LMFDB - Embedded newform 4725.2.a.bw.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)

$f(q)$	$=$	$q-0.874032 q^{2} -1.23607 q^{4} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})$	$q-0.874032 q^{2} -1.23607 q^{4} +1.00000 q^{7} +2.82843 q^{8} -2.28825 q^{11} -0.236068 q^{13} -0.874032 q^{14} -5.99070 q^{17} +4.47214 q^{19} +2.00000 q^{22} +2.28825 q^{23} +0.206331 q^{26} -1.23607 q^{28} +4.78282 q^{29} -6.70820 q^{31} -5.65685 q^{32} +5.23607 q^{34} +3.00000 q^{37} -3.90879 q^{38} +2.49458 q^{41} -3.47214 q^{43} +2.82843 q^{44} -2.00000 q^{46} +7.94510 q^{47} +1.00000 q^{49} +0.291796 q^{52} -3.36861 q^{53} +2.82843 q^{56} -4.18034 q^{58} -7.94510 q^{59} -3.70820 q^{61} +5.86319 q^{62} +4.94427 q^{64} +6.23607 q^{67} +7.40492 q^{68} -14.8098 q^{71} +5.47214 q^{73} -2.62210 q^{74} -5.52786 q^{76} -2.28825 q^{77} +7.47214 q^{79} -2.18034 q^{82} +14.2697 q^{83} +3.03476 q^{86} -6.47214 q^{88} +1.20788 q^{89} -0.236068 q^{91} -2.82843 q^{92} -6.94427 q^{94} +10.4721 q^{97} -0.874032 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^4 + 4 * q^7	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.874032	−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
$11$	−2.28825	−0.689932	−0.344966	−	0.938615i	$-0.612110\pi$
$11$	−2.28825	−0.689932	−0.344966	+	0.938615i	$0.612110\pi$
$12$	0	0
$13$	−0.236068	−0.0654735	−0.0327367	−	0.999464i	$-0.510422\pi$
$13$	−0.236068	−0.0654735	−0.0327367	+	0.999464i	$0.510422\pi$
$14$	−0.874032	−0.233595
$15$	0	0
$16$	0	0
$17$	−5.99070	−1.45296	−0.726480	−	0.687188i	$-0.758845\pi$
$17$	−5.99070	−1.45296	−0.726480	+	0.687188i	$0.758845\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	2.28825	0.477132	0.238566	−	0.971126i	$-0.423323\pi$
$23$	2.28825	0.477132	0.238566	+	0.971126i	$0.423323\pi$
$24$	0	0
$25$	0	0
$26$	0.206331	0.0404648
$27$	0	0
$28$	−1.23607	−0.233595
$29$	4.78282	0.888148	0.444074	−	0.895990i	$-0.353533\pi$
$29$	4.78282	0.888148	0.444074	+	0.895990i	$0.353533\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	5.23607	0.897978
$35$	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	$0.420687\pi$
$38$	−3.90879	−0.634089
$39$	0	0
$40$	0	0
$41$	2.49458	0.389587	0.194794	−	0.980844i	$-0.437596\pi$
$41$	2.49458	0.389587	0.194794	+	0.980844i	$0.437596\pi$
$42$	0	0
$43$	−3.47214	−0.529496	−0.264748	−	0.964318i	$-0.585289\pi$
$43$	−3.47214	−0.529496	−0.264748	+	0.964318i	$0.585289\pi$
$44$	2.82843	0.426401
$45$	0	0
$46$	−2.00000	−0.294884
$47$	7.94510	1.15891	0.579456	−	0.815004i	$-0.303265\pi$
$47$	7.94510	1.15891	0.579456	+	0.815004i	$0.303265\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0.291796	0.0404648
$53$	−3.36861	−0.462714	−0.231357	−	0.972869i	$-0.574317\pi$
$53$	−3.36861	−0.462714	−0.231357	+	0.972869i	$0.574317\pi$
$54$	0	0
$55$	0	0
$56$	2.82843	0.377964
$57$	0	0
$58$	−4.18034	−0.548906
$59$	−7.94510	−1.03436	−0.517182	−	0.855875i	$-0.673019\pi$
$59$	−7.94510	−1.03436	−0.517182	+	0.855875i	$0.673019\pi$
$60$	0	0
$61$	−3.70820	−0.474787	−0.237393	−	0.971414i	$-0.576293\pi$
$61$	−3.70820	−0.474787	−0.237393	+	0.971414i	$0.576293\pi$
$62$	5.86319	0.744625
$63$	0	0
$64$	4.94427	0.618034
$65$	0	0
$66$	0	0
$67$	6.23607	0.761857	0.380928	−	0.924605i	$-0.375604\pi$
$67$	6.23607	0.761857	0.380928	+	0.924605i	$0.375604\pi$
$68$	7.40492	0.897978
$69$	0	0
$70$	0	0
$71$	−14.8098	−1.75760	−0.878802	−	0.477186i	$-0.841657\pi$
$71$	−14.8098	−1.75760	−0.878802	+	0.477186i	$0.841657\pi$
$72$	0	0
$73$	5.47214	0.640465	0.320233	−	0.947339i	$-0.396239\pi$
$73$	5.47214	0.640465	0.320233	+	0.947339i	$0.396239\pi$
$74$	−2.62210	−0.304812
$75$	0	0
$76$	−5.52786	−0.634089
$77$	−2.28825	−0.260770
$78$	0	0
$79$	7.47214	0.840681	0.420340	−	0.907366i	$-0.361911\pi$
$79$	7.47214	0.840681	0.420340	+	0.907366i	$0.361911\pi$
$80$	0	0
$81$	0	0
$82$	−2.18034	−0.240778
$83$	14.2697	1.56630	0.783149	−	0.621834i	$-0.213612\pi$
$83$	14.2697	1.56630	0.783149	+	0.621834i	$0.213612\pi$
$84$	0	0
$85$	0	0
$86$	3.03476	0.327246
$87$	0	0
$88$	−6.47214	−0.689932
$89$	1.20788	0.128035	0.0640176	−	0.997949i	$-0.479609\pi$
$89$	1.20788	0.128035	0.0640176	+	0.997949i	$0.479609\pi$
$90$	0	0
$91$	−0.236068	−0.0247466
$92$	−2.82843	−0.294884
$93$	0	0
$94$	−6.94427	−0.716247
$95$	0	0
$96$	0	0
$97$	10.4721	1.06328	0.531642	−	0.846969i	$-0.321575\pi$
$97$	10.4721	1.06328	0.531642	+	0.846969i	$0.321575\pi$
