LMFDB - Embedded newform 4725.2.a.bw.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4725,2,Mod(1,4725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,4,0,0,4,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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			1.4
Root			$2.28825$ 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)

$f(q)$	$=$	$q+2.28825 q^{2} +3.23607 q^{4} +1.00000 q^{7} +2.82843 q^{8} +0.874032 q^{11} +4.23607 q^{13} +2.28825 q^{14} +0.333851 q^{17} -4.47214 q^{19} +2.00000 q^{22} -0.874032 q^{23} +9.69316 q^{26} +3.23607 q^{28} +7.94510 q^{29} +6.70820 q^{31} -5.65685 q^{32} +0.763932 q^{34} +3.00000 q^{37} -10.2333 q^{38} +8.81913 q^{41} +5.47214 q^{43} +2.82843 q^{44} -2.00000 q^{46} +4.78282 q^{47} +1.00000 q^{49} +13.7082 q^{52} -6.53089 q^{53} +2.82843 q^{56} +18.1803 q^{58} -4.78282 q^{59} +9.70820 q^{61} +15.3500 q^{62} -12.9443 q^{64} +1.76393 q^{67} +1.08036 q^{68} -2.16073 q^{71} -3.47214 q^{73} +6.86474 q^{74} -14.4721 q^{76} +0.874032 q^{77} -1.47214 q^{79} +20.1803 q^{82} -1.54173 q^{83} +12.5216 q^{86} +2.47214 q^{88} -8.27895 q^{89} +4.23607 q^{91} -2.82843 q^{92} +10.9443 q^{94} +1.52786 q^{97} +2.28825 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 4 q^{7} + 8 q^{13} + 8 q^{22} + 4 q^{28} + 12 q^{34} + 12 q^{37} + 4 q^{43} - 8 q^{46} + 4 q^{49} + 28 q^{52} + 28 q^{58} + 12 q^{61} - 16 q^{64} + 16 q^{67} + 4 q^{73} - 40 q^{76} + 12 q^{79}+ \cdots + 24 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 4 * q^7 + 8 * q^13 + 8 * q^22 + 4 * q^28 + 12 * q^34 + 12 * q^37 + 4 * q^43 - 8 * q^46 + 4 * q^49 + 28 * q^52 + 28 * q^58 + 12 * q^61 - 16 * q^64 + 16 * q^67 + 4 * q^73 - 40 * q^76 + 12 * q^79 + 36 * q^82 - 8 * q^88 + 8 * q^91 + 8 * q^94 + 24 * 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 $

$2$	2.28825	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$2$	2.28825	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$3$	0	0
$3$	0	0
$4$	3.23607	1.61803
$5$	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
$11$	0.874032	0.263531	0.131765	−	0.991281i	$-0.457935\pi$
$11$	0.874032	0.263531	0.131765	+	0.991281i	$0.457935\pi$
$12$	0	0
$13$	4.23607	1.17487	0.587437	−	0.809270i	$-0.300137\pi$
$13$	4.23607	1.17487	0.587437	+	0.809270i	$0.300137\pi$
$14$	2.28825	0.611559
$15$	0	0
$16$	0	0
$17$	0.333851	0.0809706	0.0404853	−	0.999180i	$-0.487110\pi$
$17$	0.333851	0.0809706	0.0404853	+	0.999180i	$0.487110\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−0.874032	−0.182248	−0.0911241	−	0.995840i	$-0.529046\pi$
$23$	−0.874032	−0.182248	−0.0911241	+	0.995840i	$0.529046\pi$
$24$	0	0
$25$	0	0
$26$	9.69316	1.90099
$27$	0	0
$28$	3.23607	0.611559
$29$	7.94510	1.47537	0.737684	−	0.675146i	$-0.235919\pi$
$29$	7.94510	1.47537	0.737684	+	0.675146i	$0.235919\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	0.763932	0.131013
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−10.2333	−1.66007
$39$	0	0
$40$	0	0
$41$	8.81913	1.37732	0.688659	−	0.725086i	$-0.258200\pi$
$41$	8.81913	1.37732	0.688659	+	0.725086i	$0.258200\pi$
$42$	0	0
$43$	5.47214	0.834493	0.417246	−	0.908793i	$-0.362995\pi$
$43$	5.47214	0.834493	0.417246	+	0.908793i	$0.362995\pi$
$44$	2.82843	0.426401
$45$	0	0
$46$	−2.00000	−0.294884
$47$	4.78282	0.697646	0.348823	−	0.937189i	$-0.386581\pi$
$47$	4.78282	0.697646	0.348823	+	0.937189i	$0.386581\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	13.7082	1.90099
$53$	−6.53089	−0.897086	−0.448543	−	0.893761i	$-0.648057\pi$
$53$	−6.53089	−0.897086	−0.448543	+	0.893761i	$0.648057\pi$
$54$	0	0
$55$	0	0
$56$	2.82843	0.377964
$57$	0	0
$58$	18.1803	2.38720
$59$	−4.78282	−0.622670	−0.311335	−	0.950300i	$-0.600776\pi$
$59$	−4.78282	−0.622670	−0.311335	+	0.950300i	$0.600776\pi$
$60$	0	0
$61$	9.70820	1.24301	0.621504	−	0.783411i	$-0.286522\pi$
$61$	9.70820	1.24301	0.621504	+	0.783411i	$0.286522\pi$
$62$	15.3500	1.94945
$63$	0	0
$64$	−12.9443	−1.61803
$65$	0	0
$66$	0	0
$67$	1.76393	0.215499	0.107749	−	0.994178i	$-0.465636\pi$
$67$	1.76393	0.215499	0.107749	+	0.994178i	$0.465636\pi$
$68$	1.08036	0.131013
$69$	0	0
$70$	0	0
$71$	−2.16073	−0.256431	−0.128216	−	0.991746i	$-0.540925\pi$
$71$	−2.16073	−0.256431	−0.128216	+	0.991746i	$0.540925\pi$
$72$	0	0
$73$	−3.47214	−0.406383	−0.203191	−	0.979139i	$-0.565131\pi$
$73$	−3.47214	−0.406383	−0.203191	+	0.979139i	$0.565131\pi$
$74$	6.86474	0.798009
$75$	0	0
$76$	−14.4721	−1.66007
$77$	0.874032	0.0996052
$78$	0	0
$79$	−1.47214	−0.165628	−0.0828141	−	0.996565i	$-0.526391\pi$
$79$	−1.47214	−0.165628	−0.0828141	+	0.996565i	$0.526391\pi$
