LMFDB - Embedded newform 4851.2.a.bp.1.3

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:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
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			1.3
Root			$-2.36147$ 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)

$f(q)$	$=$	$q+2.36147 q^{2} +3.57653 q^{4} -3.93800 q^{5} +3.72294 q^{8} +O(q^{10})$	$q+2.36147 q^{2} +3.57653 q^{4} -3.93800 q^{5} +3.72294 q^{8} -9.29947 q^{10} -1.00000 q^{11} +3.93800 q^{13} +1.63853 q^{16} +4.72294 q^{17} -4.78493 q^{19} -14.0844 q^{20} -2.36147 q^{22} -2.72294 q^{23} +10.5079 q^{25} +9.29947 q^{26} -7.93800 q^{29} -1.15307 q^{31} -3.57653 q^{32} +11.1531 q^{34} -5.50787 q^{37} -11.2995 q^{38} -14.6609 q^{40} +0.430132 q^{41} +6.72294 q^{43} -3.57653 q^{44} -6.43013 q^{46} -8.78493 q^{47} +24.8140 q^{50} +14.0844 q^{52} +3.15307 q^{53} +3.93800 q^{55} -18.7453 q^{58} -15.0911 q^{59} -6.00000 q^{61} -2.72294 q^{62} -11.7229 q^{64} -15.5079 q^{65} +3.21507 q^{67} +16.8918 q^{68} -13.4459 q^{71} -11.9380 q^{73} -13.0067 q^{74} -17.1135 q^{76} -5.44588 q^{79} -6.45254 q^{80} +1.01574 q^{82} -2.84693 q^{83} -18.5989 q^{85} +15.8760 q^{86} -3.72294 q^{88} +12.3061 q^{89} -9.73868 q^{92} -20.7453 q^{94} +18.8431 q^{95} -11.1531 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{8}+O(q^{10}) $ 3 * q + 6 * q^4 - 3 * q^8	$ 3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^8 - 9 * q^10 - 3 * q^11 + 12 * q^16 - 12 * q^19 - 21 * q^20 + 6 * q^23 + 15 * q^25 + 9 * q^26 - 12 * q^29 + 6 * q^31 - 6 * q^32 + 24 * q^34 - 15 * q^38 - 18 * q^40 + 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 - 24 * q^47 + 39 * q^50 + 21 * q^52 - 9 * q^58 - 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^73 - 39 * q^74 + 3 * q^76 + 12 * q^79 + 9 * q^80 - 30 * q^82 - 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 + 18 * q^89 + 18 * q^92 - 15 * q^94 - 12 * q^95 - 24 * 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 $

$2$	2.36147	1.66981	0.834905	−	0.550394i	$-0.185522\pi$
$2$	2.36147	1.66981	0.834905	+	0.550394i	$0.185522\pi$
$3$	0	0
$3$	0	0
$4$	3.57653	1.78827
$5$	−3.93800	−1.76113	−0.880564	−	0.473927i	$-0.842836\pi$
$5$	−3.93800	−1.76113	−0.880564	+	0.473927i	$0.842836\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.72294	1.31626
$9$	0	0
$10$	−9.29947	−2.94075
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.93800	1.09221	0.546103	−	0.837718i	$-0.316111\pi$
$13$	3.93800	1.09221	0.546103	+	0.837718i	$0.316111\pi$
$14$	0	0
$15$	0	0
$16$	1.63853	0.409633
$17$	4.72294	1.14548	0.572740	−	0.819737i	$-0.305880\pi$
$17$	4.72294	1.14548	0.572740	+	0.819737i	$0.305880\pi$
$18$	0	0
$19$	−4.78493	−1.09774	−0.548870	−	0.835908i	$-0.684942\pi$
$19$	−4.78493	−1.09774	−0.548870	+	0.835908i	$0.684942\pi$
$20$	−14.0844	−3.14937
$21$	0	0
$22$	−2.36147	−0.503467
$23$	−2.72294	−0.567772	−0.283886	−	0.958858i	$-0.591624\pi$
$23$	−2.72294	−0.567772	−0.283886	+	0.958858i	$0.591624\pi$
$24$	0	0
$25$	10.5079	2.10157
$26$	9.29947	1.82378
$27$	0	0
$28$	0	0
$29$	−7.93800	−1.47405	−0.737025	−	0.675865i	$-0.763770\pi$
$29$	−7.93800	−1.47405	−0.737025	+	0.675865i	$0.763770\pi$
$30$	0	0
$31$	−1.15307	−0.207097	−0.103549	−	0.994624i	$-0.533020\pi$
$31$	−1.15307	−0.207097	−0.103549	+	0.994624i	$0.533020\pi$
$32$	−3.57653	−0.632248
$33$	0	0
$34$	11.1531	1.91274
$35$	0	0
$36$	0	0
$37$	−5.50787	−0.905489	−0.452744	−	0.891640i	$-0.649555\pi$
$37$	−5.50787	−0.905489	−0.452744	+	0.891640i	$0.649555\pi$
$38$	−11.2995	−1.83302
$39$	0	0
$40$	−14.6609	−2.31810
$41$	0.430132	0.0671753	0.0335877	−	0.999436i	$-0.489307\pi$
$41$	0.430132	0.0671753	0.0335877	+	0.999436i	$0.489307\pi$
$42$	0	0
$43$	6.72294	1.02524	0.512619	−	0.858616i	$-0.328675\pi$
$43$	6.72294	1.02524	0.512619	+	0.858616i	$0.328675\pi$
$44$	−3.57653	−0.539183
$45$	0	0
$46$	−6.43013	−0.948071
$47$	−8.78493	−1.28141	−0.640707	−	0.767785i	$-0.721359\pi$
$47$	−8.78493	−1.28141	−0.640707	+	0.767785i	$0.721359\pi$
$48$	0	0
$49$	0	0
$50$	24.8140	3.50923
$51$	0	0
$52$	14.0844	1.95316
$53$	3.15307	0.433107	0.216554	−	0.976271i	$-0.430518\pi$
$53$	3.15307	0.433107	0.216554	+	0.976271i	$0.430518\pi$
$54$	0	0
$55$	3.93800	0.531000
$56$	0	0
$57$	0	0
$58$	−18.7453	−2.46138
$59$	−15.0911	−1.96469	−0.982345	−	0.187077i	$-0.940098\pi$
$59$	−15.0911	−1.96469	−0.982345	+	0.187077i	$0.940098\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−2.72294	−0.345813
$63$	0	0
$64$	−11.7229	−1.46537
$65$	−15.5079	−1.92351
$66$	0	0
$67$	3.21507	0.392783	0.196391	−	0.980526i	$-0.437078\pi$
$67$	3.21507	0.392783	0.196391	+	0.980526i	$0.437078\pi$
$68$	16.8918	2.04843
$69$	0	0
$70$	0	0
$71$	−13.4459	−1.59573	−0.797866	−	0.602835i	$-0.794038\pi$
$71$	−13.4459	−1.59573	−0.797866	+	0.602835i	$0.794038\pi$
