$
$601$	12.1558	0.495843	0.247922	+	0.968780i	$0.420252\pi$
$602$	0	0
$603$	0	0
$604$	−25.7659	−1.04840
$605$	−3.56885	−0.145094
$606$	0	0
$607$	−13.9672	−0.566912	−0.283456	−	0.958985i	$-0.591481\pi$
$607$	−13.9672	−0.566912	−0.283456	+	0.958985i	$0.591481\pi$
$608$	8.49649	0.344578
$609$	0	0
$610$	15.6696	0.634444
$611$	−9.79432	−0.396236
$612$	0	0
$613$	4.64189	0.187484	0.0937421	−	0.995597i	$-0.470117\pi$
$613$	4.64189	0.187484	0.0937421	+	0.995597i	$0.470117\pi$
$614$	−10.8150	−0.436459
$615$	0	0
$616$	0	0
$617$	26.3960	1.06266	0.531331	−	0.847165i	$-0.321692\pi$
$617$	26.3960	1.06266	0.531331	+	0.847165i	$0.321692\pi$
$618$	0	0
$619$	−14.6815	−0.590098	−0.295049	−	0.955482i	$-0.595336\pi$
$619$	−14.6815	−0.590098	−0.295049	+	0.955482i	$0.595336\pi$
$620$	39.6412	1.59203
$621$	0	0
$622$	−14.2021	−0.569453
$623$	0	0
$624$	0	0
$625$	−3.82710	−0.153084
$626$	10.7609	0.430090
$627$	0	0
$628$	−17.9648	−0.716874
$629$	7.47338	0.297983
$630$	0	0
$631$	−30.1498	−1.20024	−0.600122	−	0.799908i	$-0.704881\pi$
$631$	−30.1498	−1.20024	−0.600122	+	0.799908i	$0.704881\pi$
$632$	−14.9819	−0.595950
$633$	0	0
$634$	2.92937	0.116340
$635$	23.6309	0.937762
$636$	0	0
$637$	0	0
$638$	2.02243	0.0800690
$639$	0	0
$640$	−39.6137	−1.56587
$641$	16.1782	0.639000	0.319500	−	0.947586i	$-0.396485\pi$
$641$	16.1782	0.639000	0.319500	+	0.947586i	$0.396485\pi$
$642$	0	0
$643$	2.33568	0.0921101	0.0460550	−	0.998939i	$-0.485335\pi$
$643$	2.33568	0.0921101	0.0460550	+	0.998939i	$0.485335\pi$
$644$	0	0
$645$	0	0
$646$	−1.61107	−0.0633869
$647$	14.9622	0.588223	0.294112	−	0.955771i	$-0.404976\pi$
$647$	14.9622	0.588223	0.294112	+	0.955771i	$0.404976\pi$
$648$	0	0
$649$	8.85195	0.347470
$650$	−30.0345	−1.17805
$651$	0	0
$652$	14.0251	0.549265
$653$	22.8898	0.895747	0.447873	−	0.894097i	$-0.352182\pi$
$653$	22.8898	0.895747	0.447873	+	0.894097i	$0.352182\pi$
$654$	0	0
$655$	21.6188	0.844716
$656$	−2.05590	−0.0802692
$657$	0	0
$658$	0	0
$659$	2.20568	0.0859211	0.0429606	−	0.999077i	$-0.486321\pi$
$659$	2.20568	0.0859211	0.0429606	+	0.999077i	$0.486321\pi$
$660$	0	0
$661$	0.682377	0.0265414	0.0132707	−	0.999912i	$-0.495776\pi$
$661$	0.682377	0.0265414	0.0132707	+	0.999912i	$0.495776\pi$
$662$	−12.5018	−0.485895
$663$	0	0
$664$	0.393308	0.0152633
$665$	0	0
$666$	0	0
$667$	10.2978	0.398734
$668$	28.5809	1.10583
$669$	0	0
$670$	−23.0224	−0.889434
$671$	6.68676	0.258139
$672$	0	0
$673$	−10.8865	−0.419643	−0.209821	−	0.977740i	$-0.567288\pi$
$673$	−10.8865	−0.419643	−0.209821	+	0.977740i	$0.567288\pi$
