LMFDB - Embedded newform 4725.2.a.bu.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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.874032$ 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)

$f(q)$	$=$	$q-0.874032 q^{2} -1.23607 q^{4} -1.00000 q^{7} +2.82843 q^{8} -0.333851 q^{11} -3.47214 q^{13} +0.874032 q^{14} +5.11667 q^{17} -5.23607 q^{19} +0.291796 q^{22} -8.81913 q^{23} +3.03476 q^{26} +1.23607 q^{28} +3.70246 q^{29} -3.00000 q^{31} -5.65685 q^{32} -4.47214 q^{34} +6.70820 q^{37} +4.57649 q^{38} -5.11667 q^{41} +5.76393 q^{43} +0.412662 q^{44} +7.70820 q^{46} +10.5672 q^{47} +1.00000 q^{49} +4.29180 q^{52} -11.2349 q^{53} -2.82843 q^{56} -3.23607 q^{58} +5.32300 q^{59} +3.70820 q^{61} +2.62210 q^{62} +4.94427 q^{64} +5.76393 q^{67} -6.32456 q^{68} -7.40492 q^{71} -9.18034 q^{73} -5.86319 q^{74} +6.47214 q^{76} +0.333851 q^{77} -2.23607 q^{79} +4.47214 q^{82} +3.16228 q^{83} -5.03786 q^{86} -0.944272 q^{88} -15.5563 q^{89} +3.47214 q^{91} +10.9010 q^{92} -9.23607 q^{94} -0.763932 q^{97} -0.874032 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{7} + 4 q^{13} - 12 q^{19} + 28 q^{22} - 4 q^{28} - 12 q^{31} + 32 q^{43} + 4 q^{46} + 4 q^{49} + 44 q^{52} - 4 q^{58} - 12 q^{61} - 16 q^{64} + 32 q^{67} + 8 q^{73} + 8 q^{76} + 32 q^{88}+ \cdots - 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^7 + 4 * q^13 - 12 * q^19 + 28 * q^22 - 4 * q^28 - 12 * q^31 + 32 * q^43 + 4 * q^46 + 4 * q^49 + 44 * q^52 - 4 * q^58 - 12 * q^61 - 16 * q^64 + 32 * q^67 + 8 * q^73 + 8 * q^76 + 32 * q^88 - 4 * q^91 - 28 * q^94 - 12 * 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 $

$2$	−0.874032	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$2$	−0.874032	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$3$	0	0
$3$	0	0
$4$	−1.23607	−0.618034
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.82843	1.00000
$9$	0	0
$10$	0	0
$11$	−0.333851	−0.100660	−0.0503299	−	0.998733i	$-0.516027\pi$
$11$	−0.333851	−0.100660	−0.0503299	+	0.998733i	$0.516027\pi$
$12$	0	0
$13$	−3.47214	−0.962997	−0.481499	−	0.876447i	$-0.659907\pi$
$13$	−3.47214	−0.962997	−0.481499	+	0.876447i	$0.659907\pi$
$14$	0.874032	0.233595
$15$	0	0
$16$	0	0
$17$	5.11667	1.24098	0.620488	−	0.784216i	$-0.286935\pi$
$17$	5.11667	1.24098	0.620488	+	0.784216i	$0.286935\pi$
$18$	0	0
$19$	−5.23607	−1.20124	−0.600618	−	0.799536i	$-0.705079\pi$
$19$	−5.23607	−1.20124	−0.600618	+	0.799536i	$0.705079\pi$
$20$	0	0
$21$	0	0
$22$	0.291796	0.0622111
$23$	−8.81913	−1.83892	−0.919458	−	0.393188i	$-0.871372\pi$
$23$	−8.81913	−1.83892	−0.919458	+	0.393188i	$0.871372\pi$
$24$	0	0
$25$	0	0
$26$	3.03476	0.595165
$27$	0	0
$28$	1.23607	0.233595
$29$	3.70246	0.687529	0.343765	−	0.939056i	$-0.388298\pi$
$29$	3.70246	0.687529	0.343765	+	0.939056i	$0.388298\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	−4.47214	−0.766965
$35$	0	0
$36$	0	0
$37$	6.70820	1.10282	0.551411	−	0.834234i	$-0.314090\pi$
$37$	6.70820	1.10282	0.551411	+	0.834234i	$0.314090\pi$
$38$	4.57649	0.742405
$39$	0	0
$40$	0	0
$41$	−5.11667	−0.799090	−0.399545	−	0.916714i	$-0.630832\pi$
$41$	−5.11667	−0.799090	−0.399545	+	0.916714i	$0.630832\pi$
$42$	0	0
$43$	5.76393	0.878991	0.439496	−	0.898245i	$-0.355157\pi$
$43$	5.76393	0.878991	0.439496	+	0.898245i	$0.355157\pi$
$44$	0.412662	0.0622111
$45$	0	0
$46$	7.70820	1.13651
$47$	10.5672	1.54138	0.770692	−	0.637208i	$-0.219911\pi$
$47$	10.5672	1.54138	0.770692	+	0.637208i	$0.219911\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	4.29180	0.595165
$53$	−11.2349	−1.54323	−0.771616	−	0.636089i	$-0.780551\pi$
$53$	−11.2349	−1.54323	−0.771616	+	0.636089i	$0.780551\pi$
$54$	0	0
$55$	0	0
$56$	−2.82843	−0.377964
$57$	0	0
$58$	−3.23607	−0.424917
$59$	5.32300	0.692996	0.346498	−	0.938051i	$-0.387371\pi$
$59$	5.32300	0.692996	0.346498	+	0.938051i	$0.387371\pi$
$60$	0	0
$61$	3.70820	0.474787	0.237393	−	0.971414i	$-0.423707\pi$
$61$	3.70820	0.474787	0.237393	+	0.971414i	$0.423707\pi$
$62$	2.62210	0.333007
$63$	0	0
$64$	4.94427	0.618034
$65$	0	0
$66$	0	0
$67$	5.76393	0.704176	0.352088	−	0.935967i	$-0.385472\pi$
$67$	5.76393	0.704176	0.352088	+	0.935967i	$0.385472\pi$
$68$	−6.32456	−0.766965
$69$	0	0
$70$	0	0
$71$	−7.40492	−0.878802	−0.439401	−	0.898291i	$-0.644809\pi$
$71$	−7.40492	−0.878802	−0.439401	+	0.898291i	$0.644809\pi$
$72$	0	0
$73$	−9.18034	−1.07448	−0.537239	−	0.843430i	$-0.680533\pi$
$73$	−9.18034	−1.07448	−0.537239	+	0.843430i	$0.680533\pi$
$74$	−5.86319	−0.681581
$75$	0	0
$76$	6.47214	0.742405
$77$	0.333851	0.0380458
$78$	0	0