$72$	0	0
$73$	−11.9380	−1.39724	−0.698619	−	0.715494i	$-0.746202\pi$
$73$	−11.9380	−1.39724	−0.698619	+	0.715494i	$0.746202\pi$
$74$	−13.0067	−1.51199
$75$	0	0
$76$	−17.1135	−1.96305
$77$	0	0
$78$	0	0
$79$	−5.44588	−0.612709	−0.306354	−	0.951918i	$-0.599109\pi$
$79$	−5.44588	−0.612709	−0.306354	+	0.951918i	$0.599109\pi$
$80$	−6.45254	−0.721416
$81$	0	0
$82$	1.01574	0.112170
$83$	−2.84693	−0.312491	−0.156246	−	0.987718i	$-0.549939\pi$
$83$	−2.84693	−0.312491	−0.156246	+	0.987718i	$0.549939\pi$
$84$	0	0
$85$	−18.5989	−2.01734
$86$	15.8760	1.71195
$87$	0	0
$88$	−3.72294	−0.396866
$89$	12.3061	1.30445	0.652224	−	0.758026i	$-0.273836\pi$
$89$	12.3061	1.30445	0.652224	+	0.758026i	$0.273836\pi$
$90$	0	0
$91$	0	0
$92$	−9.73868	−1.01533
$93$	0	0
$94$	−20.7453	−2.13972
$95$	18.8431	1.93326
$96$	0	0
$97$	−11.1531	−1.13242	−0.566211	−	0.824260i	$-0.691591\pi$
$97$	−11.1531	−1.13242	−0.566211	+	0.824260i	$0.691591\pi$
$98$	0	0
$99$	0	0
$100$	37.5818	3.75818
$101$	16.4750	1.63932	0.819659	−	0.572851i	$-0.194163\pi$
$101$	16.4750	1.63932	0.819659	+	0.572851i	$0.194163\pi$
$102$	0	0
$103$	1.56987	0.154684	0.0773418	−	0.997005i	$-0.475357\pi$
$103$	1.56987	0.154684	0.0773418	+	0.997005i	$0.475357\pi$
$104$	14.6609	1.43762
$105$	0	0
$106$	7.44588	0.723207
$107$	4.78493	0.462577	0.231289	−	0.972885i	$-0.425706\pi$
$107$	4.78493	0.462577	0.231289	+	0.972885i	$0.425706\pi$
$108$	0	0
$109$	−15.0291	−1.43952	−0.719762	−	0.694221i	$-0.755749\pi$
$109$	−15.0291	−1.43952	−0.719762	+	0.694221i	$0.755749\pi$
$110$	9.29947	0.886670
$111$	0	0
$112$	0	0
$113$	−7.44588	−0.700449	−0.350225	−	0.936666i	$-0.613895\pi$
$113$	−7.44588	−0.700449	−0.350225	+	0.936666i	$0.613895\pi$
$114$	0	0
$115$	10.7229	0.999919
$116$	−28.3905	−2.63600
$117$	0	0
$118$	−35.6371	−3.28066
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.1688	−1.28278
$123$	0	0
$124$	−4.12399	−0.370346
$125$	−21.6900	−1.94001
$126$	0	0
$127$	12.2928	1.09081	0.545405	−	0.838173i	$-0.316376\pi$
$127$	12.2928	1.09081	0.545405	+	0.838173i	$0.316376\pi$
$128$	−20.5303	−1.81464
$129$	0	0
$130$	−36.6214	−3.21191
$131$	7.13974	0.623802	0.311901	−	0.950115i	$-0.399034\pi$
$131$	7.13974	0.623802	0.311901	+	0.950115i	$0.399034\pi$
$132$	0	0
$133$	0	0
$134$	7.59228	0.655873
$135$	0	0
$136$	17.5832	1.50775
$137$	−1.58320	−0.135262	−0.0676310	−	0.997710i	$-0.521544\pi$
$137$	−1.58320	−0.135262	−0.0676310	+	0.997710i	$0.521544\pi$
$138$	0	0
$139$	17.1979	1.45871	0.729353	−	0.684138i	$-0.239821\pi$
$139$	17.1979	1.45871	0.729353	+	0.684138i	$0.239821\pi$
$140$	0	0
$141$	0	0
$142$	−31.7520	−2.66457
$143$	−3.93800	−0.329312
$144$	0	0
$145$	31.2599	2.59599
$146$	−28.1912	−2.33312
$147$	0	0
$148$	−19.6991	−1.61926
$149$	−15.9380	−1.30569	−0.652846	−	0.757491i	$-0.726425\pi$
$149$	−15.9380	−1.30569	−0.652846	+	0.757491i	$0.726425\pi$
$150$	0	0
$151$	6.59894	0.537014	0.268507	−	0.963278i	$-0.413470\pi$
$151$	6.59894	0.537014	0.268507	+	0.963278i	$0.413470\pi$
$152$	−17.8140	−1.44491
$153$	0	0
$154$	0	0
$155$	4.54079	0.364725
$156$	0	0
$157$	−7.32188	−0.584350	−0.292175	−	0.956365i	$-0.594379\pi$
$157$	−7.32188	−0.584350	−0.292175	+	0.956365i	$0.594379\pi$
$158$	−12.8603	−1.02311
$159$	0	0
$160$	14.0844	1.11347
$161$	0	0
$162$	0	0
$163$	14.1068	1.10493	0.552466	−	0.833536i	$-0.313687\pi$
$163$	14.1068	1.10493	0.552466	+	0.833536i	$0.313687\pi$
$164$	1.53838	0.120127
$165$	0	0
$166$	−6.72294	−0.521801
$167$	−0.416799	−0.0322528	−0.0161264	−	0.999870i	$-0.505133\pi$
$167$	−0.416799	−0.0322528	−0.0161264	+	0.999870i	$0.505133\pi$
$168$	0	0
$169$	2.50787	0.192913
$170$	−43.9208	−3.36857
$171$	0	0
$172$	24.0448	1.83340
$173$	−8.43013	−0.640931	−0.320466	−	0.947260i	$-0.603839\pi$
$173$	−8.43013	−0.640931	−0.320466	+	0.947260i	$0.603839\pi$
$174$	0	0
$175$	0	0
$176$	−1.63853	−0.123509
$177$	0	0
$178$	29.0606	2.17818
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−20.5989	−1.53111	−0.765554	−	0.643372i	$-0.777535\pi$
$181$	−20.5989	−1.53111	−0.765554	+	0.643372i	$0.777535\pi$
$182$	0	0
$183$	0	0
$184$	−10.1373	−0.747334
$185$	21.6900	1.59468
$186$	0	0
$187$	−4.72294	−0.345375
$188$	−31.4196	−2.29151
$189$	0	0
$190$	44.4974	3.22818
$191$	14.5989	1.05634	0.528171	−	0.849138i	$-0.322878\pi$
$191$	14.5989	1.05634	0.528171	+	0.849138i	$0.322878\pi$
$192$	0	0
$193$	13.4592	0.968815	0.484408	−	0.874842i	$-0.339035\pi$
$193$	13.4592	0.968815	0.484408	+	0.874842i	$0.339035\pi$
$194$	−26.3376	−1.89093
$195$	0	0
$196$	0	0
$197$	−11.4459	−0.815485	−0.407742	−	0.913097i	$-0.633684\pi$
$197$	−11.4459	−0.815485	−0.407742	+	0.913097i	$0.633684\pi$
$198$	0	0
$199$	13.0291	0.923607	0.461803	−	0.886982i	$-0.347203\pi$
$199$	13.0291	0.923607	0.461803	+	0.886982i	$0.347203\pi$
$200$	39.1201	2.76621
