LMFDB - Embedded newform 8649.2.a.bj.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, 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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.1697203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 5x^{6} + 12x^{5} + 9x^{4} - 12x^{3} - 5x^{2} + 3x + 1 $ x^8 - 3x^7 - 5x^6 + 12x^5 + 9x^4 - 12x^3 - 5x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 93)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.306487$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30649 q^{2} -0.293092 q^{4} +1.68211 q^{5} +2.45220 q^{7} -2.99589 q^{8} +O(q^{10})$	\(q+1.30649 q^{2} -0.293092 q^{4} +1.68211 q^{5} +2.45220 q^{7} -2.99589 q^{8} +2.19765 q^{10} -1.78855 q^{11} -0.377861 q^{13} +3.20377 q^{14} -3.32791 q^{16} +1.34747 q^{17} -5.58452 q^{19} -0.493012 q^{20} -2.33672 q^{22} -3.06837 q^{23} -2.17052 q^{25} -0.493670 q^{26} -0.718721 q^{28} +9.39471 q^{29} +1.64391 q^{32} +1.76045 q^{34} +4.12487 q^{35} +9.41287 q^{37} -7.29611 q^{38} -5.03942 q^{40} -7.35670 q^{41} +5.93180 q^{43} +0.524209 q^{44} -4.00879 q^{46} -1.70533 q^{47} -0.986708 q^{49} -2.83575 q^{50} +0.110748 q^{52} +3.77129 q^{53} -3.00853 q^{55} -7.34654 q^{56} +12.2741 q^{58} +13.3731 q^{59} +9.06449 q^{61} +8.80358 q^{64} -0.635602 q^{65} +14.6552 q^{67} -0.394933 q^{68} +5.38908 q^{70} +1.69200 q^{71} +12.8536 q^{73} +12.2978 q^{74} +1.63678 q^{76} -4.38588 q^{77} -5.85837 q^{79} -5.59791 q^{80} -9.61143 q^{82} +15.6324 q^{83} +2.26659 q^{85} +7.74982 q^{86} +5.35831 q^{88} -0.312052 q^{89} -0.926590 q^{91} +0.899315 q^{92} -2.22799 q^{94} -9.39377 q^{95} -10.8383 q^{97} -1.28912 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 6 q^{7}+O(q^{10}) $ 8 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 6 * q^7	$ 8 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 6 q^{7} + q^{10} + 12 q^{13} + 3 q^{14} + 3 q^{16} - 2 q^{17} - 8 q^{19} + 5 q^{20} + 4 q^{22} - 4 q^{23} - 12 q^{25} + 18 q^{26} - 15 q^{28} + 6 q^{29} + 17 q^{34} - 9 q^{35} + 8 q^{37} + 7 q^{38} + 13 q^{40} + 20 q^{41} + 14 q^{43} - 25 q^{44} + 9 q^{46} - 6 q^{47} - 20 q^{49} - 6 q^{50} + 30 q^{52} + 30 q^{53} + 19 q^{55} - 6 q^{56} + 19 q^{58} + 20 q^{59} + 13 q^{61} - 26 q^{64} + 27 q^{65} + 32 q^{67} + 15 q^{68} - 6 q^{70} + 17 q^{71} - 12 q^{73} + 10 q^{74} - 20 q^{76} - 18 q^{77} + 30 q^{79} + 21 q^{80} - 23 q^{82} - 12 q^{83} + 28 q^{85} + 46 q^{86} + 22 q^{88} - 6 q^{89} - 15 q^{91} + 70 q^{92} + 44 q^{94} + 36 q^{95} - 39 q^{97} - 32 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 6 * q^7 + q^10 + 12 * q^13 + 3 * q^14 + 3 * q^16 - 2 * q^17 - 8 * q^19 + 5 * q^20 + 4 * q^22 - 4 * q^23 - 12 * q^25 + 18 * q^26 - 15 * q^28 + 6 * q^29 + 17 * q^34 - 9 * q^35 + 8 * q^37 + 7 * q^38 + 13 * q^40 + 20 * q^41 + 14 * q^43 - 25 * q^44 + 9 * q^46 - 6 * q^47 - 20 * q^49 - 6 * q^50 + 30 * q^52 + 30 * q^53 + 19 * q^55 - 6 * q^56 + 19 * q^58 + 20 * q^59 + 13 * q^61 - 26 * q^64 + 27 * q^65 + 32 * q^67 + 15 * q^68 - 6 * q^70 + 17 * q^71 - 12 * q^73 + 10 * q^74 - 20 * q^76 - 18 * q^77 + 30 * q^79 + 21 * q^80 - 23 * q^82 - 12 * q^83 + 28 * q^85 + 46 * q^86 + 22 * q^88 - 6 * q^89 - 15 * q^91 + 70 * q^92 + 44 * q^94 + 36 * q^95 - 39 * q^97 - 32 * 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 $

$2$	1.30649	0.923826	0.461913	−	0.886925i	$-0.347163\pi$
$2$	1.30649	0.923826	0.461913	+	0.886925i	$0.347163\pi$
$3$	0	0
$3$	0	0
$4$	−0.293092	−0.146546
$5$	1.68211	0.752261	0.376131	−	0.926567i	$-0.377254\pi$
$5$	1.68211	0.752261	0.376131	+	0.926567i	$0.377254\pi$
$6$	0	0
$7$	2.45220	0.926845	0.463423	−	0.886137i	$-0.346621\pi$
$7$	2.45220	0.926845	0.463423	+	0.886137i	$0.346621\pi$
$8$	−2.99589	−1.05921
$9$	0	0
$10$	2.19765	0.694958
$11$	−1.78855	−0.539268	−0.269634	−	0.962963i	$-0.586903\pi$
$11$	−1.78855	−0.539268	−0.269634	+	0.962963i	$0.586903\pi$
$12$	0	0
$13$	−0.377861	−0.104800	−0.0523998	−	0.998626i	$-0.516687\pi$
$13$	−0.377861	−0.104800	−0.0523998	+	0.998626i	$0.516687\pi$
$14$	3.20377	0.856243
$15$	0	0
$16$	−3.32791	−0.831978
$17$	1.34747	0.326810	0.163405	−	0.986559i	$-0.447752\pi$
$17$	1.34747	0.326810	0.163405	+	0.986559i	$0.447752\pi$
$18$	0	0
$19$	−5.58452	−1.28118	−0.640589	−	0.767884i	$-0.721310\pi$
$19$	−5.58452	−1.28118	−0.640589	+	0.767884i	$0.721310\pi$
$20$	−0.493012	−0.110241
$21$	0	0
$22$	−2.33672	−0.498190
$23$	−3.06837	−0.639800	−0.319900	−	0.947451i	$-0.603649\pi$
$23$	−3.06837	−0.639800	−0.319900	+	0.947451i	$0.603649\pi$
$24$	0	0
$25$	−2.17052	−0.434103
$26$	−0.493670	−0.0968166
$27$	0	0
$28$	−0.718721	−0.135825
$29$	9.39471	1.74455	0.872277	−	0.489012i	$-0.162643\pi$
$29$	9.39471	1.74455	0.872277	+	0.489012i	$0.162643\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	1.64391	0.290606
$33$	0	0
$34$	1.76045	0.301915
$35$	4.12487	0.697230
$36$	0	0
$37$	9.41287	1.54747	0.773733	−	0.633512i	$-0.218387\pi$
$37$	9.41287	1.54747	0.773733	+	0.633512i	$0.218387\pi$
$38$	−7.29611	−1.18358
$39$	0	0
$40$	−5.03942	−0.796802
$41$	−7.35670	−1.14892	−0.574462	−	0.818531i	$-0.694789\pi$
$41$	−7.35670	−1.14892	−0.574462	+	0.818531i	$0.694789\pi$
$42$	0	0
$43$	5.93180	0.904591	0.452296	−	0.891868i	$-0.350605\pi$
$43$	5.93180	0.904591	0.452296	+	0.891868i	$0.350605\pi$
$44$	0.524209	0.0790275
