LMFDB - Embedded newform 5202.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5202,2,Mod(1,5202)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5202, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5202.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5202 = 2 \cdot 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5202.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:	$41.5381791315$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1734)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	5202.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.53209 q^{5} -2.75877 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -3.53209 q^{5} -2.75877 q^{7} +1.00000 q^{8} -3.53209 q^{10} +2.94356 q^{11} +5.06418 q^{13} -2.75877 q^{14} +1.00000 q^{16} -4.36959 q^{19} -3.53209 q^{20} +2.94356 q^{22} -6.00000 q^{23} +7.47565 q^{25} +5.06418 q^{26} -2.75877 q^{28} +4.80066 q^{29} +0.716881 q^{31} +1.00000 q^{32} +9.74422 q^{35} +10.2121 q^{37} -4.36959 q^{38} -3.53209 q^{40} -12.5817 q^{41} +10.9067 q^{43} +2.94356 q^{44} -6.00000 q^{46} -5.14796 q^{47} +0.610815 q^{49} +7.47565 q^{50} +5.06418 q^{52} -4.57398 q^{53} -10.3969 q^{55} -2.75877 q^{56} +4.80066 q^{58} -5.53209 q^{59} -6.24123 q^{61} +0.716881 q^{62} +1.00000 q^{64} -17.8871 q^{65} +5.67499 q^{67} +9.74422 q^{70} +9.80335 q^{71} -9.04189 q^{73} +10.2121 q^{74} -4.36959 q^{76} -8.12061 q^{77} -6.61856 q^{79} -3.53209 q^{80} -12.5817 q^{82} +3.32501 q^{83} +10.9067 q^{86} +2.94356 q^{88} -13.8425 q^{89} -13.9709 q^{91} -6.00000 q^{92} -5.14796 q^{94} +15.4338 q^{95} -11.0273 q^{97} +0.610815 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 18 q^{23} + 3 q^{25} + 6 q^{26} + 3 q^{28} - 6 q^{31} + 3 q^{32} + 6 q^{37} - 6 q^{38} - 6 q^{40} - 6 q^{41} + 6 q^{43} - 6 q^{44} - 18 q^{46} + 6 q^{49} + 3 q^{50} + 6 q^{52} - 6 q^{53} - 3 q^{55} + 3 q^{56} - 12 q^{59} - 30 q^{61} - 6 q^{62} + 3 q^{64} - 24 q^{65} + 12 q^{67} + 6 q^{71} - 24 q^{73} + 6 q^{74} - 6 q^{76} - 30 q^{77} - 6 q^{80} - 6 q^{82} + 15 q^{83} + 6 q^{86} - 6 q^{88} - 24 q^{89} - 6 q^{91} - 18 q^{92} + 30 q^{95} - 12 q^{97} + 6 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^19 - 6 * q^20 - 6 * q^22 - 18 * q^23 + 3 * q^25 + 6 * q^26 + 3 * q^28 - 6 * q^31 + 3 * q^32 + 6 * q^37 - 6 * q^38 - 6 * q^40 - 6 * q^41 + 6 * q^43 - 6 * q^44 - 18 * q^46 + 6 * q^49 + 3 * q^50 + 6 * q^52 - 6 * q^53 - 3 * q^55 + 3 * q^56 - 12 * q^59 - 30 * q^61 - 6 * q^62 + 3 * q^64 - 24 * q^65 + 12 * q^67 + 6 * q^71 - 24 * q^73 + 6 * q^74 - 6 * q^76 - 30 * q^77 - 6 * q^80 - 6 * q^82 + 15 * q^83 + 6 * q^86 - 6 * q^88 - 24 * q^89 - 6 * q^91 - 18 * q^92 + 30 * q^95 - 12 * q^97 + 6 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.53209	−1.57960	−0.789799	−	0.613366i	$-0.789815\pi$
$5$	−3.53209	−1.57960	−0.789799	+	0.613366i	$0.789815\pi$
$6$	0	0
$7$	−2.75877	−1.04272	−0.521359	−	0.853338i	$-0.674575\pi$
$7$	−2.75877	−1.04272	−0.521359	+	0.853338i	$0.674575\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.53209	−1.11694
$11$	2.94356	0.887518	0.443759	−	0.896146i	$-0.353645\pi$
$11$	2.94356	0.887518	0.443759	+	0.896146i	$0.353645\pi$
$12$	0	0
$13$	5.06418	1.40455	0.702275	−	0.711906i	$-0.252168\pi$
$13$	5.06418	1.40455	0.702275	+	0.711906i	$0.252168\pi$
$14$	−2.75877	−0.737312
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−4.36959	−1.00245	−0.501226	−	0.865317i	$-0.667117\pi$
$19$	−4.36959	−1.00245	−0.501226	+	0.865317i	$0.667117\pi$
$20$	−3.53209	−0.789799
$21$	0	0
$22$	2.94356	0.627570
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	7.47565	1.49513
$26$	5.06418	0.993167
$27$	0	0
$28$	−2.75877	−0.521359
$29$	4.80066	0.891460	0.445730	−	0.895167i	$-0.352944\pi$
$29$	4.80066	0.891460	0.445730	+	0.895167i	$0.352944\pi$
$30$	0	0
$31$	0.716881	0.128756	0.0643779	−	0.997926i	$-0.479494\pi$
$31$	0.716881	0.128756	0.0643779	+	0.997926i	$0.479494\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	9.74422	1.64707
$36$	0	0
$37$	10.2121	1.67886	0.839432	−	0.543464i	$-0.182888\pi$
$37$	10.2121	1.67886	0.839432	+	0.543464i	$0.182888\pi$
$38$	−4.36959	−0.708840
$39$	0	0
$40$	−3.53209	−0.558472
$41$	−12.5817	−1.96493	−0.982467	−	0.186436i	$-0.940306\pi$
$41$	−12.5817	−1.96493	−0.982467	+	0.186436i	$0.940306\pi$
$42$	0	0
$43$	10.9067	1.66326	0.831630	−	0.555330i	$-0.187408\pi$
$43$	10.9067	1.66326	0.831630	+	0.555330i	$0.187408\pi$
$44$	2.94356	0.443759
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−5.14796	−0.750907	−0.375453	−	0.926841i	$-0.622513\pi$
$47$	−5.14796	−0.750907	−0.375453	+	0.926841i	$0.622513\pi$
$48$	0	0
$49$	0.610815	0.0872592
$50$	7.47565	1.05722
$51$	0	0
$52$	5.06418	0.702275
$53$	−4.57398	−0.628284	−0.314142	−	0.949376i	$-0.601717\pi$
$53$	−4.57398	−0.628284	−0.314142	+	0.949376i	$0.601717\pi$
$54$	0	0
$55$	−10.3969	−1.40192
$56$	−2.75877	−0.368656
$57$	0	0
$58$	4.80066	0.630357
$59$	−5.53209	−0.720217	−0.360108	−	0.932911i	$-0.617260\pi$
$59$	−5.53209	−0.720217	−0.360108	+	0.932911i	$0.617260\pi$
$60$	0	0
$61$	−6.24123	−0.799108	−0.399554	−	0.916710i	$-0.630835\pi$