$201$	0	0
$202$	38.9051	2.73735
$203$	0	0
$204$	0	0
$205$	−1.69386	−0.118304
$206$	3.70719	0.258292
$207$	0	0
$208$	6.45254	0.447403
$209$	4.78493	0.330981
$210$	0	0
$211$	2.43013	0.167297	0.0836486	−	0.996495i	$-0.473343\pi$
$211$	2.43013	0.167297	0.0836486	+	0.996495i	$0.473343\pi$
$212$	11.2771	0.774512
$213$	0	0
$214$	11.2995	0.772416
$215$	−26.4750	−1.80558
$216$	0	0
$217$	0	0
$218$	−35.4907	−2.40373
$219$	0	0
$220$	14.0844	0.949570
$221$	18.5989	1.25110
$222$	0	0
$223$	−22.7678	−1.52464	−0.762321	−	0.647199i	$-0.775940\pi$
$223$	−22.7678	−1.52464	−0.762321	+	0.647199i	$0.775940\pi$
$224$	0	0
$225$	0	0
$226$	−17.5832	−1.16962
$227$	−15.8760	−1.05373	−0.526864	−	0.849950i	$-0.676632\pi$
$227$	−15.8760	−1.05373	−0.526864	+	0.849950i	$0.676632\pi$
$228$	0	0
$229$	3.15307	0.208361	0.104180	−	0.994558i	$-0.466778\pi$
$229$	3.15307	0.208361	0.104180	+	0.994558i	$0.466778\pi$
$230$	25.3219	1.66968
$231$	0	0
$232$	−29.5527	−1.94023
$233$	−13.1397	−0.860813	−0.430406	−	0.902635i	$-0.641630\pi$
$233$	−13.1397	−0.860813	−0.430406	+	0.902635i	$0.641630\pi$
$234$	0	0
$235$	34.5951	2.25674
$236$	−53.9737	−3.51339
$237$	0	0
$238$	0	0
$239$	10.1068	0.653756	0.326878	−	0.945067i	$-0.394003\pi$
$239$	10.1068	0.653756	0.326878	+	0.945067i	$0.394003\pi$
$240$	0	0
$241$	−2.36814	−0.152545	−0.0762725	−	0.997087i	$-0.524302\pi$
$241$	−2.36814	−0.152545	−0.0762725	+	0.997087i	$0.524302\pi$
$242$	2.36147	0.151801
$243$	0	0
$244$	−21.4592	−1.37379
$245$	0	0
$246$	0	0
$247$	−18.8431	−1.19896
$248$	−4.29281	−0.272593
$249$	0	0
$250$	−51.2203	−3.23946
$251$	10.2308	0.645763	0.322881	−	0.946439i	$-0.395348\pi$
$251$	10.2308	0.645763	0.322881	+	0.946439i	$0.395348\pi$
$252$	0	0
$253$	2.72294	0.171190
$254$	29.0291	1.82145
$255$	0	0
$256$	−25.0357	−1.56473
$257$	−11.0777	−0.691010	−0.345505	−	0.938417i	$-0.612292\pi$
$257$	−11.0777	−0.691010	−0.345505	+	0.938417i	$0.612292\pi$
$258$	0	0
$259$	0	0
$260$	−55.4644	−3.43976
$261$	0	0
$262$	16.8603	1.04163
$263$	8.78493	0.541702	0.270851	−	0.962621i	$-0.412695\pi$
$263$	8.78493	0.541702	0.270851	+	0.962621i	$0.412695\pi$
$264$	0	0
$265$	−12.4168	−0.762758
$266$	0	0
$267$	0	0
$268$	11.4988	0.702401
$269$	5.75201	0.350706	0.175353	−	0.984506i	$-0.443893\pi$
$269$	5.75201	0.350706	0.175353	+	0.984506i	$0.443893\pi$
$270$	0	0
$271$	−0.784934	−0.0476813	−0.0238407	−	0.999716i	$-0.507589\pi$
$271$	−0.784934	−0.0476813	−0.0238407	+	0.999716i	$0.507589\pi$
$272$	7.73868	0.469226
$273$	0	0
$274$	−3.73868	−0.225862
$275$	−10.5079	−0.633648
$276$	0	0
$277$	−11.5699	−0.695166	−0.347583	−	0.937649i	$-0.612998\pi$
$277$	−11.5699	−0.695166	−0.347583	+	0.937649i	$0.612998\pi$
$278$	40.6123	2.43576
$279$	0	0
$280$	0	0
$281$	25.2599	1.50688	0.753439	−	0.657518i	$-0.228393\pi$
$281$	25.2599	1.50688	0.753439	+	0.657518i	$0.228393\pi$
$282$	0	0
$283$	10.9671	0.651925	0.325963	−	0.945383i	$-0.394312\pi$
$283$	10.9671	0.651925	0.325963	+	0.945383i	$0.394312\pi$
$284$	−48.0896	−2.85360
$285$	0	0
$286$	−9.29947	−0.549889
$287$	0	0
$288$	0	0
$289$	5.30614	0.312126
$290$	73.8192	4.33482
$291$	0	0
$292$	−42.6967	−2.49863
$293$	14.7363	0.860902	0.430451	−	0.902614i	$-0.358355\pi$
$293$	14.7363	0.860902	0.430451	+	0.902614i	$0.358355\pi$
$294$	0	0
$295$	59.4287	3.46007
$296$	−20.5055	−1.19186
$297$	0	0
$298$	−37.6371	−2.18026
$299$	−10.7229	−0.620123
$300$	0	0
$301$	0	0
$302$	15.5832	0.896712
$303$	0	0
$304$	−7.84026	−0.449670
$305$	23.6280	1.35294
$306$	0	0
$307$	−0.860264	−0.0490979	−0.0245489	−	0.999699i	$-0.507815\pi$
$307$	−0.860264	−0.0490979	−0.0245489	+	0.999699i	$0.507815\pi$
$308$	0	0
$309$	0	0
$310$	10.7229	0.609022
$311$	26.8918	1.52489	0.762446	−	0.647052i	$-0.223998\pi$
$311$	26.8918	1.52489	0.762446	+	0.647052i	$0.223998\pi$
$312$	0	0
$313$	−10.1688	−0.574775	−0.287388	−	0.957814i	$-0.592787\pi$
$313$	−10.1688	−0.574775	−0.287388	+	0.957814i	$0.592787\pi$
$314$	−17.2904	−0.975753
$315$	0	0
$316$	−19.4774	−1.09569
$317$	27.4907	1.54403	0.772016	−	0.635604i	$-0.219249\pi$
$317$	27.4907	1.54403	0.772016	+	0.635604i	$0.219249\pi$
$318$	0	0
$319$	7.93800	0.444443
$320$	46.1650	2.58070
$321$	0	0
$322$	0	0
$323$	−22.5989	−1.25744
$324$	0	0
$325$	41.3800	2.29535
$326$	33.3128	1.84503
$327$	0	0
$328$	1.60135	0.0884200
$329$	0	0
$330$	0	0
$331$	−17.1979	−0.945281	−0.472641	−	0.881255i	$-0.656699\pi$
$331$	−17.1979	−0.945281	−0.472641	+	0.881255i	$0.656699\pi$
$332$	−10.1821	−0.558818
$333$	0	0
$334$	−0.984257	−0.0538561
$335$	−12.6609	−0.691741
$336$	0	0
$337$	19.7387	1.07523	0.537617	−	0.843189i	$-0.319325\pi$
$337$	19.7387	1.07523	0.537617	+	0.843189i	$0.319325\pi$
$338$	5.92226	0.322128
$339$	0	0
$340$	−66.5198	−3.60754
$341$	1.15307	0.0624422
$342$	0	0
$343$	0	0
$344$	25.0291	1.34948
$345$	0	0