$98$	−0.874032	−0.0882906
$99$	0	0
$100$	0	0
$101$	−3.36861	−0.335189	−0.167595	−	0.985856i	$-0.553600\pi$
$101$	−3.36861	−0.335189	−0.167595	+	0.985856i	$0.553600\pi$
$102$	0	0
$103$	12.2361	1.20566	0.602828	−	0.797871i	$-0.294041\pi$
$103$	12.2361	1.20566	0.602828	+	0.797871i	$0.294041\pi$
$104$	−0.667701	−0.0654735
$105$	0	0
$106$	2.94427	0.285973
$107$	−8.94665	−0.864905	−0.432453	−	0.901657i	$-0.642352\pi$
$107$	−8.94665	−0.864905	−0.432453	+	0.901657i	$0.642352\pi$
$108$	0	0
$109$	4.70820	0.450964	0.225482	−	0.974247i	$-0.427604\pi$
$109$	4.70820	0.450964	0.225482	+	0.974247i	$0.427604\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.8083	−1.29897	−0.649487	−	0.760373i	$-0.725016\pi$
$113$	−13.8083	−1.29897	−0.649487	+	0.760373i	$0.725016\pi$
$114$	0	0
$115$	0	0
$116$	−5.91189	−0.548906
$117$	0	0
$118$	6.94427	0.639272
$119$	−5.99070	−0.549167
$120$	0	0
$121$	−5.76393	−0.523994
$122$	3.24109	0.293434
$123$	0	0
$124$	8.29180	0.744625
$125$	0	0
$126$	0	0
$127$	14.4721	1.28419	0.642097	−	0.766623i	$-0.278065\pi$
$127$	14.4721	1.28419	0.642097	+	0.766623i	$0.278065\pi$
$128$	6.99226	0.618034
$129$	0	0
$130$	0	0
$131$	−18.0996	−1.58137	−0.790686	−	0.612222i	$-0.790276\pi$
$131$	−18.0996	−1.58137	−0.790686	+	0.612222i	$0.790276\pi$
$132$	0	0
$133$	4.47214	0.387783
$134$	−5.45052	−0.470853
$135$	0	0
$136$	−16.9443	−1.45296
$137$	7.73877	0.661168	0.330584	−	0.943777i	$-0.392754\pi$
$137$	7.73877	0.661168	0.330584	+	0.943777i	$0.392754\pi$
$138$	0	0
$139$	7.76393	0.658528	0.329264	−	0.944238i	$-0.393199\pi$
$139$	7.76393	0.658528	0.329264	+	0.944238i	$0.393199\pi$
$140$	0	0
$141$	0	0
$142$	12.9443	1.08626
$143$	0.540182	0.0451722
$144$	0	0
$145$	0	0
$146$	−4.78282	−0.395829
$147$	0	0
$148$	−3.70820	−0.304812
$149$	−0.952843	−0.0780600	−0.0390300	−	0.999238i	$-0.512427\pi$
$149$	−0.952843	−0.0780600	−0.0390300	+	0.999238i	$0.512427\pi$
$150$	0	0
$151$	−4.47214	−0.363937	−0.181969	−	0.983304i	$-0.558247\pi$
$151$	−4.47214	−0.363937	−0.181969	+	0.983304i	$0.558247\pi$
$152$	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	$-0.471560\pi$
$157$	2.23607	0.178458	0.0892288	+	0.996011i	$0.471560\pi$
$158$	−6.53089	−0.519569
$159$	0	0
$160$	0	0
$161$	2.28825	0.180339
$162$	0	0
$163$	14.2361	1.11505	0.557527	−	0.830159i	$-0.311750\pi$
$163$	14.2361	1.11505	0.557527	+	0.830159i	$0.311750\pi$
$164$	−3.08347	−0.240778
$165$	0	0
$166$	−12.4721	−0.968025
$167$	−22.0084	−1.70306	−0.851531	−	0.524303i	$-0.824326\pi$
$167$	−22.0084	−1.70306	−0.851531	+	0.524303i	$0.824326\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	0	0
$171$	0	0
$172$	4.29180	0.327246
$173$	9.82068	0.746653	0.373326	−	0.927700i	$-0.378217\pi$
$173$	9.82068	0.746653	0.373326	+	0.927700i	$0.378217\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−1.05573	−0.0791302
$179$	−0.0788114	−0.00589064	−0.00294532	−	0.999996i	$-0.500938\pi$
$179$	−0.0788114	−0.00589064	−0.00294532	+	0.999996i	$0.500938\pi$
$180$	0	0
$181$	19.3607	1.43907	0.719534	−	0.694457i	$-0.244355\pi$
$181$	19.3607	1.43907	0.719534	+	0.694457i	$0.244355\pi$
$182$	0.206331	0.0152943
$183$	0	0
$184$	6.47214	0.477132
$185$	0	0
$186$	0	0
$187$	13.7082	1.00244
$188$	−9.82068	−0.716247
$189$	0	0
$190$	0	0
$191$	1.95440	0.141415	0.0707075	−	0.997497i	$-0.477474\pi$
$191$	1.95440	0.141415	0.0707075	+	0.997497i	$0.477474\pi$
$192$	0	0
$193$	21.4164	1.54159	0.770793	−	0.637085i	$-0.219860\pi$
$193$	21.4164	1.54159	0.770793	+	0.637085i	$0.219860\pi$
$194$	−9.15298	−0.657146
$195$	0	0
$196$	−1.23607	−0.0882906
$197$	−16.1452	−1.15030	−0.575150	−	0.818048i	$-0.695056\pi$
$197$	−16.1452	−1.15030	−0.575150	+	0.818048i	$0.695056\pi$
$198$	0	0
$199$	9.47214	0.671462	0.335731	−	0.941958i	$-0.391017\pi$
$199$	9.47214	0.671462	0.335731	+	0.941958i	$0.391017\pi$
$200$	0	0
$201$	0	0
$202$	2.94427	0.207158
$203$	4.78282	0.335688
$204$	0	0
$205$	0	0
$206$	−10.6947	−0.745136
$207$	0	0
$208$	0	0
$209$	−10.2333	−0.707855
$210$	0	0
$211$	−24.7082	−1.70098	−0.850491	−	0.525989i	$-0.823695\pi$
$211$	−24.7082	−1.70098	−0.850491	+	0.525989i	$0.823695\pi$
$212$	4.16383	0.285973
$213$	0	0
$214$	7.81966	0.534541
$215$	0	0
$216$	0	0
$217$	−6.70820	−0.455383
$218$	−4.11512	−0.278711
$219$	0	0
$220$	0	0
$221$	1.41421	0.0951303
$222$	0	0
$223$	−5.18034	−0.346901	−0.173451	−	0.984843i	$-0.555492\pi$