$79$	−2.23607	−0.251577	−0.125789	−	0.992057i	$-0.540146\pi$
$79$	−2.23607	−0.251577	−0.125789	+	0.992057i	$0.540146\pi$
$80$	0	0
$81$	0	0
$82$	4.47214	0.493865
$83$	3.16228	0.347105	0.173553	−	0.984825i	$-0.444475\pi$
$83$	3.16228	0.347105	0.173553	+	0.984825i	$0.444475\pi$
$84$	0	0
$85$	0	0
$86$	−5.03786	−0.543247
$87$	0	0
$88$	−0.944272	−0.100660
$89$	−15.5563	−1.64897	−0.824485	−	0.565884i	$-0.808535\pi$
$89$	−15.5563	−1.64897	−0.824485	+	0.565884i	$0.808535\pi$
$90$	0	0
$91$	3.47214	0.363979
$92$	10.9010	1.13651
$93$	0	0
$94$	−9.23607	−0.952628
$95$	0	0
$96$	0	0
$97$	−0.763932	−0.0775655	−0.0387828	−	0.999248i	$-0.512348\pi$
$97$	−0.763932	−0.0775655	−0.0387828	+	0.999248i	$0.512348\pi$
$98$	−0.874032	−0.0882906
$99$	0	0
$100$	0	0
$101$	−7.73877	−0.770036	−0.385018	−	0.922909i	$-0.625805\pi$
$101$	−7.73877	−0.770036	−0.385018	+	0.922909i	$0.625805\pi$
$102$	0	0
$103$	13.1803	1.29870	0.649349	−	0.760491i	$-0.275042\pi$
$103$	13.1803	1.29870	0.649349	+	0.760491i	$0.275042\pi$
$104$	−9.82068	−0.962997
$105$	0	0
$106$	9.81966	0.953770
$107$	−12.1877	−1.17823	−0.589117	−	0.808048i	$-0.700524\pi$
$107$	−12.1877	−1.17823	−0.589117	+	0.808048i	$0.700524\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.5107	1.64727	0.823636	−	0.567119i	$-0.191942\pi$
$113$	17.5107	1.64727	0.823636	+	0.567119i	$0.191942\pi$
$114$	0	0
$115$	0	0
$116$	−4.57649	−0.424917
$117$	0	0
$118$	−4.65248	−0.428295
$119$	−5.11667	−0.469045
$120$	0	0
$121$	−10.8885	−0.989868
$122$	−3.24109	−0.293434
$123$	0	0
$124$	3.70820	0.333007
$125$	0	0
$126$	0	0
$127$	16.9443	1.50356	0.751780	−	0.659413i	$-0.229195\pi$
$127$	16.9443	1.50356	0.751780	+	0.659413i	$0.229195\pi$
$128$	6.99226	0.618034
$129$	0	0
$130$	0	0
$131$	4.37016	0.381823	0.190911	−	0.981607i	$-0.438856\pi$
$131$	4.37016	0.381823	0.190911	+	0.981607i	$0.438856\pi$
$132$	0	0
$133$	5.23607	0.454025
$134$	−5.03786	−0.435205
$135$	0	0
$136$	14.4721	1.24098
$137$	−8.61280	−0.735841	−0.367921	−	0.929857i	$-0.619930\pi$
$137$	−8.61280	−0.735841	−0.367921	+	0.929857i	$0.619930\pi$
$138$	0	0
$139$	−17.6525	−1.49726	−0.748632	−	0.662986i	$-0.769289\pi$
$139$	−17.6525	−1.49726	−0.748632	+	0.662986i	$0.769289\pi$
$140$	0	0
$141$	0	0
$142$	6.47214	0.543130
$143$	1.15917	0.0969350
$144$	0	0
$145$	0	0
$146$	8.02391	0.664064
$147$	0	0
$148$	−8.29180	−0.681581
$149$	0.333851	0.0273501	0.0136751	−	0.999906i	$-0.495647\pi$
$149$	0.333851	0.0273501	0.0136751	+	0.999906i	$0.495647\pi$
$150$	0	0
$151$	17.2361	1.40265	0.701326	−	0.712841i	$-0.252592\pi$
$151$	17.2361	1.40265	0.701326	+	0.712841i	$0.252592\pi$
$152$	−14.8098	−1.20124
$153$	0	0
$154$	−0.291796	−0.0235136
$155$	0	0
$156$	0	0
$157$	15.7639	1.25810	0.629049	−	0.777365i	$-0.283444\pi$
$157$	15.7639	1.25810	0.629049	+	0.777365i	$0.283444\pi$
$158$	1.95440	0.155483
$159$	0	0
$160$	0	0
$161$	8.81913	0.695045
$162$	0	0
$163$	19.4721	1.52518	0.762588	−	0.646885i	$-0.223929\pi$
$163$	19.4721	1.52518	0.762588	+	0.646885i	$0.223929\pi$
$164$	6.32456	0.493865
$165$	0	0
$166$	−2.76393	−0.214523
$167$	10.6947	0.827582	0.413791	−	0.910372i	$-0.364204\pi$
$167$	10.6947	0.827582	0.413791	+	0.910372i	$0.364204\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	0	0
$172$	−7.12461	−0.543247
$173$	−17.6383	−1.34101	−0.670506	−	0.741904i	$-0.733923\pi$
$173$	−17.6383	−1.34101	−0.670506	+	0.741904i	$0.733923\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	13.5967	1.01912
$179$	21.6746	1.62003	0.810017	−	0.586407i	$-0.199458\pi$
$179$	21.6746	1.62003	0.810017	+	0.586407i	$0.199458\pi$
$180$	0	0
$181$	5.94427	0.441834	0.220917	−	0.975293i	$-0.429095\pi$
$181$	5.94427	0.441834	0.220917	+	0.975293i	$0.429095\pi$
$182$	−3.03476	−0.224951
$183$	0	0
$184$	−24.9443	−1.83892
$185$	0	0
$186$	0	0
$187$	−1.70820	−0.124916
$188$	−13.0618	−0.952628
$189$	0	0
$190$	0	0
$191$	−7.19859	−0.520872	−0.260436	−	0.965491i	$-0.583866\pi$
$191$	−7.19859	−0.520872	−0.260436	+	0.965491i	$0.583866\pi$
$192$	0	0
$193$	5.41641	0.389882	0.194941	−	0.980815i	$-0.437549\pi$
$193$	5.41641	0.389882	0.194941	+	0.980815i	$0.437549\pi$
$194$	0.667701	0.0479381
$195$	0	0
$196$	−1.23607	−0.0882906
$197$	16.5579	1.17970	0.589851	−	0.807512i	$-0.299187\pi$
$197$	16.5579	1.17970	0.589851	+	0.807512i	$0.299187\pi$
$198$	0	0
$199$	13.1803	0.934330	0.467165	−	0.884170i	$-0.345275\pi$
$199$	13.1803	0.934330	0.467165	+	0.884170i	$0.345275\pi$
$200$	0	0
$201$	0	0
$202$	6.76393	0.475909
$203$	−3.70246	−0.259862
$204$	0	0
$205$	0	0
$206$	−11.5200	−0.802639
$207$	0	0
$208$	0	0
$209$	1.74806	0.120916