$674$	−17.7384	−0.683259
$675$	0	0
$676$	−34.4432	−1.32474
$677$	45.6654	1.75506	0.877532	−	0.479519i	$-0.159189\pi$
$677$	45.6654	1.75506	0.877532	+	0.479519i	$0.159189\pi$
$678$	0	0
$679$	0	0
$680$	13.8546	0.531300
$681$	0	0
$682$	−4.64892	−0.178016
$683$	−6.48372	−0.248093	−0.124046	−	0.992276i	$-0.539587\pi$
$683$	−6.48372	−0.248093	−0.124046	+	0.992276i	$0.539587\pi$
$684$	0	0
$685$	−26.4932	−1.01225
$686$	0	0
$687$	0	0
$688$	2.55676	0.0974757
$689$	−54.5431	−2.07793
$690$	0	0
$691$	−10.9468	−0.416434	−0.208217	−	0.978083i	$-0.566766\pi$
$691$	−10.9468	−0.416434	−0.208217	+	0.978083i	$0.566766\pi$
$692$	−30.6885	−1.16660
$693$	0	0
$694$	−13.2754	−0.503927
$695$	−38.6309	−1.46535
$696$	0	0
$697$	2.13000	0.0806793
$698$	−7.89418	−0.298799
$699$	0	0
$700$	0	0
$701$	0.914874	0.0345543	0.0172772	−	0.999851i	$-0.494500\pi$
$701$	0.914874	0.0345543	0.0172772	+	0.999851i	$0.494500\pi$
$702$	0	0
$703$	6.68147	0.251996
$704$	0.568850	0.0214393
$705$	0	0
$706$	7.06466	0.265882
$707$	0	0
$708$	0	0
$709$	44.3123	1.66418	0.832092	−	0.554637i	$-0.187143\pi$
$709$	44.3123	1.66418	0.832092	+	0.554637i	$0.187143\pi$
$710$	20.1971	0.757982
$711$	0	0
$712$	−6.01979	−0.225601
$713$	−23.6714	−0.886499
$714$	0	0
$715$	−21.0999	−0.789090
$716$	5.09547	0.190427
$717$	0	0
$718$	15.9494	0.595226
$719$	−5.20236	−0.194015	−0.0970076	−	0.995284i	$-0.530927\pi$
$719$	−5.20236	−0.194015	−0.0970076	+	0.995284i	$0.530927\pi$
$720$	0	0
$721$	0	0
$722$	11.0354	0.410696
$723$	0	0
$724$	−16.2231	−0.602925
$725$	−23.8295	−0.885006
$726$	0	0
$727$	−50.0871	−1.85763	−0.928814	−	0.370547i	$-0.879170\pi$
$727$	−50.0871	−1.85763	−0.928814	+	0.370547i	$0.879170\pi$
$728$	0	0
$729$	0	0
$730$	−10.6885	−0.395599
$731$	−2.64892	−0.0979737
$732$	0	0
$733$	47.0697	1.73856	0.869280	−	0.494320i	$-0.164583\pi$
$733$	47.0697	1.73856	0.869280	+	0.494320i	$0.164583\pi$
$734$	7.12055	0.262824
$735$	0	0
$736$	19.1799	0.706981
$737$	−9.82446	−0.361889
$738$	0	0
$739$	46.9914	1.72861	0.864303	−	0.502971i	$-0.167760\pi$
$739$	46.9914	1.72861	0.864303	+	0.502971i	$0.167760\pi$
$740$	−25.2583	−0.928512
$741$	0	0
$742$	0	0
$743$	5.19533	0.190598	0.0952991	−	0.995449i	$-0.469619\pi$
$743$	5.19533	0.190598	0.0952991	+	0.995449i	$0.469619\pi$
$744$	0	0
$745$	−3.56885	−0.130753
$746$	21.6843	0.793920
$747$	0	0
$748$	2.59899	0.0950284
$749$	0	0
$750$	0	0
$751$	33.4536	1.22074	0.610369	−	0.792117i	$-0.291021\pi$
$751$	33.4536	1.22074	0.610369	+	0.792117i	$0.291021\pi$
$752$	−2.64892	−0.0965961
$753$	0	0
$754$	11.9571	0.435452