$61$	−6.24123	−0.799108	−0.399554	+	0.916710i	$0.630835\pi$
$62$	0.716881	0.0910440
$63$	0	0
$64$	1.00000	0.125000
$65$	−17.8871	−2.21862
$66$	0	0
$67$	5.67499	0.693311	0.346655	−	0.937993i	$-0.387317\pi$
$67$	5.67499	0.693311	0.346655	+	0.937993i	$0.387317\pi$
$68$	0	0
$69$	0	0
$70$	9.74422	1.16466
$71$	9.80335	1.16344	0.581722	−	0.813388i	$-0.302379\pi$
$71$	9.80335	1.16344	0.581722	+	0.813388i	$0.302379\pi$
$72$	0	0
$73$	−9.04189	−1.05827	−0.529137	−	0.848537i	$-0.677484\pi$
$73$	−9.04189	−1.05827	−0.529137	+	0.848537i	$0.677484\pi$
$74$	10.2121	1.18714
$75$	0	0
$76$	−4.36959	−0.501226
$77$	−8.12061	−0.925430
$78$	0	0
$79$	−6.61856	−0.744646	−0.372323	−	0.928103i	$-0.621439\pi$
$79$	−6.61856	−0.744646	−0.372323	+	0.928103i	$0.621439\pi$
$80$	−3.53209	−0.394900
$81$	0	0
$82$	−12.5817	−1.38942
$83$	3.32501	0.364967	0.182484	−	0.983209i	$-0.441586\pi$
$83$	3.32501	0.364967	0.182484	+	0.983209i	$0.441586\pi$
$84$	0	0
$85$	0	0
$86$	10.9067	1.17610
$87$	0	0
$88$	2.94356	0.313785
$89$	−13.8425	−1.46731	−0.733654	−	0.679524i	$-0.762186\pi$
$89$	−13.8425	−1.46731	−0.733654	+	0.679524i	$0.762186\pi$
$90$	0	0
$91$	−13.9709	−1.46455
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−5.14796	−0.530971
$95$	15.4338	1.58347
$96$	0	0
$97$	−11.0273	−1.11966	−0.559828	−	0.828609i	$-0.689133\pi$
$97$	−11.0273	−1.11966	−0.559828	+	0.828609i	$0.689133\pi$
$98$	0.610815	0.0617016
$99$	0	0
$100$	7.47565	0.747565
$101$	−0.921274	−0.0916702	−0.0458351	−	0.998949i	$-0.514595\pi$
$101$	−0.921274	−0.0916702	−0.0458351	+	0.998949i	$0.514595\pi$
$102$	0	0
$103$	−16.9736	−1.67246	−0.836229	−	0.548381i	$-0.815245\pi$
$103$	−16.9736	−1.67246	−0.836229	+	0.548381i	$0.815245\pi$
$104$	5.06418	0.496583
$105$	0	0
$106$	−4.57398	−0.444264
$107$	3.08647	0.298380	0.149190	−	0.988809i	$-0.452333\pi$
$107$	3.08647	0.298380	0.149190	+	0.988809i	$0.452333\pi$
$108$	0	0
$109$	−2.24123	−0.214671	−0.107335	−	0.994223i	$-0.534232\pi$
$109$	−2.24123	−0.214671	−0.107335	+	0.994223i	$0.534232\pi$
$110$	−10.3969	−0.991308
$111$	0	0
$112$	−2.75877	−0.260679
$113$	−11.4338	−1.07560	−0.537799	−	0.843073i	$-0.680744\pi$
$113$	−11.4338	−1.07560	−0.537799	+	0.843073i	$0.680744\pi$
$114$	0	0
$115$	21.1925	1.97621
$116$	4.80066	0.445730
$117$	0	0
$118$	−5.53209	−0.509270
$119$	0	0
$120$	0	0
$121$	−2.33544	−0.212312
$122$	−6.24123	−0.565054
$123$	0	0
$124$	0.716881	0.0643779
$125$	−8.74422	−0.782107
$126$	0	0
$127$	−8.75877	−0.777215	−0.388608	−	0.921403i	$-0.627044\pi$
$127$	−8.75877	−0.777215	−0.388608	+	0.921403i	$0.627044\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−17.8871	−1.56880
$131$	−1.28581	−0.112341	−0.0561707	−	0.998421i	$-0.517889\pi$
$131$	−1.28581	−0.112341	−0.0561707	+	0.998421i	$0.517889\pi$
$132$	0	0
$133$	12.0547	1.04527
$134$	5.67499	0.490245
$135$	0	0
$136$	0	0
$137$	−2.36959	−0.202447	−0.101224	−	0.994864i	$-0.532276\pi$
$137$	−2.36959	−0.202447	−0.101224	+	0.994864i	$0.532276\pi$
$138$	0	0
$139$	5.64590	0.478879	0.239439	−	0.970911i	$-0.423036\pi$
$139$	5.64590	0.478879	0.239439	+	0.970911i	$0.423036\pi$
$140$	9.74422	0.823537
$141$	0	0
$142$	9.80335	0.822679
$143$	14.9067	1.24656
$144$	0	0
$145$	−16.9564	−1.40815
$146$	−9.04189	−0.748312
$147$	0	0
$148$	10.2121	0.839432
$149$	−11.4260	−0.936056	−0.468028	−	0.883714i	$-0.655035\pi$
$149$	−11.4260	−0.936056	−0.468028	+	0.883714i	$0.655035\pi$
$150$	0	0
$151$	11.7510	0.956285	0.478143	−	0.878282i	$-0.341310\pi$
$151$	11.7510	0.956285	0.478143	+	0.878282i	$0.341310\pi$
$152$	−4.36959	−0.354420
$153$	0	0
$154$	−8.12061	−0.654378
$155$	−2.53209	−0.203382
$156$	0	0
$157$	−3.51754	−0.280730	−0.140365	−	0.990100i	$-0.544828\pi$
$157$	−3.51754	−0.280730	−0.140365	+	0.990100i	$0.544828\pi$
$158$	−6.61856	−0.526544
$159$	0	0
$160$	−3.53209	−0.279236
$161$	16.5526	1.30453
$162$	0	0
$163$	−6.32501	−0.495413	−0.247706	−	0.968835i	$-0.579677\pi$
$163$	−6.32501	−0.495413	−0.247706	+	0.968835i	$0.579677\pi$
$164$	−12.5817	−0.982467
$165$	0	0
$166$	3.32501	0.258071
$167$	14.3405	1.10970	0.554850	−	0.831950i	$-0.312776\pi$
$167$	14.3405	1.10970	0.554850	+	0.831950i	$0.312776\pi$
$168$	0	0
$169$	12.6459	0.972761
$170$	0	0
$171$	0	0
$172$	10.9067	0.831630
$173$	0.815207	0.0619791	0.0309895	−	0.999520i	$-0.490134\pi$
$173$	0.815207	0.0619791	0.0309895	+	0.999520i	$0.490134\pi$
$174$	0	0
$175$	−20.6236	−1.55900
$176$	2.94356	0.221879
$177$	0	0
$178$	−13.8425	−1.03754
$179$	−14.0642	−1.05121	−0.525603	−	0.850730i	$-0.676160\pi$
$179$	−14.0642	−1.05121	−0.525603	+	0.850730i	$0.676160\pi$
$180$	0	0
$181$	−20.3405	−1.51190	−0.755948	−	0.654631i	$-0.772824\pi$
$181$	−20.3405	−1.51190	−0.755948	+	0.654631i	$0.772824\pi$
$182$	−13.9709	−1.03559
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−36.0702	−2.65193
$186$	0	0
$187$	0	0
$188$	−5.14796	−0.375453
$189$	0	0
$190$	15.4338	1.11968
$191$	−15.6459	−1.13210	−0.566049	−	0.824372i	$-0.691529\pi$
$191$	−15.6459	−1.13210	−0.566049	+	0.824372i	$0.691529\pi$
$192$	0	0
$193$	10.2713	0.739341	0.369671	−	0.929163i	$-0.379471\pi$
$193$	10.2713	0.739341	0.369671	+	0.929163i	$0.379471\pi$