$346$	−19.9075	−1.07023
$347$	−16.6123	−0.891794	−0.445897	−	0.895084i	$-0.647115\pi$
$347$	−16.6123	−0.891794	−0.445897	+	0.895084i	$0.647115\pi$
$348$	0	0
$349$	−6.95375	−0.372226	−0.186113	−	0.982528i	$-0.559589\pi$
$349$	−6.95375	−0.372226	−0.186113	+	0.982528i	$0.559589\pi$
$350$	0	0
$351$	0	0
$352$	3.57653	0.190630
$353$	−3.22840	−0.171830	−0.0859152	−	0.996302i	$-0.527381\pi$
$353$	−3.22840	−0.171830	−0.0859152	+	0.996302i	$0.527381\pi$
$354$	0	0
$355$	52.9499	2.81029
$356$	44.0133	2.33270
$357$	0	0
$358$	28.3376	1.49769
$359$	−10.3061	−0.543937	−0.271969	−	0.962306i	$-0.587675\pi$
$359$	−10.3061	−0.543937	−0.271969	+	0.962306i	$0.587675\pi$
$360$	0	0
$361$	3.89559	0.205031
$362$	−48.6438	−2.55666
$363$	0	0
$364$	0	0
$365$	47.0119	2.46072
$366$	0	0
$367$	11.8760	0.619923	0.309961	−	0.950749i	$-0.399684\pi$
$367$	11.8760	0.619923	0.309961	+	0.950749i	$0.399684\pi$
$368$	−4.46162	−0.232578
$369$	0	0
$370$	51.2203	2.66282
$371$	0	0
$372$	0	0
$373$	16.7229	0.865881	0.432940	−	0.901423i	$-0.357476\pi$
$373$	16.7229	0.865881	0.432940	+	0.901423i	$0.357476\pi$
$374$	−11.1531	−0.576711
$375$	0	0
$376$	−32.7058	−1.68667
$377$	−31.2599	−1.60997
$378$	0	0
$379$	−10.9671	−0.563341	−0.281671	−	0.959511i	$-0.590889\pi$
$379$	−10.9671	−0.563341	−0.281671	+	0.959511i	$0.590889\pi$
$380$	67.3930	3.45719
$381$	0	0
$382$	34.4750	1.76389
$383$	15.7520	0.804890	0.402445	−	0.915444i	$-0.368160\pi$
$383$	15.7520	0.804890	0.402445	+	0.915444i	$0.368160\pi$
$384$	0	0
$385$	0	0
$386$	31.7835	1.61774
$387$	0	0
$388$	−39.8893	−2.02507
$389$	−36.7678	−1.86420	−0.932100	−	0.362202i	$-0.882025\pi$
$389$	−36.7678	−1.86420	−0.932100	+	0.362202i	$0.882025\pi$
$390$	0	0
$391$	−12.8603	−0.650371
$392$	0	0
$393$	0	0
$394$	−27.0291	−1.36171
$395$	21.4459	1.07906
$396$	0	0
$397$	−11.8627	−0.595371	−0.297685	−	0.954664i	$-0.596215\pi$
$397$	−11.8627	−0.595371	−0.297685	+	0.954664i	$0.596215\pi$
$398$	30.7678	1.54225
$399$	0	0
$400$	17.2175	0.860874
$401$	−3.40106	−0.169841	−0.0849203	−	0.996388i	$-0.527064\pi$
$401$	−3.40106	−0.169841	−0.0849203	+	0.996388i	$0.527064\pi$
$402$	0	0
$403$	−4.54079	−0.226193
$404$	58.9232	2.93154
$405$	0	0
$406$	0	0
$407$	5.50787	0.273015
$408$	0	0
$409$	28.0315	1.38607	0.693034	−	0.720905i	$-0.256274\pi$
$409$	28.0315	1.38607	0.693034	+	0.720905i	$0.256274\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	5.61469	0.276616
$413$	0	0
$414$	0	0
$415$	11.2112	0.550337
$416$	−14.0844	−0.690545
$417$	0	0
$418$	11.2995	0.552675
$419$	28.2890	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$419$	28.2890	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$420$	0	0
$421$	−14.4921	−0.706303	−0.353152	−	0.935566i	$-0.614890\pi$
$421$	−14.4921	−0.706303	−0.353152	+	0.935566i	$0.614890\pi$
$422$	5.73868	0.279355
$423$	0	0
$424$	11.7387	0.570081
$425$	49.6280	2.40731
$426$	0	0
$427$	0	0
$428$	17.1135	0.827211
$429$	0	0
$430$	−62.5198	−3.01497
$431$	24.5369	1.18190	0.590952	−	0.806707i	$-0.298752\pi$
$431$	24.5369	1.18190	0.590952	+	0.806707i	$0.298752\pi$
$432$	0	0
$433$	7.32188	0.351867	0.175934	−	0.984402i	$-0.443706\pi$
$433$	7.32188	0.351867	0.175934	+	0.984402i	$0.443706\pi$
$434$	0	0
$435$	0	0
$436$	−53.7520	−2.57425
$437$	13.0291	0.623265
$438$	0	0
$439$	−16.5369	−0.789265	−0.394633	−	0.918839i	$-0.629128\pi$
$439$	−16.5369	−0.789265	−0.394633	+	0.918839i	$0.629128\pi$
$440$	14.6609	0.698933
$441$	0	0
$442$	43.9208	2.08910
$443$	−25.1979	−1.19719	−0.598594	−	0.801053i	$-0.704274\pi$
$443$	−25.1979	−1.19719	−0.598594	+	0.801053i	$0.704274\pi$
$444$	0	0
$445$	−48.4616	−2.29730
$446$	−53.7653	−2.54586
$447$	0	0
$448$	0	0
$449$	36.5198	1.72347	0.861737	−	0.507355i	$-0.169377\pi$
$449$	36.5198	1.72347	0.861737	+	0.507355i	$0.169377\pi$
$450$	0	0
$451$	−0.430132	−0.0202541
$452$	−26.6304	−1.25259
$453$	0	0
$454$	−37.4907	−1.75953
$455$	0	0
$456$	0	0
$457$	3.15307	0.147494	0.0737472	−	0.997277i	$-0.476504\pi$
$457$	3.15307	0.147494	0.0737472	+	0.997277i	$0.476504\pi$
$458$	7.44588	0.347923
$459$	0	0
$460$	38.3510	1.78812
$461$	−24.7678	−1.15355	−0.576775	−	0.816903i	$-0.695689\pi$
$461$	−24.7678	−1.15355	−0.576775	+	0.816903i	$0.695689\pi$
$462$	0	0
$463$	−35.4287	−1.64651	−0.823256	−	0.567671i	$-0.807845\pi$
$463$	−35.4287	−1.64651	−0.823256	+	0.567671i	$0.807845\pi$
$464$	−13.0067	−0.603819
$465$	0	0
$466$	−31.0291	−1.43739
$467$	−23.9247	−1.10710	−0.553551	−	0.832815i	$-0.686728\pi$
$467$	−23.9247	−1.10710	−0.553551	+	0.832815i	$0.686728\pi$
$468$	0	0
$469$	0	0
$470$	81.6953	3.76832
$471$	0	0
$472$	−56.1831	−2.58604
$473$	−6.72294	−0.309121
$474$	0	0
$475$	−50.2795	−2.30698
$476$	0	0
$477$	0	0
$478$	23.8669	1.09165
$479$	4.54079	0.207474	0.103737	−	0.994605i	$-0.466920\pi$