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	13.8871	0.953770
$213$	0	0
$214$	10.6525	0.728188
$215$	0	0
$216$	0	0
$217$	3.00000	0.203653
$218$	4.37016	0.295985
$219$	0	0
$220$	0	0
$221$	−17.7658	−1.19506
$222$	0	0
$223$	23.1803	1.55227	0.776135	−	0.630567i	$-0.217177\pi$
$223$	23.1803	1.55227	0.776135	+	0.630567i	$0.217177\pi$
$224$	5.65685	0.377964
$225$	0	0
$226$	−15.3050	−1.01807
$227$	6.78593	0.450398	0.225199	−	0.974313i	$-0.427697\pi$
$227$	6.78593	0.450398	0.225199	+	0.974313i	$0.427697\pi$
$228$	0	0
$229$	−10.5279	−0.695701	−0.347850	−	0.937550i	$-0.613088\pi$
$229$	−10.5279	−0.695701	−0.347850	+	0.937550i	$0.613088\pi$
$230$	0	0
$231$	0	0
$232$	10.4721	0.687529
$233$	−7.73877	−0.506984	−0.253492	−	0.967338i	$-0.581579\pi$
$233$	−7.73877	−0.506984	−0.253492	+	0.967338i	$0.581579\pi$
$234$	0	0
$235$	0	0
$236$	−6.57959	−0.428295
$237$	0	0
$238$	4.47214	0.289886
$239$	−0.333851	−0.0215950	−0.0107975	−	0.999942i	$-0.503437\pi$
$239$	−0.333851	−0.0215950	−0.0107975	+	0.999942i	$0.503437\pi$
$240$	0	0
$241$	17.3607	1.11830	0.559150	−	0.829067i	$-0.311128\pi$
$241$	17.3607	1.11830	0.559150	+	0.829067i	$0.311128\pi$
$242$	9.51694	0.611772
$243$	0	0
$244$	−4.58359	−0.293434
$245$	0	0
$246$	0	0
$247$	18.1803	1.15679
$248$	−8.48528	−0.538816
$249$	0	0
$250$	0	0
$251$	18.6398	1.17653	0.588267	−	0.808667i	$-0.299810\pi$
$251$	18.6398	1.17653	0.588267	+	0.808667i	$0.299810\pi$
$252$	0	0
$253$	2.94427	0.185105
$254$	−14.8098	−0.929252
$255$	0	0
$256$	−16.0000	−1.00000
$257$	−10.6947	−0.667118	−0.333559	−	0.942729i	$-0.608250\pi$
$257$	−10.6947	−0.667118	−0.333559	+	0.942729i	$0.608250\pi$
$258$	0	0
$259$	−6.70820	−0.416828
$260$	0	0
$261$	0	0
$262$	−3.81966	−0.235979
$263$	21.3407	1.31593	0.657963	−	0.753051i	$-0.271418\pi$
$263$	21.3407	1.31593	0.657963	+	0.753051i	$0.271418\pi$
$264$	0	0
$265$	0	0
$266$	−4.57649	−0.280603
$267$	0	0
$268$	−7.12461	−0.435205
$269$	−24.9157	−1.51913	−0.759567	−	0.650429i	$-0.774589\pi$
$269$	−24.9157	−1.51913	−0.759567	+	0.650429i	$0.774589\pi$
$270$	0	0
$271$	7.29180	0.442945	0.221473	−	0.975167i	$-0.428914\pi$
$271$	7.29180	0.442945	0.221473	+	0.975167i	$0.428914\pi$
$272$	0	0
$273$	0	0
$274$	7.52786	0.454775
$275$	0	0
$276$	0	0
$277$	15.3607	0.922934	0.461467	−	0.887157i	$-0.347323\pi$
$277$	15.3607	0.922934	0.461467	+	0.887157i	$0.347323\pi$
$278$	15.4288	0.925360
$279$	0	0
$280$	0	0
$281$	−7.73877	−0.461656	−0.230828	−	0.972995i	$-0.574144\pi$
$281$	−7.73877	−0.461656	−0.230828	+	0.972995i	$0.574144\pi$
$282$	0	0
$283$	8.70820	0.517649	0.258824	−	0.965924i	$-0.416665\pi$
$283$	8.70820	0.517649	0.258824	+	0.965924i	$0.416665\pi$
$284$	9.15298	0.543130
$285$	0	0
$286$	−1.01316	−0.0599091
$287$	5.11667	0.302028
$288$	0	0
$289$	9.18034	0.540020
$290$	0	0
$291$	0	0
$292$	11.3475	0.664064
$293$	−8.02391	−0.468762	−0.234381	−	0.972145i	$-0.575306\pi$
$293$	−8.02391	−0.468762	−0.234381	+	0.972145i	$0.575306\pi$
$294$	0	0
$295$	0	0
$296$	18.9737	1.10282
$297$	0	0
$298$	−0.291796	−0.0169033
$299$	30.6212	1.77087
$300$	0	0
$301$	−5.76393	−0.332228
$302$	−15.0649	−0.866886
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.7082	1.41017	0.705086	−	0.709122i	$-0.250908\pi$
$307$	24.7082	1.41017	0.705086	+	0.709122i	$0.250908\pi$
$308$	−0.412662	−0.0235136
$309$	0	0
$310$	0	0
$311$	24.0903	1.36604	0.683019	−	0.730401i	$-0.260667\pi$
$311$	24.0903	1.36604	0.683019	+	0.730401i	$0.260667\pi$
$312$	0	0
$313$	20.2361	1.14381	0.571905	−	0.820320i	$-0.306205\pi$
$313$	20.2361	1.14381	0.571905	+	0.820320i	$0.306205\pi$
$314$	−13.7782	−0.777548
$315$	0	0
$316$	2.76393	0.155483
$317$	−3.03476	−0.170449	−0.0852245	−	0.996362i	$-0.527161\pi$
$317$	−3.03476	−0.170449	−0.0852245	+	0.996362i	$0.527161\pi$
$318$	0	0
$319$	−1.23607	−0.0692065
$320$	0	0
$321$	0	0
$322$	−7.70820	−0.429561
$323$	−26.7912	−1.49070
$324$	0	0
$325$	0	0
$326$	−17.0193	−0.942610
$327$	0	0
$328$	−14.4721	−0.799090
$329$	−10.5672	−0.582588
$330$	0	0
$331$	11.2918	0.620653	0.310327	−	0.950630i	$-0.399562\pi$
$331$	11.2918	0.620653	0.310327	+	0.950630i	$0.399562\pi$
$332$	−3.90879	−0.214523
$333$	0	0
$334$	−9.34752	−0.511474
$335$	0	0
$336$	0	0
$337$	35.0689	1.91032	0.955162	−	0.296084i	$-0.0956808\pi$
$337$	35.0689	1.91032	0.955162	+	0.296084i	$0.0956808\pi$
$338$	0.825324	0.0448917
$339$	0	0
$340$	0	0
$341$	1.00155	0.0542371
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	16.3029	0.878991
$345$	0	0
$346$	15.4164	0.828791
$347$	−1.41421	−0.0759190	−0.0379595	−	0.999279i	$-0.512086\pi$
$347$	−1.41421	−0.0759190	−0.0379595	+	0.999279i	$0.512086\pi$