$80$	0	0
$81$	0	0
$82$	20.1803	2.22855
$83$	−1.54173	−0.169227	−0.0846136	−	0.996414i	$-0.526966\pi$
$83$	−1.54173	−0.169227	−0.0846136	+	0.996414i	$0.526966\pi$
$84$	0	0
$85$	0	0
$86$	12.5216	1.35024
$87$	0	0
$88$	2.47214	0.263531
$89$	−8.27895	−0.877567	−0.438783	−	0.898593i	$-0.644590\pi$
$89$	−8.27895	−0.877567	−0.438783	+	0.898593i	$0.644590\pi$
$90$	0	0
$91$	4.23607	0.444061
$92$	−2.82843	−0.294884
$93$	0	0
$94$	10.9443	1.12882
$95$	0	0
$96$	0	0
$97$	1.52786	0.155131	0.0775655	−	0.996987i	$-0.475285\pi$
$97$	1.52786	0.155131	0.0775655	+	0.996987i	$0.475285\pi$
$98$	2.28825	0.231148
$99$	0	0
$100$	0	0
$101$	−6.53089	−0.649847	−0.324924	−	0.945740i	$-0.605339\pi$
$101$	−6.53089	−0.649847	−0.324924	+	0.945740i	$0.605339\pi$
$102$	0	0
$103$	7.76393	0.765003	0.382501	−	0.923955i	$-0.375063\pi$
$103$	7.76393	0.765003	0.382501	+	0.923955i	$0.375063\pi$
$104$	11.9814	1.17487
$105$	0	0
$106$	−14.9443	−1.45152
$107$	13.1893	1.27506	0.637528	−	0.770427i	$-0.279957\pi$
$107$	13.1893	1.27506	0.637528	+	0.770427i	$0.279957\pi$
$108$	0	0
$109$	−8.70820	−0.834095	−0.417047	−	0.908885i	$-0.636935\pi$
$109$	−8.70820	−0.834095	−0.417047	+	0.908885i	$0.636935\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.1328	−1.89394	−0.946969	−	0.321324i	$-0.895872\pi$
$113$	−20.1328	−1.89394	−0.946969	+	0.321324i	$0.895872\pi$
$114$	0	0
$115$	0	0
$116$	25.7109	2.38720
$117$	0	0
$118$	−10.9443	−1.00750
$119$	0.333851	0.0306040
$120$	0	0
$121$	−10.2361	−0.930552
$122$	22.2148	2.01123
$123$	0	0
$124$	21.7082	1.94945
$125$	0	0
$126$	0	0
$127$	5.52786	0.490519	0.245259	−	0.969458i	$-0.421127\pi$
$127$	5.52786	0.490519	0.245259	+	0.969458i	$0.421127\pi$
$128$	−18.3060	−1.61803
$129$	0	0
$130$	0	0
$131$	16.6854	1.45781	0.728906	−	0.684614i	$-0.240029\pi$
$131$	16.6854	1.45781	0.728906	+	0.684614i	$0.240029\pi$
$132$	0	0
$133$	−4.47214	−0.387783
$134$	4.03631	0.348684
$135$	0	0
$136$	0.944272	0.0809706
$137$	−4.91034	−0.419519	−0.209759	−	0.977753i	$-0.567268\pi$
$137$	−4.91034	−0.419519	−0.209759	+	0.977753i	$0.567268\pi$
$138$	0	0
$139$	12.2361	1.03785	0.518925	−	0.854820i	$-0.326332\pi$
$139$	12.2361	1.03785	0.518925	+	0.854820i	$0.326332\pi$
$140$	0	0
$141$	0	0
$142$	−4.94427	−0.414914
$143$	3.70246	0.309615
$144$	0	0
$145$	0	0
$146$	−7.94510	−0.657541
$147$	0	0
$148$	9.70820	0.798009
$149$	−23.0888	−1.89151	−0.945754	−	0.324885i	$-0.894674\pi$
$149$	−23.0888	−1.89151	−0.945754	+	0.324885i	$0.894674\pi$
$150$	0	0
$151$	4.47214	0.363937	0.181969	−	0.983304i	$-0.441753\pi$
$151$	4.47214	0.363937	0.181969	+	0.983304i	$0.441753\pi$
$152$	−12.6491	−1.02598
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	−2.23607	−0.178458	−0.0892288	−	0.996011i	$-0.528440\pi$
$157$	−2.23607	−0.178458	−0.0892288	+	0.996011i	$0.528440\pi$
$158$	−3.36861	−0.267992
$159$	0	0
$160$	0	0
$161$	−0.874032	−0.0688834
$162$	0	0
$163$	9.76393	0.764770	0.382385	−	0.924003i	$-0.375103\pi$
$163$	9.76393	0.764770	0.382385	+	0.924003i	$0.375103\pi$
$164$	28.5393	2.22855
$165$	0	0
$166$	−3.52786	−0.273815
$167$	6.45207	0.499277	0.249638	−	0.968339i	$-0.419688\pi$
$167$	6.45207	0.499277	0.249638	+	0.968339i	$0.419688\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	0	0
$171$	0	0
$172$	17.7082	1.35024
$173$	−15.4775	−1.17674	−0.588368	−	0.808593i	$-0.700229\pi$
$173$	−15.4775	−1.17674	−0.588368	+	0.808593i	$0.700229\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−18.9443	−1.41993
$179$	−25.3770	−1.89677	−0.948384	−	0.317124i	$-0.897283\pi$
$179$	−25.3770	−1.89677	−0.948384	+	0.317124i	$0.897283\pi$
$180$	0	0
$181$	−25.3607	−1.88504	−0.942522	−	0.334143i	$-0.891553\pi$
$181$	−25.3607	−1.88504	−0.942522	+	0.334143i	$0.891553\pi$
$182$	9.69316	0.718505
$183$	0	0
$184$	−2.47214	−0.182248
$185$	0	0
$186$	0	0
$187$	0.291796	0.0213382
$188$	15.4775	1.12882
$189$	0	0
$190$	0	0
$191$	5.11667	0.370229	0.185115	−	0.982717i	$-0.440734\pi$
$191$	5.11667	0.370229	0.185115	+	0.982717i	$0.440734\pi$
$192$	0	0
$193$	−5.41641	−0.389882	−0.194941	−	0.980815i	$-0.562451\pi$
$193$	−5.41641	−0.389882	−0.194941	+	0.980815i	$0.562451\pi$
$194$	3.49613	0.251007
$195$	0	0
$196$	3.23607	0.231148
$197$	21.8021	1.55333	0.776667	−	0.629911i	$-0.216909\pi$
$197$	21.8021	1.55333	0.776667	+	0.629911i	$0.216909\pi$
$198$	0	0
$199$	0.527864	0.0374193	0.0187096	−	0.999825i	$-0.494044\pi$
$199$	0.527864	0.0374193	0.0187096	+	0.999825i	$0.494044\pi$
$200$	0	0
$201$	0	0
$202$	−14.9443	−1.05148
$203$	7.94510	0.557637
$204$	0	0
$205$	0	0
$206$	17.7658	1.23780
$207$	0	0
$208$	0	0
