LMFDB - Embedded newform 4725.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4725,2,Mod(1,4725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4725, 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("4725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4725 = 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4725.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:	$37.7293149551$
Analytic rank:	$1$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.91223$ of defining polynomial
Character	$\chi$	$=$	4725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.656620 q^{2} -1.56885 q^{4} -1.00000 q^{7} -2.34338 q^{8} +1.91223 q^{11} -0.255609 q^{13} -0.656620 q^{14} +1.59899 q^{16} -6.16784 q^{17} +7.08007 q^{19} +1.25561 q^{22} +2.56885 q^{23} -0.167838 q^{26} +1.56885 q^{28} -2.56885 q^{29} -1.05763 q^{31} +5.73669 q^{32} -4.04993 q^{34} -8.39331 q^{37} +4.64892 q^{38} +10.8744 q^{41} +5.08777 q^{43} -3.00000 q^{44} +1.68676 q^{46} -3.05763 q^{47} +1.00000 q^{49} +0.401012 q^{52} -2.77453 q^{53} +2.34338 q^{56} -1.68676 q^{58} -8.36317 q^{59} -12.3055 q^{61} -0.694463 q^{62} +0.568850 q^{64} -8.93972 q^{67} +9.67641 q^{68} +12.5611 q^{71} +9.96216 q^{73} -5.51122 q^{74} -11.1076 q^{76} -1.91223 q^{77} -9.96216 q^{79} +7.14034 q^{82} -3.71425 q^{83} +3.34073 q^{86} -4.48108 q^{88} -6.33568 q^{89} +0.255609 q^{91} -4.03014 q^{92} -2.00770 q^{94} +5.64892 q^{97} +0.656620 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{4} - 3 q^{7} - 9 q^{8} - q^{11} + 4 q^{13} + 2 q^{16} - 7 q^{17} + 3 q^{19} - q^{22} - q^{23} + 11 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} - 3 q^{32} + 12 q^{34} - 3 q^{37} - 13 q^{38} - 5 q^{41}+ \cdots - 10 q^{97}+O(q^{100}) $ 3 * q + 4 * q^4 - 3 * q^7 - 9 * q^8 - q^11 + 4 * q^13 + 2 * q^16 - 7 * q^17 + 3 * q^19 - q^22 - q^23 + 11 * q^26 - 4 * q^28 + q^29 - 4 * q^31 - 3 * q^32 + 12 * q^34 - 3 * q^37 - 13 * q^38 - 5 * q^41 + 22 * q^43 - 9 * q^44 + 9 * q^46 - 10 * q^47 + 3 * q^49 + 4 * q^52 - 19 * q^53 + 9 * q^56 - 9 * q^58 + 3 * q^59 - 8 * q^61 - 31 * q^62 - 7 * q^64 - 15 * q^67 - 3 * q^68 + 4 * q^71 - q^73 - 7 * q^74 - 10 * q^76 + q^77 + q^79 + 15 * q^82 - 10 * q^83 + q^86 + 2 * q^88 + 4 * q^89 - 4 * q^91 - 18 * q^92 - 31 * q^94 - 10 * q^97

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	$-0.425424\pi$
$2$	0.656620	0.464301	0.232150	+	0.972680i	$0.425424\pi$
$3$	0	0
$3$	0	0
$4$	−1.56885	−0.784425
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.34338	−0.828510
$9$	0	0
$10$	0	0
$11$	1.91223	0.576559	0.288279	−	0.957546i	$-0.406917\pi$
$11$	1.91223	0.576559	0.288279	+	0.957546i	$0.406917\pi$
$12$	0	0
$13$	−0.255609	−0.0708931	−0.0354466	−	0.999372i	$-0.511285\pi$
$13$	−0.255609	−0.0708931	−0.0354466	+	0.999372i	$0.511285\pi$
$14$	−0.656620	−0.175489
$15$	0	0
$16$	1.59899	0.399747
$17$	−6.16784	−1.49592	−0.747960	−	0.663744i	$-0.768967\pi$
$17$	−6.16784	−1.49592	−0.747960	+	0.663744i	$0.768967\pi$
$18$	0	0
$19$	7.08007	1.62428	0.812139	−	0.583463i	$-0.198303\pi$
$19$	7.08007	1.62428	0.812139	+	0.583463i	$0.198303\pi$
$20$	0	0
$21$	0	0
$22$	1.25561	0.267697
$23$	2.56885	0.535642	0.267821	−	0.963469i	$-0.413696\pi$
$23$	2.56885	0.535642	0.267821	+	0.963469i	$0.413696\pi$
$24$	0	0
$25$	0	0
$26$	−0.167838	−0.0329157
$27$	0	0
$28$	1.56885	0.296485
$29$	−2.56885	−0.477023	−0.238512	−	0.971140i	$-0.576660\pi$
$29$	−2.56885	−0.477023	−0.238512	+	0.971140i	$0.576660\pi$
$30$	0	0
$31$	−1.05763	−0.189956	−0.0949782	−	0.995479i	$-0.530278\pi$
$31$	−1.05763	−0.189956	−0.0949782	+	0.995479i	$0.530278\pi$
$32$	5.73669	1.01411
$33$	0	0
$34$	−4.04993	−0.694557
$35$	0	0
$36$	0	0
$37$	−8.39331	−1.37985	−0.689926	−	0.723880i	$-0.742357\pi$
$37$	−8.39331	−1.37985	−0.689926	+	0.723880i	$0.742357\pi$
$38$	4.64892	0.754154
$39$	0	0
$40$	0	0
$41$	10.8744	1.69829	0.849147	−	0.528157i	$-0.177117\pi$
$41$	10.8744	1.69829	0.849147	+	0.528157i	$0.177117\pi$
$42$	0	0
$43$	5.08777	0.775878	0.387939	−	0.921685i	$-0.373187\pi$
$43$	5.08777	0.775878	0.387939	+	0.921685i	$0.373187\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	1.68676	0.248699
$47$	−3.05763	−0.446001	−0.223001	−	0.974818i	$-0.571585\pi$
$47$	−3.05763	−0.446001	−0.223001	+	0.974818i	$0.571585\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0.401012	0.0556103
$53$	−2.77453	−0.381111	−0.190555	−	0.981676i	$-0.561029\pi$
$53$	−2.77453	−0.381111	−0.190555	+	0.981676i	$0.561029\pi$
$54$	0	0
$55$	0	0
$56$	2.34338	0.313147
$57$	0	0
$58$	−1.68676	−0.221482
$59$	−8.36317	−1.08879	−0.544396	−	0.838828i	$-0.683241\pi$
$59$	−8.36317	−1.08879	−0.544396	+	0.838828i	$0.683241\pi$
$60$	0	0
$61$	−12.3055	−1.57556	−0.787781	−	0.615955i	$-0.788770\pi$
$61$	−12.3055	−1.57556	−0.787781	+	0.615955i	$0.788770\pi$
$62$	−0.694463	−0.0881969