$45$	0	0
$46$	−4.00879	−0.591063
$47$	−1.70533	−0.248748	−0.124374	−	0.992235i	$-0.539692\pi$
$47$	−1.70533	−0.248748	−0.124374	+	0.992235i	$0.539692\pi$
$48$	0	0
$49$	−0.986708	−0.140958
$50$	−2.83575	−0.401036
$51$	0	0
$52$	0.110748	0.0153580
$53$	3.77129	0.518027	0.259013	−	0.965874i	$-0.416603\pi$
$53$	3.77129	0.518027	0.259013	+	0.965874i	$0.416603\pi$
$54$	0	0
$55$	−3.00853	−0.405670
$56$	−7.34654	−0.981722
$57$	0	0
$58$	12.2741	1.61166
$59$	13.3731	1.74103	0.870517	−	0.492139i	$-0.163785\pi$
$59$	13.3731	1.74103	0.870517	+	0.492139i	$0.163785\pi$
$60$	0	0
$61$	9.06449	1.16059	0.580295	−	0.814407i	$-0.302937\pi$
$61$	9.06449	1.16059	0.580295	+	0.814407i	$0.302937\pi$
$62$	0	0
$63$	0	0
$64$	8.80358	1.10045
$65$	−0.635602	−0.0788367
$66$	0	0
$67$	14.6552	1.79041	0.895206	−	0.445652i	$-0.147028\pi$
$67$	14.6552	1.79041	0.895206	+	0.445652i	$0.147028\pi$
$68$	−0.394933	−0.0478926
$69$	0	0
$70$	5.38908	0.644119
$71$	1.69200	0.200804	0.100402	−	0.994947i	$-0.467987\pi$
$71$	1.69200	0.200804	0.100402	+	0.994947i	$0.467987\pi$
$72$	0	0
$73$	12.8536	1.50440	0.752202	−	0.658933i	$-0.228992\pi$
$73$	12.8536	1.50440	0.752202	+	0.658933i	$0.228992\pi$
$74$	12.2978	1.42959
$75$	0	0
$76$	1.63678	0.187751
$77$	−4.38588	−0.499818
$78$	0	0
$79$	−5.85837	−0.659118	−0.329559	−	0.944135i	$-0.606900\pi$
$79$	−5.85837	−0.659118	−0.329559	+	0.944135i	$0.606900\pi$
$80$	−5.59791	−0.625865
$81$	0	0
$82$	−9.61143	−1.06141
$83$	15.6324	1.71588	0.857941	−	0.513748i	$-0.171743\pi$
$83$	15.6324	1.71588	0.857941	+	0.513748i	$0.171743\pi$
$84$	0	0
$85$	2.26659	0.245846
$86$	7.74982	0.835685
$87$	0	0
$88$	5.35831	0.571197
$89$	−0.312052	−0.0330775	−0.0165387	−	0.999863i	$-0.505265\pi$
$89$	−0.312052	−0.0330775	−0.0165387	+	0.999863i	$0.505265\pi$
$90$	0	0
$91$	−0.926590	−0.0971331
$92$	0.899315	0.0937601
$93$	0	0
$94$	−2.22799	−0.229800
$95$	−9.39377	−0.963780
$96$	0	0
$97$	−10.8383	−1.10046	−0.550230	−	0.835013i	$-0.685460\pi$
$97$	−10.8383	−1.10046	−0.550230	+	0.835013i	$0.685460\pi$
$98$	−1.28912	−0.130221
$99$	0	0
$100$	0.636161	0.0636161
$101$	−5.21183	−0.518596	−0.259298	−	0.965797i	$-0.583491\pi$
$101$	−5.21183	−0.518596	−0.259298	+	0.965797i	$0.583491\pi$
$102$	0	0
$103$	9.11510	0.898138	0.449069	−	0.893497i	$-0.351756\pi$
$103$	9.11510	0.898138	0.449069	+	0.893497i	$0.351756\pi$
$104$	1.13203	0.111005
$105$	0	0
$106$	4.92714	0.478566
$107$	10.4954	1.01463	0.507315	−	0.861761i	$-0.330638\pi$
$107$	10.4954	1.01463	0.507315	+	0.861761i	$0.330638\pi$
$108$	0	0
$109$	−1.75401	−0.168004	−0.0840021	−	0.996466i	$-0.526770\pi$
$109$	−1.75401	−0.168004	−0.0840021	+	0.996466i	$0.526770\pi$
$110$	−3.93061	−0.374769
$111$	0	0
$112$	−8.16071	−0.771115
$113$	9.69481	0.912011	0.456005	−	0.889977i	$-0.349280\pi$
$113$	9.69481	0.912011	0.456005	+	0.889977i	$0.349280\pi$
$114$	0	0
$115$	−5.16133	−0.481296
$116$	−2.75351	−0.255657
$117$	0	0
$118$	17.4718	1.60841
$119$	3.30427	0.302902
$120$	0	0
$121$	−7.80109	−0.709190
$122$	11.8426	1.07218
$123$	0	0
$124$	0	0
$125$	−12.0616	−1.07882
$126$	0	0
$127$	−12.5860	−1.11683	−0.558413	−	0.829563i	$-0.688590\pi$
$127$	−12.5860	−1.11683	−0.558413	+	0.829563i	$0.688590\pi$
$128$	8.21393	0.726016
$129$	0	0
$130$	−0.830406	−0.0728314
$131$	9.31191	0.813585	0.406792	−	0.913521i	$-0.366647\pi$
$131$	9.31191	0.813585	0.406792	+	0.913521i	$0.366647\pi$
$132$	0	0
$133$	−13.6944	−1.18745
$134$	19.1468	1.65403
$135$	0	0
$136$	−4.03688	−0.346160
$137$	−10.3140	−0.881184	−0.440592	−	0.897707i	$-0.645231\pi$
$137$	−10.3140	−0.881184	−0.440592	+	0.897707i	$0.645231\pi$
$138$	0	0
$139$	1.98790	0.168612	0.0843059	−	0.996440i	$-0.473133\pi$
$139$	1.98790	0.168612	0.0843059	+	0.996440i	$0.473133\pi$
$140$	−1.20896	−0.102176
$141$	0	0
$142$	2.21058	0.185508
$143$	0.675822	0.0565151
$144$	0	0
$145$	15.8029	1.31236
$146$	16.7931	1.38981
$147$	0	0
$148$	−2.75884	−0.226775
$149$	13.4332	1.10049	0.550247	−	0.835002i	$-0.314534\pi$
$149$	13.4332	1.10049	0.550247	+	0.835002i	$0.314534\pi$
$150$	0	0
$151$	1.30450	0.106158	0.0530792	−	0.998590i	$-0.483096\pi$
$151$	1.30450	0.106158	0.0530792	+	0.998590i	$0.483096\pi$
$152$	16.7306	1.35703
$153$	0	0
$154$	−5.73010	−0.461745
$155$	0	0
$156$	0	0
$157$	−1.09266	−0.0872040	−0.0436020	−	0.999049i	$-0.513883\pi$
$157$	−1.09266	−0.0872040	−0.0436020	+	0.999049i	$0.513883\pi$
$158$	−7.65388	−0.608910
$159$	0	0
$160$	2.76524	0.218611
$161$	−7.52426	−0.592995
$162$	0	0
$163$	−15.1403	−1.18588	−0.592941	−	0.805246i	$-0.702033\pi$
$163$	−15.1403	−1.18588	−0.592941	+	0.805246i	$0.702033\pi$
$164$	2.15619	0.168370
$165$	0	0
$166$	20.4236	1.58518
$167$	−3.89474	−0.301384	−0.150692	−	0.988581i	$-0.548150\pi$
$167$	−3.89474	−0.301384	−0.150692	+	0.988581i	$0.548150\pi$
$168$	0	0
$169$	−12.8572	−0.989017
$170$	2.96127	0.227119
$171$	0	0
$172$	−1.73856	−0.132564
$173$	4.83740	0.367781	0.183890	−	0.982947i	$-0.441131\pi$
$173$	4.83740	0.367781	0.183890	+	0.982947i	$0.441131\pi$
$174$	0	0
$175$	−5.32254	−0.402346
$176$	5.95214	0.448659
$177$	0	0
$178$	−0.407692	−0.0305578
$179$	−8.69719	−0.650059	−0.325029	−	0.945704i	$-0.605374\pi$
$179$	−8.69719	−0.650059	−0.325029	+	0.945704i	$0.605374\pi$
$180$	0	0
$181$	18.3812	1.36626	0.683132	−	0.730295i	$-0.260617\pi$