$194$	−11.0273	−0.791717
$195$	0	0
$196$	0.610815	0.0436296
$197$	8.72967	0.621964	0.310982	−	0.950416i	$-0.399342\pi$
$197$	8.72967	0.621964	0.310982	+	0.950416i	$0.399342\pi$
$198$	0	0
$199$	16.8922	1.19745	0.598727	−	0.800953i	$-0.295673\pi$
$199$	16.8922	1.19745	0.598727	+	0.800953i	$0.295673\pi$
$200$	7.47565	0.528608
$201$	0	0
$202$	−0.921274	−0.0648206
$203$	−13.2439	−0.929541
$204$	0	0
$205$	44.4397	3.10381
$206$	−16.9736	−1.18261
$207$	0	0
$208$	5.06418	0.351138
$209$	−12.8621	−0.889693
$210$	0	0
$211$	−14.0993	−0.970633	−0.485317	−	0.874339i	$-0.661296\pi$
$211$	−14.0993	−0.970633	−0.485317	+	0.874339i	$0.661296\pi$
$212$	−4.57398	−0.314142
$213$	0	0
$214$	3.08647	0.210987
$215$	−38.5235	−2.62728
$216$	0	0
$217$	−1.97771	−0.134256
$218$	−2.24123	−0.151795
$219$	0	0
$220$	−10.3969	−0.700961
$221$	0	0
$222$	0	0
$223$	−26.3037	−1.76142	−0.880711	−	0.473653i	$-0.842935\pi$
$223$	−26.3037	−1.76142	−0.880711	+	0.473653i	$0.842935\pi$
$224$	−2.75877	−0.184328
$225$	0	0
$226$	−11.4338	−0.760563
$227$	6.06418	0.402494	0.201247	−	0.979541i	$-0.435501\pi$
$227$	6.06418	0.402494	0.201247	+	0.979541i	$0.435501\pi$
$228$	0	0
$229$	8.56624	0.566073	0.283036	−	0.959109i	$-0.408658\pi$
$229$	8.56624	0.566073	0.283036	+	0.959109i	$0.408658\pi$
$230$	21.1925	1.39739
$231$	0	0
$232$	4.80066	0.315179
$233$	−12.2121	−0.800043	−0.400022	−	0.916506i	$-0.630997\pi$
$233$	−12.2121	−0.800043	−0.400022	+	0.916506i	$0.630997\pi$
$234$	0	0
$235$	18.1830	1.18613
$236$	−5.53209	−0.360108
$237$	0	0
$238$	0	0
$239$	−11.8716	−0.767913	−0.383956	−	0.923351i	$-0.625439\pi$
$239$	−11.8716	−0.767913	−0.383956	+	0.923351i	$0.625439\pi$
$240$	0	0
$241$	−23.9172	−1.54064	−0.770320	−	0.637658i	$-0.779903\pi$
$241$	−23.9172	−1.54064	−0.770320	+	0.637658i	$0.779903\pi$
$242$	−2.33544	−0.150128
$243$	0	0
$244$	−6.24123	−0.399554
$245$	−2.15745	−0.137835
$246$	0	0
$247$	−22.1284	−1.40799
$248$	0.716881	0.0455220
$249$	0	0
$250$	−8.74422	−0.553033
$251$	18.6040	1.17427	0.587137	−	0.809487i	$-0.300255\pi$
$251$	18.6040	1.17427	0.587137	+	0.809487i	$0.300255\pi$
$252$	0	0
$253$	−17.6614	−1.11036
$254$	−8.75877	−0.549574
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.16756	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$257$	4.16756	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$258$	0	0
$259$	−28.1729	−1.75058
$260$	−17.8871	−1.10931
$261$	0	0
$262$	−1.28581	−0.0794374
$263$	−7.97090	−0.491507	−0.245754	−	0.969332i	$-0.579035\pi$
$263$	−7.97090	−0.491507	−0.245754	+	0.969332i	$0.579035\pi$
$264$	0	0
$265$	16.1557	0.992437
$266$	12.0547	0.739120
$267$	0	0
$268$	5.67499	0.346655
$269$	5.96316	0.363580	0.181790	−	0.983337i	$-0.441811\pi$
$269$	5.96316	0.363580	0.181790	+	0.983337i	$0.441811\pi$
$270$	0	0
$271$	−3.07098	−0.186549	−0.0932745	−	0.995640i	$-0.529733\pi$
$271$	−3.07098	−0.186549	−0.0932745	+	0.995640i	$0.529733\pi$
$272$	0	0
$273$	0	0
$274$	−2.36959	−0.143152
$275$	22.0051	1.32695
$276$	0	0
$277$	−23.2472	−1.39679	−0.698395	−	0.715713i	$-0.746102\pi$
$277$	−23.2472	−1.39679	−0.698395	+	0.715713i	$0.746102\pi$
$278$	5.64590	0.338618
$279$	0	0
$280$	9.74422	0.582329
$281$	−21.7297	−1.29628	−0.648142	−	0.761520i	$-0.724454\pi$
$281$	−21.7297	−1.29628	−0.648142	+	0.761520i	$0.724454\pi$
$282$	0	0
$283$	−17.1925	−1.02199	−0.510995	−	0.859584i	$-0.670723\pi$
$283$	−17.1925	−1.02199	−0.510995	+	0.859584i	$0.670723\pi$
$284$	9.80335	0.581722
$285$	0	0
$286$	14.9067	0.881453
$287$	34.7101	2.04887
$288$	0	0
$289$	0	0
$290$	−16.9564	−0.995712
$291$	0	0
$292$	−9.04189	−0.529137
$293$	9.20708	0.537883	0.268942	−	0.963156i	$-0.413326\pi$
$293$	9.20708	0.537883	0.268942	+	0.963156i	$0.413326\pi$
$294$	0	0
$295$	19.5398	1.13765
$296$	10.2121	0.593568
$297$	0	0
$298$	−11.4260	−0.661892
$299$	−30.3851	−1.75721
$300$	0	0
$301$	−30.0892	−1.73431
$302$	11.7510	0.676196
$303$	0	0
$304$	−4.36959	−0.250613
$305$	22.0446	1.26227
$306$	0	0
$307$	2.90673	0.165896	0.0829478	−	0.996554i	$-0.473567\pi$
$307$	2.90673	0.165896	0.0829478	+	0.996554i	$0.473567\pi$
$308$	−8.12061	−0.462715
$309$	0	0
$310$	−2.53209	−0.143813
$311$	13.0351	0.739152	0.369576	−	0.929201i	$-0.379503\pi$
$311$	13.0351	0.739152	0.369576	+	0.929201i	$0.379503\pi$
$312$	0	0
$313$	3.22163	0.182097	0.0910486	−	0.995846i	$-0.470978\pi$
$313$	3.22163	0.182097	0.0910486	+	0.995846i	$0.470978\pi$
$314$	−3.51754	−0.198506
$315$	0	0
$316$	−6.61856	−0.372323
$317$	19.0838	1.07185	0.535926	−	0.844265i	$-0.319963\pi$
$317$	19.0838	1.07185	0.535926	+	0.844265i	$0.319963\pi$
$318$	0	0
$319$	14.1310	0.791187
$320$	−3.53209	−0.197450
$321$	0	0
$322$	16.5526	0.922442
$323$	0	0
$324$	0	0
$325$	37.8580	2.09999
$326$	−6.32501	−0.350310
$327$	0	0
$328$	−12.5817	−0.694709
$329$	14.2020	0.782983
$330$	0	0
$331$	3.43376	0.188737	0.0943683	−	0.995537i	$-0.469917\pi$
$331$	3.43376	0.188737	0.0943683	+	0.995537i	$0.469917\pi$
$332$	3.32501	0.182484
$333$	0	0
$334$	14.3405	0.784677
$335$	−20.0446	−1.09515
$336$	0	0
$337$	−35.7202	−1.94580	−0.972901	−	0.231222i	$-0.925728\pi$
$337$	−35.7202	−1.94580	−0.972901	+	0.231222i	$0.925728\pi$