$209$	−3.90879	−0.270377
$210$	0	0
$211$	−11.2918	−0.777359	−0.388680	−	0.921373i	$-0.627069\pi$
$211$	−11.2918	−0.777359	−0.388680	+	0.921373i	$0.627069\pi$
$212$	−21.1344	−1.45152
$213$	0	0
$214$	30.1803	2.06309
$215$	0	0
$216$	0	0
$217$	6.70820	0.455383
$218$	−19.9265	−1.34959
$219$	0	0
$220$	0	0
$221$	1.41421	0.0951303
$222$	0	0
$223$	17.1803	1.15048	0.575240	−	0.817984i	$-0.304909\pi$
$223$	17.1803	1.15048	0.575240	+	0.817984i	$0.304909\pi$
$224$	−5.65685	−0.377964
$225$	0	0
$226$	−46.0689	−3.06446
$227$	−0.540182	−0.0358531	−0.0179266	−	0.999839i	$-0.505707\pi$
$227$	−0.540182	−0.0358531	−0.0179266	+	0.999839i	$0.505707\pi$
$228$	0	0
$229$	21.6525	1.43084	0.715418	−	0.698697i	$-0.246236\pi$
$229$	21.6525	1.43084	0.715418	+	0.698697i	$0.246236\pi$
$230$	0	0
$231$	0	0
$232$	22.4721	1.47537
$233$	28.7456	1.88319	0.941595	−	0.336748i	$-0.109327\pi$
$233$	28.7456	1.88319	0.941595	+	0.336748i	$0.109327\pi$
$234$	0	0
$235$	0	0
$236$	−15.4775	−1.00750
$237$	0	0
$238$	0.763932	0.0495184
$239$	12.6004	0.815052	0.407526	−	0.913194i	$-0.366392\pi$
$239$	12.6004	0.815052	0.407526	+	0.913194i	$0.366392\pi$
$240$	0	0
$241$	17.4721	1.12548	0.562740	−	0.826634i	$-0.309747\pi$
$241$	17.4721	1.12548	0.562740	+	0.826634i	$0.309747\pi$
$242$	−23.4226	−1.50566
$243$	0	0
$244$	31.4164	2.01123
$245$	0	0
$246$	0	0
$247$	−18.9443	−1.20540
$248$	18.9737	1.20483
$249$	0	0
$250$	0	0
$251$	4.11512	0.259744	0.129872	−	0.991531i	$-0.458543\pi$
$251$	4.11512	0.259744	0.129872	+	0.991531i	$0.458543\pi$
$252$	0	0
$253$	−0.763932	−0.0480280
$254$	12.6491	0.793676
$255$	0	0
$256$	−16.0000	−1.00000
$257$	12.5216	0.781075	0.390538	−	0.920587i	$-0.372289\pi$
$257$	12.5216	0.781075	0.390538	+	0.920587i	$0.372289\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	38.1803	2.35879
$263$	−30.4149	−1.87546	−0.937731	−	0.347361i	$-0.887078\pi$
$263$	−30.4149	−1.87546	−0.937731	+	0.347361i	$0.887078\pi$
$264$	0	0
$265$	0	0
$266$	−10.2333	−0.627447
$267$	0	0
$268$	5.70820	0.348684
$269$	28.6181	1.74488	0.872439	−	0.488723i	$-0.162537\pi$
$269$	28.6181	1.74488	0.872439	+	0.488723i	$0.162537\pi$
$270$	0	0
$271$	18.4164	1.11872	0.559359	−	0.828926i	$-0.311048\pi$
$271$	18.4164	1.11872	0.559359	+	0.828926i	$0.311048\pi$
$272$	0	0
$273$	0	0
$274$	−11.2361	−0.678796
$275$	0	0
$276$	0	0
$277$	18.2361	1.09570	0.547850	−	0.836577i	$-0.315447\pi$
$277$	18.2361	1.09570	0.547850	+	0.836577i	$0.315447\pi$
$278$	27.9991	1.67928
$279$	0	0
$280$	0	0
$281$	−17.0193	−1.01528	−0.507642	−	0.861568i	$-0.669483\pi$
$281$	−17.0193	−1.01528	−0.507642	+	0.861568i	$0.669483\pi$
$282$	0	0
$283$	−12.4164	−0.738079	−0.369039	−	0.929414i	$-0.620313\pi$
$283$	−12.4164	−0.738079	−0.369039	+	0.929414i	$0.620313\pi$
$284$	−6.99226	−0.414914
$285$	0	0
$286$	8.47214	0.500968
$287$	8.81913	0.520577
$288$	0	0
$289$	−16.8885	−0.993444
$290$	0	0
$291$	0	0
$292$	−11.2361	−0.657541
$293$	9.02546	0.527273	0.263637	−	0.964622i	$-0.415078\pi$
$293$	9.02546	0.527273	0.263637	+	0.964622i	$0.415078\pi$
$294$	0	0
$295$	0	0
$296$	8.48528	0.493197
$297$	0	0
$298$	−52.8328	−3.06052
$299$	−3.70246	−0.214119
$300$	0	0
$301$	5.47214	0.315409
$302$	10.2333	0.588863
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.4164	−0.936934	−0.468467	−	0.883481i	$-0.655193\pi$
$307$	−16.4164	−0.936934	−0.468467	+	0.883481i	$0.655193\pi$
$308$	2.82843	0.161165
$309$	0	0
$310$	0	0
$311$	−27.7928	−1.57598	−0.787992	−	0.615685i	$-0.788879\pi$
$311$	−27.7928	−1.57598	−0.787992	+	0.615685i	$0.788879\pi$
$312$	0	0
$313$	−20.8885	−1.18069	−0.590345	−	0.807151i	$-0.701008\pi$
$313$	−20.8885	−1.18069	−0.590345	+	0.807151i	$0.701008\pi$
$314$	−5.11667	−0.288751
$315$	0	0
$316$	−4.76393	−0.267992
$317$	−12.5216	−0.703283	−0.351641	−	0.936135i	$-0.614376\pi$
$317$	−12.5216	−0.703283	−0.351641	+	0.936135i	$0.614376\pi$
$318$	0	0
$319$	6.94427	0.388805
$320$	0	0
$321$	0	0
$322$	−2.00000	−0.111456
$323$	−1.49302	−0.0830741
$324$	0	0
$325$	0	0
$326$	22.3423	1.23742
$327$	0	0
$328$	24.9443	1.37732
$329$	4.78282	0.263686
$330$	0	0
$331$	−33.5410	−1.84358	−0.921791	−	0.387688i	$-0.873274\pi$
$331$	−33.5410	−1.84358	−0.921791	+	0.387688i	$0.873274\pi$
$332$	−4.98915	−0.273815
$333$	0	0
$334$	14.7639	0.807846
$335$	0	0
$336$	0	0
$337$	0.819660	0.0446497	0.0223249	−	0.999751i	$-0.492893\pi$
$337$	0.819660	0.0446497	0.0223249	+	0.999751i	$0.492893\pi$
$338$	11.3137	0.615385
$339$	0	0
$340$	0	0
$341$	5.86319	0.317509
$342$	0	0
$343$	1.00000	0.0539949
$344$	15.4775	0.834493
$345$	0	0
$346$	−35.4164	−1.90400
$347$	−32.4968	−1.74452	−0.872260	−	0.489042i	$-0.837346\pi$