$755$	−58.6128	−2.13314
$756$	0	0
$757$	40.0440	1.45542	0.727711	−	0.685884i	$-0.240584\pi$
$757$	40.0440	1.45542	0.727711	+	0.685884i	$0.240584\pi$
$758$	14.4010	0.523068
$759$	0	0
$760$	12.3865	0.449306
$761$	−7.55850	−0.273995	−0.136998	−	0.990571i	$-0.543745\pi$
$761$	−7.55850	−0.273995	−0.136998	+	0.990571i	$0.543745\pi$
$762$	0	0
$763$	0	0
$764$	15.7237	0.568863
$765$	0	0
$766$	−23.9424	−0.865073
$767$	52.3348	1.88970
$768$	0	0
$769$	−51.5407	−1.85860	−0.929302	−	0.369320i	$-0.879591\pi$
$769$	−51.5407	−1.85860	−0.929302	+	0.369320i	$0.879591\pi$
$770$	0	0
$771$	0	0
$772$	−39.7316	−1.42997
$773$	−15.4657	−0.556262	−0.278131	−	0.960543i	$-0.589715\pi$
$773$	−15.4657	−0.556262	−0.278131	+	0.960543i	$0.589715\pi$
$774$	0	0
$775$	54.7763	1.96762
$776$	−22.8168	−0.819074
$777$	0	0
$778$	12.7884	0.458485
$779$	1.90429	0.0682284
$780$	0	0
$781$	8.61878	0.308404
$782$	−3.63683	−0.130053
$783$	0	0
$784$	0	0
$785$	−40.8667	−1.45859
$786$	0	0
$787$	−21.9518	−0.782497	−0.391249	−	0.920285i	$-0.627957\pi$
$787$	−21.9518	−0.782497	−0.391249	+	0.920285i	$0.627957\pi$
$788$	−38.5638	−1.37378
$789$	0	0
$790$	−14.9819	−0.533034
$791$	0	0
$792$	0	0
$793$	39.5337	1.40388
$794$	−5.14782	−0.182689
$795$	0	0
$796$	8.77188	0.310911
$797$	42.6258	1.50988	0.754942	−	0.655792i	$-0.227665\pi$
$797$	42.6258	1.50988	0.754942	+	0.655792i	$0.227665\pi$
$798$	0	0
$799$	2.74439	0.0970896
$800$	−44.3830	−1.56917
$801$	0	0
$802$	−16.0517	−0.566804
$803$	−4.56115	−0.160959
$804$	0	0
$805$	0	0
$806$	−27.4855	−0.968134
$807$	0	0
$808$	4.34602	0.152893
$809$	−3.96745	−0.139488	−0.0697440	−	0.997565i	$-0.522218\pi$
$809$	−3.96745	−0.139488	−0.0697440	+	0.997565i	$0.522218\pi$
$810$	0	0
$811$	46.9217	1.64764	0.823821	−	0.566850i	$-0.191838\pi$
$811$	46.9217	1.64764	0.823821	+	0.566850i	$0.191838\pi$
$812$	0	0
$813$	0	0
$814$	2.96216	0.103824
$815$	31.9045	1.11757
$816$	0	0
$817$	−2.36823	−0.0828538
$818$	−1.54551	−0.0540374
$819$	0	0
$820$	−7.19888	−0.251396
$821$	50.6000	1.76595	0.882977	−	0.469416i	$-0.155536\pi$
$821$	50.6000	1.76595	0.882977	+	0.469416i	$0.155536\pi$
$822$	0	0
$823$	0.398599	0.0138943	0.00694714	−	0.999976i	$-0.497789\pi$
$823$	0.398599	0.0138943	0.00694714	+	0.999976i	$0.497789\pi$
$824$	7.42345	0.258608
$825$	0	0
$826$	0	0
$827$	20.4234	0.710193	0.355096	−	0.934830i	$-0.384448\pi$
$827$	20.4234	0.710193	0.355096	+	0.934830i	$0.384448\pi$
$828$	0	0
$829$	23.2633	0.807968	0.403984	−	0.914766i	$-0.367625\pi$
$829$	23.2633	0.807968	0.403984	+	0.914766i	$0.367625\pi$