$338$	12.6459	0.687846
$339$	0	0
$340$	0	0
$341$	2.11019	0.114273
$342$	0	0
$343$	17.6263	0.951731
$344$	10.9067	0.588051
$345$	0	0
$346$	0.815207	0.0438258
$347$	1.10782	0.0594710	0.0297355	−	0.999558i	$-0.490534\pi$
$347$	1.10782	0.0594710	0.0297355	+	0.999558i	$0.490534\pi$
$348$	0	0
$349$	11.2371	0.601509	0.300754	−	0.953702i	$-0.402762\pi$
$349$	11.2371	0.601509	0.300754	+	0.953702i	$0.402762\pi$
$350$	−20.6236	−1.10238
$351$	0	0
$352$	2.94356	0.156892
$353$	11.2608	0.599353	0.299677	−	0.954041i	$-0.403121\pi$
$353$	11.2608	0.599353	0.299677	+	0.954041i	$0.403121\pi$
$354$	0	0
$355$	−34.6263	−1.83777
$356$	−13.8425	−0.733654
$357$	0	0
$358$	−14.0642	−0.743315
$359$	−2.04458	−0.107909	−0.0539543	−	0.998543i	$-0.517183\pi$
$359$	−2.04458	−0.107909	−0.0539543	+	0.998543i	$0.517183\pi$
$360$	0	0
$361$	0.0932736	0.00490914
$362$	−20.3405	−1.06907
$363$	0	0
$364$	−13.9709	−0.732274
$365$	31.9368	1.67165
$366$	0	0
$367$	24.4097	1.27418	0.637088	−	0.770791i	$-0.280139\pi$
$367$	24.4097	1.27418	0.637088	+	0.770791i	$0.280139\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−36.0702	−1.87520
$371$	12.6186	0.655123
$372$	0	0
$373$	27.3756	1.41745	0.708727	−	0.705483i	$-0.249270\pi$
$373$	27.3756	1.41745	0.708727	+	0.705483i	$0.249270\pi$
$374$	0	0
$375$	0	0
$376$	−5.14796	−0.265486
$377$	24.3114	1.25210
$378$	0	0
$379$	22.4979	1.15564	0.577821	−	0.816164i	$-0.303903\pi$
$379$	22.4979	1.15564	0.577821	+	0.816164i	$0.303903\pi$
$380$	15.4338	0.791735
$381$	0	0
$382$	−15.6459	−0.800514
$383$	−16.5972	−0.848077	−0.424039	−	0.905644i	$-0.639388\pi$
$383$	−16.5972	−0.848077	−0.424039	+	0.905644i	$0.639388\pi$
$384$	0	0
$385$	28.6827	1.46181
$386$	10.2713	0.522793
$387$	0	0
$388$	−11.0273	−0.559828
$389$	17.2249	0.873338	0.436669	−	0.899622i	$-0.356158\pi$
$389$	17.2249	0.873338	0.436669	+	0.899622i	$0.356158\pi$
$390$	0	0
$391$	0	0
$392$	0.610815	0.0308508
$393$	0	0
$394$	8.72967	0.439795
$395$	23.3773	1.17624
$396$	0	0
$397$	4.49794	0.225745	0.112873	−	0.993609i	$-0.463995\pi$
$397$	4.49794	0.225745	0.112873	+	0.993609i	$0.463995\pi$
$398$	16.8922	0.846728
$399$	0	0
$400$	7.47565	0.373783
$401$	−2.04458	−0.102101	−0.0510507	−	0.998696i	$-0.516257\pi$
$401$	−2.04458	−0.102101	−0.0510507	+	0.998696i	$0.516257\pi$
$402$	0	0
$403$	3.63041	0.180844
$404$	−0.921274	−0.0458351
$405$	0	0
$406$	−13.2439	−0.657285
$407$	30.0601	1.49002
$408$	0	0
$409$	23.2841	1.15132	0.575661	−	0.817688i	$-0.304745\pi$
$409$	23.2841	1.15132	0.575661	+	0.817688i	$0.304745\pi$
$410$	44.4397	2.19472
$411$	0	0
$412$	−16.9736	−0.836229
$413$	15.2618	0.750982
$414$	0	0
$415$	−11.7442	−0.576501
$416$	5.06418	0.248292
$417$	0	0
$418$	−12.8621	−0.629108
$419$	−1.50299	−0.0734260	−0.0367130	−	0.999326i	$-0.511689\pi$
$419$	−1.50299	−0.0734260	−0.0367130	+	0.999326i	$0.511689\pi$
$420$	0	0
$421$	18.1438	0.884277	0.442138	−	0.896947i	$-0.354220\pi$
$421$	18.1438	0.884277	0.442138	+	0.896947i	$0.354220\pi$
$422$	−14.0993	−0.686341
$423$	0	0
$424$	−4.57398	−0.222132
$425$	0	0
$426$	0	0
$427$	17.2181	0.833243
$428$	3.08647	0.149190
$429$	0	0
$430$	−38.5235	−1.85777
$431$	−26.2276	−1.26334	−0.631670	−	0.775237i	$-0.717630\pi$
$431$	−26.2276	−1.26334	−0.631670	+	0.775237i	$0.717630\pi$
$432$	0	0
$433$	−27.6604	−1.32928	−0.664638	−	0.747165i	$-0.731414\pi$
$433$	−27.6604	−1.32928	−0.664638	+	0.747165i	$0.731414\pi$
$434$	−1.97771	−0.0949332
$435$	0	0
$436$	−2.24123	−0.107335
$437$	26.2175	1.25415
$438$	0	0
$439$	29.6459	1.41492	0.707461	−	0.706753i	$-0.249841\pi$
$439$	29.6459	1.41492	0.707461	+	0.706753i	$0.249841\pi$
$440$	−10.3969	−0.495654
$441$	0	0
$442$	0	0
$443$	34.3209	1.63063	0.815317	−	0.579014i	$-0.196563\pi$
$443$	34.3209	1.63063	0.815317	+	0.579014i	$0.196563\pi$
$444$	0	0
$445$	48.8931	2.31776
$446$	−26.3037	−1.24551
$447$	0	0
$448$	−2.75877	−0.130340
$449$	23.8188	1.12408	0.562040	−	0.827110i	$-0.310017\pi$
$449$	23.8188	1.12408	0.562040	+	0.827110i	$0.310017\pi$
$450$	0	0
$451$	−37.0351	−1.74391
$452$	−11.4338	−0.537799
$453$	0	0
$454$	6.06418	0.284606
$455$	49.3465	2.31340
$456$	0	0
$457$	−21.4662	−1.00414	−0.502072	−	0.864826i	$-0.667429\pi$
$457$	−21.4662	−1.00414	−0.502072	+	0.864826i	$0.667429\pi$
$458$	8.56624	0.400274
$459$	0	0
$460$	21.1925	0.988107
$461$	16.8898	0.786637	0.393319	−	0.919402i	$-0.371327\pi$
$461$	16.8898	0.786637	0.393319	+	0.919402i	$0.371327\pi$
$462$	0	0
$463$	−3.75784	−0.174641	−0.0873207	−	0.996180i	$-0.527831\pi$
$463$	−3.75784	−0.174641	−0.0873207	+	0.996180i	$0.527831\pi$
$464$	4.80066	0.222865
$465$	0	0
$466$	−12.2121	−0.565716
$467$	−12.4311	−0.575242	−0.287621	−	0.957744i	$-0.592864\pi$
$467$	−12.4311	−0.575242	−0.287621	+	0.957744i	$0.592864\pi$
$468$	0	0
$469$	−15.6560	−0.722927
$470$	18.1830	0.838721
$471$	0	0
$472$	−5.53209	−0.254635
$473$	32.1046	1.47617
$474$	0	0
$475$	−32.6655	−1.49880
$476$	0	0
$477$	0	0
$478$	−11.8716	−0.542996
$479$	−11.6905	−0.534151	−0.267076	−	0.963676i	$-0.586057\pi$
$479$	−11.6905	−0.534151	−0.267076	+	0.963676i	$0.586057\pi$
$480$	0	0
$481$	51.7161	2.35805
$482$	−23.9172	−1.08940