$347$	−32.4968	−1.74452	−0.872260	+	0.489042i	$0.837346\pi$
$348$	0	0
$349$	−35.0689	−1.87719	−0.938597	−	0.345015i	$-0.887874\pi$
$349$	−35.0689	−1.87719	−0.938597	+	0.345015i	$0.887874\pi$
$350$	0	0
$351$	0	0
$352$	−4.94427	−0.263531
$353$	−19.3863	−1.03183	−0.515915	−	0.856640i	$-0.672548\pi$
$353$	−19.3863	−1.03183	−0.515915	+	0.856640i	$0.672548\pi$
$354$	0	0
$355$	0	0
$356$	−26.7912	−1.41993
$357$	0	0
$358$	−58.0689	−3.06904
$359$	−6.78593	−0.358147	−0.179074	−	0.983836i	$-0.557310\pi$
$359$	−6.78593	−0.358147	−0.179074	+	0.983836i	$0.557310\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−58.0315	−3.05007
$363$	0	0
$364$	13.7082	0.718505
$365$	0	0
$366$	0	0
$367$	25.0000	1.30499	0.652495	−	0.757793i	$-0.273722\pi$
$367$	25.0000	1.30499	0.652495	+	0.757793i	$0.273722\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.53089	−0.339067
$372$	0	0
$373$	29.4164	1.52312	0.761562	−	0.648092i	$-0.224433\pi$
$373$	29.4164	1.52312	0.761562	+	0.648092i	$0.224433\pi$
$374$	0.667701	0.0345260
$375$	0	0
$376$	13.5279	0.697646
$377$	33.6560	1.73337
$378$	0	0
$379$	3.65248	0.187615	0.0938075	−	0.995590i	$-0.470096\pi$
$379$	3.65248	0.187615	0.0938075	+	0.995590i	$0.470096\pi$
$380$	0	0
$381$	0	0
$382$	11.7082	0.599044
$383$	−11.2349	−0.574076	−0.287038	−	0.957919i	$-0.592671\pi$
$383$	−11.2349	−0.574076	−0.287038	+	0.957919i	$0.592671\pi$
$384$	0	0
$385$	0	0
$386$	−12.3941	−0.630842
$387$	0	0
$388$	4.94427	0.251007
$389$	−21.8021	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$389$	−21.8021	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$390$	0	0
$391$	−0.291796	−0.0147568
$392$	2.82843	0.142857
$393$	0	0
$394$	49.8885	2.51335
$395$	0	0
$396$	0	0
$397$	1.47214	0.0738844	0.0369422	−	0.999317i	$-0.488238\pi$
$397$	1.47214	0.0738844	0.0369422	+	0.999317i	$0.488238\pi$
$398$	1.20788	0.0605457
$399$	0	0
$400$	0	0
$401$	−19.4651	−0.972043	−0.486021	−	0.873947i	$-0.661552\pi$
$401$	−19.4651	−0.972043	−0.486021	+	0.873947i	$0.661552\pi$
$402$	0	0
$403$	28.4164	1.41552
$404$	−21.1344	−1.05148
$405$	0	0
$406$	18.1803	0.902275
$407$	2.62210	0.129972
$408$	0	0
$409$	−14.5279	−0.718357	−0.359178	−	0.933269i	$-0.616943\pi$
$409$	−14.5279	−0.718357	−0.359178	+	0.933269i	$0.616943\pi$
$410$	0	0
$411$	0	0
$412$	25.1246	1.23780
$413$	−4.78282	−0.235347
$414$	0	0
$415$	0	0
$416$	−23.9628	−1.17487
$417$	0	0
$418$	−8.94427	−0.437479
$419$	−7.27740	−0.355524	−0.177762	−	0.984073i	$-0.556886\pi$
$419$	−7.27740	−0.355524	−0.177762	+	0.984073i	$0.556886\pi$
$420$	0	0
$421$	−4.70820	−0.229464	−0.114732	−	0.993396i	$-0.536601\pi$
$421$	−4.70820	−0.229464	−0.114732	+	0.993396i	$0.536601\pi$
$422$	−25.8384	−1.25779
$423$	0	0
$424$	−18.4721	−0.897086
$425$	0	0
$426$	0	0
$427$	9.70820	0.469813
$428$	42.6814	2.06309
$429$	0	0
$430$	0	0
$431$	−23.7078	−1.14196	−0.570982	−	0.820963i	$-0.693437\pi$
$431$	−23.7078	−1.14196	−0.570982	+	0.820963i	$0.693437\pi$
$432$	0	0
$433$	−9.65248	−0.463868	−0.231934	−	0.972731i	$-0.574505\pi$
$433$	−9.65248	−0.463868	−0.231934	+	0.972731i	$0.574505\pi$
$434$	15.3500	0.736824
$435$	0	0
$436$	−28.1803	−1.34959
$437$	3.90879	0.186983
$438$	0	0
$439$	9.00000	0.429547	0.214773	−	0.976664i	$-0.431099\pi$
$439$	9.00000	0.429547	0.214773	+	0.976664i	$0.431099\pi$
$440$	0	0
$441$	0	0
$442$	3.23607	0.153924
$443$	−19.5927	−0.930875	−0.465438	−	0.885081i	$-0.654103\pi$
$443$	−19.5927	−0.930875	−0.465438	+	0.885081i	$0.654103\pi$
$444$	0	0
$445$	0	0
$446$	39.3128	1.86152
$447$	0	0
$448$	−12.9443	−0.611559
$449$	−37.4373	−1.76677	−0.883387	−	0.468645i	$-0.844742\pi$
$449$	−37.4373	−1.76677	−0.883387	+	0.468645i	$0.844742\pi$
$450$	0	0
$451$	7.70820	0.362965
$452$	−65.1512	−3.06446
$453$	0	0
$454$	−1.23607	−0.0580115
$455$	0	0
$456$	0	0
$457$	−30.3607	−1.42021	−0.710106	−	0.704094i	$-0.751353\pi$
$457$	−30.3607	−1.42021	−0.710106	+	0.704094i	$0.751353\pi$
$458$	49.5462	2.31514
$459$	0	0
$460$	0	0
$461$	−38.3600	−1.78660	−0.893302	−	0.449458i	$-0.851617\pi$
$461$	−38.3600	−1.78660	−0.893302	+	0.449458i	$0.851617\pi$
$462$	0	0
$463$	21.9443	1.01984	0.509918	−	0.860223i	$-0.329676\pi$
$463$	21.9443	1.01984	0.509918	+	0.860223i	$0.329676\pi$
$464$	0	0
$465$	0	0
$466$	65.7771	3.04706
$467$	−17.8446	−0.825749	−0.412875	−	0.910788i	$-0.635475\pi$
$467$	−17.8446	−0.825749	−0.412875	+	0.910788i	$0.635475\pi$
$468$	0	0
$469$	1.76393	0.0814508
$470$	0	0
$471$	0	0
$472$	−13.5279	−0.622670
$473$	4.78282	0.219914
$474$	0	0
$475$	0	0
$476$	1.08036	0.0495184
$477$	0	0
$478$	28.8328	1.31878