$181$	18.3812	1.36626	0.683132	+	0.730295i	$0.260617\pi$
$182$	−1.21058	−0.0897340
$183$	0	0
$184$	9.19252	0.677681
$185$	15.8335	1.16410
$186$	0	0
$187$	−2.41002	−0.176238
$188$	0.499819	0.0364530
$189$	0	0
$190$	−12.2728	−0.890365
$191$	20.1026	1.45457	0.727286	−	0.686335i	$-0.240781\pi$
$191$	20.1026	1.45457	0.727286	+	0.686335i	$0.240781\pi$
$192$	0	0
$193$	5.11792	0.368396	0.184198	−	0.982889i	$-0.441031\pi$
$193$	5.11792	0.368396	0.184198	+	0.982889i	$0.441031\pi$
$194$	−14.1601	−1.01663
$195$	0	0
$196$	0.289196	0.0206569
$197$	18.7857	1.33843	0.669213	−	0.743071i	$-0.266631\pi$
$197$	18.7857	1.33843	0.669213	+	0.743071i	$0.266631\pi$
$198$	0	0
$199$	21.4779	1.52253	0.761265	−	0.648440i	$-0.224578\pi$
$199$	21.4779	1.52253	0.761265	+	0.648440i	$0.224578\pi$
$200$	6.50264	0.459806
$201$	0	0
$202$	−6.80918	−0.479092
$203$	23.0377	1.61693
$204$	0	0
$205$	−12.3748	−0.864291
$206$	11.9088	0.829723
$207$	0	0
$208$	1.25749	0.0871911
$209$	9.98820	0.690898
$210$	0	0
$211$	−24.1276	−1.66101	−0.830505	−	0.557012i	$-0.811948\pi$
$211$	−24.1276	−1.66101	−0.830505	+	0.557012i	$0.811948\pi$
$212$	−1.10533	−0.0759147
$213$	0	0
$214$	13.7121	0.937342
$215$	9.97793	0.680489
$216$	0	0
$217$	0	0
$218$	−2.29160	−0.155207
$219$	0	0
$220$	0.881776	0.0594494
$221$	−0.509156	−0.0342496
$222$	0	0
$223$	12.6033	0.843982	0.421991	−	0.906600i	$-0.361331\pi$
$223$	12.6033	0.843982	0.421991	+	0.906600i	$0.361331\pi$
$224$	4.03121	0.269346
$225$	0	0
$226$	12.6661	0.842539
$227$	6.16746	0.409349	0.204674	−	0.978830i	$-0.434386\pi$
$227$	6.16746	0.409349	0.204674	+	0.978830i	$0.434386\pi$
$228$	0	0
$229$	−21.6555	−1.43104	−0.715518	−	0.698595i	$-0.753809\pi$
$229$	−21.6555	−1.43104	−0.715518	+	0.698595i	$0.753809\pi$
$230$	−6.74321	−0.444634
$231$	0	0
$232$	−28.1456	−1.84785
$233$	4.09915	0.268544	0.134272	−	0.990944i	$-0.457130\pi$
$233$	4.09915	0.268544	0.134272	+	0.990944i	$0.457130\pi$
$234$	0	0
$235$	−2.86855	−0.187124
$236$	−3.91956	−0.255141
$237$	0	0
$238$	4.31699	0.279829
$239$	8.12753	0.525726	0.262863	−	0.964833i	$-0.415333\pi$
$239$	8.12753	0.525726	0.262863	+	0.964833i	$0.415333\pi$
$240$	0	0
$241$	−13.4478	−0.866250	−0.433125	−	0.901334i	$-0.642589\pi$
$241$	−13.4478	−0.866250	−0.433125	+	0.901334i	$0.642589\pi$
$242$	−10.1920	−0.655168
$243$	0	0
$244$	−2.65673	−0.170080
$245$	−1.65975	−0.106037
$246$	0	0
$247$	2.11017	0.134267
$248$	0	0
$249$	0	0
$250$	−15.7583	−0.996642
$251$	−10.4348	−0.658638	−0.329319	−	0.944219i	$-0.606819\pi$
$251$	−10.4348	−0.658638	−0.329319	+	0.944219i	$0.606819\pi$
$252$	0	0
$253$	5.48793	0.345023
$254$	−16.4434	−1.03175
$255$	0	0
$256$	−6.87576	−0.429735
$257$	−23.2697	−1.45152	−0.725762	−	0.687946i	$-0.758512\pi$
$257$	−23.2697	−1.45152	−0.725762	+	0.687946i	$0.758512\pi$
$258$	0	0
$259$	23.0822	1.43426
$260$	0.186290	0.0115532
$261$	0	0
$262$	12.1659	0.751611
$263$	2.32340	0.143267	0.0716333	−	0.997431i	$-0.477179\pi$
$263$	2.32340	0.143267	0.0716333	+	0.997431i	$0.477179\pi$
$264$	0	0
$265$	6.34371	0.389691
$266$	−17.8915	−1.09700
$267$	0	0
$268$	−4.29531	−0.262378
$269$	−30.6869	−1.87101	−0.935507	−	0.353309i	$-0.885056\pi$
$269$	−30.6869	−1.87101	−0.935507	+	0.353309i	$0.885056\pi$
$270$	0	0
$271$	−6.02634	−0.366074	−0.183037	−	0.983106i	$-0.558593\pi$
$271$	−6.02634	−0.366074	−0.183037	+	0.983106i	$0.558593\pi$
$272$	−4.48427	−0.271899
$273$	0	0
$274$	−13.4751	−0.814061
$275$	3.88207	0.234098
$276$	0	0
$277$	1.43090	0.0859743	0.0429871	−	0.999076i	$-0.486313\pi$
$277$	1.43090	0.0859743	0.0429871	+	0.999076i	$0.486313\pi$
$278$	2.59717	0.155768
$279$	0	0
$280$	−12.3577	−0.738512
$281$	−27.5518	−1.64360	−0.821802	−	0.569773i	$-0.807031\pi$
$281$	−27.5518	−1.64360	−0.821802	+	0.569773i	$0.807031\pi$
$282$	0	0
$283$	10.9180	0.649009	0.324504	−	0.945884i	$-0.394802\pi$
$283$	10.9180	0.649009	0.324504	+	0.945884i	$0.394802\pi$
$284$	−0.495913	−0.0294270
$285$	0	0
$286$	0.882953	0.0522101
$287$	−18.0401	−1.06487
$288$	0	0
$289$	−15.1843	−0.893195
$290$	20.6463	1.21239
$291$	0	0
$292$	−3.76730	−0.220464
$293$	25.6197	1.49672	0.748359	−	0.663294i	$-0.230842\pi$
$293$	25.6197	1.49672	0.748359	+	0.663294i	$0.230842\pi$
$294$	0	0
$295$	22.4950	1.30971
$296$	−28.2000	−1.63909
$297$	0	0
$298$	17.5504	1.01666
$299$	1.15942	0.0670508
$300$	0	0
$301$	14.5460	0.838416
$302$	1.70431	0.0980718
$303$	0	0
$304$	18.5848	1.06591
$305$	15.2474	0.873066
$306$	0	0
$307$	8.26420	0.471663	0.235831	−	0.971794i	$-0.424219\pi$
$307$	8.26420	0.471663	0.235831	+	0.971794i	$0.424219\pi$
$308$	1.28547	0.0732463
$309$	0	0
$310$	0	0
$311$	6.92587	0.392730	0.196365	−	0.980531i	$-0.437086\pi$
$311$	6.92587	0.392730	0.196365	+	0.980531i	$0.437086\pi$
$312$	0	0
$313$	5.26772	0.297749	0.148874	−	0.988856i	$-0.452435\pi$
$313$	5.26772	0.297749	0.148874	+	0.988856i	$0.452435\pi$
$314$	−1.42755	−0.0805613
$315$	0	0
$316$	1.71704	0.0965911
$317$	29.5365	1.65893	0.829466	−	0.558557i	$-0.188645\pi$
$317$	29.5365	1.65893	0.829466	+	0.558557i	$0.188645\pi$
$318$	0	0
$319$	−16.8029	−0.940782
$320$	14.8086	0.827824
$321$	0	0
$322$	−9.83035	−0.547824
$323$	−7.52498	−0.418701
$324$	0	0
$325$	0.820152	0.0454939
$326$	−19.7806	−1.09555
$327$	0	0
$328$	22.0399	1.21695
$329$	−4.18182	−0.230551
$330$	0	0
$331$	10.9518	0.601964	0.300982	−	0.953630i	$-0.402686\pi$