$483$	0	0
$484$	−2.33544	−0.106156
$485$	38.9495	1.76861
$486$	0	0
$487$	−7.33544	−0.332400	−0.166200	−	0.986092i	$-0.553150\pi$
$487$	−7.33544	−0.332400	−0.166200	+	0.986092i	$0.553150\pi$
$488$	−6.24123	−0.282527
$489$	0	0
$490$	−2.15745	−0.0974637
$491$	−2.27126	−0.102500	−0.0512502	−	0.998686i	$-0.516321\pi$
$491$	−2.27126	−0.102500	−0.0512502	+	0.998686i	$0.516321\pi$
$492$	0	0
$493$	0	0
$494$	−22.1284	−0.995602
$495$	0	0
$496$	0.716881	0.0321889
$497$	−27.0452	−1.21314
$498$	0	0
$499$	−9.71957	−0.435108	−0.217554	−	0.976048i	$-0.569808\pi$
$499$	−9.71957	−0.435108	−0.217554	+	0.976048i	$0.569808\pi$
$500$	−8.74422	−0.391054
$501$	0	0
$502$	18.6040	0.830337
$503$	17.7588	0.791824	0.395912	−	0.918288i	$-0.370428\pi$
$503$	17.7588	0.791824	0.395912	+	0.918288i	$0.370428\pi$
$504$	0	0
$505$	3.25402	0.144802
$506$	−17.6614	−0.785144
$507$	0	0
$508$	−8.75877	−0.388608
$509$	7.39786	0.327904	0.163952	−	0.986468i	$-0.447576\pi$
$509$	7.39786	0.327904	0.163952	+	0.986468i	$0.447576\pi$
$510$	0	0
$511$	24.9445	1.10348
$512$	1.00000	0.0441942
$513$	0	0
$514$	4.16756	0.183823
$515$	59.9522	2.64181
$516$	0	0
$517$	−15.1533	−0.666443
$518$	−28.1729	−1.23785
$519$	0	0
$520$	−17.8871	−0.784402
$521$	4.12298	0.180631	0.0903155	−	0.995913i	$-0.471212\pi$
$521$	4.12298	0.180631	0.0903155	+	0.995913i	$0.471212\pi$
$522$	0	0
$523$	−36.5134	−1.59662	−0.798310	−	0.602246i	$-0.794272\pi$
$523$	−36.5134	−1.59662	−0.798310	+	0.602246i	$0.794272\pi$
$524$	−1.28581	−0.0561707
$525$	0	0
$526$	−7.97090	−0.347548
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	16.1557	0.701759
$531$	0	0
$532$	12.0547	0.522637
$533$	−63.7161	−2.75985
$534$	0	0
$535$	−10.9017	−0.471320
$536$	5.67499	0.245122
$537$	0	0
$538$	5.96316	0.257090
$539$	1.79797	0.0774441
$540$	0	0
$541$	−44.7939	−1.92584	−0.962919	−	0.269790i	$-0.913046\pi$
$541$	−44.7939	−1.92584	−0.962919	+	0.269790i	$0.913046\pi$
$542$	−3.07098	−0.131910
$543$	0	0
$544$	0	0
$545$	7.91622	0.339094
$546$	0	0
$547$	−22.0993	−0.944896	−0.472448	−	0.881359i	$-0.656630\pi$
$547$	−22.0993	−0.944896	−0.472448	+	0.881359i	$0.656630\pi$
$548$	−2.36959	−0.101224
$549$	0	0
$550$	22.0051	0.938299
$551$	−20.9769	−0.893646
$552$	0	0
$553$	18.2591	0.776455
$554$	−23.2472	−0.987680
$555$	0	0
$556$	5.64590	0.239439
$557$	−21.9905	−0.931768	−0.465884	−	0.884846i	$-0.654264\pi$
$557$	−21.9905	−0.931768	−0.465884	+	0.884846i	$0.654264\pi$
$558$	0	0
$559$	55.2336	2.33613
$560$	9.74422	0.411769
$561$	0	0
$562$	−21.7297	−0.916611
$563$	31.9486	1.34647	0.673237	−	0.739427i	$-0.264903\pi$
$563$	31.9486	1.34647	0.673237	+	0.739427i	$0.264903\pi$
$564$	0	0
$565$	40.3851	1.69901
$566$	−17.1925	−0.722656
$567$	0	0
$568$	9.80335	0.411339
$569$	30.9614	1.29797	0.648985	−	0.760801i	$-0.275194\pi$
$569$	30.9614	1.29797	0.648985	+	0.760801i	$0.275194\pi$
$570$	0	0
$571$	−16.5080	−0.690840	−0.345420	−	0.938448i	$-0.612264\pi$
$571$	−16.5080	−0.690840	−0.345420	+	0.938448i	$0.612264\pi$
$572$	14.9067	0.623282
$573$	0	0
$574$	34.7101	1.44877
$575$	−44.8539	−1.87054
$576$	0	0
$577$	−18.9026	−0.786926	−0.393463	−	0.919340i	$-0.628723\pi$
$577$	−18.9026	−0.786926	−0.393463	+	0.919340i	$0.628723\pi$
$578$	0	0
$579$	0	0
$580$	−16.9564	−0.704074
$581$	−9.17293	−0.380557
$582$	0	0
$583$	−13.4638	−0.557613
$584$	−9.04189	−0.374156
$585$	0	0
$586$	9.20708	0.380341
$587$	−7.17799	−0.296267	−0.148134	−	0.988967i	$-0.547327\pi$
$587$	−7.17799	−0.296267	−0.148134	+	0.988967i	$0.547327\pi$
$588$	0	0
$589$	−3.13247	−0.129071
$590$	19.5398	0.804442
$591$	0	0
$592$	10.2121	0.419716
$593$	−26.2823	−1.07928	−0.539642	−	0.841894i	$-0.681441\pi$
$593$	−26.2823	−1.07928	−0.539642	+	0.841894i	$0.681441\pi$
$594$	0	0
$595$	0	0
$596$	−11.4260	−0.468028
$597$	0	0
$598$	−30.3851	−1.24254
$599$	28.3114	1.15677	0.578386	−	0.815763i	$-0.303683\pi$
$599$	28.3114	1.15677	0.578386	+	0.815763i	$0.303683\pi$
$600$	0	0
$601$	−11.4406	−0.466671	−0.233335	−	0.972396i	$-0.574964\pi$
$601$	−11.4406	−0.466671	−0.233335	+	0.972396i	$0.574964\pi$
$602$	−30.0892	−1.22634
$603$	0	0
$604$	11.7510	0.478143
$605$	8.24897	0.335368
$606$	0	0
$607$	32.4911	1.31877	0.659387	−	0.751804i	$-0.270816\pi$
$607$	32.4911	1.31877	0.659387	+	0.751804i	$0.270816\pi$
$608$	−4.36959	−0.177210
$609$	0	0
$610$	22.0446	0.892559
$611$	−26.0702	−1.05469
$612$	0	0
$613$	−14.2959	−0.577406	−0.288703	−	0.957419i	$-0.593224\pi$
$613$	−14.2959	−0.577406	−0.288703	+	0.957419i	$0.593224\pi$
$614$	2.90673	0.117306
$615$	0	0
$616$	−8.12061	−0.327189
$617$	−41.4356	−1.66814	−0.834068	−	0.551662i	$-0.813994\pi$
$617$	−41.4356	−1.66814	−0.834068	+	0.551662i	$0.813994\pi$
$618$	0	0
$619$	−8.05468	−0.323745	−0.161873	−	0.986812i	$-0.551753\pi$
$619$	−8.05468	−0.323745	−0.161873	+	0.986812i	$0.551753\pi$
$620$	−2.53209	−0.101691
$621$	0	0
$622$	13.0351	0.522659
$623$	38.1884	1.52999
$624$	0	0
$625$	−6.49289	−0.259716
$626$	3.22163	0.128762
$627$	0	0
$628$	−3.51754	−0.140365
$629$	0	0
$630$	0	0
$631$	10.5885	0.421523	0.210761	−	0.977538i	$-0.432406\pi$
$631$	10.5885	0.421523	0.210761	+	0.977538i	$0.432406\pi$