$63$	0	0
$64$	0.568850	0.0711062
$65$	0	0
$66$	0	0
$67$	−8.93972	−1.09216	−0.546080	−	0.837733i	$-0.683881\pi$
$67$	−8.93972	−1.09216	−0.546080	+	0.837733i	$0.683881\pi$
$68$	9.67641	1.17344
$69$	0	0
$70$	0	0
$71$	12.5611	1.49073	0.745367	−	0.666654i	$-0.232274\pi$
$71$	12.5611	1.49073	0.745367	+	0.666654i	$0.232274\pi$
$72$	0	0
$73$	9.96216	1.16598	0.582991	−	0.812478i	$-0.301882\pi$
$73$	9.96216	1.16598	0.582991	+	0.812478i	$0.301882\pi$
$74$	−5.51122	−0.640666
$75$	0	0
$76$	−11.1076	−1.27412
$77$	−1.91223	−0.217919
$78$	0	0
$79$	−9.96216	−1.12083	−0.560415	−	0.828212i	$-0.689358\pi$
$79$	−9.96216	−1.12083	−0.560415	+	0.828212i	$0.689358\pi$
$80$	0	0
$81$	0	0
$82$	7.14034	0.788519
$83$	−3.71425	−0.407692	−0.203846	−	0.979003i	$-0.565344\pi$
$83$	−3.71425	−0.407692	−0.203846	+	0.979003i	$0.565344\pi$
$84$	0	0
$85$	0	0
$86$	3.34073	0.360241
$87$	0	0
$88$	−4.48108	−0.477685
$89$	−6.33568	−0.671580	−0.335790	−	0.941937i	$-0.609003\pi$
$89$	−6.33568	−0.671580	−0.335790	+	0.941937i	$0.609003\pi$
$90$	0	0
$91$	0.255609	0.0267951
$92$	−4.03014	−0.420171
$93$	0	0
$94$	−2.00770	−0.207079
$95$	0	0
$96$	0	0
$97$	5.64892	0.573561	0.286780	−	0.957996i	$-0.407415\pi$
$97$	5.64892	0.573561	0.286780	+	0.957996i	$0.407415\pi$
$98$	0.656620	0.0663287
$99$	0	0
$100$	0	0
$101$	−14.0422	−1.39725	−0.698627	−	0.715486i	$-0.746205\pi$
$101$	−14.0422	−1.39725	−0.698627	+	0.715486i	$0.746205\pi$
$102$	0	0
$103$	2.10756	0.207664	0.103832	−	0.994595i	$-0.466890\pi$
$103$	2.10756	0.207664	0.103832	+	0.994595i	$0.466890\pi$
$104$	0.598988	0.0587356
$105$	0	0
$106$	−1.82181	−0.176950
$107$	−11.1755	−1.08038	−0.540190	−	0.841543i	$-0.681648\pi$
$107$	−11.1755	−1.08038	−0.540190	+	0.841543i	$0.681648\pi$
$108$	0	0
$109$	−1.86230	−0.178376	−0.0891880	−	0.996015i	$-0.528427\pi$
$109$	−1.86230	−0.178376	−0.0891880	+	0.996015i	$0.528427\pi$
$110$	0	0
$111$	0	0
$112$	−1.59899	−0.151090
$113$	−8.67906	−0.816457	−0.408228	−	0.912880i	$-0.633853\pi$
$113$	−8.67906	−0.816457	−0.408228	+	0.912880i	$0.633853\pi$
$114$	0	0
$115$	0	0
$116$	4.03014	0.374189
$117$	0	0
$118$	−5.49143	−0.505527
$119$	6.16784	0.565405
$120$	0	0
$121$	−7.34338	−0.667580
$122$	−8.08007	−0.731535
$123$	0	0
$124$	1.65927	0.149006
$125$	0	0
$126$	0	0
$127$	12.9424	1.14845	0.574225	−	0.818698i	$-0.305304\pi$
$127$	12.9424	1.14845	0.574225	+	0.818698i	$0.305304\pi$
$128$	−11.0999	−0.981098
$129$	0	0
$130$	0	0
$131$	−5.30554	−0.463547	−0.231773	−	0.972770i	$-0.574453\pi$
$131$	−5.30554	−0.463547	−0.231773	+	0.972770i	$0.574453\pi$
$132$	0	0
$133$	−7.08007	−0.613920
$134$	−5.87000	−0.507091
$135$	0	0
$136$	14.4536	1.23938
$137$	−19.9320	−1.70291	−0.851454	−	0.524430i	$-0.824278\pi$
$137$	−19.9320	−1.70291	−0.851454	+	0.524430i	$0.824278\pi$
$138$	0	0
$139$	9.27540	0.786729	0.393365	−	0.919383i	$-0.371311\pi$
$139$	9.27540	0.786729	0.393365	+	0.919383i	$0.371311\pi$
$140$	0	0
$141$	0	0
$142$	8.24791	0.692149
$143$	−0.488783	−0.0408740
$144$	0	0
$145$	0	0
$146$	6.54136	0.541366
$147$	0	0
$148$	13.1678	1.08239
$149$	2.08007	0.170406	0.0852029	−	0.996364i	$-0.472846\pi$
$149$	2.08007	0.170406	0.0852029	+	0.996364i	$0.472846\pi$
$150$	0	0
$151$	7.73669	0.629603	0.314801	−	0.949158i	$-0.398062\pi$
$151$	7.73669	0.629603	0.314801	+	0.949158i	$0.398062\pi$
$152$	−16.5913	−1.34573
$153$	0	0
$154$	−1.25561	−0.101180
$155$	0	0
$156$	0	0
$157$	−5.45094	−0.435032	−0.217516	−	0.976057i	$-0.569795\pi$
$157$	−5.45094	−0.435032	−0.217516	+	0.976057i	$0.569795\pi$
$158$	−6.54136	−0.520402
$159$	0	0
$160$	0	0
$161$	−2.56885	−0.202454
$162$	0	0
$163$	−22.3581	−1.75122	−0.875611	−	0.483017i	$-0.839541\pi$
$163$	−22.3581	−1.75122	−0.875611	+	0.483017i	$0.839541\pi$
$164$	−17.0603	−1.33218
$165$	0	0
$166$	−2.43885	−0.189292
$167$	−12.1678	−0.941576	−0.470788	−	0.882246i	$-0.656030\pi$
$167$	−12.1678	−0.941576	−0.470788	+	0.882246i	$0.656030\pi$
$168$	0	0
$169$	−12.9347	−0.994974
$170$	0	0
$171$	0	0
$172$	−7.98195	−0.608618
$173$	−16.0499	−1.22025	−0.610127	−	0.792304i	$-0.708881\pi$
$173$	−16.0499	−1.22025	−0.610127	+	0.792304i	$0.708881\pi$
$174$	0	0
$175$	0	0
$176$	3.05763	0.230478
$177$	0	0
$178$	−4.16013	−0.311815
$179$	−10.8546	−0.811311	−0.405655	−	0.914026i	$-0.632957\pi$
$179$	−10.8546	−0.811311	−0.405655	+	0.914026i	$0.632957\pi$
$180$	0	0
$181$	2.82446	0.209940	0.104970	−	0.994475i	$-0.466525\pi$
$181$	2.82446	0.209940	0.104970	+	0.994475i	$0.466525\pi$
$182$	0.167838	0.0124410
$183$	0	0
$184$	−6.01979	−0.443785
$185$	0	0
$186$	0	0
$187$	−11.7943	−0.862486
$188$	4.79696	0.349855
$189$	0	0
$190$	0	0
$191$	24.4855	1.77171	0.885853	−	0.463966i	$-0.153574\pi$
$191$	24.4855	1.77171	0.885853	+	0.463966i	$0.153574\pi$
$192$	0	0
$193$	10.5990	0.762932	0.381466	−	0.924383i	$-0.375419\pi$