$331$	10.9518	0.601964	0.300982	+	0.953630i	$0.402686\pi$
$332$	−4.58174	−0.251456
$333$	0	0
$334$	−5.08842	−0.278426
$335$	24.6516	1.34686
$336$	0	0
$337$	−10.4719	−0.570438	−0.285219	−	0.958462i	$-0.592066\pi$
$337$	−10.4719	−0.570438	−0.285219	+	0.958462i	$0.592066\pi$
$338$	−16.7978	−0.913679
$339$	0	0
$340$	−0.664319	−0.0360278
$341$	0	0
$342$	0	0
$343$	−19.5850	−1.05749
$344$	−17.7711	−0.958151
$345$	0	0
$346$	6.32000	0.339765
$347$	26.6451	1.43038	0.715192	−	0.698928i	$-0.246339\pi$
$347$	26.6451	1.43038	0.715192	+	0.698928i	$0.246339\pi$
$348$	0	0
$349$	−17.7616	−0.950757	−0.475379	−	0.879781i	$-0.657689\pi$
$349$	−17.7616	−0.950757	−0.475379	+	0.879781i	$0.657689\pi$
$350$	−6.95383	−0.371698
$351$	0	0
$352$	−2.94022	−0.156714
$353$	8.57748	0.456533	0.228266	−	0.973599i	$-0.426694\pi$
$353$	8.57748	0.456533	0.228266	+	0.973599i	$0.426694\pi$
$354$	0	0
$355$	2.84613	0.151057
$356$	0.0914600	0.00484737
$357$	0	0
$358$	−11.3628	−0.600541
$359$	−6.59912	−0.348288	−0.174144	−	0.984720i	$-0.555716\pi$
$359$	−6.59912	−0.348288	−0.174144	+	0.984720i	$0.555716\pi$
$360$	0	0
$361$	12.1869	0.641416
$362$	24.0148	1.26219
$363$	0	0
$364$	0.271576	0.0142345
$365$	21.6212	1.13170
$366$	0	0
$367$	15.0922	0.787805	0.393902	−	0.919152i	$-0.371125\pi$
$367$	15.0922	0.787805	0.393902	+	0.919152i	$0.371125\pi$
$368$	10.2113	0.532299
$369$	0	0
$370$	20.6862	1.07542
$371$	9.24796	0.480130
$372$	0	0
$373$	−2.85472	−0.147812	−0.0739059	−	0.997265i	$-0.523546\pi$
$373$	−2.85472	−0.147812	−0.0739059	+	0.997265i	$0.523546\pi$
$374$	−3.14866	−0.162813
$375$	0	0
$376$	5.10899	0.263476
$377$	−3.54989	−0.182829
$378$	0	0
$379$	3.85372	0.197952	0.0989762	−	0.995090i	$-0.468443\pi$
$379$	3.85372	0.197952	0.0989762	+	0.995090i	$0.468443\pi$
$380$	2.75324	0.141238
$381$	0	0
$382$	26.2638	1.34377
$383$	30.1784	1.54204	0.771022	−	0.636808i	$-0.219746\pi$
$383$	30.1784	1.54204	0.771022	+	0.636808i	$0.219746\pi$
$384$	0	0
$385$	−7.37753	−0.375994
$386$	6.68649	0.340334
$387$	0	0
$388$	3.17661	0.161268
$389$	12.1572	0.616394	0.308197	−	0.951323i	$-0.400274\pi$
$389$	12.1572	0.616394	0.308197	+	0.951323i	$0.400274\pi$
$390$	0	0
$391$	−4.13454	−0.209093
$392$	2.95607	0.149304
$393$	0	0
$394$	24.5433	1.23647
$395$	−9.85441	−0.495829
$396$	0	0
$397$	−20.4220	−1.02495	−0.512475	−	0.858702i	$-0.671271\pi$
$397$	−20.4220	−1.02495	−0.512475	+	0.858702i	$0.671271\pi$
$398$	28.0606	1.40655
$399$	0	0
$400$	7.22329	0.361164
$401$	6.77790	0.338472	0.169236	−	0.985576i	$-0.445870\pi$
$401$	6.77790	0.338472	0.169236	+	0.985576i	$0.445870\pi$
$402$	0	0
$403$	0	0
$404$	1.52754	0.0759982
$405$	0	0
$406$	30.0985	1.49376
$407$	−16.8354	−0.834499
$408$	0	0
$409$	−20.2432	−1.00096	−0.500482	−	0.865747i	$-0.666844\pi$
$409$	−20.2432	−1.00096	−0.500482	+	0.865747i	$0.666844\pi$
$410$	−16.1675	−0.798454
$411$	0	0
$412$	−2.67156	−0.131618
$413$	32.7936	1.61367
$414$	0	0
$415$	26.2954	1.29079
$416$	−0.621170	−0.0304554
$417$	0	0
$418$	13.0494	0.638269
$419$	17.4587	0.852912	0.426456	−	0.904508i	$-0.359762\pi$
$419$	17.4587	0.852912	0.426456	+	0.904508i	$0.359762\pi$
$420$	0	0
$421$	−17.0987	−0.833341	−0.416670	−	0.909058i	$-0.636803\pi$
$421$	−17.0987	−0.833341	−0.416670	+	0.909058i	$0.636803\pi$
$422$	−31.5223	−1.53448
$423$	0	0
$424$	−11.2984	−0.548698
$425$	−2.92471	−0.141869
$426$	0	0
$427$	22.2280	1.07569
$428$	−3.07612	−0.148690
$429$	0	0
$430$	13.0360	0.628653
$431$	23.4602	1.13004	0.565019	−	0.825078i	$-0.308869\pi$
$431$	23.4602	1.13004	0.565019	+	0.825078i	$0.308869\pi$
$432$	0	0
$433$	−4.19606	−0.201650	−0.100825	−	0.994904i	$-0.532148\pi$
$433$	−4.19606	−0.201650	−0.100825	+	0.994904i	$0.532148\pi$
$434$	0	0
$435$	0	0
$436$	0.514088	0.0246203
$437$	17.1354	0.819697
$438$	0	0
$439$	−9.00099	−0.429594	−0.214797	−	0.976659i	$-0.568909\pi$
$439$	−9.00099	−0.429594	−0.214797	+	0.976659i	$0.568909\pi$
$440$	9.01324	0.429690
$441$	0	0
$442$	−0.665206	−0.0316406
$443$	32.7696	1.55693	0.778464	−	0.627689i	$-0.215999\pi$
$443$	32.7696	1.55693	0.778464	+	0.627689i	$0.215999\pi$
$444$	0	0
$445$	−0.524905	−0.0248829
$446$	16.4661	0.779692
$447$	0	0
$448$	21.5881	1.01994
$449$	−24.0329	−1.13418	−0.567091	−	0.823655i	$-0.691931\pi$
$449$	−24.0329	−1.13418	−0.567091	+	0.823655i	$0.691931\pi$
$450$	0	0
$451$	13.1578	0.619578
$452$	−2.84147	−0.133652
$453$	0	0
$454$	8.05771	0.378167
$455$	−1.55862	−0.0730694
$456$	0	0
$457$	−36.1460	−1.69084	−0.845420	−	0.534103i	$-0.820650\pi$
$457$	−36.1460	−1.69084	−0.845420	+	0.534103i	$0.820650\pi$
$458$	−28.2926	−1.32203
$459$	0	0
$460$	1.51274	0.0705321
$461$	14.8777	0.692922	0.346461	−	0.938064i	$-0.387383\pi$
$461$	14.8777	0.692922	0.346461	+	0.938064i	$0.387383\pi$
$462$	0	0
$463$	24.9797	1.16091	0.580453	−	0.814293i	$-0.302875\pi$
$463$	24.9797	1.16091	0.580453	+	0.814293i	$0.302875\pi$
$464$	−31.2648	−1.45143
$465$	0	0
$466$	5.35549	0.248088
$467$	−0.362266	−0.0167637	−0.00838185	−	0.999965i	$-0.502668\pi$
$467$	−0.362266	−0.0167637	−0.00838185	+	0.999965i	$0.502668\pi$
$468$	0	0
$469$	35.9374	1.65944
$470$	−3.74772	−0.172870
$471$	0	0
$472$	−40.0645	−1.84412
$473$	−10.6093	−0.487817
$474$	0	0
$475$	12.1213	0.556163
$476$	−0.968455	−0.0443891
$477$	0	0
$478$	10.6185	0.485680
