LMFDB - Embedded newform 1341.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1341,2,Mod(1,1341)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1341, 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("1341.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1341 = 3^{2} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1341.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:	$10.7079389111$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 12x^{6} + 75x^{5} - 48x^{4} - 137x^{3} + 76x^{2} + 68x - 39 $ x^9 - x^8 - 15x^7 + 12x^6 + 75x^5 - 48x^4 - 137x^3 + 76x^2 + 68*x - 39
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 149)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37103$ of defining polynomial
Character	$\chi$	$=$	1341.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37103 q^{2} +3.62178 q^{4} +1.07225 q^{5} +2.43457 q^{7} -3.84529 q^{8} +O(q^{10})$	\(q-2.37103 q^{2} +3.62178 q^{4} +1.07225 q^{5} +2.43457 q^{7} -3.84529 q^{8} -2.54234 q^{10} +5.52141 q^{11} +5.05635 q^{13} -5.77244 q^{14} +1.87373 q^{16} +4.60209 q^{17} +3.43613 q^{19} +3.88345 q^{20} -13.0914 q^{22} -1.11473 q^{23} -3.85028 q^{25} -11.9888 q^{26} +8.81749 q^{28} +2.19038 q^{29} +2.13474 q^{31} +3.24791 q^{32} -10.9117 q^{34} +2.61047 q^{35} +5.35331 q^{37} -8.14717 q^{38} -4.12311 q^{40} -0.531188 q^{41} +3.95649 q^{43} +19.9973 q^{44} +2.64305 q^{46} -9.95650 q^{47} -1.07285 q^{49} +9.12913 q^{50} +18.3130 q^{52} -12.9544 q^{53} +5.92033 q^{55} -9.36163 q^{56} -5.19345 q^{58} -4.46458 q^{59} -1.93846 q^{61} -5.06153 q^{62} -11.4483 q^{64} +5.42167 q^{65} -12.5240 q^{67} +16.6677 q^{68} -6.18950 q^{70} -1.68191 q^{71} -7.52597 q^{73} -12.6929 q^{74} +12.4449 q^{76} +13.4423 q^{77} -2.22168 q^{79} +2.00910 q^{80} +1.25946 q^{82} -13.0111 q^{83} +4.93459 q^{85} -9.38095 q^{86} -21.2314 q^{88} +15.1250 q^{89} +12.3101 q^{91} -4.03730 q^{92} +23.6072 q^{94} +3.68439 q^{95} -18.9449 q^{97} +2.54377 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	$ 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) $ 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * 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 $

$2$	−2.37103	−1.67657	−0.838285	−	0.545232i	$-0.816442\pi$
$2$	−2.37103	−1.67657	−0.838285	+	0.545232i	$0.816442\pi$
$3$	0	0
$3$	0	0
$4$	3.62178	1.81089
$5$	1.07225	0.479525	0.239762	−	0.970832i	$-0.422930\pi$
$5$	1.07225	0.479525	0.239762	+	0.970832i	$0.422930\pi$
$6$	0	0
$7$	2.43457	0.920182	0.460091	−	0.887872i	$-0.347817\pi$
$7$	2.43457	0.920182	0.460091	+	0.887872i	$0.347817\pi$
$8$	−3.84529	−1.35951
$9$	0	0
$10$	−2.54234	−0.803957
$11$	5.52141	1.66477	0.832384	−	0.554200i	$-0.186976\pi$
$11$	5.52141	1.66477	0.832384	+	0.554200i	$0.186976\pi$
$12$	0	0
$13$	5.05635	1.40238	0.701190	−	0.712975i	$-0.252652\pi$
$13$	5.05635	1.40238	0.701190	+	0.712975i	$0.252652\pi$
$14$	−5.77244	−1.54275
$15$	0	0
$16$	1.87373	0.468432