$632$	−6.61856	−0.263272
$633$	0	0
$634$	19.0838	0.757914
$635$	30.9368	1.22769
$636$	0	0
$637$	3.09327	0.122560
$638$	14.1310	0.559453
$639$	0	0
$640$	−3.53209	−0.139618
$641$	32.5526	1.28575	0.642876	−	0.765971i	$-0.277741\pi$
$641$	32.5526	1.28575	0.642876	+	0.765971i	$0.277741\pi$
$642$	0	0
$643$	10.5817	0.417302	0.208651	−	0.977990i	$-0.433093\pi$
$643$	10.5817	0.417302	0.208651	+	0.977990i	$0.433093\pi$
$644$	16.5526	0.652265
$645$	0	0
$646$	0	0
$647$	25.4884	1.00205	0.501027	−	0.865432i	$-0.332956\pi$
$647$	25.4884	1.00205	0.501027	+	0.865432i	$0.332956\pi$
$648$	0	0
$649$	−16.2841	−0.639205
$650$	37.8580	1.48491
$651$	0	0
$652$	−6.32501	−0.247706
$653$	20.6108	0.806564	0.403282	−	0.915076i	$-0.367869\pi$
$653$	20.6108	0.806564	0.403282	+	0.915076i	$0.367869\pi$
$654$	0	0
$655$	4.54158	0.177454
$656$	−12.5817	−0.491234
$657$	0	0
$658$	14.2020	0.553653
$659$	41.5117	1.61706	0.808532	−	0.588452i	$-0.200262\pi$
$659$	41.5117	1.61706	0.808532	+	0.588452i	$0.200262\pi$
$660$	0	0
$661$	−17.0060	−0.661456	−0.330728	−	0.943726i	$-0.607294\pi$
$661$	−17.0060	−0.661456	−0.330728	+	0.943726i	$0.607294\pi$
$662$	3.43376	0.133457
$663$	0	0
$664$	3.32501	0.129035
$665$	−42.5782	−1.65111
$666$	0	0
$667$	−28.8040	−1.11529
$668$	14.3405	0.554850
$669$	0	0
$670$	−20.0446	−0.774390
$671$	−18.3715	−0.709222
$672$	0	0
$673$	20.0392	0.772454	0.386227	−	0.922404i	$-0.373778\pi$
$673$	20.0392	0.772454	0.386227	+	0.922404i	$0.373778\pi$
$674$	−35.7202	−1.37589
$675$	0	0
$676$	12.6459	0.486381
$677$	47.9823	1.84411	0.922054	−	0.387061i	$-0.126510\pi$
$677$	47.9823	1.84411	0.922054	+	0.387061i	$0.126510\pi$
$678$	0	0
$679$	30.4219	1.16749
$680$	0	0
$681$	0	0
$682$	2.11019	0.0808032
$683$	−33.4739	−1.28084	−0.640422	−	0.768024i	$-0.721240\pi$
$683$	−33.4739	−1.28084	−0.640422	+	0.768024i	$0.721240\pi$
$684$	0	0
$685$	8.36959	0.319785
$686$	17.6263	0.672975
$687$	0	0
$688$	10.9067	0.415815
$689$	−23.1634	−0.882457
$690$	0	0
$691$	20.4979	0.779778	0.389889	−	0.920862i	$-0.372513\pi$
$691$	20.4979	0.779778	0.389889	+	0.920862i	$0.372513\pi$
$692$	0.815207	0.0309895
$693$	0	0
$694$	1.10782	0.0420523
$695$	−19.9418	−0.756436
$696$	0	0
$697$	0	0
$698$	11.2371	0.425331
$699$	0	0
$700$	−20.6236	−0.779499
$701$	−51.4415	−1.94292	−0.971459	−	0.237206i	$-0.923768\pi$
$701$	−51.4415	−1.94292	−0.971459	+	0.237206i	$0.923768\pi$
$702$	0	0
$703$	−44.6228	−1.68298
$704$	2.94356	0.110940
$705$	0	0
$706$	11.2608	0.423807
$707$	2.54158	0.0955861
$708$	0	0
$709$	−10.1438	−0.380960	−0.190480	−	0.981691i	$-0.561004\pi$
$709$	−10.1438	−0.380960	−0.190480	+	0.981691i	$0.561004\pi$
$710$	−34.6263	−1.29950
$711$	0	0
$712$	−13.8425	−0.518771
$713$	−4.30129	−0.161085
$714$	0	0
$715$	−52.6519	−1.96907
$716$	−14.0642	−0.525603
$717$	0	0
$718$	−2.04458	−0.0763030
$719$	15.4047	0.574497	0.287249	−	0.957856i	$-0.407259\pi$
$719$	15.4047	0.574497	0.287249	+	0.957856i	$0.407259\pi$
$720$	0	0
$721$	46.8262	1.74390
$722$	0.0932736	0.00347128
$723$	0	0
$724$	−20.3405	−0.755948
$725$	35.8881	1.33285
$726$	0	0
$727$	−23.1097	−0.857091	−0.428545	−	0.903520i	$-0.640974\pi$
$727$	−23.1097	−0.857091	−0.428545	+	0.903520i	$0.640974\pi$
$728$	−13.9709	−0.517796
$729$	0	0
$730$	31.9368	1.18203
$731$	0	0
$732$	0	0
$733$	−4.20676	−0.155380	−0.0776901	−	0.996978i	$-0.524754\pi$
$733$	−4.20676	−0.155380	−0.0776901	+	0.996978i	$0.524754\pi$
$734$	24.4097	0.900979
$735$	0	0
$736$	−6.00000	−0.221163
$737$	16.7047	0.615325
$738$	0	0
$739$	7.79797	0.286853	0.143427	−	0.989661i	$-0.454188\pi$
$739$	7.79797	0.286853	0.143427	+	0.989661i	$0.454188\pi$
$740$	−36.0702	−1.32597
$741$	0	0
$742$	12.6186	0.463242
$743$	11.5567	0.423976	0.211988	−	0.977272i	$-0.432006\pi$
$743$	11.5567	0.423976	0.211988	+	0.977272i	$0.432006\pi$
$744$	0	0
$745$	40.3577	1.47859
$746$	27.3756	1.00229
$747$	0	0
$748$	0	0
$749$	−8.51485	−0.311126
$750$	0	0
$751$	−13.9172	−0.507844	−0.253922	−	0.967225i	$-0.581721\pi$
$751$	−13.9172	−0.507844	−0.253922	+	0.967225i	$0.581721\pi$
$752$	−5.14796	−0.187727
$753$	0	0
$754$	24.3114	0.885369
$755$	−41.5057	−1.51055
$756$	0	0
$757$	−18.0702	−0.656771	−0.328386	−	0.944544i	$-0.606505\pi$
$757$	−18.0702	−0.656771	−0.328386	+	0.944544i	$0.606505\pi$
$758$	22.4979	0.817162
$759$	0	0
$760$	15.4338	0.559841
$761$	0.241230	0.00874456	0.00437228	−	0.999990i	$-0.498608\pi$
$761$	0.241230	0.00874456	0.00437228	+	0.999990i	$0.498608\pi$
$762$	0	0
$763$	6.18304	0.223841
$764$	−15.6459	−0.566049
$765$	0	0
$766$	−16.5972	−0.599681
$767$	−28.0155	−1.01158
$768$	0	0
$769$	12.5963	0.454233	0.227116	−	0.973868i	$-0.427070\pi$
$769$	12.5963	0.454233	0.227116	+	0.973868i	$0.427070\pi$
$770$	28.6827	1.03365
$771$	0	0
$772$	10.2713	0.369671
$773$	−33.4807	−1.20422	−0.602109	−	0.798414i	$-0.705673\pi$
$773$	−33.4807	−1.20422	−0.602109	+	0.798414i	$0.705673\pi$
$774$	0	0
$775$	5.35916	0.192507
$776$	−11.0273	−0.395858
$777$	0	0
$778$	17.2249	0.617544
$779$	54.9769	1.96975
$780$	0	0
$781$	28.8568	1.03258
$782$	0	0
$783$	0	0
$784$	0.610815	0.0218148
$785$	12.4243	0.443441
$786$	0	0
$787$	25.9181	0.923880	0.461940	−	0.886911i	$-0.347154\pi$