$193$	10.5990	0.762932	0.381466	+	0.924383i	$0.375419\pi$
$194$	3.70919	0.266305
$195$	0	0
$196$	−1.56885	−0.112061
$197$	10.4811	0.746746	0.373373	−	0.927681i	$-0.378201\pi$
$197$	10.4811	0.746746	0.373373	+	0.927681i	$0.378201\pi$
$198$	0	0
$199$	−0.827104	−0.0586318	−0.0293159	−	0.999570i	$-0.509333\pi$
$199$	−0.827104	−0.0586318	−0.0293159	+	0.999570i	$0.509333\pi$
$200$	0	0
$201$	0	0
$202$	−9.22041	−0.648746
$203$	2.56885	0.180298
$204$	0	0
$205$	0	0
$206$	1.38387	0.0964186
$207$	0	0
$208$	−0.408715	−0.0283393
$209$	13.5387	0.936492
$210$	0	0
$211$	−4.88209	−0.336097	−0.168048	−	0.985779i	$-0.553747\pi$
$211$	−4.88209	−0.336097	−0.168048	+	0.985779i	$0.553747\pi$
$212$	4.35282	0.298953
$213$	0	0
$214$	−7.33809	−0.501621
$215$	0	0
$216$	0	0
$217$	1.05763	0.0717967
$218$	−1.22282	−0.0828201
$219$	0	0
$220$	0	0
$221$	1.57655	0.106050
$222$	0	0
$223$	14.7263	0.986149	0.493074	−	0.869987i	$-0.335873\pi$
$223$	14.7263	0.986149	0.493074	+	0.869987i	$0.335873\pi$
$224$	−5.73669	−0.383299
$225$	0	0
$226$	−5.69885	−0.379082
$227$	−9.73404	−0.646071	−0.323036	−	0.946387i	$-0.604703\pi$
$227$	−9.73404	−0.646071	−0.323036	+	0.946387i	$0.604703\pi$
$228$	0	0
$229$	−3.24791	−0.214628	−0.107314	−	0.994225i	$-0.534225\pi$
$229$	−3.24791	−0.214628	−0.107314	+	0.994225i	$0.534225\pi$
$230$	0	0
$231$	0	0
$232$	6.01979	0.395219
$233$	−9.58094	−0.627668	−0.313834	−	0.949478i	$-0.601614\pi$
$233$	−9.58094	−0.627668	−0.313834	+	0.949478i	$0.601614\pi$
$234$	0	0
$235$	0	0
$236$	13.1206	0.854075
$237$	0	0
$238$	4.04993	0.262518
$239$	−21.2831	−1.37669	−0.688345	−	0.725384i	$-0.741662\pi$
$239$	−21.2831	−1.37669	−0.688345	+	0.725384i	$0.741662\pi$
$240$	0	0
$241$	−18.7565	−1.20821	−0.604105	−	0.796904i	$-0.706469\pi$
$241$	−18.7565	−1.20821	−0.604105	+	0.796904i	$0.706469\pi$
$242$	−4.82181	−0.309958
$243$	0	0
$244$	19.3055	1.23591
$245$	0	0
$246$	0	0
$247$	−1.80973	−0.115150
$248$	2.47843	0.157381
$249$	0	0
$250$	0	0
$251$	−6.27804	−0.396267	−0.198133	−	0.980175i	$-0.563488\pi$
$251$	−6.27804	−0.396267	−0.198133	+	0.980175i	$0.563488\pi$
$252$	0	0
$253$	4.91223	0.308829
$254$	8.49822	0.533226
$255$	0	0
$256$	−8.42609	−0.526631
$257$	−25.5009	−1.59070	−0.795350	−	0.606150i	$-0.792713\pi$
$257$	−25.5009	−1.59070	−0.795350	+	0.606150i	$0.792713\pi$
$258$	0	0
$259$	8.39331	0.521535
$260$	0	0
$261$	0	0
$262$	−3.48372	−0.215225
$263$	1.08271	0.0667629	0.0333815	−	0.999443i	$-0.489372\pi$
$263$	1.08271	0.0667629	0.0333815	+	0.999443i	$0.489372\pi$
$264$	0	0
$265$	0	0
$266$	−4.64892	−0.285043
$267$	0	0
$268$	14.0251	0.856718
$269$	13.1403	0.801181	0.400590	−	0.916257i	$-0.368805\pi$
$269$	13.1403	0.801181	0.400590	+	0.916257i	$0.368805\pi$
$270$	0	0
$271$	25.8717	1.57160	0.785798	−	0.618483i	$-0.212252\pi$
$271$	25.8717	1.57160	0.785798	+	0.618483i	$0.212252\pi$
$272$	−9.86230	−0.597990
$273$	0	0
$274$	−13.0878	−0.790661
$275$	0	0
$276$	0	0
$277$	−12.6540	−0.760304	−0.380152	−	0.924924i	$-0.624128\pi$
$277$	−12.6540	−0.760304	−0.380152	+	0.924924i	$0.624128\pi$
$278$	6.09042	0.365279
$279$	0	0
$280$	0	0
$281$	−26.6988	−1.59272	−0.796360	−	0.604823i	$-0.793244\pi$
$281$	−26.6988	−1.59272	−0.796360	+	0.604823i	$0.793244\pi$
$282$	0	0
$283$	11.4613	0.681303	0.340651	−	0.940190i	$-0.389352\pi$
$283$	11.4613	0.681303	0.340651	+	0.940190i	$0.389352\pi$
$284$	−19.7065	−1.16937
$285$	0	0
$286$	−0.320945	−0.0189779
$287$	−10.8744	−0.641895
$288$	0	0
$289$	21.0422	1.23778
$290$	0	0
$291$	0	0
$292$	−15.6291	−0.914625
$293$	−9.44588	−0.551834	−0.275917	−	0.961181i	$-0.588982\pi$
$293$	−9.44588	−0.551834	−0.275917	+	0.961181i	$0.588982\pi$
$294$	0	0
$295$	0	0
$296$	19.6687	1.14322
$297$	0	0
$298$	1.36581	0.0791195
$299$	−0.656620	−0.0379733
$300$	0	0
$301$	−5.08777	−0.293254
$302$	5.08007	0.292325
$303$	0	0
$304$	11.3209	0.649301
$305$	0	0
$306$	0	0
$307$	20.7642	1.18507	0.592537	−	0.805543i	$-0.298126\pi$
$307$	20.7642	1.18507	0.592537	+	0.805543i	$0.298126\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	0	0
$311$	13.1454	0.745407	0.372704	−	0.927950i	$-0.378431\pi$
$311$	13.1454	0.745407	0.372704	+	0.927950i	$0.378431\pi$
$312$	0	0
$313$	−27.1799	−1.53630	−0.768150	−	0.640270i	$-0.778823\pi$
$313$	−27.1799	−1.53630	−0.768150	+	0.640270i	$0.778823\pi$
$314$	−3.57920	−0.201986
$315$	0	0
$316$	15.6291	0.879207
$317$	2.67906	0.150471	0.0752354	−	0.997166i	$-0.476029\pi$
$317$	2.67906	0.150471	0.0752354	+	0.997166i	$0.476029\pi$
$318$	0	0
$319$	−4.91223	−0.275032
$320$	0	0
$321$	0	0
$322$	−1.68676	−0.0939994
$323$	−43.6687	−2.42979
$324$	0	0
$325$	0	0
$326$	−14.6808	−0.813094
$327$	0	0
$328$	−25.4828	−1.40705
$329$	3.05763	0.168573
$330$	0	0
$331$	1.68411	0.0925673	0.0462836	−	0.998928i	$-0.485262\pi$
$331$	1.68411	0.0925673	0.0462836	+	0.998928i	$0.485262\pi$
$332$	5.82710	0.319804