$479$	4.54079	0.207474	0.103737	+	0.994605i	$0.466920\pi$
$480$	0	0
$481$	−21.6900	−0.988980
$482$	−5.59228	−0.254721
$483$	0	0
$484$	3.57653	0.162570
$485$	43.9208	1.99434
$486$	0	0
$487$	27.1397	1.22982	0.614909	−	0.788598i	$-0.289193\pi$
$487$	27.1397	1.22982	0.614909	+	0.788598i	$0.289193\pi$
$488$	−22.3376	−1.01118
$489$	0	0
$490$	0	0
$491$	−3.46305	−0.156285	−0.0781427	−	0.996942i	$-0.524899\pi$
$491$	−3.46305	−0.156285	−0.0781427	+	0.996942i	$0.524899\pi$
$492$	0	0
$493$	−37.4907	−1.68850
$494$	−44.4974	−2.00203
$495$	0	0
$496$	−1.88934	−0.0848339
$497$	0	0
$498$	0	0
$499$	−18.9671	−0.849083	−0.424542	−	0.905408i	$-0.639565\pi$
$499$	−18.9671	−0.849083	−0.424542	+	0.905408i	$0.639565\pi$
$500$	−77.5751	−3.46926
$501$	0	0
$502$	24.1597	1.07830
$503$	5.86267	0.261404	0.130702	−	0.991422i	$-0.458277\pi$
$503$	5.86267	0.261404	0.130702	+	0.991422i	$0.458277\pi$
$504$	0	0
$505$	−64.8784	−2.88705
$506$	6.43013	0.285854
$507$	0	0
$508$	43.9656	1.95066
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	0	0
$512$	−18.0606	−0.798172
$513$	0	0
$514$	−26.1597	−1.15386
$515$	−6.18215	−0.272418
$516$	0	0
$517$	8.78493	0.386361
$518$	0	0
$519$	0	0
$520$	−57.7348	−2.53184
$521$	−2.24414	−0.0983177	−0.0491588	−	0.998791i	$-0.515654\pi$
$521$	−2.24414	−0.0983177	−0.0491588	+	0.998791i	$0.515654\pi$
$522$	0	0
$523$	17.9828	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$523$	17.9828	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$524$	25.5355	1.11552
$525$	0	0
$526$	20.7453	0.904540
$527$	−5.44588	−0.237226
$528$	0	0
$529$	−15.5856	−0.677635
$530$	−29.3219	−1.27366
$531$	0	0
$532$	0	0
$533$	1.69386	0.0733693
$534$	0	0
$535$	−18.8431	−0.814658
$536$	11.9695	0.517003
$537$	0	0
$538$	13.5832	0.585613
$539$	0	0
$540$	0	0
$541$	28.7678	1.23682	0.618411	−	0.785855i	$-0.287777\pi$
$541$	28.7678	1.23682	0.618411	+	0.785855i	$0.287777\pi$
$542$	−1.85360	−0.0796188
$543$	0	0
$544$	−16.8918	−0.724228
$545$	59.1846	2.53519
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−5.66237	−0.241885
$549$	0	0
$550$	−24.8140	−1.05807
$551$	37.9828	1.61812
$552$	0	0
$553$	0	0
$554$	−27.3219	−1.16080
$555$	0	0
$556$	61.5088	2.60856
$557$	13.5079	0.572347	0.286173	−	0.958178i	$-0.407617\pi$
$557$	13.5079	0.572347	0.286173	+	0.958178i	$0.407617\pi$
$558$	0	0
$559$	26.4750	1.11977
$560$	0	0
$561$	0	0
$562$	59.6504	2.51620
$563$	24.7811	1.04440	0.522199	−	0.852824i	$-0.325112\pi$
$563$	24.7811	1.04440	0.522199	+	0.852824i	$0.325112\pi$
$564$	0	0
$565$	29.3219	1.23358
$566$	25.8984	1.08859
$567$	0	0
$568$	−50.0582	−2.10039
$569$	−1.75201	−0.0734482	−0.0367241	−	0.999325i	$-0.511692\pi$
$569$	−1.75201	−0.0734482	−0.0367241	+	0.999325i	$0.511692\pi$
$570$	0	0
$571$	−3.16640	−0.132510	−0.0662549	−	0.997803i	$-0.521105\pi$
$571$	−3.16640	−0.132510	−0.0662549	+	0.997803i	$0.521105\pi$
$572$	−14.0844	−0.588899
$573$	0	0
$574$	0	0
$575$	−28.6123	−1.19321
$576$	0	0
$577$	−20.4750	−0.852383	−0.426192	−	0.904633i	$-0.640145\pi$
$577$	−20.4750	−0.852383	−0.426192	+	0.904633i	$0.640145\pi$
$578$	12.5303	0.521191
$579$	0	0
$580$	111.802	4.64233
$581$	0	0
$582$	0	0
$583$	−3.15307	−0.130587
$584$	−44.4444	−1.83912
$585$	0	0
$586$	34.7992	1.43754
$587$	−16.6609	−0.687671	−0.343835	−	0.939030i	$-0.611726\pi$
$587$	−16.6609	−0.687671	−0.343835	+	0.939030i	$0.611726\pi$
$588$	0	0
$589$	5.51736	0.227339
$590$	140.339	5.77767
$591$	0	0
$592$	−9.02482	−0.370918
$593$	27.9075	1.14602	0.573012	−	0.819547i	$-0.305775\pi$
$593$	27.9075	1.14602	0.573012	+	0.819547i	$0.305775\pi$
$594$	0	0
$595$	0	0
$596$	−57.0028	−2.33493
$597$	0	0
$598$	−25.3219	−1.03549
$599$	24.3376	0.994408	0.497204	−	0.867634i	$-0.334360\pi$
$599$	24.3376	0.994408	0.497204	+	0.867634i	$0.334360\pi$
$600$	0	0
$601$	9.50787	0.387834	0.193917	−	0.981018i	$-0.437881\pi$
$601$	9.50787	0.387834	0.193917	+	0.981018i	$0.437881\pi$
$602$	0	0
$603$	0	0
$604$	23.6014	0.960325
$605$	−3.93800	−0.160103
$606$	0	0
$607$	−26.8431	−1.08953	−0.544764	−	0.838590i	$-0.683381\pi$
$607$	−26.8431	−1.08953	−0.544764	+	0.838590i	$0.683381\pi$
$608$	17.1135	0.694043
$609$	0	0
$610$	55.7968	2.25915
$611$	−34.5951	−1.39957
$612$	0	0
$613$	−23.0291	−0.930136	−0.465068	−	0.885275i	$-0.653970\pi$
$613$	−23.0291	−0.930136	−0.465068	+	0.885275i	$0.653970\pi$
$614$	−2.03149	−0.0819841
$615$	0	0
$616$	0	0
$617$	−31.4459	−1.26596	−0.632982	−	0.774167i	$-0.718169\pi$
$617$	−31.4459	−1.26596	−0.632982	+	0.774167i	$0.718169\pi$
$618$	0	0
$619$	−2.26132	−0.0908901	−0.0454450	−	0.998967i	$-0.514471\pi$
$619$	−2.26132	−0.0908901	−0.0454450	+	0.998967i	$0.514471\pi$
$620$	16.2403	0.652226
$621$	0	0
$622$	63.5040	2.54628
$623$	0	0
$624$	0	0
$625$	32.8760	1.31504
$626$	−24.0133	−0.959766
$627$	0	0
$628$	−26.1870	−1.04497
$629$	−26.0133	−1.03722