$787$	25.9181	0.923880	0.461940	+	0.886911i	$0.347154\pi$
$788$	8.72967	0.310982
$789$	0	0
$790$	23.3773	0.831728
$791$	31.5431	1.12154
$792$	0	0
$793$	−31.6067	−1.12239
$794$	4.49794	0.159626
$795$	0	0
$796$	16.8922	0.598727
$797$	−55.5580	−1.96797	−0.983983	−	0.178264i	$-0.942952\pi$
$797$	−55.5580	−1.96797	−0.983983	+	0.178264i	$0.942952\pi$
$798$	0	0
$799$	0	0
$800$	7.47565	0.264304
$801$	0	0
$802$	−2.04458	−0.0721965
$803$	−26.6154	−0.939236
$804$	0	0
$805$	−58.4653	−2.06063
$806$	3.63041	0.127876
$807$	0	0
$808$	−0.921274	−0.0324103
$809$	17.9317	0.630445	0.315223	−	0.949018i	$-0.397921\pi$
$809$	17.9317	0.630445	0.315223	+	0.949018i	$0.397921\pi$
$810$	0	0
$811$	−25.2472	−0.886550	−0.443275	−	0.896386i	$-0.646183\pi$
$811$	−25.2472	−0.886550	−0.443275	+	0.896386i	$0.646183\pi$
$812$	−13.2439	−0.464770
$813$	0	0
$814$	30.0601	1.05360
$815$	22.3405	0.782553
$816$	0	0
$817$	−47.6579	−1.66734
$818$	23.2841	0.814108
$819$	0	0
$820$	44.4397	1.55190
$821$	−2.68098	−0.0935668	−0.0467834	−	0.998905i	$-0.514897\pi$
$821$	−2.68098	−0.0935668	−0.0467834	+	0.998905i	$0.514897\pi$
$822$	0	0
$823$	25.1712	0.877412	0.438706	−	0.898631i	$-0.355437\pi$
$823$	25.1712	0.877412	0.438706	+	0.898631i	$0.355437\pi$
$824$	−16.9736	−0.591303
$825$	0	0
$826$	15.2618	0.531025
$827$	11.4638	0.398635	0.199318	−	0.979935i	$-0.436127\pi$
$827$	11.4638	0.398635	0.199318	+	0.979935i	$0.436127\pi$
$828$	0	0
$829$	49.4593	1.71779	0.858897	−	0.512148i	$-0.171150\pi$
$829$	49.4593	1.71779	0.858897	+	0.512148i	$0.171150\pi$
$830$	−11.7442	−0.407648
$831$	0	0
$832$	5.06418	0.175569
$833$	0	0
$834$	0	0
$835$	−50.6519	−1.75288
$836$	−12.8621	−0.444847
$837$	0	0
$838$	−1.50299	−0.0519200
$839$	33.6560	1.16193	0.580967	−	0.813927i	$-0.302674\pi$
$839$	33.6560	1.16193	0.580967	+	0.813927i	$0.302674\pi$
$840$	0	0
$841$	−5.95367	−0.205299
$842$	18.1438	0.625278
$843$	0	0
$844$	−14.0993	−0.485317
$845$	−44.6664	−1.53657
$846$	0	0
$847$	6.44293	0.221382
$848$	−4.57398	−0.157071
$849$	0	0
$850$	0	0
$851$	−61.2728	−2.10040
$852$	0	0
$853$	−2.42427	−0.0830053	−0.0415027	−	0.999138i	$-0.513215\pi$
$853$	−2.42427	−0.0830053	−0.0415027	+	0.999138i	$0.513215\pi$
$854$	17.2181	0.589192
$855$	0	0
$856$	3.08647	0.105493
$857$	−3.21625	−0.109865	−0.0549325	−	0.998490i	$-0.517494\pi$
$857$	−3.21625	−0.109865	−0.0549325	+	0.998490i	$0.517494\pi$
$858$	0	0
$859$	23.4831	0.801232	0.400616	−	0.916246i	$-0.368796\pi$
$859$	23.4831	0.801232	0.400616	+	0.916246i	$0.368796\pi$
$860$	−38.5235	−1.31364
$861$	0	0
$862$	−26.2276	−0.893316
$863$	−44.4344	−1.51256	−0.756282	−	0.654246i	$-0.772986\pi$
$863$	−44.4344	−1.51256	−0.756282	+	0.654246i	$0.772986\pi$
$864$	0	0
$865$	−2.87939	−0.0979020
$866$	−27.6604	−0.939940
$867$	0	0
$868$	−1.97771	−0.0671279
$869$	−19.4821	−0.660886
$870$	0	0
$871$	28.7392	0.973790
$872$	−2.24123	−0.0758976
$873$	0	0
$874$	26.2175	0.886821
$875$	24.1233	0.815516
$876$	0	0
$877$	9.07966	0.306598	0.153299	−	0.988180i	$-0.451010\pi$
$877$	9.07966	0.306598	0.153299	+	0.988180i	$0.451010\pi$
$878$	29.6459	1.00050
$879$	0	0
$880$	−10.3969	−0.350480
$881$	−25.5912	−0.862190	−0.431095	−	0.902307i	$-0.641873\pi$
$881$	−25.5912	−0.862190	−0.431095	+	0.902307i	$0.641873\pi$
$882$	0	0
$883$	−22.3506	−0.752157	−0.376079	−	0.926588i	$-0.622728\pi$
$883$	−22.3506	−0.752157	−0.376079	+	0.926588i	$0.622728\pi$
$884$	0	0
$885$	0	0
$886$	34.3209	1.15303
$887$	−26.4688	−0.888737	−0.444368	−	0.895844i	$-0.646572\pi$
$887$	−26.4688	−0.888737	−0.444368	+	0.895844i	$0.646572\pi$
$888$	0	0
$889$	24.1634	0.810416
$890$	48.8931	1.63890
$891$	0	0
$892$	−26.3037	−0.880711
$893$	22.4944	0.752747
$894$	0	0
$895$	49.6759	1.66048
$896$	−2.75877	−0.0921641
$897$	0	0
$898$	23.8188	0.794845
$899$	3.44150	0.114781
$900$	0	0
$901$	0	0
$902$	−37.0351	−1.23313
$903$	0	0
$904$	−11.4338	−0.380281
$905$	71.8444	2.38819
$906$	0	0
$907$	55.5722	1.84525	0.922623	−	0.385704i	$-0.126041\pi$
$907$	55.5722	1.84525	0.922623	+	0.385704i	$0.126041\pi$
$908$	6.06418	0.201247
$909$	0	0
$910$	49.3465	1.63582
$911$	49.0506	1.62512	0.812559	−	0.582879i	$-0.198074\pi$
$911$	49.0506	1.62512	0.812559	+	0.582879i	$0.198074\pi$
$912$	0	0
$913$	9.78737	0.323915
$914$	−21.4662	−0.710037
$915$	0	0
$916$	8.56624	0.283036
$917$	3.54725	0.117140
$918$	0	0
$919$	−15.2020	−0.501469	−0.250734	−	0.968056i	$-0.580672\pi$
$919$	−15.2020	−0.501469	−0.250734	+	0.968056i	$0.580672\pi$
$920$	21.1925	0.698697
$921$	0	0
$922$	16.8898	0.556236
$923$	49.6459	1.63411
$924$	0	0
$925$	76.3424	2.51012
$926$	−3.75784	−0.123490
$927$	0	0
$928$	4.80066	0.157589
$929$	−22.8966	−0.751214	−0.375607	−	0.926779i	$-0.622566\pi$
$929$	−22.8966	−0.751214	−0.375607	+	0.926779i	$0.622566\pi$
$930$	0	0
$931$	−2.66901	−0.0874731
$932$	−12.2121	−0.400022
$933$	0	0
$934$	−12.4311	−0.406757
$935$	0	0
$936$	0	0
$937$	3.34554	0.109294	0.0546470	−	0.998506i	$-0.482597\pi$
$937$	3.34554	0.109294	0.0546470	+	0.998506i	$0.482597\pi$
$938$	−15.6560	−0.511187
$939$	0	0
$940$	18.1830	0.593065
$941$	−11.5877	−0.377748	−0.188874	−	0.982001i	$-0.560484\pi$