$333$	0	0
$334$	−7.98965	−0.437174
$335$	0	0
$336$	0	0
$337$	−15.4760	−0.843033	−0.421516	−	0.906821i	$-0.638502\pi$
$337$	−15.4760	−0.843033	−0.421516	+	0.906821i	$0.638502\pi$
$338$	−8.49316	−0.461967
$339$	0	0
$340$	0	0
$341$	−2.02243	−0.109521
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−11.9226	−0.642822
$345$	0	0
$346$	−10.5387	−0.566565
$347$	−20.4683	−1.09880	−0.549398	−	0.835561i	$-0.685143\pi$
$347$	−20.4683	−1.09880	−0.549398	+	0.835561i	$0.685143\pi$
$348$	0	0
$349$	−17.0647	−0.913450	−0.456725	−	0.889608i	$-0.650978\pi$
$349$	−17.0647	−0.913450	−0.456725	+	0.889608i	$0.650978\pi$
$350$	0	0
$351$	0	0
$352$	10.9699	0.584696
$353$	−36.6188	−1.94902	−0.974510	−	0.224342i	$-0.927977\pi$
$353$	−36.6188	−1.94902	−0.974510	+	0.224342i	$0.927977\pi$
$354$	0	0
$355$	0	0
$356$	9.93972	0.526804
$357$	0	0
$358$	−7.12735	−0.376692
$359$	25.3856	1.33980	0.669900	−	0.742451i	$-0.266337\pi$
$359$	25.3856	1.33980	0.669900	+	0.742451i	$0.266337\pi$
$360$	0	0
$361$	31.1274	1.63828
$362$	1.85460	0.0974755
$363$	0	0
$364$	−0.401012	−0.0210187
$365$	0	0
$366$	0	0
$367$	26.1274	1.36384	0.681918	−	0.731428i	$-0.261146\pi$
$367$	26.1274	1.36384	0.681918	+	0.731428i	$0.261146\pi$
$368$	4.10756	0.214121
$369$	0	0
$370$	0	0
$371$	2.77453	0.144046
$372$	0	0
$373$	20.7840	1.07615	0.538077	−	0.842896i	$-0.319151\pi$
$373$	20.7840	1.07615	0.538077	+	0.842896i	$0.319151\pi$
$374$	−7.74439	−0.400453
$375$	0	0
$376$	7.16519	0.369517
$377$	0.656620	0.0338177
$378$	0	0
$379$	27.8942	1.43283	0.716414	−	0.697676i	$-0.245782\pi$
$379$	27.8942	1.43283	0.716414	+	0.697676i	$0.245782\pi$
$380$	0	0
$381$	0	0
$382$	16.0777	0.822604
$383$	19.1850	0.980307	0.490153	−	0.871636i	$-0.336941\pi$
$383$	19.1850	0.980307	0.490153	+	0.871636i	$0.336941\pi$
$384$	0	0
$385$	0	0
$386$	6.95951	0.354230
$387$	0	0
$388$	−8.86230	−0.449915
$389$	18.7092	0.948594	0.474297	−	0.880365i	$-0.342702\pi$
$389$	18.7092	0.948594	0.474297	+	0.880365i	$0.342702\pi$
$390$	0	0
$391$	−15.8442	−0.801278
$392$	−2.34338	−0.118359
$393$	0	0
$394$	6.88209	0.346715
$395$	0	0
$396$	0	0
$397$	−11.2453	−0.564383	−0.282192	−	0.959358i	$-0.591061\pi$
$397$	−11.2453	−0.564383	−0.282192	+	0.959358i	$0.591061\pi$
$398$	−0.543093	−0.0272228
$399$	0	0
$400$	0	0
$401$	26.2875	1.31273	0.656367	−	0.754442i	$-0.272092\pi$
$401$	26.2875	1.31273	0.656367	+	0.754442i	$0.272092\pi$
$402$	0	0
$403$	0.270340	0.0134666
$404$	22.0301	1.09604
$405$	0	0
$406$	1.68676	0.0837125
$407$	−16.0499	−0.795565
$408$	0	0
$409$	9.16519	0.453190	0.226595	−	0.973989i	$-0.427241\pi$
$409$	9.16519	0.453190	0.226595	+	0.973989i	$0.427241\pi$
$410$	0	0
$411$	0	0
$412$	−3.30645	−0.162897
$413$	8.36317	0.411525
$414$	0	0
$415$	0	0
$416$	−1.46635	−0.0718936
$417$	0	0
$418$	8.88979	0.434814
$419$	−8.08007	−0.394737	−0.197369	−	0.980329i	$-0.563240\pi$
$419$	−8.08007	−0.394737	−0.197369	+	0.980329i	$0.563240\pi$
$420$	0	0
$421$	6.37087	0.310497	0.155249	−	0.987875i	$-0.450382\pi$
$421$	6.37087	0.310497	0.155249	+	0.987875i	$0.450382\pi$
$422$	−3.20568	−0.156050
$423$	0	0
$424$	6.50178	0.315754
$425$	0	0
$426$	0	0
$427$	12.3055	0.595507
$428$	17.5327	0.847477
$429$	0	0
$430$	0	0
$431$	−36.5284	−1.75951	−0.879755	−	0.475428i	$-0.842293\pi$
$431$	−36.5284	−1.75951	−0.879755	+	0.475428i	$0.842293\pi$
$432$	0	0
$433$	−10.9622	−0.526808	−0.263404	−	0.964686i	$-0.584845\pi$
$433$	−10.9622	−0.526808	−0.263404	+	0.964686i	$0.584845\pi$
$434$	0.694463	0.0333353
$435$	0	0
$436$	2.92167	0.139923
$437$	18.1876	0.870032
$438$	0	0
$439$	9.29519	0.443635	0.221818	−	0.975088i	$-0.428801\pi$
$439$	9.29519	0.443635	0.221818	+	0.975088i	$0.428801\pi$
$440$	0	0
$441$	0	0
$442$	1.03520	0.0492393
$443$	−16.3709	−0.777804	−0.388902	−	0.921279i	$-0.627145\pi$
$443$	−16.3709	−0.777804	−0.388902	+	0.921279i	$0.627145\pi$
$444$	0	0
$445$	0	0
$446$	9.66961	0.457870
$447$	0	0
$448$	−0.568850	−0.0268756
$449$	−34.3427	−1.62073	−0.810366	−	0.585924i	$-0.800732\pi$
$449$	−34.3427	−1.62073	−0.810366	+	0.585924i	$0.800732\pi$
$450$	0	0
$451$	20.7943	0.979166
$452$	13.6161	0.640449
$453$	0	0
$454$	−6.39157	−0.299971
$455$	0	0
$456$	0	0
$457$	−1.92432	−0.0900157	−0.0450078	−	0.998987i	$-0.514331\pi$
$457$	−1.92432	−0.0900157	−0.0450078	+	0.998987i	$0.514331\pi$
$458$	−2.13264	−0.0996518
$459$	0	0
$460$	0	0
$461$	37.3676	1.74038	0.870190	−	0.492716i	$-0.163996\pi$
$461$	37.3676	1.74038	0.870190	+	0.492716i	$0.163996\pi$
$462$	0	0
$463$	13.2074	0.613801	0.306900	−	0.951742i	$-0.400708\pi$
$463$	13.2074	0.613801	0.306900	+	0.951742i	$0.400708\pi$
$464$	−4.10756	−0.190689
$465$	0	0
$466$	−6.29104	−0.291427
$467$	−7.14034	−0.330416	−0.165208	−	0.986259i	$-0.552830\pi$
$467$	−7.14034	−0.330416	−0.165208	+	0.986259i	$0.552830\pi$
$468$	0	0
$469$	8.93972	0.412798
$470$	0	0