$223$	−5.18034	−0.346901	−0.173451	+	0.984843i	$0.555492\pi$
$224$	−5.65685	−0.377964
$225$	0	0
$226$	12.0689	0.802810
$227$	−3.70246	−0.245741	−0.122870	−	0.992423i	$-0.539210\pi$
$227$	−3.70246	−0.245741	−0.122870	+	0.992423i	$0.539210\pi$
$228$	0	0
$229$	−9.65248	−0.637854	−0.318927	−	0.947779i	$-0.603322\pi$
$229$	−9.65248	−0.637854	−0.318927	+	0.947779i	$0.603322\pi$
$230$	0	0
$231$	0	0
$232$	13.5279	0.888148
$233$	6.60970	0.433016	0.216508	−	0.976281i	$-0.430533\pi$
$233$	6.60970	0.433016	0.216508	+	0.976281i	$0.430533\pi$
$234$	0	0
$235$	0	0
$236$	9.82068	0.639272
$237$	0	0
$238$	5.23607	0.339404
$239$	28.4118	1.83781	0.918903	−	0.394484i	$-0.129077\pi$
$239$	28.4118	1.83781	0.918903	+	0.394484i	$0.129077\pi$
$240$	0	0
$241$	8.52786	0.549328	0.274664	−	0.961540i	$-0.411433\pi$
$241$	8.52786	0.549328	0.274664	+	0.961540i	$0.411433\pi$
$242$	5.03786	0.323846
$243$	0	0
$244$	4.58359	0.293434
$245$	0	0
$246$	0	0
$247$	−1.05573	−0.0671744
$248$	−18.9737	−1.20483
$249$	0	0
$250$	0	0
$251$	19.9265	1.25775	0.628875	−	0.777506i	$-0.283516\pi$
$251$	19.9265	1.25775	0.628875	+	0.777506i	$0.283516\pi$
$252$	0	0
$253$	−5.23607	−0.329189
$254$	−12.6491	−0.793676
$255$	0	0
$256$	−16.0000	−1.00000
$257$	3.03476	0.189303	0.0946515	−	0.995510i	$-0.469826\pi$
$257$	3.03476	0.189303	0.0946515	+	0.995510i	$0.469826\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	15.8197	0.977342
$263$	23.3438	1.43944	0.719721	−	0.694263i	$-0.244270\pi$
$263$	23.3438	1.43944	0.719721	+	0.694263i	$0.244270\pi$
$264$	0	0
$265$	0	0
$266$	−3.90879	−0.239663
$267$	0	0
$268$	−7.70820	−0.470853
$269$	22.2936	1.35926	0.679631	−	0.733554i	$-0.262140\pi$
$269$	22.2936	1.35926	0.679631	+	0.733554i	$0.262140\pi$
$270$	0	0
$271$	−8.41641	−0.511260	−0.255630	−	0.966775i	$-0.582283\pi$
$271$	−8.41641	−0.511260	−0.255630	+	0.966775i	$0.582283\pi$
$272$	0	0
$273$	0	0
$274$	−6.76393	−0.408624
$275$	0	0
$276$	0	0
$277$	13.7639	0.826995	0.413497	−	0.910505i	$-0.364307\pi$
$277$	13.7639	0.826995	0.413497	+	0.910505i	$0.364307\pi$
$278$	−6.78593	−0.406993
$279$	0	0
$280$	0	0
$281$	24.0903	1.43711	0.718555	−	0.695471i	$-0.244804\pi$
$281$	24.0903	1.43711	0.718555	+	0.695471i	$0.244804\pi$
$282$	0	0
$283$	14.4164	0.856966	0.428483	−	0.903550i	$-0.359048\pi$
$283$	14.4164	0.856966	0.428483	+	0.903550i	$0.359048\pi$
$284$	18.3060	1.08626
$285$	0	0
$286$	−0.472136	−0.0279180
$287$	2.49458	0.147250
$288$	0	0
$289$	18.8885	1.11109
$290$	0	0
$291$	0	0
$292$	−6.76393	−0.395829
$293$	12.1877	0.712015	0.356008	−	0.934483i	$-0.384138\pi$
$293$	12.1877	0.712015	0.356008	+	0.934483i	$0.384138\pi$
$294$	0	0
$295$	0	0
$296$	8.48528	0.493197
$297$	0	0
$298$	0.832816	0.0482437
$299$	−0.540182	−0.0312395
$300$	0	0
$301$	−3.47214	−0.200131
$302$	3.90879	0.224926
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	10.4164	0.594496	0.297248	−	0.954800i	$-0.403931\pi$
$307$	10.4164	0.594496	0.297248	+	0.954800i	$0.403931\pi$
$308$	2.82843	0.161165
$309$	0	0
$310$	0	0
$311$	16.4791	0.934443	0.467222	−	0.884140i	$-0.345255\pi$
$311$	16.4791	0.934443	0.467222	+	0.884140i	$0.345255\pi$
$312$	0	0
$313$	14.8885	0.841550	0.420775	−	0.907165i	$-0.361758\pi$
$313$	14.8885	0.841550	0.420775	+	0.907165i	$0.361758\pi$
$314$	−1.95440	−0.110293
$315$	0	0
$316$	−9.23607	−0.519569
$317$	−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
$319$	−10.9443	−0.612762
$320$	0	0
$321$	0	0
$322$	−2.00000	−0.111456
$323$	−26.7912	−1.49070
$324$	0	0
$325$	0	0
$326$	−12.4428	−0.689142
$327$	0	0
$328$	7.05573	0.389587
$329$	7.94510	0.438028
$330$	0	0
$331$	33.5410	1.84358	0.921791	−	0.387688i	$-0.126726\pi$
$331$	33.5410	1.84358	0.921791	+	0.387688i	$0.126726\pi$
$332$	−17.6383	−0.968025
$333$	0	0
$334$	19.2361	1.05255
$335$	0	0
$336$	0	0
$337$	23.1803	1.26271	0.631357	−	0.775492i	$-0.282498\pi$
$337$	23.1803	1.26271	0.631357	+	0.775492i	$0.282498\pi$
$338$	11.3137	0.615385
$339$	0	0
$340$	0	0
$341$	15.3500	0.831250
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.82068	−0.529496
$345$	0	0
$346$	−8.58359	−0.461457
$347$	33.9110	1.82044	0.910220	−	0.414126i	$-0.135913\pi$
$347$	33.9110	1.82044	0.910220	+	0.414126i	$0.135913\pi$
$348$	0	0
$349$	23.0689	1.23485	0.617425	−	0.786630i	$-0.288176\pi$
$349$	23.0689	1.23485	0.617425	+	0.786630i	$0.288176\pi$
$350$	0	0
$351$	0	0
$352$	12.9443	0.689932
$353$	−0.412662	−0.0219638	−0.0109819	−	0.999940i	$-0.503496\pi$