$830$	0.393308	0.0136519
$831$	0	0
$832$	3.36317	0.116597
$833$	0	0
$834$	0	0
$835$	65.0165	2.24999
$836$	2.32359	0.0803630
$837$	0	0
$838$	−6.10053	−0.210739
$839$	−16.4861	−0.569165	−0.284582	−	0.958652i	$-0.591855\pi$
$839$	−16.4861	−0.569165	−0.284582	+	0.958652i	$0.591855\pi$
$840$	0	0
$841$	−19.5132	−0.672869
$842$	25.6291	0.883238
$843$	0	0
$844$	−18.9829	−0.653417
$845$	−78.3521	−2.69540
$846$	0	0
$847$	0	0
$848$	−14.7514	−0.506566
$849$	0	0
$850$	8.41574	0.288658
$851$	15.0827	0.517029
$852$	0	0
$853$	14.8315	0.507820	0.253910	−	0.967228i	$-0.418283\pi$
$853$	14.8315	0.507820	0.253910	+	0.967228i	$0.418283\pi$
$854$	0	0
$855$	0	0
$856$	11.1705	0.381799
$857$	−5.03188	−0.171886	−0.0859428	−	0.996300i	$-0.527390\pi$
$857$	−5.03188	−0.171886	−0.0859428	+	0.996300i	$0.527390\pi$
$858$	0	0
$859$	56.0363	1.91193	0.955966	−	0.293477i	$-0.0948123\pi$
$859$	56.0363	1.91193	0.955966	+	0.293477i	$0.0948123\pi$
$860$	8.95272	0.305285
$861$	0	0
$862$	−2.49649	−0.0850307
$863$	36.4760	1.24166	0.620829	−	0.783946i	$-0.286796\pi$
$863$	36.4760	1.24166	0.620829	+	0.783946i	$0.286796\pi$
$864$	0	0
$865$	−69.8108	−2.37364
$866$	5.39769	0.183421
$867$	0	0
$868$	0	0
$869$	−6.39331	−0.216878
$870$	0	0
$871$	−58.0844	−1.96812
$872$	−34.8744	−1.18100
$873$	0	0
$874$	−3.25146	−0.109982
$875$	0	0
$876$	0	0
$877$	−6.61613	−0.223411	−0.111705	−	0.993741i	$-0.535631\pi$
$877$	−6.61613	−0.223411	−0.111705	+	0.993741i	$0.535631\pi$
$878$	2.98195	0.100636
$879$	0	0
$880$	−5.70655	−0.192368
$881$	−22.5286	−0.759008	−0.379504	−	0.925190i	$-0.623905\pi$
$881$	−22.5286	−0.759008	−0.379504	+	0.925190i	$0.623905\pi$
$882$	0	0
$883$	−51.1652	−1.72185	−0.860923	−	0.508735i	$-0.830113\pi$
$883$	−51.1652	−1.72185	−0.860923	+	0.508735i	$0.830113\pi$
$884$	15.3658	0.516808
$885$	0	0
$886$	9.02485	0.303196
$887$	−6.33809	−0.212812	−0.106406	−	0.994323i	$-0.533934\pi$
$887$	−6.33809	−0.212812	−0.106406	+	0.994323i	$0.533934\pi$
$888$	0	0
$889$	0	0
$890$	−6.01979	−0.201784
$891$	0	0
$892$	24.5156	0.820843
$893$	2.45359	0.0821061
$894$	0	0
$895$	11.5913	0.387454
$896$	0	0
$897$	0	0
$898$	14.1608	0.472552
$899$	−21.8071	−0.727307
$900$	0	0
$901$	15.2831	0.509154
$902$	0.844248	0.0281104
$903$	0	0
$904$	−29.1592	−0.969821
$905$	−36.9045	−1.22675
$906$	0	0
$907$	−6.88474	−0.228604	−0.114302	−	0.993446i	$-0.536463\pi$
$907$	−6.88474	−0.228604	−0.114302	+	0.993446i	$0.536463\pi$
$908$	24.5818	0.815777
$909$	0	0
$910$	0	0
$911$	41.0818	1.36110	0.680550	−	0.732701i	$-0.261741\pi$
$911$	41.0818	1.36110	0.680550	+	0.732701i	$0.261741\pi$