$479$	−21.9269	−1.00187	−0.500933	−	0.865486i	$-0.667010\pi$
$479$	−21.9269	−1.00187	−0.500933	+	0.865486i	$0.667010\pi$
$480$	0	0
$481$	−3.55675	−0.162174
$482$	−17.5694	−0.800264
$483$	0	0
$484$	2.28644	0.103929
$485$	−18.2311	−0.827834
$486$	0	0
$487$	−14.5669	−0.660090	−0.330045	−	0.943965i	$-0.607064\pi$
$487$	−14.5669	−0.660090	−0.330045	+	0.943965i	$0.607064\pi$
$488$	−27.1563	−1.22931
$489$	0	0
$490$	−2.16844	−0.0979601
$491$	28.7793	1.29879	0.649397	−	0.760450i	$-0.275021\pi$
$491$	28.7793	1.29879	0.649397	+	0.760450i	$0.275021\pi$
$492$	0	0
$493$	12.6591	0.570137
$494$	2.75691	0.124039
$495$	0	0
$496$	0	0
$497$	4.14913	0.186114
$498$	0	0
$499$	−18.3478	−0.821358	−0.410679	−	0.911780i	$-0.634708\pi$
$499$	−18.3478	−0.821358	−0.410679	+	0.911780i	$0.634708\pi$
$500$	3.53515	0.158097
$501$	0	0
$502$	−13.6329	−0.608466
$503$	−37.8433	−1.68735	−0.843675	−	0.536855i	$-0.819612\pi$
$503$	−37.8433	−1.68735	−0.843675	+	0.536855i	$0.819612\pi$
$504$	0	0
$505$	−8.76685	−0.390120
$506$	7.16991	0.318741
$507$	0	0
$508$	3.68886	0.163667
$509$	9.93767	0.440480	0.220240	−	0.975446i	$-0.429316\pi$
$509$	9.93767	0.440480	0.220240	+	0.975446i	$0.429316\pi$
$510$	0	0
$511$	31.5197	1.39435
$512$	−25.4110	−1.12302
$513$	0	0
$514$	−30.4015	−1.34095
$515$	15.3326	0.675634
$516$	0	0
$517$	3.05007	0.134142
$518$	30.1567	1.32501
$519$	0	0
$520$	1.90420	0.0835046
$521$	26.5695	1.16403	0.582016	−	0.813178i	$-0.302264\pi$
$521$	26.5695	1.16403	0.582016	+	0.813178i	$0.302264\pi$
$522$	0	0
$523$	−12.6422	−0.552805	−0.276403	−	0.961042i	$-0.589142\pi$
$523$	−12.6422	−0.552805	−0.276403	+	0.961042i	$0.589142\pi$
$524$	−2.72924	−0.119228
$525$	0	0
$526$	3.03549	0.132353
$527$	0	0
$528$	0	0
$529$	−13.5851	−0.590657
$530$	8.28798	0.360007
$531$	0	0
$532$	4.01371	0.174016
$533$	2.77981	0.120407
$534$	0	0
$535$	17.6544	0.763267
$536$	−43.9053	−1.89642
$537$	0	0
$538$	−40.0920	−1.72849
$539$	1.76478	0.0760143
$540$	0	0
$541$	−3.00729	−0.129293	−0.0646467	−	0.997908i	$-0.520592\pi$
$541$	−3.00729	−0.129293	−0.0646467	+	0.997908i	$0.520592\pi$
$542$	−7.87333	−0.338188
$543$	0	0
$544$	2.21513	0.0949728
$545$	−2.95044	−0.126383
$546$	0	0
$547$	16.4211	0.702114	0.351057	−	0.936354i	$-0.385822\pi$
$547$	16.4211	0.702114	0.351057	+	0.936354i	$0.385822\pi$
$548$	3.02295	0.129134
$549$	0	0
$550$	5.07188	0.216266
$551$	−52.4650	−2.23508
$552$	0	0
$553$	−14.3659	−0.610900
$554$	1.86945	0.0794252
$555$	0	0
$556$	−0.582639	−0.0247094
$557$	−29.2708	−1.24024	−0.620122	−	0.784505i	$-0.712917\pi$
$557$	−29.2708	−1.24024	−0.620122	+	0.784505i	$0.712917\pi$
$558$	0	0
$559$	−2.24139	−0.0948009
$560$	−13.7272	−0.580080
$561$	0	0
$562$	−35.9961	−1.51840
$563$	14.0720	0.593063	0.296531	−	0.955023i	$-0.404170\pi$
$563$	14.0720	0.593063	0.296531	+	0.955023i	$0.404170\pi$
$564$	0	0
$565$	16.3077	0.686070
$566$	14.2643	0.599571
$567$	0	0
$568$	−5.06906	−0.212693
$569$	15.5487	0.651834	0.325917	−	0.945398i	$-0.394327\pi$
$569$	15.5487	0.651834	0.325917	+	0.945398i	$0.394327\pi$
$570$	0	0
$571$	−46.2045	−1.93360	−0.966798	−	0.255540i	$-0.917747\pi$
$571$	−46.2045	−1.93360	−0.966798	+	0.255540i	$0.917747\pi$
$572$	−0.198078	−0.00828206
$573$	0	0
$574$	−23.5692	−0.983758
$575$	6.65995	0.277739
$576$	0	0
$577$	−8.77588	−0.365345	−0.182672	−	0.983174i	$-0.558475\pi$
$577$	−8.77588	−0.365345	−0.182672	+	0.983174i	$0.558475\pi$
$578$	−19.8381	−0.825157
$579$	0	0
$580$	−4.63171	−0.192321
$581$	38.3339	1.59036
$582$	0	0
$583$	−6.74514	−0.279355
$584$	−38.5081	−1.59348
$585$	0	0
$586$	33.4718	1.38271
$587$	36.3512	1.50038	0.750188	−	0.661225i	$-0.229963\pi$
$587$	36.3512	1.50038	0.750188	+	0.661225i	$0.229963\pi$
$588$	0	0
$589$	0	0
$590$	29.3895	1.20995
$591$	0	0
$592$	−31.3252	−1.28746
$593$	20.6338	0.847330	0.423665	−	0.905819i	$-0.360743\pi$
$593$	20.6338	0.847330	0.423665	+	0.905819i	$0.360743\pi$
$594$	0	0
$595$	5.55814	0.227861
$596$	−3.93718	−0.161273
$597$	0	0
$598$	1.51476	0.0619432
$599$	−29.0555	−1.18718	−0.593588	−	0.804769i	$-0.702289\pi$
$599$	−29.0555	−1.18718	−0.593588	+	0.804769i	$0.702289\pi$
$600$	0	0
$601$	13.0827	0.533656	0.266828	−	0.963744i	$-0.414024\pi$
$601$	13.0827	0.533656	0.266828	+	0.963744i	$0.414024\pi$
$602$	19.0041	0.774550
$603$	0	0
$604$	−0.382337	−0.0155571
$605$	−13.1223	−0.533496
$606$	0	0
$607$	−17.9732	−0.729510	−0.364755	−	0.931104i	$-0.618847\pi$
$607$	−17.9732	−0.729510	−0.364755	+	0.931104i	$0.618847\pi$
$608$	−9.18048	−0.372318
$609$	0	0
$610$	19.9206	0.806561
$611$	0.644378	0.0260687
$612$	0	0
$613$	−5.36329	−0.216621	−0.108311	−	0.994117i	$-0.534544\pi$
$613$	−5.36329	−0.216621	−0.108311	+	0.994117i	$0.534544\pi$
$614$	10.7971	0.435734
$615$	0	0
$616$	13.1396	0.529411
$617$	−16.0127	−0.644647	−0.322323	−	0.946630i	$-0.604464\pi$
$617$	−16.0127	−0.644647	−0.322323	+	0.946630i	$0.604464\pi$
$618$	0	0
$619$	18.9305	0.760882	0.380441	−	0.924805i	$-0.375772\pi$
$619$	18.9305	0.760882	0.380441	+	0.924805i	$0.375772\pi$
$620$	0	0
$621$	0	0
$622$	9.04856	0.362814
$623$	−0.765215	−0.0306577
$624$	0	0
$625$	−9.43629	−0.377451
$626$	6.88220	0.275068
$627$	0	0
$628$	0.320251	0.0127794
$629$	12.6836	0.505727
$630$	0	0
$631$	−0.832672	−0.0331481	−0.0165741	−	0.999863i	$-0.505276\pi$
$631$	−0.832672	−0.0331481	−0.0165741	+	0.999863i	$0.505276\pi$