$348$	0	0
$349$	−9.76393	−0.522651	−0.261326	−	0.965251i	$-0.584160\pi$
$349$	−9.76393	−0.522651	−0.261326	+	0.965251i	$0.584160\pi$
$350$	0	0
$351$	0	0
$352$	1.88854	0.100660
$353$	21.8021	1.16041	0.580204	−	0.814471i	$-0.302973\pi$
$353$	21.8021	1.16041	0.580204	+	0.814471i	$0.302973\pi$
$354$	0	0
$355$	0	0
$356$	19.2287	1.01912
$357$	0	0
$358$	−18.9443	−1.00124
$359$	7.32611	0.386657	0.193329	−	0.981134i	$-0.438072\pi$
$359$	7.32611	0.386657	0.193329	+	0.981134i	$0.438072\pi$
$360$	0	0
$361$	8.41641	0.442969
$362$	−5.19548	−0.273069
$363$	0	0
$364$	−4.29180	−0.224951
$365$	0	0
$366$	0	0
$367$	−12.1246	−0.632900	−0.316450	−	0.948609i	$-0.602491\pi$
$367$	−12.1246	−0.632900	−0.316450	+	0.948609i	$0.602491\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.2349	0.583287
$372$	0	0
$373$	−17.4164	−0.901787	−0.450894	−	0.892578i	$-0.648895\pi$
$373$	−17.4164	−0.901787	−0.450894	+	0.892578i	$0.648895\pi$
$374$	1.49302	0.0772025
$375$	0	0
$376$	29.8885	1.54138
$377$	−12.8554	−0.662089
$378$	0	0
$379$	25.4721	1.30842	0.654208	−	0.756315i	$-0.273002\pi$
$379$	25.4721	1.30842	0.654208	+	0.756315i	$0.273002\pi$
$380$	0	0
$381$	0	0
$382$	6.29180	0.321916
$383$	−0.285142	−0.0145701	−0.00728505	−	0.999973i	$-0.502319\pi$
$383$	−0.285142	−0.0145701	−0.00728505	+	0.999973i	$0.502319\pi$
$384$	0	0
$385$	0	0
$386$	−4.73411	−0.240960
$387$	0	0
$388$	0.944272	0.0479381
$389$	27.0463	1.37130	0.685651	−	0.727931i	$-0.259518\pi$
$389$	27.0463	1.37130	0.685651	+	0.727931i	$0.259518\pi$
$390$	0	0
$391$	−45.1246	−2.28205
$392$	2.82843	0.142857
$393$	0	0
$394$	−14.4721	−0.729096
$395$	0	0
$396$	0	0
$397$	−25.3607	−1.27282	−0.636408	−	0.771353i	$-0.719581\pi$
$397$	−25.3607	−1.27282	−0.636408	+	0.771353i	$0.719581\pi$
$398$	−11.5200	−0.577447
$399$	0	0
$400$	0	0
$401$	14.6823	0.733200	0.366600	−	0.930379i	$-0.380522\pi$
$401$	14.6823	0.733200	0.366600	+	0.930379i	$0.380522\pi$
$402$	0	0
$403$	10.4164	0.518878
$404$	9.56564	0.475909
$405$	0	0
$406$	3.23607	0.160603
$407$	−2.23954	−0.111010
$408$	0	0
$409$	25.9443	1.28286	0.641431	−	0.767181i	$-0.278341\pi$
$409$	25.9443	1.28286	0.641431	+	0.767181i	$0.278341\pi$
$410$	0	0
$411$	0	0
$412$	−16.2918	−0.802639
$413$	−5.32300	−0.261928
$414$	0	0
$415$	0	0
$416$	19.6414	0.962997
$417$	0	0
$418$	−1.52786	−0.0747303
$419$	−8.69161	−0.424613	−0.212307	−	0.977203i	$-0.568098\pi$
$419$	−8.69161	−0.424613	−0.212307	+	0.977203i	$0.568098\pi$
$420$	0	0
$421$	6.41641	0.312717	0.156358	−	0.987700i	$-0.450025\pi$
$421$	6.41641	0.312717	0.156358	+	0.987700i	$0.450025\pi$
$422$	2.62210	0.127642
$423$	0	0
$424$	−31.7771	−1.54323
$425$	0	0
$426$	0	0
$427$	−3.70820	−0.179453
$428$	15.0649	0.728188
$429$	0	0
$430$	0	0
$431$	7.81758	0.376560	0.188280	−	0.982115i	$-0.439709\pi$
$431$	7.81758	0.376560	0.188280	+	0.982115i	$0.439709\pi$
$432$	0	0
$433$	8.34752	0.401156	0.200578	−	0.979678i	$-0.435718\pi$
$433$	8.34752	0.401156	0.200578	+	0.979678i	$0.435718\pi$
$434$	−2.62210	−0.125865
$435$	0	0
$436$	6.18034	0.295985
$437$	46.1776	2.20897
$438$	0	0
$439$	−21.0000	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$440$	0	0
$441$	0	0
$442$	15.5279	0.738585
$443$	6.24574	0.296744	0.148372	−	0.988932i	$-0.452597\pi$
$443$	6.24574	0.296744	0.148372	+	0.988932i	$0.452597\pi$
$444$	0	0
$445$	0	0
$446$	−20.2604	−0.959356
$447$	0	0
$448$	−4.94427	−0.233595
$449$	13.0618	0.616423	0.308212	−	0.951318i	$-0.400270\pi$
$449$	13.0618	0.616423	0.308212	+	0.951318i	$0.400270\pi$
$450$	0	0
$451$	1.70820	0.0804362
$452$	−21.6445	−1.01807
$453$	0	0
$454$	−5.93112	−0.278361
$455$	0	0
$456$	0	0
$457$	−18.9443	−0.886176	−0.443088	−	0.896478i	$-0.646117\pi$
$457$	−18.9443	−0.886176	−0.443088	+	0.896478i	$0.646117\pi$
$458$	9.20169	0.429967
$459$	0	0
$460$	0	0
$461$	−19.7990	−0.922131	−0.461065	−	0.887366i	$-0.652533\pi$
$461$	−19.7990	−0.922131	−0.461065	+	0.887366i	$0.652533\pi$
$462$	0	0
$463$	−28.5967	−1.32900	−0.664502	−	0.747287i	$-0.731356\pi$
$463$	−28.5967	−1.32900	−0.664502	+	0.747287i	$0.731356\pi$
$464$	0	0
$465$	0	0
$466$	6.76393	0.313333
$467$	−24.5517	−1.13612	−0.568059	−	0.822988i	$-0.692305\pi$
$467$	−24.5517	−1.13612	−0.568059	+	0.822988i	$0.692305\pi$
$468$	0	0
$469$	−5.76393	−0.266154
$470$	0	0
$471$	0	0
$472$	15.0557	0.692996
$473$	−1.92429	−0.0884790
$474$	0	0
$475$	0	0
$476$	6.32456	0.289886
$477$	0	0
$478$	0.291796	0.0133464
$479$	−9.69316	−0.442892	−0.221446	−	0.975173i	$-0.571078\pi$
$479$	−9.69316	−0.442892	−0.221446	+	0.975173i	$0.571078\pi$
$480$	0	0
$481$	−23.2918	−1.06201
$482$	−15.1738	−0.691147