$630$	0	0
$631$	10.3061	0.410281	0.205140	−	0.978733i	$-0.434235\pi$
$631$	10.3061	0.410281	0.205140	+	0.978733i	$0.434235\pi$
$632$	−20.2747	−0.806482
$633$	0	0
$634$	64.9184	2.57824
$635$	−48.4091	−1.92106
$636$	0	0
$637$	0	0
$638$	18.7453	0.742135
$639$	0	0
$640$	80.8483	3.19581
$641$	−4.13733	−0.163415	−0.0817073	−	0.996656i	$-0.526037\pi$
$641$	−4.13733	−0.163415	−0.0817073	+	0.996656i	$0.526037\pi$
$642$	0	0
$643$	8.44347	0.332978	0.166489	−	0.986043i	$-0.446757\pi$
$643$	8.44347	0.332978	0.166489	+	0.986043i	$0.446757\pi$
$644$	0	0
$645$	0	0
$646$	−53.3667	−2.09968
$647$	−25.2732	−0.993593	−0.496796	−	0.867867i	$-0.665490\pi$
$647$	−25.2732	−0.993593	−0.496796	+	0.867867i	$0.665490\pi$
$648$	0	0
$649$	15.0911	0.592376
$650$	97.7177	3.83280
$651$	0	0
$652$	50.4535	1.97591
$653$	−47.9075	−1.87477	−0.937383	−	0.348302i	$-0.886759\pi$
$653$	−47.9075	−1.87477	−0.937383	+	0.348302i	$0.886759\pi$
$654$	0	0
$655$	−28.1163	−1.09859
$656$	0.704785	0.0275172
$657$	0	0
$658$	0	0
$659$	28.2890	1.10198	0.550991	−	0.834511i	$-0.314250\pi$
$659$	28.2890	1.10198	0.550991	+	0.834511i	$0.314250\pi$
$660$	0	0
$661$	−19.7653	−0.768783	−0.384391	−	0.923170i	$-0.625589\pi$
$661$	−19.7653	−0.768783	−0.384391	+	0.923170i	$0.625589\pi$
$662$	−40.6123	−1.57844
$663$	0	0
$664$	−10.5989	−0.411319
$665$	0	0
$666$	0	0
$667$	21.6147	0.836924
$668$	−1.49069	−0.0576767
$669$	0	0
$670$	−29.8984	−1.15508
$671$	6.00000	0.231627
$672$	0	0
$673$	−51.5174	−1.98585	−0.992924	−	0.118750i	$-0.962111\pi$
$673$	−51.5174	−1.98585	−0.992924	+	0.118750i	$0.962111\pi$
$674$	46.6123	1.79544
$675$	0	0
$676$	8.96949	0.344980
$677$	46.3376	1.78090	0.890450	−	0.455081i	$-0.150390\pi$
$677$	46.3376	1.78090	0.890450	+	0.455081i	$0.150390\pi$
$678$	0	0
$679$	0	0
$680$	−69.2427	−2.65534
$681$	0	0
$682$	2.72294	0.104267
$683$	39.2112	1.50038	0.750188	−	0.661225i	$-0.229963\pi$
$683$	39.2112	1.50038	0.750188	+	0.661225i	$0.229963\pi$
$684$	0	0
$685$	6.23465	0.238214
$686$	0	0
$687$	0	0
$688$	11.0157	0.419971
$689$	12.4168	0.473042
$690$	0	0
$691$	−7.62802	−0.290184	−0.145092	−	0.989418i	$-0.546348\pi$
$691$	−7.62802	−0.290184	−0.145092	+	0.989418i	$0.546348\pi$
$692$	−30.1507	−1.14616
$693$	0	0
$694$	−39.2294	−1.48913
$695$	−67.7253	−2.56897
$696$	0	0
$697$	2.03149	0.0769480
$698$	−16.4211	−0.621546
$699$	0	0
$700$	0	0
$701$	−2.61228	−0.0986644	−0.0493322	−	0.998782i	$-0.515709\pi$
$701$	−2.61228	−0.0986644	−0.0493322	+	0.998782i	$0.515709\pi$
$702$	0	0
$703$	26.3548	0.993990
$704$	11.7229	0.441825
$705$	0	0
$706$	−7.62376	−0.286924
$707$	0	0
$708$	0	0
$709$	36.1516	1.35770	0.678852	−	0.734276i	$-0.262478\pi$
$709$	36.1516	1.35770	0.678852	+	0.734276i	$0.262478\pi$
$710$	125.040	4.69265
$711$	0	0
$712$	45.8150	1.71699
$713$	3.13974	0.117584
$714$	0	0
$715$	15.5079	0.579962
$716$	42.9184	1.60394
$717$	0	0
$718$	−24.3376	−0.908272
$719$	19.6767	0.733816	0.366908	−	0.930257i	$-0.380416\pi$
$719$	19.6767	0.733816	0.366908	+	0.930257i	$0.380416\pi$
$720$	0	0
$721$	0	0
$722$	9.19932	0.342363
$723$	0	0
$724$	−73.6728	−2.73803
$725$	−83.4115	−3.09783
$726$	0	0
$727$	−2.47495	−0.0917909	−0.0458954	−	0.998946i	$-0.514614\pi$
$727$	−2.47495	−0.0917909	−0.0458954	+	0.998946i	$0.514614\pi$
$728$	0	0
$729$	0	0
$730$	111.017	4.10893
$731$	31.7520	1.17439
$732$	0	0
$733$	40.0315	1.47860	0.739298	−	0.673378i	$-0.235157\pi$
$733$	40.0315	1.47860	0.739298	+	0.673378i	$0.235157\pi$
$734$	28.0448	1.03515
$735$	0	0
$736$	9.73868	0.358973
$737$	−3.21507	−0.118428
$738$	0	0
$739$	25.2771	0.929832	0.464916	−	0.885355i	$-0.346085\pi$
$739$	25.2771	0.929832	0.464916	+	0.885355i	$0.346085\pi$
$740$	77.5751	2.85172
$741$	0	0
$742$	0	0
$743$	−2.10682	−0.0772916	−0.0386458	−	0.999253i	$-0.512304\pi$
$743$	−2.10682	−0.0772916	−0.0386458	+	0.999253i	$0.512304\pi$
$744$	0	0
$745$	62.7639	2.29949
$746$	39.4907	1.44586
$747$	0	0
$748$	−16.8918	−0.617624
$749$	0	0
$750$	0	0
$751$	−45.9828	−1.67794	−0.838969	−	0.544180i	$-0.816841\pi$
$751$	−45.9828	−1.67794	−0.838969	+	0.544180i	$0.816841\pi$
$752$	−14.3944	−0.524909
$753$	0	0
$754$	−73.8192	−2.68834
$755$	−25.9867	−0.945752
$756$	0	0
$757$	−29.2599	−1.06347	−0.531734	−	0.846911i	$-0.678460\pi$
$757$	−29.2599	−1.06347	−0.531734	+	0.846911i	$0.678460\pi$
$758$	−25.8984	−0.940673
$759$	0	0
$760$	70.1516	2.54467
$761$	21.8760	0.793005	0.396502	−	0.918034i	$-0.370224\pi$
$761$	21.8760	0.793005	0.396502	+	0.918034i	$0.370224\pi$
$762$	0	0
$763$	0	0
$764$	52.2136	1.88902
$765$	0	0
$766$	37.1979	1.34401
$767$	−59.4287	−2.14585
$768$	0	0
$769$	16.2756	0.586914	0.293457	−	0.955972i	$-0.405194\pi$
$769$	16.2756	0.586914	0.293457	+	0.955972i	$0.405194\pi$
$770$	0	0
$771$	0	0
$772$	48.1373	1.73250
$773$	−30.4921	−1.09673	−0.548363	−	0.836241i	$-0.684749\pi$