$471$	0	0
$472$	19.5981	0.902075
$473$	9.72898	0.447339
$474$	0	0
$475$	0	0
$476$	−9.67641	−0.443518
$477$	0	0
$478$	−13.9749	−0.639198
$479$	−33.8770	−1.54788	−0.773941	−	0.633258i	$-0.781717\pi$
$479$	−33.8770	−1.54788	−0.773941	+	0.633258i	$0.781717\pi$
$480$	0	0
$481$	2.14540	0.0978220
$482$	−12.3159	−0.560973
$483$	0	0
$484$	11.5207	0.523666
$485$	0	0
$486$	0	0
$487$	30.4881	1.38155	0.690774	−	0.723071i	$-0.257270\pi$
$487$	30.4881	1.38155	0.690774	+	0.723071i	$0.257270\pi$
$488$	28.8365	1.30537
$489$	0	0
$490$	0	0
$491$	−14.1755	−0.639733	−0.319867	−	0.947463i	$-0.603638\pi$
$491$	−14.1755	−0.639733	−0.319867	+	0.947463i	$0.603638\pi$
$492$	0	0
$493$	15.8442	0.713589
$494$	−1.18830	−0.0534643
$495$	0	0
$496$	−1.69114	−0.0759345
$497$	−12.5611	−0.563444
$498$	0	0
$499$	−6.19533	−0.277341	−0.138671	−	0.990339i	$-0.544283\pi$
$499$	−6.19533	−0.277341	−0.138671	+	0.990339i	$0.544283\pi$
$500$	0	0
$501$	0	0
$502$	−4.12229	−0.183987
$503$	−1.62913	−0.0726392	−0.0363196	−	0.999340i	$-0.511563\pi$
$503$	−1.62913	−0.0726392	−0.0363196	+	0.999340i	$0.511563\pi$
$504$	0	0
$505$	0	0
$506$	3.22547	0.143390
$507$	0	0
$508$	−20.3046	−0.900872
$509$	5.47670	0.242750	0.121375	−	0.992607i	$-0.461270\pi$
$509$	5.47670	0.242750	0.121375	+	0.992607i	$0.461270\pi$
$510$	0	0
$511$	−9.96216	−0.440700
$512$	16.6670	0.736583
$513$	0	0
$514$	−16.7444	−0.738563
$515$	0	0
$516$	0	0
$517$	−5.84689	−0.257146
$518$	5.51122	0.242149
$519$	0	0
$520$	0	0
$521$	−2.08513	−0.0913510	−0.0456755	−	0.998956i	$-0.514544\pi$
$521$	−2.08513	−0.0913510	−0.0456755	+	0.998956i	$0.514544\pi$
$522$	0	0
$523$	−18.1652	−0.794308	−0.397154	−	0.917752i	$-0.630002\pi$
$523$	−18.1652	−0.794308	−0.397154	+	0.917752i	$0.630002\pi$
$524$	8.32359	0.363618
$525$	0	0
$526$	0.710931	0.0309981
$527$	6.52330	0.284160
$528$	0	0
$529$	−16.4010	−0.713087
$530$	0	0
$531$	0	0
$532$	11.1076	0.481574
$533$	−2.77959	−0.120397
$534$	0	0
$535$	0	0
$536$	20.9492	0.904866
$537$	0	0
$538$	8.62822	0.371989
$539$	1.91223	0.0823655
$540$	0	0
$541$	42.9716	1.84749	0.923747	−	0.383004i	$-0.125110\pi$
$541$	42.9716	1.84749	0.923747	+	0.383004i	$0.125110\pi$
$542$	16.9879	0.729693
$543$	0	0
$544$	−35.3830	−1.51703
$545$	0	0
$546$	0	0
$547$	−28.4355	−1.21582	−0.607908	−	0.794008i	$-0.707991\pi$
$547$	−28.4355	−1.21582	−0.607908	+	0.794008i	$0.707991\pi$
$548$	31.2703	1.33580
$549$	0	0
$550$	0	0
$551$	−18.1876	−0.774819
$552$	0	0
$553$	9.96216	0.423634
$554$	−8.30886	−0.353010
$555$	0	0
$556$	−14.5517	−0.617130
$557$	24.7290	1.04780	0.523900	−	0.851780i	$-0.324476\pi$
$557$	24.7290	1.04780	0.523900	+	0.851780i	$0.324476\pi$
$558$	0	0
$559$	−1.30048	−0.0550044
$560$	0	0
$561$	0	0
$562$	−17.5310	−0.739501
$563$	−6.46899	−0.272636	−0.136318	−	0.990665i	$-0.543527\pi$
$563$	−6.46899	−0.272636	−0.136318	+	0.990665i	$0.543527\pi$
$564$	0	0
$565$	0	0
$566$	7.52572	0.316329
$567$	0	0
$568$	−29.4355	−1.23509
$569$	5.02243	0.210551	0.105276	−	0.994443i	$-0.466427\pi$
$569$	5.02243	0.210551	0.105276	+	0.994443i	$0.466427\pi$
$570$	0	0
$571$	−21.0372	−0.880378	−0.440189	−	0.897905i	$-0.645089\pi$
$571$	−21.0372	−0.880378	−0.440189	+	0.897905i	$0.645089\pi$
$572$	0.766826	0.0320626
$573$	0	0
$574$	−7.14034	−0.298032
$575$	0	0
$576$	0	0
$577$	30.2677	1.26006	0.630030	−	0.776571i	$-0.283042\pi$
$577$	30.2677	1.26006	0.630030	+	0.776571i	$0.283042\pi$
$578$	13.8168	0.574701
$579$	0	0
$580$	0	0
$581$	3.71425	0.154093
$582$	0	0
$583$	−5.30554	−0.219733
$584$	−23.3451	−0.966028
$585$	0	0
$586$	−6.20236	−0.256217
$587$	−13.8020	−0.569670	−0.284835	−	0.958576i	$-0.591939\pi$
$587$	−13.8020	−0.569670	−0.284835	+	0.958576i	$0.591939\pi$
$588$	0	0
$589$	−7.48811	−0.308542
$590$	0	0
$591$	0	0
$592$	−13.4208	−0.551592
$593$	12.6335	0.518796	0.259398	−	0.965771i	$-0.416476\pi$
$593$	12.6335	0.518796	0.259398	+	0.965771i	$0.416476\pi$
$594$	0	0
$595$	0	0
$596$	−3.26331	−0.133670
$597$	0	0
$598$	−0.431150	−0.0176311
$599$	−7.39066	−0.301974	−0.150987	−	0.988536i	$-0.548245\pi$
$599$	−7.39066	−0.301974	−0.150987	+	0.988536i	$0.548245\pi$
$600$	0	0
$601$	35.5053	1.44829	0.724145	−	0.689648i	$-0.242235\pi$
$601$	35.5053	1.44829	0.724145	+	0.689648i	$0.242235\pi$
$602$	−3.34073	−0.136158
$603$	0	0
$604$	−12.1377	−0.493876
$605$	0	0
$606$	0	0
$607$	−18.7116	−0.759481	−0.379740	−	0.925093i	$-0.623987\pi$
$607$	−18.7116	−0.759481	−0.379740	+	0.925093i	$0.623987\pi$
$608$	40.6161	1.64720
$609$	0	0
$610$	0	0
$611$	0.781558	0.0316184
$612$	0	0
$613$	−16.6265	−0.671537	−0.335769	−	0.941944i	$-0.608996\pi$
$613$	−16.6265	−0.671537	−0.335769	+	0.941944i	$0.608996\pi$
$614$	13.6342	0.550231
$615$	0	0
$616$	4.48108	0.180548
$617$	45.8064	1.84410	0.922048	−	0.387075i	$-0.126515\pi$
$617$	45.8064	1.84410	0.922048	+	0.387075i	$0.126515\pi$