$17$	4.60209	1.11617	0.558085	−	0.829784i	$-0.311536\pi$
$17$	4.60209	1.11617	0.558085	+	0.829784i	$0.311536\pi$
$18$	0	0
$19$	3.43613	0.788303	0.394152	−	0.919045i	$-0.371039\pi$
$19$	3.43613	0.788303	0.394152	+	0.919045i	$0.371039\pi$
$20$	3.88345	0.868366
$21$	0	0
$22$	−13.0914	−2.79110
$23$	−1.11473	−0.232437	−0.116218	−	0.993224i	$-0.537077\pi$
$23$	−1.11473	−0.232437	−0.116218	+	0.993224i	$0.537077\pi$
$24$	0	0
$25$	−3.85028	−0.770056
$26$	−11.9888	−2.35119
$27$	0	0
$28$	8.81749	1.66635
$29$	2.19038	0.406743	0.203371	−	0.979102i	$-0.434810\pi$
$29$	2.19038	0.406743	0.203371	+	0.979102i	$0.434810\pi$
$30$	0	0
$31$	2.13474	0.383410	0.191705	−	0.981453i	$-0.438598\pi$
$31$	2.13474	0.383410	0.191705	+	0.981453i	$0.438598\pi$
$32$	3.24791	0.574155
$33$	0	0
$34$	−10.9117	−1.87134
$35$	2.61047	0.441250
$36$	0	0
$37$	5.35331	0.880078	0.440039	−	0.897979i	$-0.354965\pi$
$37$	5.35331	0.880078	0.440039	+	0.897979i	$0.354965\pi$
$38$	−8.14717	−1.32165
$39$	0	0
$40$	−4.12311	−0.651921
$41$	−0.531188	−0.0829576	−0.0414788	−	0.999139i	$-0.513207\pi$
$41$	−0.531188	−0.0829576	−0.0414788	+	0.999139i	$0.513207\pi$
$42$	0	0
$43$	3.95649	0.603359	0.301679	−	0.953409i	$-0.402453\pi$
$43$	3.95649	0.603359	0.301679	+	0.953409i	$0.402453\pi$
$44$	19.9973	3.01471
$45$	0	0
$46$	2.64305	0.389697
$47$	−9.95650	−1.45230	−0.726152	−	0.687534i	$-0.758693\pi$
$47$	−9.95650	−1.45230	−0.726152	+	0.687534i	$0.758693\pi$
$48$	0	0
$49$	−1.07285	−0.153265
$50$	9.12913	1.29105
$51$	0	0
$52$	18.3130	2.53956
$53$	−12.9544	−1.77943	−0.889714	−	0.456518i	$-0.849096\pi$
$53$	−12.9544	−1.77943	−0.889714	+	0.456518i	$0.849096\pi$
$54$	0	0
$55$	5.92033	0.798297
$56$	−9.36163	−1.25100
$57$	0	0
$58$	−5.19345	−0.681933
$59$	−4.46458	−0.581239	−0.290620	−	0.956839i	$-0.593861\pi$
$59$	−4.46458	−0.581239	−0.290620	+	0.956839i	$0.593861\pi$
$60$	0	0
$61$	−1.93846	−0.248195	−0.124097	−	0.992270i	$-0.539604\pi$
$61$	−1.93846	−0.248195	−0.124097	+	0.992270i	$0.539604\pi$
$62$	−5.06153	−0.642815
$63$	0	0
$64$	−11.4483	−1.43104
$65$	5.42167	0.672476
$66$	0	0
$67$	−12.5240	−1.53004	−0.765022	−	0.644004i	$-0.777272\pi$
$67$	−12.5240	−1.53004	−0.765022	+	0.644004i	$0.777272\pi$
$68$	16.6677	2.02126
$69$	0	0
$70$	−6.18950	−0.739787
$71$	−1.68191	−0.199606	−0.0998031	−	0.995007i	$-0.531821\pi$
$71$	−1.68191	−0.199606	−0.0998031	+	0.995007i	$0.531821\pi$
$72$	0	0
$73$	−7.52597	−0.880848	−0.440424	−	0.897790i	$-0.645172\pi$
$73$	−7.52597	−0.880848	−0.440424	+	0.897790i	$0.645172\pi$
$74$	−12.6929	−1.47551
$75$	0	0
$76$	12.4449	1.42753
$77$	13.4423	1.53189
$78$	0	0
$79$	−2.22168	−0.249959	−0.124979	−	0.992159i	$-0.539886\pi$
$79$	−2.22168	−0.249959	−0.124979	+	0.992159i	$0.539886\pi$
$80$	2.00910	0.224625
$81$	0	0
$82$	1.25946	0.139084
$83$	−13.0111	−1.42815	−0.714075	−	0.700069i	$-0.753152\pi$
$83$	−13.0111	−1.42815	−0.714075	+	0.700069i	$0.753152\pi$
$84$	0	0
$85$	4.93459	0.535231