$773$	−30.4921	−1.09673	−0.548363	+	0.836241i	$0.684749\pi$
$774$	0	0
$775$	−12.1163	−0.435231
$776$	−41.5222	−1.49056
$777$	0	0
$778$	−86.8259	−3.11286
$779$	−2.05815	−0.0737410
$780$	0	0
$781$	13.4459	0.481131
$782$	−30.3691	−1.08600
$783$	0	0
$784$	0	0
$785$	28.8336	1.02912
$786$	0	0
$787$	−35.2151	−1.25528	−0.627641	−	0.778503i	$-0.715979\pi$
$787$	−35.2151	−1.25528	−0.627641	+	0.778503i	$0.715979\pi$
$788$	−40.9366	−1.45830
$789$	0	0
$790$	50.6438	1.80182
$791$	0	0
$792$	0	0
$793$	−23.6280	−0.839056
$794$	−28.0133	−0.994156
$795$	0	0
$796$	46.5989	1.65166
$797$	−14.9804	−0.530633	−0.265317	−	0.964161i	$-0.585477\pi$
$797$	−14.9804	−0.530633	−0.265317	+	0.964161i	$0.585477\pi$
$798$	0	0
$799$	−41.4907	−1.46784
$800$	−37.5818	−1.32872
$801$	0	0
$802$	−8.03149	−0.283602
$803$	11.9380	0.421283
$804$	0	0
$805$	0	0
$806$	−10.7229	−0.377699
$807$	0	0
$808$	61.3352	2.15777
$809$	25.2599	0.888090	0.444045	−	0.896004i	$-0.353543\pi$
$809$	25.2599	0.888090	0.444045	+	0.896004i	$0.353543\pi$
$810$	0	0
$811$	−29.5212	−1.03663	−0.518315	−	0.855190i	$-0.673440\pi$
$811$	−29.5212	−1.03663	−0.518315	+	0.855190i	$0.673440\pi$
$812$	0	0
$813$	0	0
$814$	13.0067	0.455883
$815$	−55.5527	−1.94593
$816$	0	0
$817$	−32.1688	−1.12544
$818$	66.1955	2.31447
$819$	0	0
$820$	−6.05815	−0.211560
$821$	−23.7873	−0.830184	−0.415092	−	0.909779i	$-0.636251\pi$
$821$	−23.7873	−0.830184	−0.415092	+	0.909779i	$0.636251\pi$
$822$	0	0
$823$	17.7692	0.619395	0.309698	−	0.950835i	$-0.399772\pi$
$823$	17.7692	0.619395	0.309698	+	0.950835i	$0.399772\pi$
$824$	5.84452	0.203604
$825$	0	0
$826$	0	0
$827$	−8.90893	−0.309794	−0.154897	−	0.987931i	$-0.549505\pi$
$827$	−8.90893	−0.309794	−0.154897	+	0.987931i	$0.549505\pi$
$828$	0	0
$829$	32.7678	1.13807	0.569036	−	0.822313i	$-0.307317\pi$
$829$	32.7678	1.13807	0.569036	+	0.822313i	$0.307317\pi$
$830$	26.4750	0.918959
$831$	0	0
$832$	−46.1650	−1.60048
$833$	0	0
$834$	0	0
$835$	1.64135	0.0568014
$836$	17.1135	0.591882
$837$	0	0
$838$	66.8035	2.30769
$839$	6.96708	0.240530	0.120265	−	0.992742i	$-0.461626\pi$
$839$	6.96708	0.240530	0.120265	+	0.992742i	$0.461626\pi$
$840$	0	0
$841$	34.0119	1.17282
$842$	−34.2227	−1.17939
$843$	0	0
$844$	8.69145	0.299172
$845$	−9.87601	−0.339745
$846$	0	0
$847$	0	0
$848$	5.16640	0.177415
$849$	0	0
$850$	117.195	4.01976
$851$	14.9976	0.514111
$852$	0	0
$853$	−11.1979	−0.383408	−0.191704	−	0.981453i	$-0.561401\pi$
$853$	−11.1979	−0.383408	−0.191704	+	0.981453i	$0.561401\pi$
$854$	0	0
$855$	0	0
$856$	17.8140	0.608870
$857$	−44.0315	−1.50409	−0.752043	−	0.659114i	$-0.770932\pi$
$857$	−44.0315	−1.50409	−0.752043	+	0.659114i	$0.770932\pi$
$858$	0	0
$859$	−7.13974	−0.243605	−0.121802	−	0.992554i	$-0.538867\pi$
$859$	−7.13974	−0.243605	−0.121802	+	0.992554i	$0.538867\pi$
$860$	−94.6886	−3.22885
$861$	0	0
$862$	57.9432	1.97355
$863$	5.27706	0.179633	0.0898166	−	0.995958i	$-0.471372\pi$
$863$	5.27706	0.179633	0.0898166	+	0.995958i	$0.471372\pi$
$864$	0	0
$865$	33.1979	1.12876
$866$	17.2904	0.587552
$867$	0	0
$868$	0	0
$869$	5.44588	0.184739
$870$	0	0
$871$	12.6609	0.429000
$872$	−55.9523	−1.89478
$873$	0	0
$874$	30.7678	1.04073
$875$	0	0
$876$	0	0
$877$	−32.4301	−1.09509	−0.547544	−	0.836777i	$-0.684437\pi$
$877$	−32.4301	−1.09509	−0.547544	+	0.836777i	$0.684437\pi$
$878$	−39.0515	−1.31792
$879$	0	0
$880$	6.45254	0.217515
$881$	39.5660	1.33301	0.666507	−	0.745499i	$-0.267789\pi$
$881$	39.5660	1.33301	0.666507	+	0.745499i	$0.267789\pi$
$882$	0	0
$883$	−24.4130	−0.821561	−0.410781	−	0.911734i	$-0.634744\pi$
$883$	−24.4130	−0.821561	−0.410781	+	0.911734i	$0.634744\pi$
$884$	66.5198	2.23730
$885$	0	0
$886$	−59.5040	−1.99908
$887$	36.2928	1.21859	0.609297	−	0.792942i	$-0.291452\pi$
$887$	36.2928	1.21859	0.609297	+	0.792942i	$0.291452\pi$
$888$	0	0
$889$	0	0
$890$	−114.441	−3.83606
$891$	0	0
$892$	−81.4297	−2.72647
$893$	42.0353	1.40666
$894$	0	0
$895$	−47.2560	−1.57960
$896$	0	0
$897$	0	0
$898$	86.2403	2.87788
$899$	9.15307	0.305272
$900$	0	0
$901$	14.8918	0.496116
$902$	−1.01574	−0.0338205
$903$	0	0
$904$	−27.7205	−0.921971
$905$	81.1187	2.69648
$906$	0	0
$907$	−30.3958	−1.00928	−0.504638	−	0.863331i	$-0.668374\pi$
$907$	−30.3958	−1.00928	−0.504638	+	0.863331i	$0.668374\pi$
$908$	−56.7811	−1.88435
$909$	0	0
$910$	0	0
$911$	56.8259	1.88273	0.941363	−	0.337395i	$-0.109546\pi$
$911$	56.8259	1.88273	0.941363	+	0.337395i	$0.109546\pi$
$912$	0	0
$913$	2.84693	0.0942196
$914$	7.44588	0.246288
$915$	0	0
$916$	11.2771	0.372605
$917$	0	0
$918$	0	0
$919$	−40.1688	−1.32505	−0.662523	−	0.749041i	$-0.730514\pi$
$919$	−40.1688	−1.32505	−0.662523	+	0.749041i	$0.730514\pi$
$920$	39.9208	1.31615
$921$	0	0
$922$	−58.4883	−1.92621
$923$	−52.9499	−1.74287
$924$	0	0
$925$	−57.8760	−1.90295