$483$	0	0
$484$	13.4590	0.611772
$485$	0	0
$486$	0	0
$487$	27.1803	1.23166	0.615829	−	0.787880i	$-0.288821\pi$
$487$	27.1803	1.23166	0.615829	+	0.787880i	$0.288821\pi$
$488$	10.4884	0.474787
$489$	0	0
$490$	0	0
$491$	37.2796	1.68241	0.841203	−	0.540719i	$-0.181848\pi$
$491$	37.2796	1.68241	0.841203	+	0.540719i	$0.181848\pi$
$492$	0	0
$493$	18.9443	0.853207
$494$	−15.8902	−0.714934
$495$	0	0
$496$	0	0
$497$	7.40492	0.332156
$498$	0	0
$499$	−12.8885	−0.576970	−0.288485	−	0.957484i	$-0.593152\pi$
$499$	−12.8885	−0.576970	−0.288485	+	0.957484i	$0.593152\pi$
$500$	0	0
$501$	0	0
$502$	−16.2918	−0.727138
$503$	28.9520	1.29090	0.645452	−	0.763801i	$-0.276669\pi$
$503$	28.9520	1.29090	0.645452	+	0.763801i	$0.276669\pi$
$504$	0	0
$505$	0	0
$506$	−2.57339	−0.114401
$507$	0	0
$508$	−20.9443	−0.929252
$509$	3.95750	0.175413	0.0877065	−	0.996146i	$-0.472046\pi$
$509$	3.95750	0.175413	0.0877065	+	0.996146i	$0.472046\pi$
$510$	0	0
$511$	9.18034	0.406114
$512$	0	0
$513$	0	0
$514$	9.34752	0.412302
$515$	0	0
$516$	0	0
$517$	−3.52786	−0.155155
$518$	5.86319	0.257614
$519$	0	0
$520$	0	0
$521$	15.2712	0.669044	0.334522	−	0.942388i	$-0.391425\pi$
$521$	15.2712	0.669044	0.334522	+	0.942388i	$0.391425\pi$
$522$	0	0
$523$	38.0689	1.66464	0.832318	−	0.554298i	$-0.187013\pi$
$523$	38.0689	1.66464	0.832318	+	0.554298i	$0.187013\pi$
$524$	−5.40182	−0.235979
$525$	0	0
$526$	−18.6525	−0.813287
$527$	−15.3500	−0.668657
$528$	0	0
$529$	54.7771	2.38161
$530$	0	0
$531$	0	0
$532$	−6.47214	−0.280603
$533$	17.7658	0.769521
$534$	0	0
$535$	0	0
$536$	16.3029	0.704176
$537$	0	0
$538$	21.7771	0.938877
$539$	−0.333851	−0.0143800
$540$	0	0
$541$	−20.6525	−0.887919	−0.443960	−	0.896047i	$-0.646427\pi$
$541$	−20.6525	−0.887919	−0.443960	+	0.896047i	$0.646427\pi$
$542$	−6.37326	−0.273755
$543$	0	0
$544$	−28.9443	−1.24098
$545$	0	0
$546$	0	0
$547$	−14.8885	−0.636588	−0.318294	−	0.947992i	$-0.603110\pi$
$547$	−14.8885	−0.636588	−0.318294	+	0.947992i	$0.603110\pi$
$548$	10.6460	0.454775
$549$	0	0
$550$	0	0
$551$	−19.3863	−0.825885
$552$	0	0
$553$	2.23607	0.0950873
$554$	−13.4257	−0.570404
$555$	0	0
$556$	21.8197	0.925360
$557$	−27.0463	−1.14599	−0.572994	−	0.819560i	$-0.694218\pi$
$557$	−27.0463	−1.14599	−0.572994	+	0.819560i	$0.694218\pi$
$558$	0	0
$559$	−20.0132	−0.846466
$560$	0	0
$561$	0	0
$562$	6.76393	0.285319
$563$	−10.0270	−0.422588	−0.211294	−	0.977423i	$-0.567768\pi$
$563$	−10.0270	−0.422588	−0.211294	+	0.977423i	$0.567768\pi$
$564$	0	0
$565$	0	0
$566$	−7.61125	−0.319925
$567$	0	0
$568$	−20.9443	−0.878802
$569$	40.2356	1.68676	0.843382	−	0.537315i	$-0.180561\pi$
$569$	40.2356	1.68676	0.843382	+	0.537315i	$0.180561\pi$
$570$	0	0
$571$	−3.05573	−0.127878	−0.0639391	−	0.997954i	$-0.520366\pi$
$571$	−3.05573	−0.127878	−0.0639391	+	0.997954i	$0.520366\pi$
$572$	−1.43282	−0.0599091
$573$	0	0
$574$	−4.47214	−0.186663
$575$	0	0
$576$	0	0
$577$	−15.1246	−0.629646	−0.314823	−	0.949150i	$-0.601945\pi$
$577$	−15.1246	−0.629646	−0.314823	+	0.949150i	$0.601945\pi$
$578$	−8.02391	−0.333751
$579$	0	0
$580$	0	0
$581$	−3.16228	−0.131193
$582$	0	0
$583$	3.75078	0.155341
$584$	−25.9659	−1.07448
$585$	0	0
$586$	7.01316	0.289711
$587$	37.1521	1.53343	0.766716	−	0.641987i	$-0.221890\pi$
$587$	37.1521	1.53343	0.766716	+	0.641987i	$0.221890\pi$
$588$	0	0
$589$	15.7082	0.647245
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.8763	−1.26794	−0.633968	−	0.773359i	$-0.718575\pi$
$593$	−30.8763	−1.26794	−0.633968	+	0.773359i	$0.718575\pi$
$594$	0	0
$595$	0	0
$596$	−0.412662	−0.0169033
$597$	0	0
$598$	−26.7639	−1.09446
$599$	−35.1490	−1.43615	−0.718075	−	0.695966i	$-0.754976\pi$
$599$	−35.1490	−1.43615	−0.718075	+	0.695966i	$0.754976\pi$
$600$	0	0
$601$	−22.5967	−0.921741	−0.460870	−	0.887468i	$-0.652463\pi$
$601$	−22.5967	−0.921741	−0.460870	+	0.887468i	$0.652463\pi$
$602$	5.03786	0.205328
$603$	0	0
$604$	−21.3050	−0.866886
$605$	0	0
$606$	0	0
$607$	−5.58359	−0.226631	−0.113316	−	0.993559i	$-0.536147\pi$
$607$	−5.58359	−0.226631	−0.113316	+	0.993559i	$0.536147\pi$
$608$	29.6197	1.20124
$609$	0	0
$610$	0	0
$611$	−36.6907	−1.48435
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	−21.5958	−0.871534
$615$	0	0
$616$	0.944272	0.0380458
$617$	−9.31061	−0.374831	−0.187415	−	0.982281i	$-0.560011\pi$
$617$	−9.31061	−0.374831	−0.187415	+	0.982281i	$0.560011\pi$
$618$	0	0
$619$	17.4164	0.700025	0.350012	−	0.936745i	$-0.386177\pi$
$619$	17.4164	0.700025	0.350012	+	0.936745i	$0.386177\pi$
$620$	0	0
$621$	0	0
$622$	−21.0557	−0.844258
$623$	15.5563	0.623252
$624$	0	0
$625$	0	0