$479$	−18.3848	−0.840022	−0.420011	−	0.907519i	$-0.637974\pi$
$479$	−18.3848	−0.840022	−0.420011	+	0.907519i	$0.637974\pi$
$480$	0	0
$481$	12.7082	0.579444
$482$	39.9805	1.82106
$483$	0	0
$484$	−33.1246	−1.50566
$485$	0	0
$486$	0	0
$487$	7.18034	0.325372	0.162686	−	0.986678i	$-0.447984\pi$
$487$	7.18034	0.325372	0.162686	+	0.986678i	$0.447984\pi$
$488$	27.4589	1.24301
$489$	0	0
$490$	0	0
$491$	4.98915	0.225157	0.112579	−	0.993643i	$-0.464089\pi$
$491$	4.98915	0.225157	0.112579	+	0.993643i	$0.464089\pi$
$492$	0	0
$493$	2.65248	0.119462
$494$	−43.3491	−1.95037
$495$	0	0
$496$	0	0
$497$	−2.16073	−0.0969218
$498$	0	0
$499$	−35.3607	−1.58296	−0.791481	−	0.611194i	$-0.790689\pi$
$499$	−35.3607	−1.58296	−0.791481	+	0.611194i	$0.790689\pi$
$500$	0	0
$501$	0	0
$502$	9.41641	0.420275
$503$	23.5502	1.05005	0.525025	−	0.851087i	$-0.324056\pi$
$503$	23.5502	1.05005	0.525025	+	0.851087i	$0.324056\pi$
$504$	0	0
$505$	0	0
$506$	−1.74806	−0.0777109
$507$	0	0
$508$	17.8885	0.793676
$509$	−27.3314	−1.21144	−0.605722	−	0.795676i	$-0.707116\pi$
$509$	−27.3314	−1.21144	−0.605722	+	0.795676i	$0.707116\pi$
$510$	0	0
$511$	−3.47214	−0.153598
$512$	0	0
$513$	0	0
$514$	28.6525	1.26381
$515$	0	0
$516$	0	0
$517$	4.18034	0.183851
$518$	6.86474	0.301619
$519$	0	0
$520$	0	0
$521$	−18.4335	−0.807586	−0.403793	−	0.914850i	$-0.632308\pi$
$521$	−18.4335	−0.807586	−0.403793	+	0.914850i	$0.632308\pi$
$522$	0	0
$523$	36.6525	1.60270	0.801350	−	0.598195i	$-0.204115\pi$
$523$	36.6525	1.60270	0.801350	+	0.598195i	$0.204115\pi$
$524$	53.9952	2.35879
$525$	0	0
$526$	−69.5967	−3.03456
$527$	2.23954	0.0975558
$528$	0	0
$529$	−22.2361	−0.966786
$530$	0	0
$531$	0	0
$532$	−14.4721	−0.627447
$533$	37.3584	1.61817
$534$	0	0
$535$	0	0
$536$	4.98915	0.215499
$537$	0	0
$538$	65.4853	2.82327
$539$	0.874032	0.0376472
$540$	0	0
$541$	34.6525	1.48983	0.744913	−	0.667161i	$-0.232491\pi$
$541$	34.6525	1.48983	0.744913	+	0.667161i	$0.232491\pi$
$542$	42.1413	1.81012
$543$	0	0
$544$	−1.88854	−0.0809706
$545$	0	0
$546$	0	0
$547$	18.8197	0.804671	0.402335	−	0.915492i	$-0.368199\pi$
$547$	18.8197	0.804671	0.402335	+	0.915492i	$0.368199\pi$
$548$	−15.8902	−0.678796
$549$	0	0
$550$	0	0
$551$	−35.5316	−1.51370
$552$	0	0
$553$	−1.47214	−0.0626016
$554$	41.7286	1.77288
$555$	0	0
$556$	39.5967	1.67928
$557$	−4.83153	−0.204719	−0.102359	−	0.994747i	$-0.532639\pi$
$557$	−4.83153	−0.204719	−0.102359	+	0.994747i	$0.532639\pi$
$558$	0	0
$559$	23.1803	0.980424
$560$	0	0
$561$	0	0
$562$	−38.9443	−1.64276
$563$	−24.9157	−1.05007	−0.525035	−	0.851081i	$-0.675948\pi$
$563$	−24.9157	−1.05007	−0.525035	+	0.851081i	$0.675948\pi$
$564$	0	0
$565$	0	0
$566$	−28.4118	−1.19424
$567$	0	0
$568$	−6.11146	−0.256431
$569$	−5.99070	−0.251143	−0.125572	−	0.992085i	$-0.540077\pi$
$569$	−5.99070	−0.251143	−0.125572	+	0.992085i	$0.540077\pi$
$570$	0	0
$571$	−6.11146	−0.255756	−0.127878	−	0.991790i	$-0.540817\pi$
$571$	−6.11146	−0.255756	−0.127878	+	0.991790i	$0.540817\pi$
$572$	11.9814	0.500968
$573$	0	0
$574$	20.1803	0.842311
$575$	0	0
$576$	0	0
$577$	−13.1246	−0.546385	−0.273192	−	0.961959i	$-0.588080\pi$
$577$	−13.1246	−0.546385	−0.273192	+	0.961959i	$0.588080\pi$
$578$	−38.6451	−1.60743
$579$	0	0
$580$	0	0
$581$	−1.54173	−0.0639619
$582$	0	0
$583$	−5.70820	−0.236410
$584$	−9.82068	−0.406383
$585$	0	0
$586$	20.6525	0.853146
$587$	15.5563	0.642079	0.321040	−	0.947066i	$-0.395968\pi$
$587$	15.5563	0.642079	0.321040	+	0.947066i	$0.395968\pi$
$588$	0	0
$589$	−30.0000	−1.23613
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.7642	−0.688424	−0.344212	−	0.938892i	$-0.611854\pi$
$593$	−16.7642	−0.688424	−0.344212	+	0.938892i	$0.611854\pi$
$594$	0	0
$595$	0	0
$596$	−74.7169	−3.06052
$597$	0	0
$598$	−8.47214	−0.346451
$599$	−25.6321	−1.04730	−0.523649	−	0.851934i	$-0.675430\pi$
$599$	−25.6321	−1.04730	−0.523649	+	0.851934i	$0.675430\pi$
$600$	0	0
$601$	33.4721	1.36536	0.682678	−	0.730719i	$-0.260815\pi$
$601$	33.4721	1.36536	0.682678	+	0.730719i	$0.260815\pi$
$602$	12.5216	0.510342
$603$	0	0
$604$	14.4721	0.588863
$605$	0	0
$606$	0	0
$607$	25.5410	1.03668	0.518339	−	0.855175i	$-0.326551\pi$
$607$	25.5410	1.03668	0.518339	+	0.855175i	$0.326551\pi$
$608$	25.2982	1.02598
$609$	0	0
$610$	0	0
$611$	20.2604	0.819646
$612$	0	0
$613$	24.4164	0.986169	0.493085	−	0.869981i	$-0.335869\pi$
$613$	24.4164	0.986169	0.493085	+	0.869981i	$0.335869\pi$
$614$	−37.5648	−1.51599
$615$	0	0
$616$	2.47214	0.0996052
$617$	−19.7990	−0.797077	−0.398539	−	0.917152i	$-0.630483\pi$
$617$	−19.7990	−0.797077	−0.398539	+	0.917152i	$0.630483\pi$
$618$	0	0