$353$	−0.412662	−0.0219638	−0.0109819	+	0.999940i	$0.503496\pi$
$354$	0	0
$355$	0	0
$356$	−1.49302	−0.0791302
$357$	0	0
$358$	0.0688837	0.00364062
$359$	27.9991	1.47774	0.738869	−	0.673849i	$-0.235360\pi$
$359$	27.9991	1.47774	0.738869	+	0.673849i	$0.235360\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−16.9219	−0.889393
$363$	0	0
$364$	0.291796	0.0152943
$365$	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
$371$	−3.36861	−0.174889
$372$	0	0
$373$	2.58359	0.133773	0.0668867	−	0.997761i	$-0.478693\pi$
$373$	2.58359	0.133773	0.0668867	+	0.997761i	$0.478693\pi$
$374$	−11.9814	−0.619544
$375$	0	0
$376$	22.4721	1.15891
$377$	−1.12907	−0.0581501
$378$	0	0
$379$	−27.6525	−1.42041	−0.710206	−	0.703994i	$-0.751398\pi$
$379$	−27.6525	−1.42041	−0.710206	+	0.703994i	$0.751398\pi$
$380$	0	0
$381$	0	0
$382$	−1.70820	−0.0873993
$383$	14.0633	0.718602	0.359301	−	0.933222i	$-0.383015\pi$
$383$	14.0633	0.718602	0.359301	+	0.933222i	$0.383015\pi$
$384$	0	0
$385$	0	0
$386$	−18.7186	−0.952753
$387$	0	0
$388$	−12.9443	−0.657146
$389$	16.1452	0.818596	0.409298	−	0.912401i	$-0.365774\pi$
$389$	16.1452	0.818596	0.409298	+	0.912401i	$0.365774\pi$
$390$	0	0
$391$	−13.7082	−0.693254
$392$	2.82843	0.142857
$393$	0	0
$394$	14.1115	0.710925
$395$	0	0
$396$	0	0
$397$	−7.47214	−0.375016	−0.187508	−	0.982263i	$-0.560041\pi$
$397$	−7.47214	−0.375016	−0.187508	+	0.982263i	$0.560041\pi$
$398$	−8.27895	−0.414986
$399$	0	0
$400$	0	0
$401$	−25.7897	−1.28788	−0.643938	−	0.765078i	$-0.722701\pi$
$401$	−25.7897	−1.28788	−0.643938	+	0.765078i	$0.722701\pi$
$402$	0	0
$403$	1.58359	0.0788843
$404$	4.16383	0.207158
$405$	0	0
$406$	−4.18034	−0.207467
$407$	−6.86474	−0.340272
$408$	0	0
$409$	−23.4721	−1.16062	−0.580311	−	0.814395i	$-0.697069\pi$
$409$	−23.4721	−1.16062	−0.580311	+	0.814395i	$0.697069\pi$
$410$	0	0
$411$	0	0
$412$	−15.1246	−0.745136
$413$	−7.94510	−0.390953
$414$	0	0
$415$	0	0
$416$	1.33540	0.0654735
$417$	0	0
$418$	8.94427	0.437479
$419$	−16.7642	−0.818986	−0.409493	−	0.912313i	$-0.634294\pi$
$419$	−16.7642	−0.818986	−0.409493	+	0.912313i	$0.634294\pi$
$420$	0	0
$421$	8.70820	0.424412	0.212206	−	0.977225i	$-0.431935\pi$
$421$	8.70820	0.424412	0.212206	+	0.977225i	$0.431935\pi$
$422$	21.5958	1.05127
$423$	0	0
$424$	−9.52786	−0.462714
$425$	0	0
$426$	0	0
$427$	−3.70820	−0.179453
$428$	11.0587	0.534541
$429$	0	0
$430$	0	0
$431$	−30.0323	−1.44661	−0.723303	−	0.690530i	$-0.757377\pi$
$431$	−30.0323	−1.44661	−0.723303	+	0.690530i	$0.757377\pi$
$432$	0	0
$433$	21.6525	1.04055	0.520276	−	0.853998i	$-0.325829\pi$
$433$	21.6525	1.04055	0.520276	+	0.853998i	$0.325829\pi$
$434$	5.86319	0.281442
$435$	0	0
$436$	−5.81966	−0.278711
$437$	10.2333	0.489527
$438$	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
$442$	−1.23607	−0.0587938
$443$	−10.1058	−0.480142	−0.240071	−	0.970755i	$-0.577171\pi$
$443$	−10.1058	−0.480142	−0.240071	+	0.970755i	$0.577171\pi$
$444$	0	0
$445$	0	0
$446$	4.52778	0.214397
$447$	0	0
$448$	4.94427	0.233595
$449$	−24.7881	−1.16982	−0.584912	−	0.811096i	$-0.698871\pi$
$449$	−24.7881	−1.16982	−0.584912	+	0.811096i	$0.698871\pi$
$450$	0	0
$451$	−5.70820	−0.268789
$452$	17.0680	0.802810
$453$	0	0
$454$	3.23607	0.151876
$455$	0	0
$456$	0	0
$457$	14.3607	0.671764	0.335882	−	0.941904i	$-0.390966\pi$
$457$	14.3607	0.671764	0.335882	+	0.941904i	$0.390966\pi$
$458$	8.43657	0.394215
$459$	0	0
$460$	0	0
$461$	18.5610	0.864472	0.432236	−	0.901760i	$-0.357725\pi$
$461$	18.5610	0.864472	0.432236	+	0.901760i	$0.357725\pi$
$462$	0	0
$463$	4.05573	0.188486	0.0942428	−	0.995549i	$-0.469957\pi$
$463$	4.05573	0.188486	0.0942428	+	0.995549i	$0.469957\pi$
$464$	0	0
$465$	0	0
$466$	−5.77709	−0.267618
$467$	−14.6823	−0.679417	−0.339708	−	0.940531i	$-0.610328\pi$
$467$	−14.6823	−0.679417	−0.339708	+	0.940531i	$0.610328\pi$
$468$	0	0
$469$	6.23607	0.287955
$470$	0	0
$471$	0	0
$472$	−22.4721	−1.03436
$473$	7.94510	0.365316
$474$	0	0
$475$	0	0
$476$	7.40492	0.339404
$477$	0	0
$478$	−24.8328	−1.13583
$479$	−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
$481$	−0.708204	−0.0322913
$482$	−7.45363	−0.339503
$483$	0	0
$484$	7.12461	0.323846
$485$	0	0
$486$	0	0
$487$	−15.1803	−0.687887	−0.343943	−	0.938990i	$-0.611763\pi$
$487$	−15.1803	−0.687887	−0.343943	+	0.938990i	$0.611763\pi$
$488$	−10.4884	−0.474787
$489$	0	0
$490$	0	0