$86$	−9.38095	−1.01157
$87$	0	0
$88$	−21.2314	−2.26328
$89$	15.1250	1.60324	0.801622	−	0.597831i	$-0.203971\pi$
$89$	15.1250	1.60324	0.801622	+	0.597831i	$0.203971\pi$
$90$	0	0
$91$	12.3101	1.29044
$92$	−4.03730	−0.420917
$93$	0	0
$94$	23.6072	2.43489
$95$	3.68439	0.378011
$96$	0	0
$97$	−18.9449	−1.92357	−0.961783	−	0.273811i	$-0.911716\pi$
$97$	−18.9449	−1.92357	−0.961783	+	0.273811i	$0.911716\pi$
$98$	2.54377	0.256960
$99$	0	0
$100$	−13.9449	−1.39449
$101$	−2.89823	−0.288385	−0.144193	−	0.989550i	$-0.546058\pi$
$101$	−2.89823	−0.288385	−0.144193	+	0.989550i	$0.546058\pi$
$102$	0	0
$103$	14.8220	1.46045	0.730227	−	0.683205i	$-0.239414\pi$
$103$	14.8220	1.46045	0.730227	+	0.683205i	$0.239414\pi$
$104$	−19.4431	−1.90656
$105$	0	0
$106$	30.7153	2.98334
$107$	−0.930628	−0.0899672	−0.0449836	−	0.998988i	$-0.514324\pi$
$107$	−0.930628	−0.0899672	−0.0449836	+	0.998988i	$0.514324\pi$
$108$	0	0
$109$	−11.6825	−1.11898	−0.559492	−	0.828835i	$-0.689004\pi$
$109$	−11.6825	−1.11898	−0.559492	+	0.828835i	$0.689004\pi$
$110$	−14.0373	−1.33840
$111$	0	0
$112$	4.56173	0.431043
$113$	18.3381	1.72510	0.862551	−	0.505969i	$-0.168865\pi$
$113$	18.3381	1.72510	0.862551	+	0.505969i	$0.168865\pi$
$114$	0	0
$115$	−1.19527	−0.111459
$116$	7.93306	0.736566
$117$	0	0
$118$	10.5857	0.974489
$119$	11.2041	1.02708
$120$	0	0
$121$	19.4860	1.77145
$122$	4.59615	0.416116
$123$	0	0
$124$	7.73156	0.694314
$125$	−9.48971	−0.848786
$126$	0	0
$127$	3.47204	0.308094	0.154047	−	0.988064i	$-0.450769\pi$
$127$	3.47204	0.308094	0.154047	+	0.988064i	$0.450769\pi$
$128$	20.6485	1.82509
$129$	0	0
$130$	−12.8549	−1.12745
$131$	−2.46789	−0.215620	−0.107810	−	0.994171i	$-0.534384\pi$
$131$	−2.46789	−0.215620	−0.107810	+	0.994171i	$0.534384\pi$
$132$	0	0
$133$	8.36552	0.725382
$134$	29.6947	2.56523
$135$	0	0
$136$	−17.6963	−1.51745
$137$	3.78777	0.323611	0.161806	−	0.986823i	$-0.448268\pi$
$137$	3.78777	0.323611	0.161806	+	0.986823i	$0.448268\pi$
$138$	0	0
$139$	13.5035	1.14535	0.572674	−	0.819783i	$-0.305906\pi$
$139$	13.5035	1.14535	0.572674	+	0.819783i	$0.305906\pi$
$140$	9.45455	0.799055
$141$	0	0
$142$	3.98786	0.334654
$143$	27.9182	2.33464
$144$	0	0
$145$	2.34863	0.195043
$146$	17.8443	1.47680
$147$	0	0
$148$	19.3885	1.59373
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−4.00428	−0.325863	−0.162932	−	0.986637i	$-0.552095\pi$
$151$	−4.00428	−0.325863	−0.162932	+	0.986637i	$0.552095\pi$
$152$	−13.2129	−1.07171
$153$	0	0
$154$	−31.8720	−2.56832
$155$	2.28897	0.183855
$156$	0	0
$157$	−19.1992	−1.53226	−0.766130	−	0.642686i	$-0.777820\pi$
$157$	−19.1992	−1.53226	−0.766130	+	0.642686i	$0.777820\pi$
$158$	5.26768	0.419074
$159$	0	0
$160$	3.48257	0.275321
$161$	−2.71388	−0.213884
$162$	0	0
$163$	−21.2008	−1.66057	−0.830286	−	0.557338i	$-0.811823\pi$
$163$	−21.2008	−1.66057	−0.830286	+	0.557338i	$0.811823\pi$
$164$	−1.92384	−0.150227
$165$	0	0
$166$	30.8496	2.39439