$618$	0	0
$619$	18.3528	0.737662	0.368831	−	0.929496i	$-0.379758\pi$
$619$	18.3528	0.737662	0.368831	+	0.929496i	$0.379758\pi$
$620$	0	0
$621$	0	0
$622$	8.63154	0.346093
$623$	6.33568	0.253833
$624$	0	0
$625$	0	0
$626$	−17.8469	−0.713305
$627$	0	0
$628$	8.55171	0.341250
$629$	51.7686	2.06415
$630$	0	0
$631$	−22.5759	−0.898731	−0.449366	−	0.893348i	$-0.648350\pi$
$631$	−22.5759	−0.898731	−0.449366	+	0.893348i	$0.648350\pi$
$632$	23.3451	0.928619
$633$	0	0
$634$	1.75912	0.0698637
$635$	0	0
$636$	0	0
$637$	−0.255609	−0.0101276
$638$	−3.22547	−0.127698
$639$	0	0
$640$	0	0
$641$	−1.22109	−0.0482301	−0.0241150	−	0.999709i	$-0.507677\pi$
$641$	−1.22109	−0.0482301	−0.0241150	+	0.999709i	$0.507677\pi$
$642$	0	0
$643$	35.0878	1.38373	0.691863	−	0.722029i	$-0.256790\pi$
$643$	35.0878	1.38373	0.691863	+	0.722029i	$0.256790\pi$
$644$	4.03014	0.158810
$645$	0	0
$646$	−28.6738	−1.12815
$647$	−14.1179	−0.555032	−0.277516	−	0.960721i	$-0.589511\pi$
$647$	−14.1179	−0.555032	−0.277516	+	0.960721i	$0.589511\pi$
$648$	0	0
$649$	−15.9923	−0.627753
$650$	0	0
$651$	0	0
$652$	35.0765	1.37370
$653$	5.73669	0.224494	0.112247	−	0.993680i	$-0.464195\pi$
$653$	5.73669	0.224494	0.112247	+	0.993680i	$0.464195\pi$
$654$	0	0
$655$	0	0
$656$	17.3880	0.678888
$657$	0	0
$658$	2.00770	0.0782684
$659$	−27.8838	−1.08620	−0.543100	−	0.839668i	$-0.682749\pi$
$659$	−27.8838	−1.08620	−0.543100	+	0.839668i	$0.682749\pi$
$660$	0	0
$661$	−18.7943	−0.731014	−0.365507	−	0.930809i	$-0.619104\pi$
$661$	−18.7943	−0.731014	−0.365507	+	0.930809i	$0.619104\pi$
$662$	1.10582	0.0429791
$663$	0	0
$664$	8.70390	0.337777
$665$	0	0
$666$	0	0
$667$	−6.59899	−0.255514
$668$	19.0895	0.738595
$669$	0	0
$670$	0	0
$671$	−23.5310	−0.908404
$672$	0	0
$673$	−29.1628	−1.12414	−0.562071	−	0.827089i	$-0.689995\pi$
$673$	−29.1628	−1.12414	−0.562071	+	0.827089i	$0.689995\pi$
$674$	−10.1619	−0.391421
$675$	0	0
$676$	20.2925	0.780482
$677$	−14.9294	−0.573782	−0.286891	−	0.957963i	$-0.592622\pi$
$677$	−14.9294	−0.573782	−0.286891	+	0.957963i	$0.592622\pi$
$678$	0	0
$679$	−5.64892	−0.216786
$680$	0	0
$681$	0	0
$682$	−1.32797	−0.0508507
$683$	−7.94501	−0.304007	−0.152004	−	0.988380i	$-0.548573\pi$
$683$	−7.94501	−0.304007	−0.152004	+	0.988380i	$0.548573\pi$
$684$	0	0
$685$	0	0
$686$	−0.656620	−0.0250699
$687$	0	0
$688$	8.13529	0.310155
$689$	0.709194	0.0270181
$690$	0	0
$691$	6.98459	0.265706	0.132853	−	0.991136i	$-0.457586\pi$
$691$	6.98459	0.265706	0.132853	+	0.991136i	$0.457586\pi$
$692$	25.1799	0.957197
$693$	0	0
$694$	−13.4399	−0.510172
$695$	0	0
$696$	0	0
$697$	−67.0715	−2.54051
$698$	−11.2050	−0.424116
$699$	0	0
$700$	0	0
$701$	−21.1678	−0.799498	−0.399749	−	0.916625i	$-0.630903\pi$
$701$	−21.1678	−0.799498	−0.399749	+	0.916625i	$0.630903\pi$
$702$	0	0
$703$	−59.4252	−2.24126
$704$	1.08777	0.0409969
$705$	0	0
$706$	−24.0446	−0.904932
$707$	14.0422	0.528112
$708$	0	0
$709$	34.4701	1.29455	0.647275	−	0.762257i	$-0.275909\pi$
$709$	34.4701	1.29455	0.647275	+	0.762257i	$0.275909\pi$
$710$	0	0
$711$	0	0
$712$	14.8469	0.556411
$713$	−2.71690	−0.101749
$714$	0	0
$715$	0	0
$716$	17.0292	0.636412
$717$	0	0
$718$	16.6687	0.622071
$719$	33.9217	1.26506	0.632532	−	0.774534i	$-0.282016\pi$
$719$	33.9217	1.26506	0.632532	+	0.774534i	$0.282016\pi$
$720$	0	0
$721$	−2.10756	−0.0784897
$722$	20.4389	0.760655
$723$	0	0
$724$	−4.43115	−0.164682
$725$	0	0
$726$	0	0
$727$	10.8674	0.403048	0.201524	−	0.979484i	$-0.435411\pi$
$727$	10.8674	0.403048	0.201524	+	0.979484i	$0.435411\pi$
$728$	−0.598988	−0.0222000
$729$	0	0
$730$	0	0
$731$	−31.3805	−1.16065
$732$	0	0
$733$	14.7919	0.546352	0.273176	−	0.961964i	$-0.411926\pi$
$733$	14.7919	0.546352	0.273176	+	0.961964i	$0.411926\pi$
$734$	17.1558	0.633230
$735$	0	0
$736$	14.7367	0.543202
$737$	−17.0948	−0.629695
$738$	0	0
$739$	−6.94410	−0.255443	−0.127722	−	0.991810i	$-0.540766\pi$
$739$	−6.94410	−0.255443	−0.127722	+	0.991810i	$0.540766\pi$
$740$	0	0
$741$	0	0
$742$	1.82181	0.0668809
$743$	−48.9890	−1.79723	−0.898616	−	0.438737i	$-0.855426\pi$
$743$	−48.9890	−1.79723	−0.898616	+	0.438737i	$0.855426\pi$
$744$	0	0
$745$	0	0
$746$	13.6472	0.499659
$747$	0	0
$748$	18.5035	0.676555
$749$	11.1755	0.408345
$750$	0	0
$751$	5.66168	0.206598	0.103299	−	0.994650i	$-0.467060\pi$
$751$	5.66168	0.206598	0.103299	+	0.994650i	$0.467060\pi$
$752$	−4.88912	−0.178288
$753$	0	0
$754$	0.431150	0.0157016
$755$	0	0
$756$	0	0
$757$	45.9063	1.66849	0.834246	−	0.551393i	$-0.185903\pi$
$757$	45.9063	1.66849	0.834246	+	0.551393i	$0.185903\pi$
$758$	18.3159	0.665263
$759$	0	0
$760$	0	0
$761$	−25.9173	−0.939501	−0.469750	−	0.882799i	$-0.655656\pi$
$761$	−25.9173	−0.939501	−0.469750	+	0.882799i	$0.655656\pi$
$762$	0	0
$763$	1.86230	0.0674198
$764$	−38.4140	−1.38977
$765$	0	0
$766$	12.5973	0.455157