$941$	−11.5877	−0.377748	−0.188874	+	0.982001i	$0.560484\pi$
$942$	0	0
$943$	75.4903	2.45830
$944$	−5.53209	−0.180054
$945$	0	0
$946$	32.1046	1.04381
$947$	−33.6792	−1.09443	−0.547214	−	0.836993i	$-0.684312\pi$
$947$	−33.6792	−1.09443	−0.547214	+	0.836993i	$0.684312\pi$
$948$	0	0
$949$	−45.7897	−1.48640
$950$	−32.6655	−1.05981
$951$	0	0
$952$	0	0
$953$	13.9655	0.452388	0.226194	−	0.974082i	$-0.427372\pi$
$953$	13.9655	0.452388	0.226194	+	0.974082i	$0.427372\pi$
$954$	0	0
$955$	55.2627	1.78826
$956$	−11.8716	−0.383956
$957$	0	0
$958$	−11.6905	−0.377702
$959$	6.53714	0.211095
$960$	0	0
$961$	−30.4861	−0.983422
$962$	51.7161	1.66739
$963$	0	0
$964$	−23.9172	−0.770320
$965$	−36.2790	−1.16786
$966$	0	0
$967$	47.0515	1.51307	0.756537	−	0.653951i	$-0.226890\pi$
$967$	47.0515	1.51307	0.756537	+	0.653951i	$0.226890\pi$
$968$	−2.33544	−0.0750638
$969$	0	0
$970$	38.9495	1.25059
$971$	−30.2044	−0.969305	−0.484653	−	0.874707i	$-0.661054\pi$
$971$	−30.2044	−0.969305	−0.484653	+	0.874707i	$0.661054\pi$
$972$	0	0
$973$	−15.5757	−0.499335
$974$	−7.33544	−0.235043
$975$	0	0
$976$	−6.24123	−0.199777
$977$	−12.4534	−0.398418	−0.199209	−	0.979957i	$-0.563837\pi$
$977$	−12.4534	−0.398418	−0.199209	+	0.979957i	$0.563837\pi$
$978$	0	0
$979$	−40.7464	−1.30226
$980$	−2.15745	−0.0689173
$981$	0	0
$982$	−2.27126	−0.0724788
$983$	−42.9769	−1.37075	−0.685375	−	0.728190i	$-0.740362\pi$
$983$	−42.9769	−1.37075	−0.685375	+	0.728190i	$0.740362\pi$
$984$	0	0
$985$	−30.8340	−0.982453
$986$	0	0
$987$	0	0
$988$	−22.1284	−0.703997
$989$	−65.4404	−2.08088
$990$	0	0
$991$	−9.24123	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$991$	−9.24123	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$992$	0.716881	0.0227610
$993$	0	0
$994$	−27.0452	−0.857821
$995$	−59.6647	−1.89150
$996$	0	0
$997$	28.2567	0.894899	0.447450	−	0.894309i	$-0.352332\pi$
$997$	28.2567	0.894899	0.447450	+	0.894309i	$0.352332\pi$
$998$	−9.71957	−0.307668
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.bm.1.1		3
3.2	odd	2		1734.2.a.q.1.3	yes	3
17.16	even	2		5202.2.a.bp.1.3		3
51.2	odd	8		1734.2.f.n.1483.4		12
51.8	odd	8		1734.2.f.n.829.3		12
51.26	odd	8		1734.2.f.n.829.4		12
51.32	odd	8		1734.2.f.n.1483.3		12
51.38	odd	4		1734.2.b.j.577.1		6
51.47	odd	4		1734.2.b.j.577.6		6
51.50	odd	2		1734.2.a.p.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1734.2.a.p.1.1	✓	3	51.50	odd	2
1734.2.a.q.1.3	yes	3	3.2	odd	2
1734.2.b.j.577.1		6	51.38	odd	4
1734.2.b.j.577.6		6	51.47	odd	4
1734.2.f.n.829.3		12	51.8	odd	8
1734.2.f.n.829.4		12	51.26	odd	8
1734.2.f.n.1483.3		12	51.32	odd	8
1734.2.f.n.1483.4		12	51.2	odd	8
5202.2.a.bm.1.1		3	1.1	even	1	trivial
5202.2.a.bp.1.3		3	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5202.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.53209 q^{5} -2.75877 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.53209 q^{5} -2.75877 q^{7} +1.00000 q^{8} -3.53209 q^{10} +2.94356 q^{11} +5.06418 q^{13} -2.75877 q^{14} +1.00000 q^{16} -4.36959 q^{19} -3.53209 q^{20} +2.94356 q^{22} -6.00000 q^{23} +7.47565 q^{25} +5.06418 q^{26} -2.75877 q^{28} +4.80066 q^{29} +0.716881 q^{31} +1.00000 q^{32} +9.74422 q^{35} +10.2121 q^{37} -4.36959 q^{38} -3.53209 q^{40} -12.5817 q^{41} +10.9067 q^{43} +2.94356 q^{44} -6.00000 q^{46} -5.14796 q^{47} +0.610815 q^{49} +7.47565 q^{50} +5.06418 q^{52} -4.57398 q^{53} -10.3969 q^{55} -2.75877 q^{56} +4.80066 q^{58} -5.53209 q^{59} -6.24123 q^{61} +0.716881 q^{62} +1.00000 q^{64} -17.8871 q^{65} +5.67499 q^{67} +9.74422 q^{70} +9.80335 q^{71} -9.04189 q^{73} +10.2121 q^{74} -4.36959 q^{76} -8.12061 q^{77} -6.61856 q^{79} -3.53209 q^{80} -12.5817 q^{82} +3.32501 q^{83} +10.9067 q^{86} +2.94356 q^{88} -13.8425 q^{89} -13.9709 q^{91} -6.00000 q^{92} -5.14796 q^{94} +15.4338 q^{95} -11.0273 q^{97} +0.610815 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} + 3 q^{8} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 18 q^{23} + 3 q^{25} + 6 q^{26} + 3 q^{28} - 6 q^{31} + 3 q^{32} + 6 q^{37} - 6 q^{38} - 6 q^{40} - 6 q^{41} + 6 q^{43} - 6 q^{44} - 18 q^{46} + 6 q^{49} + 3 q^{50} + 6 q^{52} - 6 q^{53} - 3 q^{55} + 3 q^{56} - 12 q^{59} - 30 q^{61} - 6 q^{62} + 3 q^{64} - 24 q^{65} + 12 q^{67} + 6 q^{71} - 24 q^{73} + 6 q^{74} - 6 q^{76} - 30 q^{77} - 6 q^{80} - 6 q^{82} + 15 q^{83} + 6 q^{86} - 6 q^{88} - 24 q^{89} - 6 q^{91} - 18 q^{92} + 30 q^{95} - 12 q^{97} + 6 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 + 3 * q^8 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^19 - 6 * q^20 - 6 * q^22 - 18 * q^23 + 3 * q^25 + 6 * q^26 + 3 * q^28 - 6 * q^31 + 3 * q^32 + 6 * q^37 - 6 * q^38 - 6 * q^40 - 6 * q^41 + 6 * q^43 - 6 * q^44 - 18 * q^46 + 6 * q^49 + 3 * q^50 + 6 * q^52 - 6 * q^53 - 3 * q^55 + 3 * q^56 - 12 * q^59 - 30 * q^61 - 6 * q^62 + 3 * q^64 - 24 * q^65 + 12 * q^67 + 6 * q^71 - 24 * q^73 + 6 * q^74 - 6 * q^76 - 30 * q^77 - 6 * q^80 - 6 * q^82 + 15 * q^83 + 6 * q^86 - 6 * q^88 - 24 * q^89 - 6 * q^91 - 18 * q^92 + 30 * q^95 - 12 * q^97 + 6 * q^98

Introduction

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