$167$	14.1116	1.09199	0.545994	−	0.837789i	$-0.316152\pi$
$167$	14.1116	1.09199	0.545994	+	0.837789i	$0.316152\pi$
$168$	0	0
$169$	12.5667	0.966669
$170$	−11.7001	−0.897353
$171$	0	0
$172$	14.3295	1.09262
$173$	3.31717	0.252199	0.126100	−	0.992018i	$-0.459754\pi$
$173$	3.31717	0.252199	0.126100	+	0.992018i	$0.459754\pi$
$174$	0	0
$175$	−9.37379	−0.708592
$176$	10.3456	0.779831
$177$	0	0
$178$	−35.8618	−2.68795
$179$	−12.4967	−0.934045	−0.467023	−	0.884245i	$-0.654673\pi$
$179$	−12.4967	−0.934045	−0.467023	+	0.884245i	$0.654673\pi$
$180$	0	0
$181$	−6.44911	−0.479359	−0.239679	−	0.970852i	$-0.577042\pi$
$181$	−6.44911	−0.479359	−0.239679	+	0.970852i	$0.577042\pi$
$182$	−29.1875	−2.16352
$183$	0	0
$184$	4.28645	0.316001
$185$	5.74008	0.422019
$186$	0	0
$187$	25.4100	1.85816
$188$	−36.0602	−2.62996
$189$	0	0
$190$	−8.73580	−0.633762
$191$	24.2317	1.75334	0.876671	−	0.481090i	$-0.159759\pi$
$191$	24.2317	1.75334	0.876671	+	0.481090i	$0.159759\pi$
$192$	0	0
$193$	15.8649	1.14198	0.570990	−	0.820957i	$-0.306560\pi$
$193$	15.8649	1.14198	0.570990	+	0.820957i	$0.306560\pi$
$194$	44.9190	3.22500
$195$	0	0
$196$	−3.88564	−0.277546
$197$	−13.2351	−0.942965	−0.471482	−	0.881876i	$-0.656281\pi$
$197$	−13.2351	−0.942965	−0.471482	+	0.881876i	$0.656281\pi$
$198$	0	0
$199$	4.17981	0.296299	0.148149	−	0.988965i	$-0.452668\pi$
$199$	4.17981	0.296299	0.148149	+	0.988965i	$0.452668\pi$
$200$	14.8054	1.04690
$201$	0	0
$202$	6.87180	0.483498
$203$	5.33263	0.374277
$204$	0	0
$205$	−0.569566	−0.0397802
$206$	−35.1434	−2.44855
$207$	0	0
$208$	9.47423	0.656920
$209$	18.9723	1.31234
$210$	0	0
$211$	23.9544	1.64909	0.824546	−	0.565795i	$-0.191431\pi$
$211$	23.9544	1.64909	0.824546	+	0.565795i	$0.191431\pi$
$212$	−46.9181	−3.22235
$213$	0	0
$214$	2.20655	0.150836
$215$	4.24234	0.289325
$216$	0	0
$217$	5.19718	0.352807
$218$	27.6997	1.87606
$219$	0	0
$220$	21.4421	1.44563
$221$	23.2698	1.56529
$222$	0	0
$223$	−22.5833	−1.51229	−0.756144	−	0.654405i	$-0.772919\pi$
$223$	−22.5833	−1.51229	−0.756144	+	0.654405i	$0.772919\pi$
$224$	7.90727	0.528327
$225$	0	0
$226$	−43.4802	−2.89226
$227$	1.83802	0.121994	0.0609970	−	0.998138i	$-0.480572\pi$
$227$	1.83802	0.121994	0.0609970	+	0.998138i	$0.480572\pi$
$228$	0	0
$229$	19.8162	1.30949	0.654746	−	0.755849i	$-0.272776\pi$
$229$	19.8162	1.30949	0.654746	+	0.755849i	$0.272776\pi$
$230$	2.83401	0.186869
$231$	0	0
$232$	−8.42263	−0.552972
$233$	−15.8145	−1.03604	−0.518021	−	0.855368i	$-0.673331\pi$
$233$	−15.8145	−1.03604	−0.518021	+	0.855368i	$0.673331\pi$
$234$	0	0
$235$	−10.6759	−0.696416
$236$	−16.1697	−1.05256
$237$	0	0
$238$	−26.5653	−1.72197
$239$	−15.6582	−1.01285	−0.506423	−	0.862285i	$-0.669033\pi$
$239$	−15.6582	−1.01285	−0.506423	+	0.862285i	$0.669033\pi$
$240$	0	0
$241$	22.1343	1.42580	0.712899	−	0.701267i	$-0.247382\pi$
$241$	22.1343	1.42580	0.712899	+	0.701267i	$0.247382\pi$
$242$	−46.2018	−2.96996
$243$	0	0
$244$	−7.02069	−0.449454