$619$	−29.7082	−1.19407	−0.597037	−	0.802214i	$-0.703655\pi$
$619$	−29.7082	−1.19407	−0.597037	+	0.802214i	$0.703655\pi$
$620$	0	0
$621$	0	0
$622$	−63.5967	−2.55000
$623$	−8.27895	−0.331689
$624$	0	0
$625$	0	0
$626$	−47.7981	−1.91040
$627$	0	0
$628$	−7.23607	−0.288751
$629$	1.00155	0.0399345
$630$	0	0
$631$	20.2361	0.805585	0.402793	−	0.915291i	$-0.368040\pi$
$631$	20.2361	0.805585	0.402793	+	0.915291i	$0.368040\pi$
$632$	−4.16383	−0.165628
$633$	0	0
$634$	−28.6525	−1.13794
$635$	0	0
$636$	0	0
$637$	4.23607	0.167839
$638$	15.8902	0.629099
$639$	0	0
$640$	0	0
$641$	43.8893	1.73352	0.866762	−	0.498722i	$-0.166197\pi$
$641$	43.8893	1.73352	0.866762	+	0.498722i	$0.166197\pi$
$642$	0	0
$643$	23.0557	0.909229	0.454615	−	0.890688i	$-0.349777\pi$
$643$	23.0557	0.909229	0.454615	+	0.890688i	$0.349777\pi$
$644$	−2.82843	−0.111456
$645$	0	0
$646$	−3.41641	−0.134417
$647$	30.2086	1.18762	0.593811	−	0.804605i	$-0.297623\pi$
$647$	30.2086	1.18762	0.593811	+	0.804605i	$0.297623\pi$
$648$	0	0
$649$	−4.18034	−0.164093
$650$	0	0
$651$	0	0
$652$	31.5967	1.23742
$653$	2.33695	0.0914521	0.0457260	−	0.998954i	$-0.485440\pi$
$653$	2.33695	0.0914521	0.0457260	+	0.998954i	$0.485440\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	10.9443	0.426652
$659$	−22.4211	−0.873402	−0.436701	−	0.899607i	$-0.643853\pi$
$659$	−22.4211	−0.873402	−0.436701	+	0.899607i	$0.643853\pi$
$660$	0	0
$661$	49.0689	1.90856	0.954279	−	0.298917i	$-0.0966253\pi$
$661$	49.0689	1.90856	0.954279	+	0.298917i	$0.0966253\pi$
$662$	−76.7501	−2.98298
$663$	0	0
$664$	−4.36068	−0.169227
$665$	0	0
$666$	0	0
$667$	−6.94427	−0.268883
$668$	20.8794	0.807846
$669$	0	0
$670$	0	0
$671$	8.48528	0.327571
$672$	0	0
$673$	−27.7639	−1.07022	−0.535110	−	0.844782i	$-0.679730\pi$
$673$	−27.7639	−1.07022	−0.535110	+	0.844782i	$0.679730\pi$
$674$	1.87558	0.0722448
$675$	0	0
$676$	16.0000	0.615385
$677$	−23.3438	−0.897176	−0.448588	−	0.893739i	$-0.648073\pi$
$677$	−23.3438	−0.897176	−0.448588	+	0.893739i	$0.648073\pi$
$678$	0	0
$679$	1.52786	0.0586340
$680$	0	0
$681$	0	0
$682$	13.4164	0.513741
$683$	8.48528	0.324680	0.162340	−	0.986735i	$-0.448096\pi$
$683$	8.48528	0.324680	0.162340	+	0.986735i	$0.448096\pi$
$684$	0	0
$685$	0	0
$686$	2.28825	0.0873656
$687$	0	0
$688$	0	0
$689$	−27.6653	−1.05396
$690$	0	0
$691$	1.81966	0.0692231	0.0346116	−	0.999401i	$-0.488981\pi$
$691$	1.81966	0.0692231	0.0346116	+	0.999401i	$0.488981\pi$
$692$	−50.0864	−1.90400
$693$	0	0
$694$	−74.3607	−2.82269
$695$	0	0
$696$	0	0
$697$	2.94427	0.111522
$698$	−80.2462	−3.03736
$699$	0	0
$700$	0	0
$701$	−11.3624	−0.429153	−0.214576	−	0.976707i	$-0.568837\pi$
$701$	−11.3624	−0.429153	−0.214576	+	0.976707i	$0.568837\pi$
$702$	0	0
$703$	−13.4164	−0.506009
$704$	−11.3137	−0.426401
$705$	0	0
$706$	−44.3607	−1.66954
$707$	−6.53089	−0.245619
$708$	0	0
$709$	−19.5279	−0.733384	−0.366692	−	0.930342i	$-0.619510\pi$
$709$	−19.5279	−0.733384	−0.366692	+	0.930342i	$0.619510\pi$
$710$	0	0
$711$	0	0
$712$	−23.4164	−0.877567
$713$	−5.86319	−0.219578
$714$	0	0
$715$	0	0
$716$	−82.1218	−3.06904
$717$	0	0
$718$	−15.5279	−0.579495
$719$	18.4636	0.688576	0.344288	−	0.938864i	$-0.388120\pi$
$719$	18.4636	0.688576	0.344288	+	0.938864i	$0.388120\pi$
$720$	0	0
$721$	7.76393	0.289144
$722$	2.28825	0.0851597
$723$	0	0
$724$	−82.0689	−3.05007
$725$	0	0
$726$	0	0
$727$	37.3607	1.38563	0.692815	−	0.721115i	$-0.256370\pi$
$727$	37.3607	1.38563	0.692815	+	0.721115i	$0.256370\pi$
$728$	11.9814	0.444061
$729$	0	0
$730$	0	0
$731$	1.82688	0.0675694
$732$	0	0
$733$	8.58359	0.317042	0.158521	−	0.987356i	$-0.449327\pi$
$733$	8.58359	0.317042	0.158521	+	0.987356i	$0.449327\pi$
$734$	57.2061	2.11152
$735$	0	0
$736$	4.94427	0.182248
$737$	1.54173	0.0567905
$738$	0	0
$739$	33.7771	1.24251	0.621255	−	0.783608i	$-0.286623\pi$
$739$	33.7771	1.24251	0.621255	+	0.783608i	$0.286623\pi$
$740$	0	0
$741$	0	0
$742$	−14.9443	−0.548621
$743$	23.0401	0.845259	0.422629	−	0.906303i	$-0.361107\pi$
$743$	23.0401	0.845259	0.422629	+	0.906303i	$0.361107\pi$
$744$	0	0
$745$	0	0
$746$	67.3120	2.46447
$747$	0	0
$748$	0.944272	0.0345260
$749$	13.1893	0.481926
$750$	0	0
$751$	−47.0689	−1.71757	−0.858784	−	0.512338i	$-0.828780\pi$
$751$	−47.0689	−1.71757	−0.858784	+	0.512338i	$0.828780\pi$
$752$	0	0
$753$	0	0
$754$	77.0132	2.80465
$755$	0	0
$756$	0	0
$757$	5.12461	0.186257	0.0931286	−	0.995654i	$-0.470313\pi$
$757$	5.12461	0.186257	0.0931286	+	0.995654i	$0.470313\pi$
$758$	8.35776	0.303568
$759$	0	0
$760$	0	0
$761$	31.2101	1.13137	0.565683	−	0.824623i	$-0.308613\pi$
$761$	31.2101	1.13137	0.565683	+	0.824623i	$0.308613\pi$