$626$	−17.6870	−0.706914
$627$	0	0
$628$	−19.4853	−0.777548
$629$	34.3237	1.36857
$630$	0	0
$631$	37.4721	1.49174	0.745871	−	0.666090i	$-0.232033\pi$
$631$	37.4721	1.49174	0.745871	+	0.666090i	$0.232033\pi$
$632$	−6.32456	−0.251577
$633$	0	0
$634$	2.65248	0.105343
$635$	0	0
$636$	0	0
$637$	−3.47214	−0.137571
$638$	1.08036	0.0427720
$639$	0	0
$640$	0	0
$641$	14.1908	0.560505	0.280252	−	0.959926i	$-0.409582\pi$
$641$	14.1908	0.560505	0.280252	+	0.959926i	$0.409582\pi$
$642$	0	0
$643$	7.05573	0.278251	0.139125	−	0.990275i	$-0.455571\pi$
$643$	7.05573	0.278251	0.139125	+	0.990275i	$0.455571\pi$
$644$	−10.9010	−0.429561
$645$	0	0
$646$	23.4164	0.921306
$647$	−31.6529	−1.24440	−0.622202	−	0.782857i	$-0.713762\pi$
$647$	−31.6529	−1.24440	−0.622202	+	0.782857i	$0.713762\pi$
$648$	0	0
$649$	−1.77709	−0.0697568
$650$	0	0
$651$	0	0
$652$	−24.0689	−0.942610
$653$	−6.60970	−0.258657	−0.129329	−	0.991602i	$-0.541282\pi$
$653$	−6.60970	−0.258657	−0.129329	+	0.991602i	$0.541282\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	9.23607	0.360059
$659$	−49.0848	−1.91207	−0.956036	−	0.293248i	$-0.905264\pi$
$659$	−49.0848	−1.91207	−0.956036	+	0.293248i	$0.905264\pi$
$660$	0	0
$661$	−12.2361	−0.475928	−0.237964	−	0.971274i	$-0.576480\pi$
$661$	−12.2361	−0.475928	−0.237964	+	0.971274i	$0.576480\pi$
$662$	−9.86939	−0.383585
$663$	0	0
$664$	8.94427	0.347105
$665$	0	0
$666$	0	0
$667$	−32.6525	−1.26431
$668$	−13.2194	−0.511474
$669$	0	0
$670$	0	0
$671$	−1.23799	−0.0477919
$672$	0	0
$673$	−28.3050	−1.09108	−0.545538	−	0.838086i	$-0.683675\pi$
$673$	−28.3050	−1.09108	−0.545538	+	0.838086i	$0.683675\pi$
$674$	−30.6513	−1.18065
$675$	0	0
$676$	1.16718	0.0448917
$677$	−18.6398	−0.716386	−0.358193	−	0.933648i	$-0.616607\pi$
$677$	−18.6398	−0.716386	−0.358193	+	0.933648i	$0.616607\pi$
$678$	0	0
$679$	0.763932	0.0293170
$680$	0	0
$681$	0	0
$682$	−0.875388	−0.0335203
$683$	−25.4558	−0.974041	−0.487020	−	0.873391i	$-0.661916\pi$
$683$	−25.4558	−0.974041	−0.487020	+	0.873391i	$0.661916\pi$
$684$	0	0
$685$	0	0
$686$	0.874032	0.0333707
$687$	0	0
$688$	0	0
$689$	39.0091	1.48613
$690$	0	0
$691$	0.180340	0.00686045	0.00343023	−	0.999994i	$-0.498908\pi$
$691$	0.180340	0.00686045	0.00343023	+	0.999994i	$0.498908\pi$
$692$	21.8021	0.828791
$693$	0	0
$694$	1.23607	0.0469205
$695$	0	0
$696$	0	0
$697$	−26.1803	−0.991651
$698$	8.53399	0.323016
$699$	0	0
$700$	0	0
$701$	15.3013	0.577923	0.288961	−	0.957341i	$-0.406690\pi$
$701$	15.3013	0.577923	0.288961	+	0.957341i	$0.406690\pi$
$702$	0	0
$703$	−35.1246	−1.32475
$704$	−1.65065	−0.0622111
$705$	0	0
$706$	−19.0557	−0.717172
$707$	7.73877	0.291046
$708$	0	0
$709$	−19.3050	−0.725013	−0.362506	−	0.931981i	$-0.618079\pi$
$709$	−19.3050	−0.725013	−0.362506	+	0.931981i	$0.618079\pi$
$710$	0	0
$711$	0	0
$712$	−44.0000	−1.64897
$713$	26.4574	0.990837
$714$	0	0
$715$	0	0
$716$	−26.7912	−1.00124
$717$	0	0
$718$	−6.40325	−0.238967
$719$	12.3941	0.462221	0.231110	−	0.972928i	$-0.425764\pi$
$719$	12.3941	0.462221	0.231110	+	0.972928i	$0.425764\pi$
$720$	0	0
$721$	−13.1803	−0.490862
$722$	−7.35621	−0.273770
$723$	0	0
$724$	−7.34752	−0.273069
$725$	0	0
$726$	0	0
$727$	3.65248	0.135463	0.0677314	−	0.997704i	$-0.478424\pi$
$727$	3.65248	0.135463	0.0677314	+	0.997704i	$0.478424\pi$
$728$	9.82068	0.363979
$729$	0	0
$730$	0	0
$731$	29.4922	1.09081
$732$	0	0
$733$	−28.5410	−1.05419	−0.527093	−	0.849807i	$-0.676718\pi$
$733$	−28.5410	−1.05419	−0.527093	+	0.849807i	$0.676718\pi$
$734$	10.5973	0.391153
$735$	0	0
$736$	49.8885	1.83892
$737$	−1.92429	−0.0708822
$738$	0	0
$739$	22.7639	0.837385	0.418692	−	0.908128i	$-0.362489\pi$
$739$	22.7639	0.837385	0.418692	+	0.908128i	$0.362489\pi$
$740$	0	0
$741$	0	0
$742$	−9.81966	−0.360491
$743$	−19.3863	−0.711215	−0.355608	−	0.934635i	$-0.615726\pi$
$743$	−19.3863	−0.711215	−0.355608	+	0.934635i	$0.615726\pi$
$744$	0	0
$745$	0	0
$746$	15.2225	0.557335
$747$	0	0
$748$	2.11146	0.0772025
$749$	12.1877	0.445330
$750$	0	0
$751$	−44.8885	−1.63801	−0.819003	−	0.573789i	$-0.805473\pi$
$751$	−44.8885	−1.63801	−0.819003	+	0.573789i	$0.805473\pi$
$752$	0	0
$753$	0	0
$754$	11.2361	0.409193
$755$	0	0
$756$	0	0
$757$	39.7082	1.44322	0.721610	−	0.692300i	$-0.243403\pi$
$757$	39.7082	1.44322	0.721610	+	0.692300i	$0.243403\pi$
$758$	−22.2635	−0.808645
$759$	0	0
$760$	0	0
$761$	51.0091	1.84908	0.924539	−	0.381087i	$-0.124450\pi$
$761$	51.0091	1.84908	0.924539	+	0.381087i	$0.124450\pi$
$762$	0	0
$763$	5.00000	0.181012
$764$	8.89794	0.321916
$765$	0	0
$766$	0.249224	0.00900481
$767$	−18.4822	−0.667353
$768$	0	0