$245$	−1.15037	−0.0734943
$246$	0	0
$247$	17.3743	1.10550
$248$	−8.20869	−0.521252
$249$	0	0
$250$	22.5004	1.42305
$251$	8.45138	0.533447	0.266723	−	0.963773i	$-0.414059\pi$
$251$	8.45138	0.533447	0.266723	+	0.963773i	$0.414059\pi$
$252$	0	0
$253$	−6.15486	−0.386953
$254$	−8.23231	−0.516541
$255$	0	0
$256$	−26.0616	−1.62885
$257$	−22.4454	−1.40011	−0.700054	−	0.714090i	$-0.746841\pi$
$257$	−22.4454	−1.40011	−0.700054	+	0.714090i	$0.746841\pi$
$258$	0	0
$259$	13.0330	0.809832
$260$	19.6361	1.21778
$261$	0	0
$262$	5.85143	0.361503
$263$	−22.7197	−1.40096	−0.700479	−	0.713673i	$-0.747030\pi$
$263$	−22.7197	−1.40096	−0.700479	+	0.713673i	$0.747030\pi$
$264$	0	0
$265$	−13.8904	−0.853280
$266$	−19.8349	−1.21615
$267$	0	0
$268$	−45.3590	−2.77074
$269$	−7.11622	−0.433884	−0.216942	−	0.976185i	$-0.569608\pi$
$269$	−7.11622	−0.433884	−0.216942	+	0.976185i	$0.569608\pi$
$270$	0	0
$271$	6.22249	0.377989	0.188995	−	0.981978i	$-0.439477\pi$
$271$	6.22249	0.377989	0.188995	+	0.981978i	$0.439477\pi$
$272$	8.62306	0.522850
$273$	0	0
$274$	−8.98092	−0.542557
$275$	−21.2590	−1.28196
$276$	0	0
$277$	−16.8319	−1.01133	−0.505664	−	0.862730i	$-0.668753\pi$
$277$	−16.8319	−1.01133	−0.505664	+	0.862730i	$0.668753\pi$
$278$	−32.0171	−1.92026
$279$	0	0
$280$	−10.0380	−0.599886
$281$	7.70412	0.459589	0.229795	−	0.973239i	$-0.426195\pi$
$281$	7.70412	0.459589	0.229795	+	0.973239i	$0.426195\pi$
$282$	0	0
$283$	−7.29801	−0.433822	−0.216911	−	0.976191i	$-0.569598\pi$
$283$	−7.29801	−0.433822	−0.216911	+	0.976191i	$0.569598\pi$
$284$	−6.09151	−0.361465
$285$	0	0
$286$	−66.1948	−3.91418
$287$	−1.29322	−0.0763361
$288$	0	0
$289$	4.17921	0.245836
$290$	−5.56867	−0.327004
$291$	0	0
$292$	−27.2574	−1.59512
$293$	−21.3224	−1.24567	−0.622834	−	0.782354i	$-0.714019\pi$
$293$	−21.3224	−1.24567	−0.622834	+	0.782354i	$0.714019\pi$
$294$	0	0
$295$	−4.78715	−0.278719
$296$	−20.5850	−1.19648
$297$	0	0
$298$	2.37103	0.137350
$299$	−5.63645	−0.325965
$300$	0	0
$301$	9.63236	0.555200
$302$	9.49425	0.546333
$303$	0	0
$304$	6.43838	0.369266
$305$	−2.07852	−0.119016
$306$	0	0
$307$	−6.87669	−0.392474	−0.196237	−	0.980557i	$-0.562872\pi$
$307$	−6.87669	−0.392474	−0.196237	+	0.980557i	$0.562872\pi$
$308$	48.6850	2.77408
$309$	0	0
$310$	−5.42722	−0.308246
$311$	−16.3161	−0.925200	−0.462600	−	0.886567i	$-0.653083\pi$
$311$	−16.3161	−0.925200	−0.462600	+	0.886567i	$0.653083\pi$
$312$	0	0
$313$	2.55376	0.144347	0.0721735	−	0.997392i	$-0.477006\pi$
$313$	2.55376	0.144347	0.0721735	+	0.997392i	$0.477006\pi$
$314$	45.5218	2.56894
$315$	0	0
$316$	−8.04645	−0.452648
$317$	22.3107	1.25310	0.626548	−	0.779383i	$-0.284467\pi$
$317$	22.3107	1.25310	0.626548	+	0.779383i	$0.284467\pi$
$318$	0	0
$319$	12.0940	0.677132
$320$	−12.2755	−0.686220
$321$	0	0
$322$	6.43470	0.358592
$323$	15.8134	0.879880
$324$	0	0
$325$	−19.4684	−1.07991
$326$	50.2676	2.78407
$327$	0	0
$328$	2.04257	0.112782
$329$	−24.2398	−1.33638