$632$	17.5511	0.698144
$633$	0	0
$634$	38.5890	1.53256
$635$	−21.1710	−0.840146
$636$	0	0
$637$	0.372838	0.0147724
$638$	−21.9528	−0.869119
$639$	0	0
$640$	13.8167	0.546154
$641$	−14.4143	−0.569331	−0.284666	−	0.958627i	$-0.591883\pi$
$641$	−14.4143	−0.569331	−0.284666	+	0.958627i	$0.591883\pi$
$642$	0	0
$643$	27.1917	1.07233	0.536167	−	0.844112i	$-0.319872\pi$
$643$	27.1917	1.07233	0.536167	+	0.844112i	$0.319872\pi$
$644$	2.20530	0.0869010
$645$	0	0
$646$	−9.83129	−0.386807
$647$	14.8561	0.584054	0.292027	−	0.956410i	$-0.405670\pi$
$647$	14.8561	0.584054	0.292027	+	0.956410i	$0.405670\pi$
$648$	0	0
$649$	−23.9185	−0.938883
$650$	1.07152	0.0420284
$651$	0	0
$652$	4.43751	0.173786
$653$	12.7917	0.500579	0.250290	−	0.968171i	$-0.419474\pi$
$653$	12.7917	0.500579	0.250290	+	0.968171i	$0.419474\pi$
$654$	0	0
$655$	15.6636	0.612028
$656$	24.4825	0.955880
$657$	0	0
$658$	−5.46349	−0.212989
$659$	16.2639	0.633550	0.316775	−	0.948501i	$-0.397400\pi$
$659$	16.2639	0.633550	0.316775	+	0.948501i	$0.397400\pi$
$660$	0	0
$661$	11.9766	0.465835	0.232918	−	0.972496i	$-0.425173\pi$
$661$	11.9766	0.465835	0.232918	+	0.972496i	$0.425173\pi$
$662$	14.3083	0.556110
$663$	0	0
$664$	−46.8331	−1.81748
$665$	−23.0354	−0.893275
$666$	0	0
$667$	−28.8265	−1.11617
$668$	1.14152	0.0441666
$669$	0	0
$670$	32.2069	1.24426
$671$	−16.2123	−0.625868
$672$	0	0
$673$	16.1652	0.623122	0.311561	−	0.950226i	$-0.399148\pi$
$673$	16.1652	0.623122	0.311561	+	0.950226i	$0.399148\pi$
$674$	−13.6813	−0.526985
$675$	0	0
$676$	3.76835	0.144936
$677$	9.73936	0.374314	0.187157	−	0.982330i	$-0.440073\pi$
$677$	9.73936	0.374314	0.187157	+	0.982330i	$0.440073\pi$
$678$	0	0
$679$	−26.5776	−1.01996
$680$	−6.79047	−0.260403
$681$	0	0
$682$	0	0
$683$	−14.2908	−0.546823	−0.273412	−	0.961897i	$-0.588152\pi$
$683$	−14.2908	−0.546823	−0.273412	+	0.961897i	$0.588152\pi$
$684$	0	0
$685$	−17.3493	−0.662881
$686$	−25.5876	−0.976938
$687$	0	0
$688$	−19.7405	−0.752600
$689$	−1.42502	−0.0542890
$690$	0	0
$691$	47.3839	1.80257	0.901284	−	0.433228i	$-0.142625\pi$
$691$	47.3839	1.80257	0.901284	+	0.433228i	$0.142625\pi$
$692$	−1.41780	−0.0538968
$693$	0	0
$694$	34.8115	1.32143
$695$	3.34387	0.126840
$696$	0	0
$697$	−9.91294	−0.375479
$698$	−23.2053	−0.878334
$699$	0	0
$700$	1.55999	0.0589622
$701$	−29.6981	−1.12168	−0.560841	−	0.827924i	$-0.689522\pi$
$701$	−29.6981	−1.12168	−0.560841	+	0.827924i	$0.689522\pi$
$702$	0	0
$703$	−52.5664	−1.98258
$704$	−15.7456	−0.593436
$705$	0	0
$706$	11.2064	0.421757
$707$	−12.7804	−0.480658
$708$	0	0
$709$	−30.9023	−1.16056	−0.580279	−	0.814418i	$-0.697057\pi$
$709$	−30.9023	−1.16056	−0.580279	+	0.814418i	$0.697057\pi$
$710$	3.71843	0.139550
$711$	0	0
$712$	0.934875	0.0350359
$713$	0	0
$714$	0	0
$715$	1.13681	0.0425141
$716$	2.54908	0.0952635
$717$	0	0
$718$	−8.62167	−0.321758
$719$	−12.5179	−0.466839	−0.233420	−	0.972376i	$-0.574992\pi$
$719$	−12.5179	−0.466839	−0.233420	+	0.972376i	$0.574992\pi$
$720$	0	0
$721$	22.3521	0.832435
$722$	15.9220	0.592557
$723$	0	0
$724$	−5.38738	−0.200220
$725$	−20.3914	−0.757316
$726$	0	0
$727$	22.3542	0.829072	0.414536	−	0.910033i	$-0.363944\pi$
$727$	22.3542	0.829072	0.414536	+	0.910033i	$0.363944\pi$
$728$	2.77597	0.102884
$729$	0	0
$730$	28.2478	1.04550
$731$	7.99293	0.295629
$732$	0	0
$733$	−12.5351	−0.462996	−0.231498	−	0.972835i	$-0.574363\pi$
$733$	−12.5351	−0.462996	−0.231498	+	0.972835i	$0.574363\pi$
$734$	19.7177	0.727794
$735$	0	0
$736$	−5.04414	−0.185929
$737$	−26.2115	−0.965512
$738$	0	0
$739$	10.2711	0.377830	0.188915	−	0.981993i	$-0.439503\pi$
$739$	10.2711	0.377830	0.188915	+	0.981993i	$0.439503\pi$
$740$	−4.64066	−0.170594
$741$	0	0
$742$	12.0823	0.443557
$743$	−19.1333	−0.701931	−0.350966	−	0.936388i	$-0.614147\pi$
$743$	−19.1333	−0.701931	−0.350966	+	0.936388i	$0.614147\pi$
$744$	0	0
$745$	22.5962	0.827859
$746$	−3.72965	−0.136552
$747$	0	0
$748$	0.706357	0.0258270
$749$	25.7369	0.940405
$750$	0	0
$751$	31.5162	1.15004	0.575022	−	0.818138i	$-0.304994\pi$
$751$	31.5162	1.15004	0.575022	+	0.818138i	$0.304994\pi$
$752$	5.67519	0.206953
$753$	0	0
$754$	−4.63789	−0.168902
$755$	2.19430	0.0798588
$756$	0	0
$757$	0.236486	0.00859523	0.00429761	−	0.999991i	$-0.498632\pi$
$757$	0.236486	0.00859523	0.00429761	+	0.999991i	$0.498632\pi$
$758$	5.03484	0.182874
$759$	0	0
$760$	28.1427	1.02084
$761$	28.4148	1.03003	0.515017	−	0.857180i	$-0.327786\pi$
$761$	28.4148	1.03003	0.515017	+	0.857180i	$0.327786\pi$
$762$	0	0
$763$	−4.30120	−0.155714
$764$	−5.89190	−0.213162
$765$	0	0
$766$	39.4277	1.42458
$767$	−5.05318	−0.182460
$768$	0	0
$769$	45.2552	1.63194	0.815972	−	0.578092i	$-0.196202\pi$
$769$	45.2552	1.63194	0.815972	+	0.578092i	$0.196202\pi$
$770$	−9.63864	−0.347353
$771$	0	0
$772$	−1.50002	−0.0539869
$773$	−21.6848	−0.779947	−0.389974	−	0.920826i	$-0.627516\pi$
$773$	−21.6848	−0.779947	−0.389974	+	0.920826i	$0.627516\pi$
$774$	0	0
$775$	0	0
$776$	32.4703	1.16562
$777$	0	0
$778$	15.8832	0.569441
$779$	41.0837	1.47198
$780$	0	0
$781$	−3.02623	−0.108287
$782$	−5.40172	−0.193165
$783$	0	0
$784$	3.28368	0.117274
$785$	−1.83798	−0.0656002
$786$	0	0
$787$	23.0355	0.821128	0.410564	−	0.911832i	$-0.365332\pi$
$787$	23.0355	0.821128	0.410564	+	0.911832i	$0.365332\pi$