$769$	7.76393	0.279975	0.139987	−	0.990153i	$-0.455294\pi$
$769$	7.76393	0.279975	0.139987	+	0.990153i	$0.455294\pi$
$770$	0	0
$771$	0	0
$772$	−6.69505	−0.240960
$773$	20.0053	0.719541	0.359771	−	0.933041i	$-0.382855\pi$
$773$	20.0053	0.719541	0.359771	+	0.933041i	$0.382855\pi$
$774$	0	0
$775$	0	0
$776$	−2.16073	−0.0775655
$777$	0	0
$778$	−23.6393	−0.847511
$779$	26.7912	0.959896
$780$	0	0
$781$	2.47214	0.0884600
$782$	39.4404	1.41038
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	54.4721	1.94172	0.970861	−	0.239643i	$-0.0770305\pi$
$787$	54.4721	1.94172	0.970861	+	0.239643i	$0.0770305\pi$
$788$	−20.4667	−0.729096
$789$	0	0
$790$	0	0
$791$	−17.5107	−0.622610
$792$	0	0
$793$	−12.8754	−0.457218
$794$	22.1660	0.786644
$795$	0	0
$796$	−16.2918	−0.577447
$797$	27.3801	0.969854	0.484927	−	0.874555i	$-0.338846\pi$
$797$	27.3801	0.969854	0.484927	+	0.874555i	$0.338846\pi$
$798$	0	0
$799$	54.0689	1.91282
$800$	0	0
$801$	0	0
$802$	−12.8328	−0.453142
$803$	3.06486	0.108157
$804$	0	0
$805$	0	0
$806$	−9.10427	−0.320684
$807$	0	0
$808$	−21.8885	−0.770036
$809$	38.6938	1.36040	0.680202	−	0.733025i	$-0.261892\pi$
$809$	38.6938	1.36040	0.680202	+	0.733025i	$0.261892\pi$
$810$	0	0
$811$	16.8885	0.593037	0.296518	−	0.955027i	$-0.404174\pi$
$811$	16.8885	0.593037	0.296518	+	0.955027i	$0.404174\pi$
$812$	4.57649	0.160603
$813$	0	0
$814$	1.95743	0.0686078
$815$	0	0
$816$	0	0
$817$	−30.1803	−1.05588
$818$	−22.6761	−0.792852
$819$	0	0
$820$	0	0
$821$	50.1351	1.74973	0.874863	−	0.484370i	$-0.160951\pi$
$821$	50.1351	1.74973	0.874863	+	0.484370i	$0.160951\pi$
$822$	0	0
$823$	−31.0689	−1.08299	−0.541497	−	0.840703i	$-0.682142\pi$
$823$	−31.0689	−1.08299	−0.541497	+	0.840703i	$0.682142\pi$
$824$	37.2796	1.29870
$825$	0	0
$826$	4.65248	0.161880
$827$	22.7549	0.791267	0.395633	−	0.918409i	$-0.370525\pi$
$827$	22.7549	0.791267	0.395633	+	0.918409i	$0.370525\pi$
$828$	0	0
$829$	54.5967	1.89622	0.948112	−	0.317937i	$-0.102990\pi$
$829$	54.5967	1.89622	0.948112	+	0.317937i	$0.102990\pi$
$830$	0	0
$831$	0	0
$832$	−17.1672	−0.595165
$833$	5.11667	0.177282
$834$	0	0
$835$	0	0
$836$	−2.16073	−0.0747303
$837$	0	0
$838$	7.59675	0.262425
$839$	−18.7673	−0.647920	−0.323960	−	0.946071i	$-0.605014\pi$
$839$	−18.7673	−0.647920	−0.323960	+	0.946071i	$0.605014\pi$
$840$	0	0
$841$	−15.2918	−0.527303
$842$	−5.60815	−0.193269
$843$	0	0
$844$	3.70820	0.127642
$845$	0	0
$846$	0	0
$847$	10.8885	0.374135
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−59.1605	−2.02800
$852$	0	0
$853$	44.9574	1.53931	0.769657	−	0.638458i	$-0.220427\pi$
$853$	44.9574	1.53931	0.769657	+	0.638458i	$0.220427\pi$
$854$	3.24109	0.110908
$855$	0	0
$856$	−34.4721	−1.17823
$857$	30.4937	1.04165	0.520823	−	0.853665i	$-0.325625\pi$
$857$	30.4937	1.04165	0.520823	+	0.853665i	$0.325625\pi$
$858$	0	0
$859$	−3.11146	−0.106162	−0.0530808	−	0.998590i	$-0.516904\pi$
$859$	−3.11146	−0.106162	−0.0530808	+	0.998590i	$0.516904\pi$
$860$	0	0
$861$	0	0
$862$	−6.83282	−0.232727
$863$	28.9520	0.985537	0.492768	−	0.870161i	$-0.335985\pi$
$863$	28.9520	0.985537	0.492768	+	0.870161i	$0.335985\pi$
$864$	0	0
$865$	0	0
$866$	−7.29600	−0.247928
$867$	0	0
$868$	−3.70820	−0.125865
$869$	0.746512	0.0253237
$870$	0	0
$871$	−20.0132	−0.678120
$872$	−14.1421	−0.478913
$873$	0	0
$874$	−40.3607	−1.36522
$875$	0	0
$876$	0	0
$877$	6.94427	0.234491	0.117246	−	0.993103i	$-0.462593\pi$
$877$	6.94427	0.234491	0.117246	+	0.993103i	$0.462593\pi$
$878$	18.3547	0.619440
$879$	0	0
$880$	0	0
$881$	41.6799	1.40423	0.702116	−	0.712063i	$-0.252239\pi$
$881$	41.6799	1.40423	0.702116	+	0.712063i	$0.252239\pi$
$882$	0	0
$883$	11.5967	0.390262	0.195131	−	0.980777i	$-0.437487\pi$
$883$	11.5967	0.390262	0.195131	+	0.980777i	$0.437487\pi$
$884$	21.9597	0.738585
$885$	0	0
$886$	−5.45898	−0.183398
$887$	−32.5756	−1.09378	−0.546891	−	0.837204i	$-0.684189\pi$
$887$	−32.5756	−1.09378	−0.546891	+	0.837204i	$0.684189\pi$
$888$	0	0
$889$	−16.9443	−0.568293
$890$	0	0
$891$	0	0
$892$	−28.6525	−0.959356
$893$	−55.3306	−1.85157
$894$	0	0
$895$	0	0
$896$	−6.99226	−0.233595
$897$	0	0
$898$	−11.4164	−0.380970
$899$	−11.1074	−0.370452
$900$	0	0
$901$	−57.4853	−1.91511
$902$	−1.49302	−0.0497123
$903$	0	0
$904$	49.5279	1.64727
$905$	0	0
$906$	0	0
$907$	−49.8328	−1.65467	−0.827336	−	0.561708i	$-0.810145\pi$
$907$	−49.8328	−1.65467	−0.827336	+	0.561708i	$0.810145\pi$
$908$	−8.38787	−0.278361
$909$	0	0
$910$	0	0
$911$	19.0038	0.629623	0.314811	−	0.949154i	$-0.398059\pi$
$911$	19.0038	0.629623	0.314811	+	0.949154i	$0.398059\pi$
$912$	0	0
$913$	−1.05573	−0.0349395