$491$	17.6383	0.796004	0.398002	−	0.917385i	$-0.369704\pi$
$491$	17.6383	0.796004	0.398002	+	0.917385i	$0.369704\pi$
$492$	0	0
$493$	−28.6525	−1.29044
$494$	0.922740	0.0415160
$495$	0	0
$496$	0	0
$497$	−14.8098	−0.664312
$498$	0	0
$499$	9.36068	0.419042	0.209521	−	0.977804i	$-0.432810\pi$
$499$	9.36068	0.419042	0.209521	+	0.977804i	$0.432810\pi$
$500$	0	0
$501$	0	0
$502$	−17.4164	−0.777332
$503$	−20.7217	−0.923936	−0.461968	−	0.886897i	$-0.652857\pi$
$503$	−20.7217	−0.923936	−0.461968	+	0.886897i	$0.652857\pi$
$504$	0	0
$505$	0	0
$506$	4.57649	0.203450
$507$	0	0
$508$	−17.8885	−0.793676
$509$	−5.19548	−0.230286	−0.115143	−	0.993349i	$-0.536733\pi$
$509$	−5.19548	−0.230286	−0.115143	+	0.993349i	$0.536733\pi$
$510$	0	0
$511$	5.47214	0.242073
$512$	0	0
$513$	0	0
$514$	−2.65248	−0.116996
$515$	0	0
$516$	0	0
$517$	−18.1803	−0.799570
$518$	−2.62210	−0.115208
$519$	0	0
$520$	0	0
$521$	22.6761	0.993459	0.496730	−	0.867905i	$-0.334534\pi$
$521$	22.6761	0.993459	0.496730	+	0.867905i	$0.334534\pi$
$522$	0	0
$523$	5.34752	0.233831	0.116915	−	0.993142i	$-0.462699\pi$
$523$	5.34752	0.233831	0.116915	+	0.993142i	$0.462699\pi$
$524$	22.3724	0.977342
$525$	0	0
$526$	−20.4033	−0.889624
$527$	40.1869	1.75057
$528$	0	0
$529$	−17.7639	−0.772345
$530$	0	0
$531$	0	0
$532$	−5.52786	−0.239663
$533$	−0.588890	−0.0255076
$534$	0	0
$535$	0	0
$536$	17.6383	0.761857
$537$	0	0
$538$	−19.4853	−0.840071
$539$	−2.28825	−0.0985617
$540$	0	0
$541$	3.34752	0.143921	0.0719607	−	0.997407i	$-0.477074\pi$
$541$	3.34752	0.143921	0.0719607	+	0.997407i	$0.477074\pi$
$542$	7.35621	0.315976
$543$	0	0
$544$	33.8885	1.45296
$545$	0	0
$546$	0	0
$547$	41.1803	1.76074	0.880372	−	0.474284i	$-0.157293\pi$
$547$	41.1803	1.76074	0.880372	+	0.474284i	$0.157293\pi$
$548$	−9.56564	−0.408624
$549$	0	0
$550$	0	0
$551$	21.3894	0.911220
$552$	0	0
$553$	7.47214	0.317748
$554$	−12.0301	−0.511111
$555$	0	0
$556$	−9.59675	−0.406993
$557$	33.1158	1.40316	0.701581	−	0.712590i	$-0.252478\pi$
$557$	33.1158	1.40316	0.701581	+	0.712590i	$0.252478\pi$
$558$	0	0
$559$	0.819660	0.0346679
$560$	0	0
$561$	0	0
$562$	−21.0557	−0.888182
$563$	−21.7534	−0.916796	−0.458398	−	0.888747i	$-0.651577\pi$
$563$	−21.7534	−0.916796	−0.458398	+	0.888747i	$0.651577\pi$
$564$	0	0
$565$	0	0
$566$	−12.6004	−0.529634
$567$	0	0
$568$	−41.8885	−1.75760
$569$	0.333851	0.0139957	0.00699787	−	0.999976i	$-0.497772\pi$
$569$	0.333851	0.0139957	0.00699787	+	0.999976i	$0.497772\pi$
$570$	0	0
$571$	−41.8885	−1.75298	−0.876491	−	0.481419i	$-0.840121\pi$
$571$	−41.8885	−1.75298	−0.876491	+	0.481419i	$0.840121\pi$
$572$	−0.667701	−0.0279180
$573$	0	0
$574$	−2.18034	−0.0910056
$575$	0	0
$576$	0	0
$577$	27.1246	1.12921	0.564606	−	0.825360i	$-0.309028\pi$
$577$	27.1246	1.12921	0.564606	+	0.825360i	$0.309028\pi$
$578$	−16.5092	−0.686692
$579$	0	0
$580$	0	0
$581$	14.2697	0.592005
$582$	0	0
$583$	7.70820	0.319241
$584$	15.4775	0.640465
$585$	0	0
$586$	−10.6525	−0.440050
$587$	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
$593$	−7.27740	−0.298847	−0.149423	−	0.988773i	$-0.547742\pi$
$593$	−7.27740	−0.298847	−0.149423	+	0.988773i	$0.547742\pi$
$594$	0	0
$595$	0	0
$596$	1.17778	0.0482437
$597$	0	0
$598$	0.472136	0.0193071
$599$	31.2889	1.27843	0.639215	−	0.769028i	$-0.279259\pi$
$599$	31.2889	1.27843	0.639215	+	0.769028i	$0.279259\pi$
$600$	0	0
$601$	24.5279	1.00051	0.500256	−	0.865877i	$-0.333239\pi$
$601$	24.5279	1.00051	0.500256	+	0.865877i	$0.333239\pi$
$602$	3.03476	0.123688
$603$	0	0
$604$	5.52786	0.224926
$605$	0	0
$606$	0	0
$607$	−41.5410	−1.68610	−0.843049	−	0.537837i	$-0.819242\pi$
$607$	−41.5410	−1.68610	−0.843049	+	0.537837i	$0.819242\pi$
$608$	−25.2982	−1.02598
$609$	0	0
$610$	0	0
$611$	−1.87558	−0.0758780
$612$	0	0
$613$	−2.41641	−0.0975978	−0.0487989	−	0.998809i	$-0.515539\pi$
$613$	−2.41641	−0.0975978	−0.0487989	+	0.998809i	$0.515539\pi$
$614$	−9.10427	−0.367419
$615$	0	0
$616$	−6.47214	−0.260770
$617$	−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
$619$	−16.2918	−0.654823	−0.327411	−	0.944882i	$-0.606176\pi$
$619$	−16.2918	−0.654823	−0.327411	+	0.944882i	$0.606176\pi$
$620$	0	0
$621$	0	0
$622$	−14.4033	−0.577518
$623$	1.20788	0.0483928
$624$	0	0
$625$	0	0
$626$	−13.0131	−0.520107
$627$	0	0
$628$	−2.76393	−0.110293
$629$	−17.9721	−0.716595
$630$	0	0
$631$	15.7639	0.627552	0.313776	−	0.949497i	$-0.398406\pi$