$762$	0	0
$763$	−8.70820	−0.315258
$764$	16.5579	0.599044
$765$	0	0
$766$	−25.7082	−0.928875
$767$	−20.2604	−0.731559
$768$	0	0
$769$	−13.1803	−0.475295	−0.237648	−	0.971351i	$-0.576376\pi$
$769$	−13.1803	−0.475295	−0.237648	+	0.971351i	$0.576376\pi$
$770$	0	0
$771$	0	0
$772$	−17.5279	−0.630842
$773$	10.5185	0.378324	0.189162	−	0.981946i	$-0.439423\pi$
$773$	10.5185	0.378324	0.189162	+	0.981946i	$0.439423\pi$
$774$	0	0
$775$	0	0
$776$	4.32145	0.155131
$777$	0	0
$778$	−49.8885	−1.78859
$779$	−39.4404	−1.41310
$780$	0	0
$781$	−1.88854	−0.0675774
$782$	−0.667701	−0.0238769
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.3050	0.616855	0.308427	−	0.951248i	$-0.400197\pi$
$787$	17.3050	0.616855	0.308427	+	0.951248i	$0.400197\pi$
$788$	70.5531	2.51335
$789$	0	0
$790$	0	0
$791$	−20.1328	−0.715841
$792$	0	0
$793$	41.1246	1.46038
$794$	3.36861	0.119547
$795$	0	0
$796$	1.70820	0.0605457
$797$	19.4350	0.688424	0.344212	−	0.938892i	$-0.388146\pi$
$797$	19.4350	0.688424	0.344212	+	0.938892i	$0.388146\pi$
$798$	0	0
$799$	1.59675	0.0564889
$800$	0	0
$801$	0	0
$802$	−44.5410	−1.57280
$803$	−3.03476	−0.107094
$804$	0	0
$805$	0	0
$806$	65.0237	2.29036
$807$	0	0
$808$	−18.4721	−0.649847
$809$	24.1692	0.849742	0.424871	−	0.905254i	$-0.360319\pi$
$809$	24.1692	0.849742	0.424871	+	0.905254i	$0.360319\pi$
$810$	0	0
$811$	34.2361	1.20219	0.601095	−	0.799177i	$-0.294731\pi$
$811$	34.2361	1.20219	0.601095	+	0.799177i	$0.294731\pi$
$812$	25.7109	0.902275
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	−24.4721	−0.856172
$818$	−33.2433	−1.16233
$819$	0	0
$820$	0	0
$821$	45.8924	1.60166	0.800828	−	0.598894i	$-0.204393\pi$
$821$	45.8924	1.60166	0.800828	+	0.598894i	$0.204393\pi$
$822$	0	0
$823$	−23.3607	−0.814302	−0.407151	−	0.913361i	$-0.633478\pi$
$823$	−23.3607	−0.814302	−0.407151	+	0.913361i	$0.633478\pi$
$824$	21.9597	0.765003
$825$	0	0
$826$	−10.9443	−0.380800
$827$	30.7788	1.07029	0.535143	−	0.844762i	$-0.320258\pi$
$827$	30.7788	1.07029	0.535143	+	0.844762i	$0.320258\pi$
$828$	0	0
$829$	32.7771	1.13840	0.569198	−	0.822201i	$-0.307254\pi$
$829$	32.7771	1.13840	0.569198	+	0.822201i	$0.307254\pi$
$830$	0	0
$831$	0	0
$832$	−54.8328	−1.90099
$833$	0.333851	0.0115672
$834$	0	0
$835$	0	0
$836$	−12.6491	−0.437479
$837$	0	0
$838$	−16.6525	−0.575250
$839$	12.5216	0.432293	0.216147	−	0.976361i	$-0.430651\pi$
$839$	12.5216	0.432293	0.216147	+	0.976361i	$0.430651\pi$
$840$	0	0
$841$	34.1246	1.17671
$842$	−10.7735	−0.371280
$843$	0	0
$844$	−36.5410	−1.25779
$845$	0	0
$846$	0	0
$847$	−10.2361	−0.351715
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.62210	−0.0898843
$852$	0	0
$853$	39.2492	1.34387	0.671934	−	0.740611i	$-0.265464\pi$
$853$	39.2492	1.34387	0.671934	+	0.740611i	$0.265464\pi$
$854$	22.2148	0.760173
$855$	0	0
$856$	37.3050	1.27506
$857$	39.9805	1.36571	0.682855	−	0.730554i	$-0.260738\pi$
$857$	39.9805	1.36571	0.682855	+	0.730554i	$0.260738\pi$
$858$	0	0
$859$	−48.5967	−1.65810	−0.829050	−	0.559175i	$-0.811118\pi$
$859$	−48.5967	−1.65810	−0.829050	+	0.559175i	$0.811118\pi$
$860$	0	0
$861$	0	0
$862$	−54.2492	−1.84774
$863$	−3.90879	−0.133057	−0.0665284	−	0.997785i	$-0.521192\pi$
$863$	−3.90879	−0.133057	−0.0665284	+	0.997785i	$0.521192\pi$
$864$	0	0
$865$	0	0
$866$	−22.0872	−0.750555
$867$	0	0
$868$	21.7082	0.736824
$869$	−1.28669	−0.0436481
$870$	0	0
$871$	7.47214	0.253184
$872$	−24.6305	−0.834095
$873$	0	0
$874$	8.94427	0.302545
$875$	0	0
$876$	0	0
$877$	13.7771	0.465219	0.232610	−	0.972570i	$-0.425274\pi$
$877$	13.7771	0.465219	0.232610	+	0.972570i	$0.425274\pi$
$878$	20.5942	0.695021
$879$	0	0
$880$	0	0
$881$	−35.8955	−1.20935	−0.604675	−	0.796472i	$-0.706697\pi$
$881$	−35.8955	−1.20935	−0.604675	+	0.796472i	$0.706697\pi$
$882$	0	0
$883$	52.4296	1.76440	0.882198	−	0.470879i	$-0.156063\pi$
$883$	52.4296	1.76440	0.882198	+	0.470879i	$0.156063\pi$
$884$	4.57649	0.153924
$885$	0	0
$886$	−44.8328	−1.50619
$887$	−44.7634	−1.50301	−0.751503	−	0.659729i	$-0.770671\pi$
$887$	−44.7634	−1.50301	−0.751503	+	0.659729i	$0.770671\pi$
$888$	0	0
$889$	5.52786	0.185399
$890$	0	0
$891$	0	0
$892$	55.5967	1.86152
$893$	−21.3894	−0.715770
$894$	0	0
$895$	0	0
$896$	−18.3060	−0.611559
$897$	0	0
$898$	−85.6656	−2.85870
$899$	53.2974	1.77757
$900$	0	0
$901$	−2.18034	−0.0726377
$902$	17.6383	0.587290
$903$	0	0
$904$	−56.9443	−1.89394
$905$	0	0
$906$	0	0
$907$	26.7082	0.886831	0.443416	−	0.896316i	$-0.353767\pi$
$907$	26.7082	0.886831	0.443416	+	0.896316i	$0.353767\pi$
$908$	−1.74806	−0.0580115
$909$	0	0
$910$	0	0