$914$	16.5579	0.547687
$915$	0	0
$916$	13.0132	0.429967
$917$	−4.37016	−0.144315
$918$	0	0
$919$	−4.52786	−0.149360	−0.0746802	−	0.997208i	$-0.523794\pi$
$919$	−4.52786	−0.149360	−0.0746802	+	0.997208i	$0.523794\pi$
$920$	0	0
$921$	0	0
$922$	17.3050	0.569908
$923$	25.7109	0.846284
$924$	0	0
$925$	0	0
$926$	24.9945	0.821369
$927$	0	0
$928$	−20.9443	−0.687529
$929$	−46.4326	−1.52340	−0.761702	−	0.647927i	$-0.775636\pi$
$929$	−46.4326	−1.52340	−0.761702	+	0.647927i	$0.775636\pi$
$930$	0	0
$931$	−5.23607	−0.171605
$932$	9.56564	0.313333
$933$	0	0
$934$	21.4590	0.702159
$935$	0	0
$936$	0	0
$937$	−56.7214	−1.85301	−0.926503	−	0.376287i	$-0.877201\pi$
$937$	−56.7214	−1.85301	−0.926503	+	0.376287i	$0.877201\pi$
$938$	5.03786	0.164492
$939$	0	0
$940$	0	0
$941$	−15.1437	−0.493670	−0.246835	−	0.969058i	$-0.579391\pi$
$941$	−15.1437	−0.493670	−0.246835	+	0.969058i	$0.579391\pi$
$942$	0	0
$943$	45.1246	1.46946
$944$	0	0
$945$	0	0
$946$	1.68189	0.0546830
$947$	13.2681	0.431155	0.215578	−	0.976487i	$-0.430837\pi$
$947$	13.2681	0.431155	0.215578	+	0.976487i	$0.430837\pi$
$948$	0	0
$949$	31.8754	1.03472
$950$	0	0
$951$	0	0
$952$	−14.4721	−0.469045
$953$	−8.77042	−0.284102	−0.142051	−	0.989859i	$-0.545370\pi$
$953$	−8.77042	−0.284102	−0.142051	+	0.989859i	$0.545370\pi$
$954$	0	0
$955$	0	0
$956$	0.412662	0.0133464
$957$	0	0
$958$	8.47214	0.273722
$959$	8.61280	0.278122
$960$	0	0
$961$	−22.0000	−0.709677
$962$	20.3578	0.656361
$963$	0	0
$964$	−21.4590	−0.691147
$965$	0	0
$966$	0	0
$967$	43.1803	1.38859	0.694293	−	0.719692i	$-0.255717\pi$
$967$	43.1803	1.38859	0.694293	+	0.719692i	$0.255717\pi$
$968$	−30.7975	−0.989868
$969$	0	0
$970$	0	0
$971$	−44.8121	−1.43809	−0.719044	−	0.694965i	$-0.755420\pi$
$971$	−44.8121	−1.43809	−0.719044	+	0.694965i	$0.755420\pi$
$972$	0	0
$973$	17.6525	0.565912
$974$	−23.7565	−0.761207
$975$	0	0
$976$	0	0
$977$	−54.4078	−1.74066	−0.870330	−	0.492469i	$-0.836095\pi$
$977$	−54.4078	−1.74066	−0.870330	+	0.492469i	$0.836095\pi$
$978$	0	0
$979$	5.19350	0.165985
$980$	0	0
$981$	0	0
$982$	−32.5836	−1.03978
$983$	−41.8862	−1.33596	−0.667982	−	0.744178i	$-0.732842\pi$
$983$	−41.8862	−1.33596	−0.667982	+	0.744178i	$0.732842\pi$
$984$	0	0
$985$	0	0
$986$	−16.5579	−0.527311
$987$	0	0
$988$	−22.4721	−0.714934
$989$	−50.8329	−1.61639
$990$	0	0
$991$	21.5279	0.683855	0.341928	−	0.939726i	$-0.388920\pi$
$991$	21.5279	0.683855	0.341928	+	0.939726i	$0.388920\pi$
$992$	16.9706	0.538816
$993$	0	0
$994$	−6.47214	−0.205284
$995$	0	0
$996$	0	0
$997$	4.83282	0.153057	0.0765284	−	0.997067i	$-0.475616\pi$
$997$	4.83282	0.153057	0.0765284	+	0.997067i	$0.475616\pi$
$998$	11.2650	0.356587
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4725.2.a.bu.1.2	✓	4
3.2	odd	2	inner	4725.2.a.bu.1.3	yes	4
5.4	even	2		4725.2.a.bv.1.3	yes	4
15.14	odd	2		4725.2.a.bv.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4725.2.a.bu.1.2	✓	4	1.1	even	1	trivial
4725.2.a.bu.1.3	yes	4	3.2	odd	2	inner
4725.2.a.bv.1.2	yes	4	15.14	odd	2
4725.2.a.bv.1.3	yes	4	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)

\(f(q)\)	\(=\)	\(q-0.874032 q^{2} -1.23607 q^{4} -1.00000 q^{7} +2.82843 q^{8} -0.333851 q^{11} -3.47214 q^{13} +0.874032 q^{14} +5.11667 q^{17} -5.23607 q^{19} +0.291796 q^{22} -8.81913 q^{23} +3.03476 q^{26} +1.23607 q^{28} +3.70246 q^{29} -3.00000 q^{31} -5.65685 q^{32} -4.47214 q^{34} +6.70820 q^{37} +4.57649 q^{38} -5.11667 q^{41} +5.76393 q^{43} +0.412662 q^{44} +7.70820 q^{46} +10.5672 q^{47} +1.00000 q^{49} +4.29180 q^{52} -11.2349 q^{53} -2.82843 q^{56} -3.23607 q^{58} +5.32300 q^{59} +3.70820 q^{61} +2.62210 q^{62} +4.94427 q^{64} +5.76393 q^{67} -6.32456 q^{68} -7.40492 q^{71} -9.18034 q^{73} -5.86319 q^{74} +6.47214 q^{76} +0.333851 q^{77} -2.23607 q^{79} +4.47214 q^{82} +3.16228 q^{83} -5.03786 q^{86} -0.944272 q^{88} -15.5563 q^{89} +3.47214 q^{91} +10.9010 q^{92} -9.23607 q^{94} -0.763932 q^{97} -0.874032 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{7} + 4 q^{13} - 12 q^{19} + 28 q^{22} - 4 q^{28} - 12 q^{31} + 32 q^{43} + 4 q^{46} + 4 q^{49} + 44 q^{52} - 4 q^{58} - 12 q^{61} - 16 q^{64} + 32 q^{67} + 8 q^{73} + 8 q^{76} + 32 q^{88}+ \cdots - 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^7 + 4 * q^13 - 12 * q^19 + 28 * q^22 - 4 * q^28 - 12 * q^31 + 32 * q^43 + 4 * q^46 + 4 * q^49 + 44 * q^52 - 4 * q^58 - 12 * q^61 - 16 * q^64 + 32 * q^67 + 8 * q^73 + 8 * q^76 + 32 * q^88 - 4 * q^91 - 28 * q^94 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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