$912$	0	0
$913$	0.167838	0.00555462
$914$	2.17048	0.0717932
$915$	0	0
$916$	−8.74084	−0.288805
$917$	0	0
$918$	0	0
$919$	−11.6265	−0.383522	−0.191761	−	0.981442i	$-0.561420\pi$
$919$	−11.6265	−0.383522	−0.191761	+	0.981442i	$0.561420\pi$
$920$	27.9612	0.921855
$921$	0	0
$922$	21.1119	0.695285
$923$	50.9562	1.67724
$924$	0	0
$925$	−34.9019	−1.14757
$926$	−3.82272	−0.125622
$927$	0	0
$928$	17.6694	0.580026
$929$	24.7668	0.812573	0.406287	−	0.913746i	$-0.366823\pi$
$929$	24.7668	0.812573	0.406287	+	0.913746i	$0.366823\pi$
$930$	0	0
$931$	0	0
$932$	−30.2402	−0.990551
$933$	0	0
$934$	3.94940	0.129228
$935$	5.91223	0.193351
$936$	0	0
$937$	−1.15046	−0.0375839	−0.0187920	−	0.999823i	$-0.505982\pi$
$937$	−1.15046	−0.0375839	−0.0187920	+	0.999823i	$0.505982\pi$
$938$	0	0
$939$	0	0
$940$	−9.27540	−0.302530
$941$	10.1102	0.329583	0.164792	−	0.986328i	$-0.447305\pi$
$941$	10.1102	0.329583	0.164792	+	0.986328i	$0.447305\pi$
$942$	0	0
$943$	4.29874	0.139986
$944$	14.1542	0.460679
$945$	0	0
$946$	−1.04993	−0.0341361
$947$	24.4553	0.794691	0.397346	−	0.917669i	$-0.369931\pi$
$947$	24.4553	0.794691	0.397346	+	0.917669i	$0.369931\pi$
$948$	0	0
$949$	−26.9665	−0.875371
$950$	7.52398	0.244110
$951$	0	0
$952$	0	0
$953$	16.1696	0.523784	0.261892	−	0.965097i	$-0.415654\pi$
$953$	16.1696	0.523784	0.261892	+	0.965097i	$0.415654\pi$
$954$	0	0
$955$	35.7686	1.15744
$956$	−34.7618	−1.12428
$957$	0	0
$958$	−11.2512	−0.363511
$959$	0	0
$960$	0	0
$961$	19.1274	0.617011
$962$	17.5130	0.564640
$963$	0	0
$964$	−31.1799	−1.00424
$965$	−90.3823	−2.90951
$966$	0	0
$967$	−1.55941	−0.0501472	−0.0250736	−	0.999686i	$-0.507982\pi$
$967$	−1.55941	−0.0501472	−0.0250736	+	0.999686i	$0.507982\pi$
$968$	2.34338	0.0753191
$969$	0	0
$970$	−22.8168	−0.732602
$971$	−9.04728	−0.290341	−0.145171	−	0.989407i	$-0.546373\pi$
$971$	−9.04728	−0.290341	−0.145171	+	0.989407i	$0.546373\pi$
$972$	0	0
$973$	0	0
$974$	−3.75209	−0.120225
$975$	0	0
$976$	10.6920	0.342244
$977$	6.75648	0.216159	0.108079	−	0.994142i	$-0.465530\pi$
$977$	6.75648	0.216159	0.108079	+	0.994142i	$0.465530\pi$
$978$	0	0
$979$	−2.56885	−0.0821008
$980$	0	0
$981$	0	0
$982$	15.8031	0.504297
$983$	26.4305	0.843001	0.421501	−	0.906828i	$-0.361504\pi$
$983$	26.4305	0.843001	0.421501	+	0.906828i	$0.361504\pi$
$984$	0	0
$985$	−87.7257	−2.79517
$986$	−3.35041	−0.106699
$987$	0	0
$988$	13.7376	0.437051
$989$	−5.34602	−0.169994
$990$	0	0
$991$	−0.634185	−0.0201456	−0.0100728	−	0.999949i	$-0.503206\pi$