$631$	15.7639	0.627552	0.313776	+	0.949497i	$0.398406\pi$
$632$	21.1344	0.840681
$633$	0	0
$634$	2.65248	0.105343
$635$	0	0
$636$	0	0
$637$	−0.236068	−0.00935335
$638$	9.56564	0.378707
$639$	0	0
$640$	0	0
$641$	2.77972	0.109792	0.0548961	−	0.998492i	$-0.482517\pi$
$641$	2.77972	0.109792	0.0548961	+	0.998492i	$0.482517\pi$
$642$	0	0
$643$	40.9443	1.61468	0.807342	−	0.590083i	$-0.200905\pi$
$643$	40.9443	1.61468	0.807342	+	0.590083i	$0.200905\pi$
$644$	−2.82843	−0.111456
$645$	0	0
$646$	23.4164	0.921306
$647$	−33.0370	−1.29882	−0.649409	−	0.760439i	$-0.724984\pi$
$647$	−33.0370	−1.29882	−0.649409	+	0.760439i	$0.724984\pi$
$648$	0	0
$649$	18.1803	0.713641
$650$	0	0
$651$	0	0
$652$	−17.5967	−0.689142
$653$	−41.9349	−1.64104	−0.820520	−	0.571617i	$-0.806316\pi$
$653$	−41.9349	−1.64104	−0.820520	+	0.571617i	$0.806316\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−6.94427	−0.270716
$659$	−12.9343	−0.503847	−0.251923	−	0.967747i	$-0.581063\pi$
$659$	−12.9343	−0.503847	−0.251923	+	0.967747i	$0.581063\pi$
$660$	0	0
$661$	−9.06888	−0.352739	−0.176369	−	0.984324i	$-0.556435\pi$
$661$	−9.06888	−0.352739	−0.176369	+	0.984324i	$0.556435\pi$
$662$	−29.3159	−1.13940
$663$	0	0
$664$	40.3607	1.56630
$665$	0	0
$666$	0	0
$667$	10.9443	0.423764
$668$	27.2039	1.05255
$669$	0	0
$670$	0	0
$671$	8.48528	0.327571
$672$	0	0
$673$	−32.2361	−1.24261	−0.621305	−	0.783569i	$-0.713397\pi$
$673$	−32.2361	−1.24261	−0.621305	+	0.783569i	$0.713397\pi$
$674$	−20.2604	−0.780400
$675$	0	0
$676$	16.0000	0.615385
$677$	30.4149	1.16894	0.584470	−	0.811416i	$-0.301303\pi$
$677$	30.4149	1.16894	0.584470	+	0.811416i	$0.301303\pi$
$678$	0	0
$679$	10.4721	0.401884
$680$	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
$686$	−0.874032	−0.0333707
$687$	0	0
$688$	0	0
$689$	0.795221	0.0302955
$690$	0	0
$691$	24.1803	0.919863	0.459932	−	0.887954i	$-0.347874\pi$
$691$	24.1803	0.919863	0.459932	+	0.887954i	$0.347874\pi$
$692$	−12.1390	−0.461457
$693$	0	0
$694$	−29.6393	−1.12509
$695$	0	0
$696$	0	0
$697$	−14.9443	−0.566055
$698$	−20.1629	−0.763179
$699$	0	0
$700$	0	0
$701$	29.7472	1.12354	0.561768	−	0.827295i	$-0.310121\pi$
$701$	29.7472	1.12354	0.561768	+	0.827295i	$0.310121\pi$
$702$	0	0
$703$	13.4164	0.506009
$704$	−11.3137	−0.426401
$705$	0	0
$706$	0.360680	0.0135744
$707$	−3.36861	−0.126690
$708$	0	0
$709$	−28.4721	−1.06929	−0.534647	−	0.845076i	$-0.679555\pi$
$709$	−28.4721	−1.06929	−0.534647	+	0.845076i	$0.679555\pi$
$710$	0	0
$711$	0	0
$712$	3.41641	0.128035
$713$	−15.3500	−0.574863
$714$	0	0
$715$	0	0
$716$	0.0974163	0.00364062
$717$	0	0
$718$	−24.4721	−0.913292
$719$	43.7618	1.63204	0.816020	−	0.578024i	$-0.196176\pi$
$719$	43.7618	1.63204	0.816020	+	0.578024i	$0.196176\pi$
$720$	0	0
$721$	12.2361	0.455695
$722$	−0.874032	−0.0325281
$723$	0	0
$724$	−23.9311	−0.889393
$725$	0	0
$726$	0	0
$727$	−7.36068	−0.272993	−0.136496	−	0.990641i	$-0.543584\pi$
$727$	−7.36068	−0.272993	−0.136496	+	0.990641i	$0.543584\pi$
$728$	−0.667701	−0.0247466
$729$	0	0
$730$	0	0
$731$	20.8005	0.769336
$732$	0	0
$733$	35.4164	1.30813	0.654067	−	0.756436i	$-0.273061\pi$
$733$	35.4164	1.30813	0.654067	+	0.756436i	$0.273061\pi$
$734$	−21.8508	−0.806528
$735$	0	0
$736$	−12.9443	−0.477132
$737$	−14.2697	−0.525630
$738$	0	0
$739$	−37.7771	−1.38965	−0.694826	−	0.719178i	$-0.744519\pi$
$739$	−37.7771	−1.38965	−0.694826	+	0.719178i	$0.744519\pi$
$740$	0	0
$741$	0	0
$742$	2.94427	0.108088
$743$	42.0137	1.54133	0.770667	−	0.637238i	$-0.219923\pi$
$743$	42.0137	1.54133	0.770667	+	0.637238i	$0.219923\pi$
$744$	0	0
$745$	0	0
$746$	−2.25814	−0.0826765
$747$	0	0
$748$	−16.9443	−0.619544
$749$	−8.94665	−0.326904
$750$	0	0
$751$	11.0689	0.403909	0.201955	−	0.979395i	$-0.435271\pi$
$751$	11.0689	0.403909	0.201955	+	0.979395i	$0.435271\pi$
$752$	0	0
$753$	0	0
$754$	0.986844	0.0359388
$755$	0	0
$756$	0	0
$757$	−35.1246	−1.27663	−0.638313	−	0.769777i	$-0.720367\pi$
$757$	−35.1246	−1.27663	−0.638313	+	0.769777i	$0.720367\pi$
$758$	24.1692	0.877863
$759$	0	0
$760$	0	0
$761$	−51.0091	−1.84908	−0.924539	−	0.381087i	$-0.875550\pi$
$761$	−51.0091	−1.84908	−0.924539	+	0.381087i	$0.875550\pi$
$762$	0	0
$763$	4.70820	0.170448
$764$	−2.41577	−0.0873993
$765$	0	0
$766$	−12.2918	−0.444121
$767$	1.87558	0.0677234
$768$	0	0
$769$	9.18034	0.331052	0.165526	−	0.986205i	$-0.447068\pi$