$330$	0	0
$331$	10.6974	0.587983	0.293991	−	0.955808i	$-0.405016\pi$
$331$	10.6974	0.587983	0.293991	+	0.955808i	$0.405016\pi$
$332$	−47.1232	−2.58622
$333$	0	0
$334$	−33.4590	−1.83080
$335$	−13.4288	−0.733694
$336$	0	0
$337$	−6.61011	−0.360076	−0.180038	−	0.983660i	$-0.557622\pi$
$337$	−6.61011	−0.360076	−0.180038	+	0.983660i	$0.557622\pi$
$338$	−29.7960	−1.62069
$339$	0	0
$340$	17.8720	0.969245
$341$	11.7868	0.638289
$342$	0	0
$343$	−19.6540	−1.06121
$344$	−15.2138	−0.820275
$345$	0	0
$346$	−7.86510	−0.422830
$347$	23.2580	1.24855	0.624277	−	0.781203i	$-0.285394\pi$
$347$	23.2580	1.24855	0.624277	+	0.781203i	$0.285394\pi$
$348$	0	0
$349$	−3.36784	−0.180276	−0.0901380	−	0.995929i	$-0.528731\pi$
$349$	−3.36784	−0.180276	−0.0901380	+	0.995929i	$0.528731\pi$
$350$	22.2255	1.18800
$351$	0	0
$352$	17.9330	0.955834
$353$	−23.8699	−1.27047	−0.635234	−	0.772320i	$-0.719096\pi$
$353$	−23.8699	−1.27047	−0.635234	+	0.772320i	$0.719096\pi$
$354$	0	0
$355$	−1.80343	−0.0957161
$356$	54.7793	2.90330
$357$	0	0
$358$	29.6300	1.56599
$359$	16.2205	0.856084	0.428042	−	0.903759i	$-0.359204\pi$
$359$	16.2205	0.856084	0.428042	+	0.903759i	$0.359204\pi$
$360$	0	0
$361$	−7.19299	−0.378578
$362$	15.2910	0.803679
$363$	0	0
$364$	44.5843	2.33685
$365$	−8.06972	−0.422388
$366$	0	0
$367$	7.03149	0.367041	0.183520	−	0.983016i	$-0.441251\pi$
$367$	7.03149	0.367041	0.183520	+	0.983016i	$0.441251\pi$
$368$	−2.08870	−0.108881
$369$	0	0
$370$	−13.6099	−0.707545
$371$	−31.5385	−1.63740
$372$	0	0
$373$	28.8460	1.49359	0.746796	−	0.665054i	$-0.231591\pi$
$373$	28.8460	1.49359	0.746796	+	0.665054i	$0.231591\pi$
$374$	−60.2479	−3.11534
$375$	0	0
$376$	38.2856	1.97443
$377$	11.0753	0.570408
$378$	0	0
$379$	−8.68574	−0.446156	−0.223078	−	0.974801i	$-0.571611\pi$
$379$	−8.68574	−0.446156	−0.223078	+	0.974801i	$0.571611\pi$
$380$	13.3441	0.684536
$381$	0	0
$382$	−57.4540	−2.93960
$383$	12.8063	0.654371	0.327186	−	0.944960i	$-0.393900\pi$
$383$	12.8063	0.654371	0.327186	+	0.944960i	$0.393900\pi$
$384$	0	0
$385$	14.4135	0.734579
$386$	−37.6161	−1.91461
$387$	0	0
$388$	−68.6144	−3.48337
$389$	−35.6423	−1.80714	−0.903568	−	0.428446i	$-0.859061\pi$
$389$	−35.6423	−1.80714	−0.903568	+	0.428446i	$0.859061\pi$
$390$	0	0
$391$	−5.13007	−0.259439
$392$	4.12543	0.208366
$393$	0	0
$394$	31.3809	1.58095
$395$	−2.38220	−0.119861
$396$	0	0
$397$	3.86663	0.194061	0.0970303	−	0.995281i	$-0.469066\pi$
$397$	3.86663	0.194061	0.0970303	+	0.995281i	$0.469066\pi$
$398$	−9.91045	−0.496766
$399$	0	0
$400$	−7.21438	−0.360719
$401$	−5.98549	−0.298901	−0.149451	−	0.988769i	$-0.547750\pi$
$401$	−5.98549	−0.298901	−0.149451	+	0.988769i	$0.547750\pi$
$402$	0	0
$403$	10.7940	0.537687
$404$	−10.4968	−0.522234
$405$	0	0
$406$	−12.6438	−0.627502
$407$	29.5578	1.46513
$408$	0	0
$409$	−22.3703	−1.10614	−0.553071	−	0.833134i	$-0.686544\pi$
$409$	−22.3703	−1.10614	−0.553071	+	0.833134i	$0.686544\pi$
$410$	1.35046	0.0666943