$926$	−83.6638	−2.74936
$927$	0	0
$928$	28.3905	0.931965
$929$	−27.3180	−0.896276	−0.448138	−	0.893964i	$-0.647913\pi$
$929$	−27.3180	−0.896276	−0.448138	+	0.893964i	$0.647913\pi$
$930$	0	0
$931$	0	0
$932$	−46.9947	−1.53936
$933$	0	0
$934$	−56.4974	−1.84865
$935$	18.5989	0.608251
$936$	0	0
$937$	5.75201	0.187910	0.0939551	−	0.995576i	$-0.470049\pi$
$937$	5.75201	0.187910	0.0939551	+	0.995576i	$0.470049\pi$
$938$	0	0
$939$	0	0
$940$	123.731	4.03565
$941$	−27.5699	−0.898752	−0.449376	−	0.893343i	$-0.648354\pi$
$941$	−27.5699	−0.898752	−0.449376	+	0.893343i	$0.648354\pi$
$942$	0	0
$943$	−1.17122	−0.0381402
$944$	−24.7272	−0.804802
$945$	0	0
$946$	−15.8760	−0.516174
$947$	−37.0739	−1.20474	−0.602370	−	0.798217i	$-0.705777\pi$
$947$	−37.0739	−1.20474	−0.602370	+	0.798217i	$0.705777\pi$
$948$	0	0
$949$	−47.0119	−1.52607
$950$	−118.733	−3.85222
$951$	0	0
$952$	0	0
$953$	12.0620	0.390726	0.195363	−	0.980731i	$-0.437411\pi$
$953$	12.0620	0.390726	0.195363	+	0.980731i	$0.437411\pi$
$954$	0	0
$955$	−57.4907	−1.86036
$956$	36.1474	1.16909
$957$	0	0
$958$	10.7229	0.346442
$959$	0	0
$960$	0	0
$961$	−29.6704	−0.957111
$962$	−51.2203	−1.65141
$963$	0	0
$964$	−8.46972	−0.272791
$965$	−53.0024	−1.70621
$966$	0	0
$967$	6.43013	0.206779	0.103390	−	0.994641i	$-0.467031\pi$
$967$	6.43013	0.206779	0.103390	+	0.994641i	$0.467031\pi$
$968$	3.72294	0.119660
$969$	0	0
$970$	103.718	3.33017
$971$	−35.3047	−1.13298	−0.566491	−	0.824068i	$-0.691699\pi$
$971$	−35.3047	−1.13298	−0.566491	+	0.824068i	$0.691699\pi$
$972$	0	0
$973$	0	0
$974$	64.0896	2.05356
$975$	0	0
$976$	−9.83119	−0.314689
$977$	−0.261319	−0.00836035	−0.00418017	−	0.999991i	$-0.501331\pi$
$977$	−0.261319	−0.00836035	−0.00418017	+	0.999991i	$0.501331\pi$
$978$	0	0
$979$	−12.3061	−0.393306
$980$	0	0
$981$	0	0
$982$	−8.17789	−0.260967
$983$	−16.3376	−0.521089	−0.260545	−	0.965462i	$-0.583902\pi$
$983$	−16.3376	−0.521089	−0.260545	+	0.965462i	$0.583902\pi$
$984$	0	0
$985$	45.0739	1.43617
$986$	−88.5331	−2.81947
$987$	0	0
$988$	−67.3930	−2.14406
$989$	−18.3061	−0.582101
$990$	0	0
$991$	−36.2623	−1.15191	−0.575955	−	0.817481i	$-0.695370\pi$
$991$	−36.2623	−1.15191	−0.575955	+	0.817481i	$0.695370\pi$
$992$	4.12399	0.130937
$993$	0	0
$994$	0	0
$995$	−51.3085	−1.62659
$996$	0	0
$997$	3.47254	0.109977	0.0549883	−	0.998487i	$-0.482488\pi$
$997$	3.47254	0.109977	0.0549883	+	0.998487i	$0.482488\pi$
$998$	−44.7902	−1.41781
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bp.1.3		3
3.2	odd	2		1617.2.a.s.1.1		3
7.6	odd	2		693.2.a.m.1.3		3
21.20	even	2		231.2.a.d.1.1	✓	3
77.76	even	2		7623.2.a.cb.1.1		3
84.83	odd	2		3696.2.a.bp.1.1		3
105.104	even	2		5775.2.a.bw.1.3		3
231.230	odd	2		2541.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.1	✓	3	21.20	even	2
693.2.a.m.1.3		3	7.6	odd	2
1617.2.a.s.1.1		3	3.2	odd	2
2541.2.a.bi.1.3		3	231.230	odd	2
3696.2.a.bp.1.1		3	84.83	odd	2
4851.2.a.bp.1.3		3	1.1	even	1	trivial
5775.2.a.bw.1.3		3	105.104	even	2
7623.2.a.cb.1.1		3	77.76	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36147 q^{2} +3.57653 q^{4} -3.93800 q^{5} +3.72294 q^{8} +O(q^{10})\)	\(q+2.36147 q^{2} +3.57653 q^{4} -3.93800 q^{5} +3.72294 q^{8} -9.29947 q^{10} -1.00000 q^{11} +3.93800 q^{13} +1.63853 q^{16} +4.72294 q^{17} -4.78493 q^{19} -14.0844 q^{20} -2.36147 q^{22} -2.72294 q^{23} +10.5079 q^{25} +9.29947 q^{26} -7.93800 q^{29} -1.15307 q^{31} -3.57653 q^{32} +11.1531 q^{34} -5.50787 q^{37} -11.2995 q^{38} -14.6609 q^{40} +0.430132 q^{41} +6.72294 q^{43} -3.57653 q^{44} -6.43013 q^{46} -8.78493 q^{47} +24.8140 q^{50} +14.0844 q^{52} +3.15307 q^{53} +3.93800 q^{55} -18.7453 q^{58} -15.0911 q^{59} -6.00000 q^{61} -2.72294 q^{62} -11.7229 q^{64} -15.5079 q^{65} +3.21507 q^{67} +16.8918 q^{68} -13.4459 q^{71} -11.9380 q^{73} -13.0067 q^{74} -17.1135 q^{76} -5.44588 q^{79} -6.45254 q^{80} +1.01574 q^{82} -2.84693 q^{83} -18.5989 q^{85} +15.8760 q^{86} -3.72294 q^{88} +12.3061 q^{89} -9.73868 q^{92} -20.7453 q^{94} +18.8431 q^{95} -11.1531 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q + 6 * q^4 - 3 * q^8	\( 3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^8 - 9 * q^10 - 3 * q^11 + 12 * q^16 - 12 * q^19 - 21 * q^20 + 6 * q^23 + 15 * q^25 + 9 * q^26 - 12 * q^29 + 6 * q^31 - 6 * q^32 + 24 * q^34 - 15 * q^38 - 18 * q^40 + 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 - 24 * q^47 + 39 * q^50 + 21 * q^52 - 9 * q^58 - 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^73 - 39 * q^74 + 3 * q^76 + 12 * q^79 + 9 * q^80 - 30 * q^82 - 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 + 18 * q^89 + 18 * q^92 - 15 * q^94 - 12 * q^95 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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