$769$	9.18034	0.331052	0.165526	+	0.986205i	$0.447068\pi$
$770$	0	0
$771$	0	0
$772$	−26.4721	−0.952753
$773$	38.9790	1.40198	0.700988	−	0.713173i	$-0.252742\pi$
$773$	38.9790	1.40198	0.700988	+	0.713173i	$0.252742\pi$
$774$	0	0
$775$	0	0
$776$	29.6197	1.06328
$777$	0	0
$778$	−14.1115	−0.505920
$779$	11.1561	0.399708
$780$	0	0
$781$	33.8885	1.21263
$782$	11.9814	0.428454
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−45.3050	−1.61495	−0.807474	−	0.589904i	$-0.799166\pi$
$787$	−45.3050	−1.61495	−0.807474	+	0.589904i	$0.799166\pi$
$788$	19.9566	0.710925
$789$	0	0
$790$	0	0
$791$	−13.8083	−0.490966
$792$	0	0
$793$	0.875388	0.0310859
$794$	6.53089	0.231772
$795$	0	0
$796$	−11.7082	−0.414986
$797$	−40.6482	−1.43983	−0.719917	−	0.694060i	$-0.755820\pi$
$797$	−40.6482	−1.43983	−0.719917	+	0.694060i	$0.755820\pi$
$798$	0	0
$799$	−47.5967	−1.68385
$800$	0	0
$801$	0	0
$802$	22.5410	0.795951
$803$	−12.5216	−0.441877
$804$	0	0
$805$	0	0
$806$	−1.38411	−0.0487532
$807$	0	0
$808$	−9.52786	−0.335189
$809$	8.35776	0.293843	0.146922	−	0.989148i	$-0.453063\pi$
$809$	8.35776	0.293843	0.146922	+	0.989148i	$0.453063\pi$
$810$	0	0
$811$	29.7639	1.04515	0.522577	−	0.852592i	$-0.324971\pi$
$811$	29.7639	1.04515	0.522577	+	0.852592i	$0.324971\pi$
$812$	−5.91189	−0.207467
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	−15.5279	−0.543251
$818$	20.5154	0.717304
$819$	0	0
$820$	0	0
$821$	−33.1645	−1.15745	−0.578725	−	0.815523i	$-0.696449\pi$
$821$	−33.1645	−1.15745	−0.578725	+	0.815523i	$0.696449\pi$
$822$	0	0
$823$	21.3607	0.744586	0.372293	−	0.928115i	$-0.378572\pi$
$823$	21.3607	0.744586	0.372293	+	0.928115i	$0.378572\pi$
$824$	34.6088	1.20566
$825$	0	0
$826$	6.94427	0.241622
$827$	37.1034	1.29021	0.645106	−	0.764093i	$-0.276813\pi$
$827$	37.1034	1.29021	0.645106	+	0.764093i	$0.276813\pi$
$828$	0	0
$829$	−38.7771	−1.34678	−0.673392	−	0.739286i	$-0.735163\pi$
$829$	−38.7771	−1.34678	−0.673392	+	0.739286i	$0.735163\pi$
$830$	0	0
$831$	0	0
$832$	−1.16718	−0.0404648
$833$	−5.99070	−0.207566
$834$	0	0
$835$	0	0
$836$	12.6491	0.437479
$837$	0	0
$838$	14.6525	0.506161
$839$	3.03476	0.104771	0.0523857	−	0.998627i	$-0.483317\pi$
$839$	3.03476	0.104771	0.0523857	+	0.998627i	$0.483317\pi$
$840$	0	0
$841$	−6.12461	−0.211194
$842$	−7.61125	−0.262301
$843$	0	0
$844$	30.5410	1.05127
$845$	0	0
$846$	0	0
$847$	−5.76393	−0.198051
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.86474	0.235320
$852$	0	0
$853$	−41.2492	−1.41235	−0.706173	−	0.708039i	$-0.749580\pi$
$853$	−41.2492	−1.41235	−0.706173	+	0.708039i	$0.749580\pi$
$854$	3.24109	0.110908
$855$	0	0
$856$	−25.3050	−0.864905
$857$	−7.45363	−0.254611	−0.127306	−	0.991864i	$-0.540633\pi$
$857$	−7.45363	−0.254611	−0.127306	+	0.991864i	$0.540633\pi$
$858$	0	0
$859$	0.596748	0.0203608	0.0101804	−	0.999948i	$-0.496759\pi$
$859$	0.596748	0.0203608	0.0101804	+	0.999948i	$0.496759\pi$
$860$	0	0
$861$	0	0
$862$	26.2492	0.894052
$863$	−10.2333	−0.348347	−0.174174	−	0.984715i	$-0.555725\pi$
$863$	−10.2333	−0.348347	−0.174174	+	0.984715i	$0.555725\pi$
$864$	0	0
$865$	0	0
$866$	−18.9250	−0.643096
$867$	0	0
$868$	8.29180	0.281442
$869$	−17.0981	−0.580013
$870$	0	0
$871$	−1.47214	−0.0498814
$872$	13.3168	0.450964
$873$	0	0
$874$	−8.94427	−0.302545
$875$	0	0
$876$	0	0
$877$	−57.7771	−1.95099	−0.975497	−	0.220014i	$-0.929390\pi$
$877$	−57.7771	−1.95099	−0.975497	+	0.220014i	$0.929390\pi$
$878$	−7.86629	−0.265474
$879$	0	0
$880$	0	0
$881$	−39.0578	−1.31589	−0.657945	−	0.753066i	$-0.728574\pi$
$881$	−39.0578	−1.31589	−0.657945	+	0.753066i	$0.728574\pi$
$882$	0	0
$883$	−50.4296	−1.69709	−0.848545	−	0.529123i	$-0.822521\pi$
$883$	−50.4296	−1.69709	−0.848545	+	0.529123i	$0.822521\pi$
$884$	−1.74806	−0.0587938
$885$	0	0
$886$	8.83282	0.296744
$887$	−0.491473	−0.0165021	−0.00825103	−	0.999966i	$-0.502626\pi$
$887$	−0.491473	−0.0165021	−0.00825103	+	0.999966i	$0.502626\pi$
$888$	0	0
$889$	14.4721	0.485380
$890$	0	0
$891$	0	0
$892$	6.40325	0.214397
$893$	35.5316	1.18902
$894$	0	0
$895$	0	0
$896$	6.99226	0.233595
$897$	0	0
$898$	21.6656	0.722991
$899$	−32.0841	−1.07007
$900$	0	0
$901$	20.1803	0.672305
$902$	4.98915	0.166121
$903$	0	0
$904$	−39.0557	−1.29897
$905$	0	0
$906$	0	0
$907$	13.2918	0.441347	0.220673	−	0.975348i	$-0.429175\pi$
$907$	13.2918	0.441347	0.220673	+	0.975348i	$0.429175\pi$
$908$	4.57649	0.151876
$909$	0	0
$910$	0	0