$991$	−0.634185	−0.0201456	−0.0100728	+	0.999949i	$0.503206\pi$
$992$	−40.6161	−1.28956
$993$	0	0
$994$	0	0
$995$	19.9545	0.632599
$996$	0	0
$997$	−10.1274	−0.320736	−0.160368	−	0.987057i	$-0.551268\pi$
$997$	−10.1274	−0.320736	−0.160368	+	0.987057i	$0.551268\pi$
$998$	−7.08445	−0.224254
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bn.1.2		3
3.2	odd	2		539.2.a.i.1.2		3
7.3	odd	6		693.2.i.g.100.2		6
7.5	odd	6		693.2.i.g.298.2		6
7.6	odd	2		4851.2.a.bo.1.2		3
12.11	even	2		8624.2.a.ck.1.3		3
21.2	odd	6		539.2.e.l.67.2		6
21.5	even	6		77.2.e.b.67.2	yes	6
21.11	odd	6		539.2.e.l.177.2		6
21.17	even	6		77.2.e.b.23.2	✓	6
21.20	even	2		539.2.a.h.1.2		3
33.32	even	2		5929.2.a.w.1.2		3
84.47	odd	6		1232.2.q.k.529.3		6
84.59	odd	6		1232.2.q.k.177.3		6
84.83	odd	2		8624.2.a.cl.1.1		3
231.5	even	30		847.2.n.e.487.2		24
231.17	odd	30		847.2.n.d.366.2		24
231.26	even	30		847.2.n.e.753.2		24
231.38	even	30		847.2.n.e.366.2		24
231.47	even	30		847.2.n.e.130.2		24
231.59	even	30		847.2.n.e.632.2		24
231.68	odd	30		847.2.n.d.81.2		24
231.80	even	30		847.2.n.e.9.2		24
231.101	odd	30		847.2.n.d.807.2		24
231.131	odd	6		847.2.e.d.606.2		6
231.152	even	30		847.2.n.e.81.2		24
231.164	odd	6		847.2.e.d.485.2		6
231.173	odd	30		847.2.n.d.130.2		24
231.185	even	30		847.2.n.e.807.2		24
231.194	odd	30		847.2.n.d.753.2		24
231.206	odd	30		847.2.n.d.9.2		24
231.215	odd	30		847.2.n.d.487.2		24
231.227	odd	30		847.2.n.d.632.2		24
231.230	odd	2		5929.2.a.v.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.e.b.23.2	✓	6	21.17	even	6
77.2.e.b.67.2	yes	6	21.5	even	6
539.2.a.h.1.2		3	21.20	even	2
539.2.a.i.1.2		3	3.2	odd	2
539.2.e.l.67.2		6	21.2	odd	6
539.2.e.l.177.2		6	21.11	odd	6
693.2.i.g.100.2		6	7.3	odd	6
693.2.i.g.298.2		6	7.5	odd	6
847.2.e.d.485.2		6	231.164	odd	6
847.2.e.d.606.2		6	231.131	odd	6
847.2.n.d.9.2		24	231.206	odd	30
847.2.n.d.81.2		24	231.68	odd	30
847.2.n.d.130.2		24	231.173	odd	30
847.2.n.d.366.2		24	231.17	odd	30
847.2.n.d.487.2		24	231.215	odd	30
847.2.n.d.632.2		24	231.227	odd	30
847.2.n.d.753.2		24	231.194	odd	30
847.2.n.d.807.2		24	231.101	odd	30
847.2.n.e.9.2		24	231.80	even	30
847.2.n.e.81.2		24	231.152	even	30
847.2.n.e.130.2		24	231.47	even	30
847.2.n.e.366.2		24	231.38	even	30
847.2.n.e.487.2		24	231.5	even	30
847.2.n.e.632.2		24	231.59	even	30
847.2.n.e.753.2		24	231.26	even	30
847.2.n.e.807.2		24	231.185	even	30
1232.2.q.k.177.3		6	84.59	odd	6
1232.2.q.k.529.3		6	84.47	odd	6
4851.2.a.bn.1.2		3	1.1	even	1	trivial
4851.2.a.bo.1.2		3	7.6	odd	2
5929.2.a.v.1.2		3	231.230	odd	2