$788$	−5.50594	−0.196141
$789$	0	0
$790$	−12.8747	−0.458060
$791$	23.7736	0.845293
$792$	0	0
$793$	−3.42511	−0.121629
$794$	−26.6811	−0.946876
$795$	0	0
$796$	−6.29501	−0.223121
$797$	38.2970	1.35655	0.678275	−	0.734808i	$-0.262728\pi$
$797$	38.2970	1.35655	0.678275	+	0.734808i	$0.262728\pi$
$798$	0	0
$799$	−2.29788	−0.0812933
$800$	−3.56814	−0.126153
$801$	0	0
$802$	8.85523	0.312689
$803$	−22.9894	−0.811277
$804$	0	0
$805$	−12.6566	−0.446087
$806$	0	0
$807$	0	0
$808$	15.6141	0.549302
$809$	−18.5016	−0.650482	−0.325241	−	0.945631i	$-0.605445\pi$
$809$	−18.5016	−0.650482	−0.325241	+	0.945631i	$0.605445\pi$
$810$	0	0
$811$	−5.42083	−0.190351	−0.0951756	−	0.995460i	$-0.530341\pi$
$811$	−5.42083	−0.190351	−0.0951756	+	0.995460i	$0.530341\pi$
$812$	−6.75217	−0.236955
$813$	0	0
$814$	−21.9952	−0.770931
$815$	−25.4676	−0.892092
$816$	0	0
$817$	−33.1263	−1.15894
$818$	−26.4475	−0.924716
$819$	0	0
$820$	3.62694	0.126658
$821$	54.8319	1.91365	0.956824	−	0.290668i	$-0.0938775\pi$
$821$	54.8319	1.91365	0.956824	+	0.290668i	$0.0938775\pi$
$822$	0	0
$823$	−23.6241	−0.823486	−0.411743	−	0.911300i	$-0.635080\pi$
$823$	−23.6241	−0.823486	−0.411743	+	0.911300i	$0.635080\pi$
$824$	−27.3079	−0.951315
$825$	0	0
$826$	42.8444	1.49075
$827$	−1.17573	−0.0408840	−0.0204420	−	0.999791i	$-0.506507\pi$
$827$	−1.17573	−0.0408840	−0.0204420	+	0.999791i	$0.506507\pi$
$828$	0	0
$829$	22.5094	0.781783	0.390891	−	0.920437i	$-0.372167\pi$
$829$	22.5094	0.781783	0.390891	+	0.920437i	$0.372167\pi$
$830$	34.3546	1.19247
$831$	0	0
$832$	−3.32653	−0.115327
$833$	−1.32956	−0.0460665
$834$	0	0
$835$	−6.55137	−0.226719
$836$	−2.92746	−0.101248
$837$	0	0
$838$	22.8095	0.787942
$839$	−32.8958	−1.13569	−0.567845	−	0.823135i	$-0.692223\pi$
$839$	−32.8958	−1.13569	−0.567845	+	0.823135i	$0.692223\pi$
$840$	0	0
$841$	59.2606	2.04347
$842$	−22.3393	−0.769861
$843$	0	0
$844$	7.07159	0.243414
$845$	−21.6272	−0.743999
$846$	0	0
$847$	−19.1298	−0.657309
$848$	−12.5505	−0.430987
$849$	0	0
$850$	−3.82109	−0.131062
$851$	−28.8822	−0.990068
$852$	0	0
$853$	−4.39665	−0.150538	−0.0752692	−	0.997163i	$-0.523982\pi$
$853$	−4.39665	−0.150538	−0.0752692	+	0.997163i	$0.523982\pi$
$854$	29.0405	0.993747
$855$	0	0
$856$	−31.4432	−1.07471
$857$	−17.5395	−0.599136	−0.299568	−	0.954075i	$-0.596843\pi$
$857$	−17.5395	−0.599136	−0.299568	+	0.954075i	$0.596843\pi$
$858$	0	0
$859$	−28.9984	−0.989412	−0.494706	−	0.869060i	$-0.664724\pi$
$859$	−28.9984	−0.989412	−0.494706	+	0.869060i	$0.664724\pi$
$860$	−2.92445	−0.0997229
$861$	0	0
$862$	30.6504	1.04396
$863$	44.8926	1.52816	0.764081	−	0.645120i	$-0.223193\pi$
$863$	44.8926	1.52816	0.764081	+	0.645120i	$0.223193\pi$
$864$	0	0
$865$	8.13702	0.276667
$866$	−5.48209	−0.186289
$867$	0	0
$868$	0	0
$869$	10.4780	0.355441
$870$	0	0
$871$	−5.53761	−0.187635
$872$	5.25484	0.177951
$873$	0	0
$874$	22.3872	0.757257
$875$	−29.5774	−0.999899
$876$	0	0
$877$	35.5069	1.19898	0.599492	−	0.800381i	$-0.295369\pi$
$877$	35.5069	1.19898	0.599492	+	0.800381i	$0.295369\pi$
$878$	−11.7597	−0.396870
$879$	0	0
$880$	10.0121	0.337509
$881$	0.453899	0.0152922	0.00764612	−	0.999971i	$-0.497566\pi$
$881$	0.453899	0.0152922	0.00764612	+	0.999971i	$0.497566\pi$
$882$	0	0
$883$	54.2083	1.82425	0.912127	−	0.409908i	$-0.134439\pi$
$883$	54.2083	1.82425	0.912127	+	0.409908i	$0.134439\pi$
$884$	0.149230	0.00501913
$885$	0	0
$886$	42.8130	1.43833
$887$	−19.6514	−0.659830	−0.329915	−	0.944011i	$-0.607020\pi$
$887$	−19.6514	−0.659830	−0.329915	+	0.944011i	$0.607020\pi$
$888$	0	0
$889$	−30.8634	−1.03513
$890$	−0.685782	−0.0229875
$891$	0	0
$892$	−3.69394	−0.123682
$893$	9.52346	0.318691
$894$	0	0
$895$	−14.6296	−0.489014
$896$	20.1422	0.672904
$897$	0	0
$898$	−31.3987	−1.04779
$899$	0	0
$900$	0	0
$901$	5.08170	0.169296
$902$	17.1905	0.572382
$903$	0	0
$904$	−29.0446	−0.966010
$905$	30.9191	1.02779
$906$	0	0
$907$	−30.4816	−1.01212	−0.506062	−	0.862497i	$-0.668899\pi$
$907$	−30.4816	−1.01212	−0.506062	+	0.862497i	$0.668899\pi$
$908$	−1.80763	−0.0599884
$909$	0	0
$910$	−2.03632	−0.0675034
$911$	−13.6916	−0.453623	−0.226811	−	0.973939i	$-0.572830\pi$
$911$	−13.6916	−0.453623	−0.226811	+	0.973939i	$0.572830\pi$
$912$	0	0
$913$	−27.9594	−0.925320
$914$	−47.2243	−1.56204
$915$	0	0
$916$	6.34705	0.209712
$917$	22.8347	0.754067
$918$	0	0
$919$	−10.5418	−0.347743	−0.173872	−	0.984768i	$-0.555628\pi$
$919$	−10.5418	−0.347743	−0.173872	+	0.984768i	$0.555628\pi$
$920$	15.4628	0.509793
$921$	0	0
$922$	19.4375	0.640139
$923$	−0.639342	−0.0210442
$924$	0	0
$925$	−20.4308	−0.671760
$926$	32.6357	1.07248
$927$	0	0
$928$	15.4441	0.506977
$929$	−18.1098	−0.594163	−0.297081	−	0.954852i	$-0.596013\pi$
$929$	−18.1098	−0.594163	−0.297081	+	0.954852i	$0.596013\pi$
$930$	0	0
$931$	5.51030	0.180593
$932$	−1.20143	−0.0393541
$933$	0	0
$934$	−0.473296	−0.0154867
$935$	−4.05391	−0.132577
$936$	0	0
$937$	−14.9580	−0.488655	−0.244328	−	0.969693i	$-0.578567\pi$
$937$	−14.9580	−0.488655	−0.244328	+	0.969693i	$0.578567\pi$
$938$	46.9518	1.53303
$939$	0	0
$940$	0.840749	0.0274222
$941$	−51.5419	−1.68022	−0.840109	−	0.542417i	$-0.817509\pi$
$941$	−51.5419	−1.68022	−0.840109	+	0.542417i	$0.817509\pi$
$942$	0	0
$943$	22.5731	0.735081
$944$	−44.5046	−1.44850
$945$	0	0
$946$	−13.8609	−0.450658