$767$	2.13770	0.0771878
$768$	0	0
$769$	45.7609	1.65018	0.825089	−	0.565002i	$-0.191125\pi$
$769$	45.7609	1.65018	0.825089	+	0.565002i	$0.191125\pi$
$770$	0	0
$771$	0	0
$772$	−16.6282	−0.598463
$773$	37.4778	1.34798	0.673991	−	0.738740i	$-0.264579\pi$
$773$	37.4778	1.34798	0.673991	+	0.738740i	$0.264579\pi$
$774$	0	0
$775$	0	0
$776$	−13.2376	−0.475201
$777$	0	0
$778$	12.2848	0.440433
$779$	76.9914	2.75850
$780$	0	0
$781$	24.0198	0.859496
$782$	−10.4037	−0.372034
$783$	0	0
$784$	1.59899	0.0571067
$785$	0	0
$786$	0	0
$787$	−46.3350	−1.65166	−0.825832	−	0.563916i	$-0.809294\pi$
$787$	−46.3350	−1.65166	−0.825832	+	0.563916i	$0.809294\pi$
$788$	−16.4432	−0.585766
$789$	0	0
$790$	0	0
$791$	8.67906	0.308592
$792$	0	0
$793$	3.14540	0.111697
$794$	−7.38387	−0.262044
$795$	0	0
$796$	1.29760	0.0459923
$797$	20.6214	0.730448	0.365224	−	0.930920i	$-0.380992\pi$
$797$	20.6214	0.730448	0.365224	+	0.930920i	$0.380992\pi$
$798$	0	0
$799$	18.8590	0.667183
$800$	0	0
$801$	0	0
$802$	17.2609	0.609504
$803$	19.0499	0.672257
$804$	0	0
$805$	0	0
$806$	0.177511	0.00625255
$807$	0	0
$808$	32.9063	1.15764
$809$	−35.0570	−1.23254	−0.616268	−	0.787536i	$-0.711356\pi$
$809$	−35.0570	−1.23254	−0.616268	+	0.787536i	$0.711356\pi$
$810$	0	0
$811$	35.3126	1.23999	0.619996	−	0.784605i	$-0.287134\pi$
$811$	35.3126	1.23999	0.619996	+	0.784605i	$0.287134\pi$
$812$	−4.03014	−0.141430
$813$	0	0
$814$	−10.5387	−0.369382
$815$	0	0
$816$	0	0
$817$	36.0218	1.26024
$818$	6.01805	0.210416
$819$	0	0
$820$	0	0
$821$	−51.4228	−1.79467	−0.897334	−	0.441353i	$-0.854499\pi$
$821$	−51.4228	−1.79467	−0.897334	+	0.441353i	$0.854499\pi$
$822$	0	0
$823$	−31.0389	−1.08195	−0.540974	−	0.841039i	$-0.681944\pi$
$823$	−31.0389	−1.08195	−0.540974	+	0.841039i	$0.681944\pi$
$824$	−4.93881	−0.172052
$825$	0	0
$826$	5.49143	0.191071
$827$	49.8812	1.73454	0.867269	−	0.497839i	$-0.165873\pi$
$827$	49.8812	1.73454	0.867269	+	0.497839i	$0.165873\pi$
$828$	0	0
$829$	16.3434	0.567629	0.283815	−	0.958879i	$-0.408400\pi$
$829$	16.3434	0.567629	0.283815	+	0.958879i	$0.408400\pi$
$830$	0	0
$831$	0	0
$832$	−0.145403	−0.00504094
$833$	−6.16784	−0.213703
$834$	0	0
$835$	0	0
$836$	−21.2402	−0.734608
$837$	0	0
$838$	−5.30554	−0.183277
$839$	36.6714	1.26604	0.633018	−	0.774137i	$-0.281816\pi$
$839$	36.6714	1.26604	0.633018	+	0.774137i	$0.281816\pi$
$840$	0	0
$841$	−22.4010	−0.772449
$842$	4.18325	0.144164
$843$	0	0
$844$	7.65927	0.263643
$845$	0	0
$846$	0	0
$847$	7.34338	0.252322
$848$	−4.43644	−0.152348
$849$	0	0
$850$	0	0
$851$	−21.5611	−0.739107
$852$	0	0
$853$	32.6507	1.11794	0.558969	−	0.829189i	$-0.311197\pi$
$853$	32.6507	1.11794	0.558969	+	0.829189i	$0.311197\pi$
$854$	8.08007	0.276494
$855$	0	0
$856$	26.1885	0.895106
$857$	42.5990	1.45515	0.727577	−	0.686026i	$-0.240646\pi$
$857$	42.5990	1.45515	0.727577	+	0.686026i	$0.240646\pi$
$858$	0	0
$859$	5.14805	0.175649	0.0878246	−	0.996136i	$-0.472009\pi$
$859$	5.14805	0.175649	0.0878246	+	0.996136i	$0.472009\pi$
$860$	0	0
$861$	0	0
$862$	−23.9853	−0.816941
$863$	1.60140	0.0545123	0.0272562	−	0.999628i	$-0.491323\pi$
$863$	1.60140	0.0545123	0.0272562	+	0.999628i	$0.491323\pi$
$864$	0	0
$865$	0	0
$866$	−7.19798	−0.244597
$867$	0	0
$868$	−1.65927	−0.0563191
$869$	−19.0499	−0.646225
$870$	0	0
$871$	2.28507	0.0774267
$872$	4.36408	0.147786
$873$	0	0
$874$	11.9424	0.403957
$875$	0	0
$876$	0	0
$877$	13.9940	0.472545	0.236272	−	0.971687i	$-0.424074\pi$
$877$	13.9940	0.472545	0.236272	+	0.971687i	$0.424074\pi$
$878$	6.10341	0.205980
$879$	0	0
$880$	0	0
$881$	15.9098	0.536015	0.268008	−	0.963417i	$-0.413635\pi$
$881$	15.9098	0.536015	0.268008	+	0.963417i	$0.413635\pi$
$882$	0	0
$883$	1.99230	0.0670461	0.0335231	−	0.999438i	$-0.489327\pi$
$883$	1.99230	0.0670461	0.0335231	+	0.999438i	$0.489327\pi$
$884$	−2.47338	−0.0831886
$885$	0	0
$886$	−10.7494	−0.361135
$887$	0.761768	0.0255777	0.0127888	−	0.999918i	$-0.495929\pi$
$887$	0.761768	0.0255777	0.0127888	+	0.999918i	$0.495929\pi$
$888$	0	0
$889$	−12.9424	−0.434073
$890$	0	0
$891$	0	0
$892$	−23.1034	−0.773559
$893$	−21.6482	−0.724431
$894$	0	0
$895$	0	0
$896$	11.0999	0.370820
$897$	0	0
$898$	−22.5501	−0.752507
$899$	2.71690	0.0906136
$900$	0	0
$901$	17.1129	0.570112
$902$	13.6540	0.454628
$903$	0	0
$904$	20.3383	0.676442
$905$	0	0
$906$	0	0
$907$	−52.5180	−1.74383	−0.871916	−	0.489655i	$-0.837123\pi$
$907$	−52.5180	−1.74383	−0.871916	+	0.489655i	$0.837123\pi$
$908$	15.2712	0.506794
$909$	0	0
$910$	0	0
$911$	37.0070	1.22610	0.613049	−	0.790045i	$-0.289943\pi$
$911$	37.0070	1.22610	0.613049	+	0.790045i	$0.289943\pi$
$912$	0	0
$913$	−7.10250	−0.235059
$914$	−1.26354	−0.0417943
$915$	0	0
$916$	5.09547	0.168359
$917$	5.30554	0.175204
$918$	0	0
$919$	−46.1119	−1.52109	−0.760546	−	0.649284i	$-0.775069\pi$