5929.2.a.w.1.2		3	33.32	even	2
8624.2.a.ck.1.3		3	12.11	even	2
8624.2.a.cl.1.1		3	84.83	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)

\(f(q)\)	\(=\)	\(q-0.656620 q^{2} -1.56885 q^{4} -3.56885 q^{5} +2.34338 q^{8} +O(q^{10})\)	\(q-0.656620 q^{2} -1.56885 q^{4} -3.56885 q^{5} +2.34338 q^{8} +2.34338 q^{10} +1.00000 q^{11} +5.91223 q^{13} +1.59899 q^{16} -1.65662 q^{17} -1.48108 q^{19} +5.59899 q^{20} -0.656620 q^{22} -3.34338 q^{23} +7.73669 q^{25} -3.88209 q^{26} -3.08007 q^{29} +7.08007 q^{31} -5.73669 q^{32} +1.08777 q^{34} -4.51122 q^{37} +0.972507 q^{38} -8.36317 q^{40} -1.28575 q^{41} +1.59899 q^{43} -1.56885 q^{44} +2.19533 q^{46} -1.65662 q^{47} -5.08007 q^{50} -9.27540 q^{52} -9.22547 q^{53} -3.56885 q^{55} +2.02243 q^{58} +8.85195 q^{59} +6.68676 q^{61} -4.64892 q^{62} +0.568850 q^{64} -21.0999 q^{65} -9.82446 q^{67} +2.59899 q^{68} +8.61878 q^{71} -4.56115 q^{73} +2.96216 q^{74} +2.32359 q^{76} -6.39331 q^{79} -5.70655 q^{80} +0.844248 q^{82} +0.167838 q^{83} +5.91223 q^{85} -1.04993 q^{86} +2.34338 q^{88} -2.56885 q^{89} +5.24526 q^{92} +1.08777 q^{94} +5.28575 q^{95} -9.73669 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{4} - 2 q^{5} + 9 q^{8}+O(q^{10}) \) 3 * q + 4 * q^4 - 2 * q^5 + 9 * q^8	\( 3 q + 4 q^{4} - 2 q^{5} + 9 q^{8} + 9 q^{10} + 3 q^{11} + 11 q^{13} + 2 q^{16} - 3 q^{17} + 11 q^{19} + 14 q^{20} - 12 q^{23} + 3 q^{25} + q^{26} + 9 q^{29} + 3 q^{31} + 3 q^{32} + 10 q^{34} - 4 q^{37} + 8 q^{38} + 3 q^{40} - 5 q^{41} + 2 q^{43} + 4 q^{44} - 10 q^{46} - 3 q^{47} + 3 q^{50} + 7 q^{52} - 17 q^{53} - 2 q^{55} - 13 q^{58} + 8 q^{59} + 24 q^{61} + 13 q^{62} - 7 q^{64} - 15 q^{65} - 16 q^{67} + 5 q^{68} - 7 q^{71} + 20 q^{73} - 22 q^{74} + 39 q^{76} + 3 q^{79} + 9 q^{80} - 41 q^{82} - 11 q^{83} + 11 q^{85} + 21 q^{86} + 9 q^{88} + q^{89} - 25 q^{92} + 10 q^{94} + 17 q^{95} - 9 q^{97}+O(q^{100}) \) 3 * q + 4 * q^4 - 2 * q^5 + 9 * q^8 + 9 * q^10 + 3 * q^11 + 11 * q^13 + 2 * q^16 - 3 * q^17 + 11 * q^19 + 14 * q^20 - 12 * q^23 + 3 * q^25 + q^26 + 9 * q^29 + 3 * q^31 + 3 * q^32 + 10 * q^34 - 4 * q^37 + 8 * q^38 + 3 * q^40 - 5 * q^41 + 2 * q^43 + 4 * q^44 - 10 * q^46 - 3 * q^47 + 3 * q^50 + 7 * q^52 - 17 * q^53 - 2 * q^55 - 13 * q^58 + 8 * q^59 + 24 * q^61 + 13 * q^62 - 7 * q^64 - 15 * q^65 - 16 * q^67 + 5 * q^68 - 7 * q^71 + 20 * q^73 - 22 * q^74 + 39 * q^76 + 3 * q^79 + 9 * q^80 - 41 * q^82 - 11 * q^83 + 11 * q^85 + 21 * q^86 + 9 * q^88 + q^89 - 25 * q^92 + 10 * q^94 + 17 * q^95 - 9 * q^97

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