$947$	−23.7511	−0.771807	−0.385903	−	0.922539i	$-0.626110\pi$
$947$	−23.7511	−0.771807	−0.385903	+	0.922539i	$0.626110\pi$
$948$	0	0
$949$	−4.85688	−0.157661
$950$	15.8363	0.513798
$951$	0	0
$952$	−9.89925	−0.320836
$953$	43.6073	1.41258	0.706289	−	0.707924i	$-0.250368\pi$
$953$	43.6073	1.41258	0.706289	+	0.707924i	$0.250368\pi$
$954$	0	0
$955$	33.8147	1.09422
$956$	−2.38211	−0.0770431
$957$	0	0
$958$	−28.6472	−0.925550
$959$	−25.2920	−0.816721
$960$	0	0
$961$	0	0
$962$	−4.64685	−0.149820
$963$	0	0
$964$	3.94145	0.126946
$965$	8.60889	0.277130
$966$	0	0
$967$	−40.8050	−1.31220	−0.656100	−	0.754674i	$-0.727795\pi$
$967$	−40.8050	−1.31220	−0.656100	+	0.754674i	$0.727795\pi$
$968$	23.3712	0.751180
$969$	0	0
$970$	−23.8188	−0.764774
$971$	−31.7610	−1.01926	−0.509630	−	0.860394i	$-0.670218\pi$
$971$	−31.7610	−1.01926	−0.509630	+	0.860394i	$0.670218\pi$
$972$	0	0
$973$	4.87474	0.156277
$974$	−19.0315	−0.609809
$975$	0	0
$976$	−30.1658	−0.965585
$977$	2.03630	0.0651471	0.0325735	−	0.999469i	$-0.489630\pi$
$977$	2.03630	0.0651471	0.0325735	+	0.999469i	$0.489630\pi$
$978$	0	0
$979$	0.558121	0.0178376
$980$	0.486459	0.0155394
$981$	0	0
$982$	37.5998	1.19986
$983$	−19.1416	−0.610521	−0.305260	−	0.952269i	$-0.598744\pi$
$983$	−19.1416	−0.610521	−0.305260	+	0.952269i	$0.598744\pi$
$984$	0	0
$985$	31.5996	1.00685
$986$	16.5390	0.526707
$987$	0	0
$988$	−0.618474	−0.0196763
$989$	−18.2010	−0.578757
$990$	0	0
$991$	9.57396	0.304127	0.152063	−	0.988371i	$-0.451408\pi$
$991$	9.57396	0.304127	0.152063	+	0.988371i	$0.451408\pi$
$992$	0	0
$993$	0	0
$994$	5.42079	0.171937
$995$	36.1282	1.14534
$996$	0	0
$997$	0.985486	0.0312106	0.0156053	−	0.999878i	$-0.495032\pi$
$997$	0.985486	0.0312106	0.0156053	+	0.999878i	$0.495032\pi$
$998$	−23.9711	−0.758792
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bj.1.5		8
3.2	odd	2		2883.2.a.m.1.4		8
31.14	even	15		279.2.y.b.10.1		16
31.20	even	15		279.2.y.b.28.1		16
31.30	odd	2		8649.2.a.bi.1.5		8
93.14	odd	30		93.2.m.a.10.2	✓	16
93.20	odd	30		93.2.m.a.28.2	yes	16
93.92	even	2		2883.2.a.n.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.m.a.10.2	✓	16	93.14	odd	30
93.2.m.a.28.2	yes	16	93.20	odd	30
279.2.y.b.10.1		16	31.14	even	15
279.2.y.b.28.1		16	31.20	even	15
2883.2.a.m.1.4		8	3.2	odd	2
2883.2.a.n.1.4		8	93.92	even	2
8649.2.a.bi.1.5		8	31.30	odd	2
8649.2.a.bj.1.5		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30649 q^{2} -0.293092 q^{4} +1.68211 q^{5} +2.45220 q^{7} -2.99589 q^{8} +O(q^{10})\)	\(q+1.30649 q^{2} -0.293092 q^{4} +1.68211 q^{5} +2.45220 q^{7} -2.99589 q^{8} +2.19765 q^{10} -1.78855 q^{11} -0.377861 q^{13} +3.20377 q^{14} -3.32791 q^{16} +1.34747 q^{17} -5.58452 q^{19} -0.493012 q^{20} -2.33672 q^{22} -3.06837 q^{23} -2.17052 q^{25} -0.493670 q^{26} -0.718721 q^{28} +9.39471 q^{29} +1.64391 q^{32} +1.76045 q^{34} +4.12487 q^{35} +9.41287 q^{37} -7.29611 q^{38} -5.03942 q^{40} -7.35670 q^{41} +5.93180 q^{43} +0.524209 q^{44} -4.00879 q^{46} -1.70533 q^{47} -0.986708 q^{49} -2.83575 q^{50} +0.110748 q^{52} +3.77129 q^{53} -3.00853 q^{55} -7.34654 q^{56} +12.2741 q^{58} +13.3731 q^{59} +9.06449 q^{61} +8.80358 q^{64} -0.635602 q^{65} +14.6552 q^{67} -0.394933 q^{68} +5.38908 q^{70} +1.69200 q^{71} +12.8536 q^{73} +12.2978 q^{74} +1.63678 q^{76} -4.38588 q^{77} -5.85837 q^{79} -5.59791 q^{80} -9.61143 q^{82} +15.6324 q^{83} +2.26659 q^{85} +7.74982 q^{86} +5.35831 q^{88} -0.312052 q^{89} -0.926590 q^{91} +0.899315 q^{92} -2.22799 q^{94} -9.39377 q^{95} -10.8383 q^{97} -1.28912 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 6 q^{7}+O(q^{10}) \) 8 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 6 * q^7	\( 8 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 6 q^{7} + q^{10} + 12 q^{13} + 3 q^{14} + 3 q^{16} - 2 q^{17} - 8 q^{19} + 5 q^{20} + 4 q^{22} - 4 q^{23} - 12 q^{25} + 18 q^{26} - 15 q^{28} + 6 q^{29} + 17 q^{34} - 9 q^{35} + 8 q^{37} + 7 q^{38} + 13 q^{40} + 20 q^{41} + 14 q^{43} - 25 q^{44} + 9 q^{46} - 6 q^{47} - 20 q^{49} - 6 q^{50} + 30 q^{52} + 30 q^{53} + 19 q^{55} - 6 q^{56} + 19 q^{58} + 20 q^{59} + 13 q^{61} - 26 q^{64} + 27 q^{65} + 32 q^{67} + 15 q^{68} - 6 q^{70} + 17 q^{71} - 12 q^{73} + 10 q^{74} - 20 q^{76} - 18 q^{77} + 30 q^{79} + 21 q^{80} - 23 q^{82} - 12 q^{83} + 28 q^{85} + 46 q^{86} + 22 q^{88} - 6 q^{89} - 15 q^{91} + 70 q^{92} + 44 q^{94} + 36 q^{95} - 39 q^{97} - 32 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 6 * q^7 + q^10 + 12 * q^13 + 3 * q^14 + 3 * q^16 - 2 * q^17 - 8 * q^19 + 5 * q^20 + 4 * q^22 - 4 * q^23 - 12 * q^25 + 18 * q^26 - 15 * q^28 + 6 * q^29 + 17 * q^34 - 9 * q^35 + 8 * q^37 + 7 * q^38 + 13 * q^40 + 20 * q^41 + 14 * q^43 - 25 * q^44 + 9 * q^46 - 6 * q^47 - 20 * q^49 - 6 * q^50 + 30 * q^52 + 30 * q^53 + 19 * q^55 - 6 * q^56 + 19 * q^58 + 20 * q^59 + 13 * q^61 - 26 * q^64 + 27 * q^65 + 32 * q^67 + 15 * q^68 - 6 * q^70 + 17 * q^71 - 12 * q^73 + 10 * q^74 - 20 * q^76 - 18 * q^77 + 30 * q^79 + 21 * q^80 - 23 * q^82 - 12 * q^83 + 28 * q^85 + 46 * q^86 + 22 * q^88 - 6 * q^89 - 15 * q^91 + 70 * q^92 + 44 * q^94 + 36 * q^95 - 39 * q^97 - 32 * q^98

Introduction

L-functions

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Database

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