$911$	12.3153	0.408023	0.204011	−	0.978969i	$-0.434602\pi$
$911$	12.3153	0.408023	0.204011	+	0.978969i	$0.434602\pi$
$912$	0	0
$913$	−1.34752	−0.0445965
$914$	−69.4727	−2.29795
$915$	0	0
$916$	70.0689	2.31514
$917$	16.6854	0.551001
$918$	0	0
$919$	34.5279	1.13897	0.569485	−	0.822002i	$-0.307143\pi$
$919$	34.5279	1.13897	0.569485	+	0.822002i	$0.307143\pi$
$920$	0	0
$921$	0	0
$922$	−87.7771	−2.89078
$923$	−9.15298	−0.301274
$924$	0	0
$925$	0	0
$926$	50.2139	1.65013
$927$	0	0
$928$	−44.9443	−1.47537
$929$	14.9675	0.491066	0.245533	−	0.969388i	$-0.421037\pi$
$929$	14.9675	0.491066	0.245533	+	0.969388i	$0.421037\pi$
$930$	0	0
$931$	−4.47214	−0.146568
$932$	93.0228	3.04706
$933$	0	0
$934$	−40.8328	−1.33609
$935$	0	0
$936$	0	0
$937$	−18.7639	−0.612991	−0.306495	−	0.951872i	$-0.599156\pi$
$937$	−18.7639	−0.612991	−0.306495	+	0.951872i	$0.599156\pi$
$938$	4.03631	0.131790
$939$	0	0
$940$	0	0
$941$	3.03476	0.0989303	0.0494651	−	0.998776i	$-0.484248\pi$
$941$	3.03476	0.0989303	0.0494651	+	0.998776i	$0.484248\pi$
$942$	0	0
$943$	−7.70820	−0.251014
$944$	0	0
$945$	0	0
$946$	10.9443	0.355829
$947$	28.9219	0.939834	0.469917	−	0.882710i	$-0.344284\pi$
$947$	28.9219	0.939834	0.469917	+	0.882710i	$0.344284\pi$
$948$	0	0
$949$	−14.7082	−0.477449
$950$	0	0
$951$	0	0
$952$	0.944272	0.0306040
$953$	16.2241	0.525549	0.262774	−	0.964857i	$-0.415363\pi$
$953$	16.2241	0.525549	0.262774	+	0.964857i	$0.415363\pi$
$954$	0	0
$955$	0	0
$956$	40.7758	1.31878
$957$	0	0
$958$	−42.0689	−1.35918
$959$	−4.91034	−0.158563
$960$	0	0
$961$	14.0000	0.451613
$962$	29.0795	0.937560
$963$	0	0
$964$	56.5410	1.82106
$965$	0	0
$966$	0	0
$967$	−7.06888	−0.227320	−0.113660	−	0.993520i	$-0.536257\pi$
$967$	−7.06888	−0.227320	−0.113660	+	0.993520i	$0.536257\pi$
$968$	−28.9520	−0.930552
$969$	0	0
$970$	0	0
$971$	−43.4280	−1.39367	−0.696835	−	0.717232i	$-0.745409\pi$
$971$	−43.4280	−1.39367	−0.696835	+	0.717232i	$0.745409\pi$
$972$	0	0
$973$	12.2361	0.392270
$974$	16.4304	0.526463
$975$	0	0
$976$	0	0
$977$	−59.4944	−1.90339	−0.951697	−	0.307039i	$-0.900662\pi$
$977$	−59.4944	−1.90339	−0.951697	+	0.307039i	$0.900662\pi$
$978$	0	0
$979$	−7.23607	−0.231266
$980$	0	0
$981$	0	0
$982$	11.4164	0.364312
$983$	24.2967	0.774943	0.387472	−	0.921882i	$-0.373348\pi$
$983$	24.2967	0.774943	0.387472	+	0.921882i	$0.373348\pi$
$984$	0	0
$985$	0	0
$986$	6.06952	0.193293
$987$	0	0
$988$	−61.3050	−1.95037
$989$	−4.78282	−0.152085
$990$	0	0
$991$	−20.3607	−0.646778	−0.323389	−	0.946266i	$-0.604822\pi$
$991$	−20.3607	−0.646778	−0.323389	+	0.946266i	$0.604822\pi$
$992$	−37.9473	−1.20483
$993$	0	0
$994$	−4.94427	−0.156823
$995$	0	0
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−80.9139	−2.56129
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4725.2.a.bw.1.4	yes	4
3.2	odd	2	inner	4725.2.a.bw.1.1	yes	4
5.4	even	2		4725.2.a.bt.1.1	✓	4
15.14	odd	2		4725.2.a.bt.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4725.2.a.bt.1.1	✓	4	5.4	even	2
4725.2.a.bt.1.4	yes	4	15.14	odd	2
4725.2.a.bw.1.1	yes	4	3.2	odd	2	inner
4725.2.a.bw.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.28825 q^{2} +3.23607 q^{4} +1.00000 q^{7} +2.82843 q^{8} +0.874032 q^{11} +4.23607 q^{13} +2.28825 q^{14} +0.333851 q^{17} -4.47214 q^{19} +2.00000 q^{22} -0.874032 q^{23} +9.69316 q^{26} +3.23607 q^{28} +7.94510 q^{29} +6.70820 q^{31} -5.65685 q^{32} +0.763932 q^{34} +3.00000 q^{37} -10.2333 q^{38} +8.81913 q^{41} +5.47214 q^{43} +2.82843 q^{44} -2.00000 q^{46} +4.78282 q^{47} +1.00000 q^{49} +13.7082 q^{52} -6.53089 q^{53} +2.82843 q^{56} +18.1803 q^{58} -4.78282 q^{59} +9.70820 q^{61} +15.3500 q^{62} -12.9443 q^{64} +1.76393 q^{67} +1.08036 q^{68} -2.16073 q^{71} -3.47214 q^{73} +6.86474 q^{74} -14.4721 q^{76} +0.874032 q^{77} -1.47214 q^{79} +20.1803 q^{82} -1.54173 q^{83} +12.5216 q^{86} +2.47214 q^{88} -8.27895 q^{89} +4.23607 q^{91} -2.82843 q^{92} +10.9443 q^{94} +1.52786 q^{97} +2.28825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 4 q^{7} + 8 q^{13} + 8 q^{22} + 4 q^{28} + 12 q^{34} + 12 q^{37} + 4 q^{43} - 8 q^{46} + 4 q^{49} + 28 q^{52} + 28 q^{58} + 12 q^{61} - 16 q^{64} + 16 q^{67} + 4 q^{73} - 40 q^{76} + 12 q^{79}+ \cdots + 24 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 4 * q^7 + 8 * q^13 + 8 * q^22 + 4 * q^28 + 12 * q^34 + 12 * q^37 + 4 * q^43 - 8 * q^46 + 4 * q^49 + 28 * q^52 + 28 * q^58 + 12 * q^61 - 16 * q^64 + 16 * q^67 + 4 * q^73 - 40 * q^76 + 12 * q^79 + 36 * q^82 - 8 * q^88 + 8 * q^91 + 8 * q^94 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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