$919$	−46.1119	−1.52109	−0.760546	+	0.649284i	$0.775069\pi$
$920$	0	0
$921$	0	0
$922$	24.5363	0.808060
$923$	−3.21074	−0.105683
$924$	0	0
$925$	0	0
$926$	8.67226	0.284988
$927$	0	0
$928$	−14.7367	−0.483755
$929$	50.0620	1.64248	0.821241	−	0.570582i	$-0.193282\pi$
$929$	50.0620	1.64248	0.821241	+	0.570582i	$0.193282\pi$
$930$	0	0
$931$	7.08007	0.232040
$932$	15.0310	0.492358
$933$	0	0
$934$	−4.68850	−0.153412
$935$	0	0
$936$	0	0
$937$	34.2074	1.11751	0.558754	−	0.829334i	$-0.311280\pi$
$937$	34.2074	1.11751	0.558754	+	0.829334i	$0.311280\pi$
$938$	5.87000	0.191662
$939$	0	0
$940$	0	0
$941$	−40.4432	−1.31841	−0.659206	−	0.751962i	$-0.729107\pi$
$941$	−40.4432	−1.31841	−0.659206	+	0.751962i	$0.729107\pi$
$942$	0	0
$943$	27.9347	0.909678
$944$	−13.3726	−0.435241
$945$	0	0
$946$	6.38825	0.207700
$947$	−16.7692	−0.544927	−0.272464	−	0.962166i	$-0.587838\pi$
$947$	−16.7692	−0.544927	−0.272464	+	0.962166i	$0.587838\pi$
$948$	0	0
$949$	−2.54641	−0.0826601
$950$	0	0
$951$	0	0
$952$	−14.4536	−0.468443
$953$	−24.6040	−0.797003	−0.398502	−	0.917168i	$-0.630470\pi$
$953$	−24.6040	−0.797003	−0.398502	+	0.917168i	$0.630470\pi$
$954$	0	0
$955$	0	0
$956$	33.3900	1.07991
$957$	0	0
$958$	−22.2444	−0.718682
$959$	19.9320	0.643638
$960$	0	0
$961$	−29.8814	−0.963917
$962$	1.40872	0.0454188
$963$	0	0
$964$	29.4261	0.947751
$965$	0	0
$966$	0	0
$967$	−7.88979	−0.253719	−0.126859	−	0.991921i	$-0.540490\pi$
$967$	−7.88979	−0.253719	−0.126859	+	0.991921i	$0.540490\pi$
$968$	17.2083	0.553097
$969$	0	0
$970$	0	0
$971$	49.7512	1.59659	0.798296	−	0.602266i	$-0.205735\pi$
$971$	49.7512	1.59659	0.798296	+	0.602266i	$0.205735\pi$
$972$	0	0
$973$	−9.27540	−0.297356
$974$	20.0191	0.641454
$975$	0	0
$976$	−19.6764	−0.629827
$977$	41.5508	1.32933	0.664664	−	0.747143i	$-0.268575\pi$
$977$	41.5508	1.32933	0.664664	+	0.747143i	$0.268575\pi$
$978$	0	0
$979$	−12.1153	−0.387206
$980$	0	0
$981$	0	0
$982$	−9.30795	−0.297029
$983$	−30.6714	−0.978264	−0.489132	−	0.872210i	$-0.662686\pi$
$983$	−30.6714	−0.978264	−0.489132	+	0.872210i	$0.662686\pi$
$984$	0	0
$985$	0	0
$986$	10.4037	0.331320
$987$	0	0
$988$	2.83919	0.0903267
$989$	13.0697	0.415593
$990$	0	0
$991$	−32.1370	−1.02087	−0.510433	−	0.859917i	$-0.670515\pi$
$991$	−32.1370	−1.02087	−0.510433	+	0.859917i	$0.670515\pi$
$992$	−6.06730	−0.192637
$993$	0	0
$994$	−8.24791	−0.261608
$995$	0	0
$996$	0	0
$997$	57.9769	1.83615	0.918073	−	0.396411i	$-0.129744\pi$
$997$	57.9769	1.83615	0.918073	+	0.396411i	$0.129744\pi$
$998$	−4.06798	−0.128770
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4725.2.a.bi.1.2	✓	3
3.2	odd	2		4725.2.a.bj.1.2	yes	3
5.4	even	2		4725.2.a.bl.1.2	yes	3
15.14	odd	2		4725.2.a.bk.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4725.2.a.bi.1.2	✓	3	1.1	even	1	trivial
4725.2.a.bj.1.2	yes	3	3.2	odd	2
4725.2.a.bk.1.2	yes	3	15.14	odd	2
4725.2.a.bl.1.2	yes	3	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4725 = 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4725.a (trivial)

\(f(q)\)	\(=\)	\(q+0.656620 q^{2} -1.56885 q^{4} -1.00000 q^{7} -2.34338 q^{8} +1.91223 q^{11} -0.255609 q^{13} -0.656620 q^{14} +1.59899 q^{16} -6.16784 q^{17} +7.08007 q^{19} +1.25561 q^{22} +2.56885 q^{23} -0.167838 q^{26} +1.56885 q^{28} -2.56885 q^{29} -1.05763 q^{31} +5.73669 q^{32} -4.04993 q^{34} -8.39331 q^{37} +4.64892 q^{38} +10.8744 q^{41} +5.08777 q^{43} -3.00000 q^{44} +1.68676 q^{46} -3.05763 q^{47} +1.00000 q^{49} +0.401012 q^{52} -2.77453 q^{53} +2.34338 q^{56} -1.68676 q^{58} -8.36317 q^{59} -12.3055 q^{61} -0.694463 q^{62} +0.568850 q^{64} -8.93972 q^{67} +9.67641 q^{68} +12.5611 q^{71} +9.96216 q^{73} -5.51122 q^{74} -11.1076 q^{76} -1.91223 q^{77} -9.96216 q^{79} +7.14034 q^{82} -3.71425 q^{83} +3.34073 q^{86} -4.48108 q^{88} -6.33568 q^{89} +0.255609 q^{91} -4.03014 q^{92} -2.00770 q^{94} +5.64892 q^{97} +0.656620 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{4} - 3 q^{7} - 9 q^{8} - q^{11} + 4 q^{13} + 2 q^{16} - 7 q^{17} + 3 q^{19} - q^{22} - q^{23} + 11 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} - 3 q^{32} + 12 q^{34} - 3 q^{37} - 13 q^{38} - 5 q^{41}+ \cdots - 10 q^{97}+O(q^{100}) \) 3 * q + 4 * q^4 - 3 * q^7 - 9 * q^8 - q^11 + 4 * q^13 + 2 * q^16 - 7 * q^17 + 3 * q^19 - q^22 - q^23 + 11 * q^26 - 4 * q^28 + q^29 - 4 * q^31 - 3 * q^32 + 12 * q^34 - 3 * q^37 - 13 * q^38 - 5 * q^41 + 22 * q^43 - 9 * q^44 + 9 * q^46 - 10 * q^47 + 3 * q^49 + 4 * q^52 - 19 * q^53 + 9 * q^56 - 9 * q^58 + 3 * q^59 - 8 * q^61 - 31 * q^62 - 7 * q^64 - 15 * q^67 - 3 * q^68 + 4 * q^71 - q^73 - 7 * q^74 - 10 * q^76 + q^77 + q^79 + 15 * q^82 - 10 * q^83 + q^86 + 2 * q^88 + 4 * q^89 - 4 * q^91 - 18 * q^92 - 31 * q^94 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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