$911$	−6.65841	−0.220603	−0.110301	−	0.993898i	$-0.535182\pi$
$911$	−6.65841	−0.220603	−0.110301	+	0.993898i	$0.535182\pi$
$912$	0	0
$913$	−32.6525	−1.08064
$914$	−12.5517	−0.415173
$915$	0	0
$916$	11.9311	0.394215
$917$	−18.0996	−0.597703
$918$	0	0
$919$	43.4721	1.43401	0.717007	−	0.697066i	$-0.245512\pi$
$919$	43.4721	1.43401	0.717007	+	0.697066i	$0.245512\pi$
$920$	0	0
$921$	0	0
$922$	−16.2229	−0.534273
$923$	3.49613	0.115076
$924$	0	0
$925$	0	0
$926$	−3.54484	−0.116491
$927$	0	0
$928$	−27.0557	−0.888148
$929$	52.9148	1.73608	0.868039	−	0.496496i	$-0.165380\pi$
$929$	52.9148	1.73608	0.868039	+	0.496496i	$0.165380\pi$
$930$	0	0
$931$	4.47214	0.146568
$932$	−8.17004	−0.267618
$933$	0	0
$934$	12.8328	0.419903
$935$	0	0
$936$	0	0
$937$	−23.2361	−0.759089	−0.379545	−	0.925173i	$-0.623919\pi$
$937$	−23.2361	−0.759089	−0.379545	+	0.925173i	$0.623919\pi$
$938$	−5.45052	−0.177966
$939$	0	0
$940$	0	0
$941$	12.5216	0.408192	0.204096	−	0.978951i	$-0.434574\pi$
$941$	12.5216	0.408192	0.204096	+	0.978951i	$0.434574\pi$
$942$	0	0
$943$	5.70820	0.185885
$944$	0	0
$945$	0	0
$946$	−6.94427	−0.225778
$947$	−50.1351	−1.62917	−0.814585	−	0.580044i	$-0.803036\pi$
$947$	−50.1351	−1.62917	−0.814585	+	0.580044i	$0.803036\pi$
$948$	0	0
$949$	−1.29180	−0.0419335
$950$	0	0
$951$	0	0
$952$	−16.9443	−0.549167
$953$	3.57494	0.115804	0.0579018	−	0.998322i	$-0.481559\pi$
$953$	3.57494	0.115804	0.0579018	+	0.998322i	$0.481559\pi$
$954$	0	0
$955$	0	0
$956$	−35.1189	−1.13583
$957$	0	0
$958$	16.0689	0.519162
$959$	7.73877	0.249898
$960$	0	0
$961$	14.0000	0.451613
$962$	0.618993	0.0199571
$963$	0	0
$964$	−10.5410	−0.339503
$965$	0	0
$966$	0	0
$967$	51.0689	1.64226	0.821132	−	0.570738i	$-0.193343\pi$
$967$	51.0689	1.64226	0.821132	+	0.570738i	$0.193343\pi$
$968$	−16.3029	−0.523994
$969$	0	0
$970$	0	0
$971$	−24.4543	−0.784776	−0.392388	−	0.919800i	$-0.628351\pi$
$971$	−24.4543	−0.784776	−0.392388	+	0.919800i	$0.628351\pi$
$972$	0	0
$973$	7.76393	0.248900
$974$	13.2681	0.425137
$975$	0	0
$976$	0	0
$977$	22.7248	0.727032	0.363516	−	0.931588i	$-0.381576\pi$
$977$	22.7248	0.727032	0.363516	+	0.931588i	$0.381576\pi$
$978$	0	0
$979$	−2.76393	−0.0883357
$980$	0	0
$981$	0	0
$982$	−15.4164	−0.491957
$983$	−7.32611	−0.233667	−0.116833	−	0.993152i	$-0.537274\pi$
$983$	−7.32611	−0.233667	−0.116833	+	0.993152i	$0.537274\pi$
$984$	0	0
$985$	0	0
$986$	25.0432	0.797537
$987$	0	0
$988$	1.30495	0.0415160
$989$	−7.94510	−0.252639
$990$	0	0
$991$	24.3607	0.773842	0.386921	−	0.922113i	$-0.373539\pi$
$991$	24.3607	0.773842	0.386921	+	0.922113i	$0.373539\pi$
$992$	37.9473	1.20483
$993$	0	0
$994$	12.9443	0.410567
$995$	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	$0.571158\pi$
$998$	−8.18153	−0.258982
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4725.2.a.bt.1.2	✓	4	15.14	odd	2
4725.2.a.bt.1.3	yes	4	5.4	even	2
4725.2.a.bw.1.2	yes	4	1.1	even	1	trivial
4725.2.a.bw.1.3	yes	4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.874032 q^{2} -1.23607 q^{4} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q-0.874032 q^{2} -1.23607 q^{4} +1.00000 q^{7} +2.82843 q^{8} -2.28825 q^{11} -0.236068 q^{13} -0.874032 q^{14} -5.99070 q^{17} +4.47214 q^{19} +2.00000 q^{22} +2.28825 q^{23} +0.206331 q^{26} -1.23607 q^{28} +4.78282 q^{29} -6.70820 q^{31} -5.65685 q^{32} +5.23607 q^{34} +3.00000 q^{37} -3.90879 q^{38} +2.49458 q^{41} -3.47214 q^{43} +2.82843 q^{44} -2.00000 q^{46} +7.94510 q^{47} +1.00000 q^{49} +0.291796 q^{52} -3.36861 q^{53} +2.82843 q^{56} -4.18034 q^{58} -7.94510 q^{59} -3.70820 q^{61} +5.86319 q^{62} +4.94427 q^{64} +6.23607 q^{67} +7.40492 q^{68} -14.8098 q^{71} +5.47214 q^{73} -2.62210 q^{74} -5.52786 q^{76} -2.28825 q^{77} +7.47214 q^{79} -2.18034 q^{82} +14.2697 q^{83} +3.03476 q^{86} -6.47214 q^{88} +1.20788 q^{89} -0.236068 q^{91} -2.82843 q^{92} -6.94427 q^{94} +10.4721 q^{97} -0.874032 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^4 + 4 * q^7	\( 4 q + 4 q^{4} + 4 q^{7} + 8 q^{13} + 8 q^{22} + 4 q^{28} + 12 q^{34} + 12 q^{37} + 4 q^{43} - 8 q^{46} + 4 q^{49} + 28 q^{52} + 28 q^{58} + 12 q^{61} - 16 q^{64} + 16 q^{67} + 4 q^{73} - 40 q^{76} + 12 q^{79} + 36 q^{82} - 8 q^{88} + 8 q^{91} + 8 q^{94} + 24 q^{97}+O(q^{100}) \) 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

Groups

Database

Properties

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