$411$	0	0
$412$	53.6820	2.64472
$413$	−10.8694	−0.534846
$414$	0	0
$415$	−13.9511	−0.684833
$416$	16.4226	0.805183
$417$	0	0
$418$	−44.9839	−2.20023
$419$	22.8630	1.11693	0.558466	−	0.829527i	$-0.311390\pi$
$419$	22.8630	1.11693	0.558466	+	0.829527i	$0.311390\pi$
$420$	0	0
$421$	13.5148	0.658671	0.329336	−	0.944213i	$-0.393175\pi$
$421$	13.5148	0.658671	0.329336	+	0.944213i	$0.393175\pi$
$422$	−56.7967	−2.76482
$423$	0	0
$424$	49.8135	2.41916
$425$	−17.7193	−0.859514
$426$	0	0
$427$	−4.71933	−0.228384
$428$	−3.37053	−0.162921
$429$	0	0
$430$	−10.0587	−0.485075
$431$	39.1639	1.88646	0.943229	−	0.332143i	$-0.107772\pi$
$431$	39.1639	1.88646	0.943229	+	0.332143i	$0.107772\pi$
$432$	0	0
$433$	−31.6679	−1.52186	−0.760930	−	0.648833i	$-0.775257\pi$
$433$	−31.6679	−1.52186	−0.760930	+	0.648833i	$0.775257\pi$
$434$	−12.3227	−0.591507
$435$	0	0
$436$	−42.3116	−2.02636
$437$	−3.83035	−0.183231
$438$	0	0
$439$	−10.4558	−0.499030	−0.249515	−	0.968371i	$-0.580271\pi$
$439$	−10.4558	−0.499030	−0.249515	+	0.968371i	$0.580271\pi$
$440$	−22.7654	−1.08530
$441$	0	0
$442$	−55.1733	−2.62433
$443$	−8.84486	−0.420232	−0.210116	−	0.977676i	$-0.567384\pi$
$443$	−8.84486	−0.420232	−0.210116	+	0.977676i	$0.567384\pi$
$444$	0	0
$445$	16.2177	0.768795
$446$	53.5456	2.53546
$447$	0	0
$448$	−27.8718	−1.31682
$449$	−8.91298	−0.420630	−0.210315	−	0.977634i	$-0.567449\pi$
$449$	−8.91298	−0.420630	−0.210315	+	0.977634i	$0.567449\pi$
$450$	0	0
$451$	−2.93290	−0.138105
$452$	66.4165	3.12397
$453$	0	0
$454$	−4.35801	−0.204531
$455$	13.1995	0.618800
$456$	0	0
$457$	−8.92952	−0.417705	−0.208853	−	0.977947i	$-0.566973\pi$
$457$	−8.92952	−0.417705	−0.208853	+	0.977947i	$0.566973\pi$
$458$	−46.9848	−2.19546
$459$	0	0
$460$	−4.32899	−0.201840
$461$	−10.9096	−0.508113	−0.254056	−	0.967189i	$-0.581765\pi$
$461$	−10.9096	−0.508113	−0.254056	+	0.967189i	$0.581765\pi$
$462$	0	0
$463$	37.9445	1.76343	0.881716	−	0.471780i	$-0.156388\pi$
$463$	37.9445	1.76343	0.881716	+	0.471780i	$0.156388\pi$
$464$	4.10417	0.190531
$465$	0	0
$466$	37.4966	1.73700
$467$	19.1382	0.885610	0.442805	−	0.896618i	$-0.353983\pi$
$467$	19.1382	0.885610	0.442805	+	0.896618i	$0.353983\pi$
$468$	0	0
$469$	−30.4905	−1.40792
$470$	25.3128	1.16759
$471$	0	0
$472$	17.1676	0.790203
$473$	21.8454	1.00445
$474$	0	0
$475$	−13.2301	−0.607038
$476$	40.5788	1.85993
$477$	0	0
$478$	37.1261	1.69811
$479$	−36.0071	−1.64521	−0.822604	−	0.568615i	$-0.807479\pi$
$479$	−36.0071	−1.64521	−0.822604	+	0.568615i	$0.807479\pi$
$480$	0	0
$481$	27.0682	1.23420
$482$	−52.4811	−2.39045
$483$	0	0
$484$	70.5738	3.20790
$485$	−20.3137	−0.922398
$486$	0	0
$487$	−8.41649	−0.381388	−0.190694	−	0.981650i	$-0.561074\pi$
$487$	−8.41649	−0.381388	−0.190694	+	0.981650i	$0.561074\pi$
$488$	7.45395	0.337424
$489$	0	0
$490$	2.72756	0.123218
$491$	3.47270	0.156721	0.0783605	−	0.996925i	$-0.475031\pi$
$491$	3.47270	0.156721	0.0783605	+	0.996925i	$0.475031\pi$
$492$	0	0
$493$	10.0803	0.453994
$494$	−41.1950	−1.85345
$495$	0	0
$496$	3.99992	0.179602
$497$	−4.09474	−0.183674
$498$	0	0
$499$	31.0994	1.39220	0.696101	−	0.717944i	$-0.254917\pi$
$499$	31.0994	1.39220	0.696101	+	0.717944i	$0.254917\pi$
$500$	−34.3696	−1.53706
$501$	0	0
$502$	−20.0385	−0.894361
$503$	−4.77437	−0.212879	−0.106439	−	0.994319i	$-0.533945\pi$
$503$	−4.77437	−0.212879	−0.106439	+	0.994319i	$0.533945\pi$
$504$	0	0
$505$	−3.10763	−0.138288
$506$	14.5934	0.648754
$507$	0	0
$508$	12.5750	0.557924
$509$	5.68193	0.251847	0.125924	−	0.992040i	$-0.459811\pi$
$509$	5.68193	0.251847	0.125924	+	0.992040i	$0.459811\pi$
$510$	0	0
$511$	−18.3225	−0.810540
$512$	20.4957	0.905793
$513$	0	0
$514$	53.2188	2.34738
$515$	15.8929	0.700324
$516$	0	0
$517$	−54.9739	−2.41775
$518$	−30.9017	−1.35774
$519$	0	0
$520$	−20.8479	−0.914240
$521$	10.6945	0.468532	0.234266	−	0.972172i	$-0.424731\pi$
$521$	10.6945	0.468532	0.234266	+	0.972172i	$0.424731\pi$
$522$	0	0
$523$	17.4393	0.762566	0.381283	−	0.924458i	$-0.375482\pi$
$523$	17.4393	0.762566	0.381283	+	0.924458i	$0.375482\pi$
$524$	−8.93814	−0.390465
$525$	0	0
$526$	53.8691	2.34880
$527$	9.82426	0.427951
$528$	0	0
$529$	−21.7574	−0.945973
$530$	32.9345	1.43058
$531$	0	0
$532$	30.2981	1.31359
$533$	−2.68587	−0.116338
$534$	0	0
$535$	−0.997866	−0.0431415
$536$	48.1582	2.08012
$537$	0	0
$538$	16.8728	0.727437
$539$	−5.92367	−0.255150
$540$	0	0
$541$	6.98497	0.300307	0.150154	−	0.988663i	$-0.452023\pi$
$541$	6.98497	0.300307	0.150154	+	0.988663i	$0.452023\pi$
$542$	−14.7537	−0.633726
$543$	0	0
$544$	14.9472	0.640854
$545$	−12.5266	−0.536581
$546$	0	0
$547$	−14.1635	−0.605585	−0.302793	−	0.953056i	$-0.597919\pi$
$547$	−14.1635	−0.605585	−0.302793	+	0.953056i	$0.597919\pi$
$548$	13.7185	0.586024
$549$	0	0
$550$	50.4056	2.14930
$551$	7.52642	0.320636
$552$	0	0
$553$	−5.40885	−0.230008
$554$	39.9088	1.69556
$555$	0	0
$556$	48.9066	2.07410
$557$	26.1352	1.10738	0.553692	−	0.832721i	$-0.313218\pi$
$557$	26.1352	1.10738	0.553692	+	0.832721i	$0.313218\pi$
$558$	0	0
$559$	20.0054	0.846138
$560$	4.89131	0.206696
$561$	0	0
$562$	−18.2667	−0.770534
$563$	−4.47843	−0.188743	−0.0943716	−	0.995537i	$-0.530084\pi$
$563$	−4.47843	−0.188743	−0.0943716	+	0.995537i	$0.530084\pi$
$564$	0	0
$565$	19.6630	0.827229
$566$	17.3038	0.727333
$567$	0	0
$568$	6.46743	0.271367
$569$	20.2645	0.849533	0.424767	−	0.905303i	$-0.360356\pi$
$569$	20.2645	0.849533	0.424767	+	0.905303i	$0.360356\pi$
$570$	0	0
$571$	−26.7833	−1.12085	−0.560423	−	0.828206i	$-0.689362\pi$
$571$	−26.7833	−1.12085	−0.560423	+	0.828206i	$0.689362\pi$
$572$	101.114	4.22777
$573$	0	0
$574$	3.06625	0.127983
$575$	4.29201	0.178989
$576$	0	0
$577$	1.35657	0.0564747	0.0282373	−	0.999601i	$-0.491011\pi$
$577$	1.35657	0.0564747	0.0282373	+	0.999601i	$0.491011\pi$
$578$	−9.90903	−0.412161
$579$	0	0
$580$	8.50622	0.353202
$581$	−31.6764	−1.31416
$582$	0	0
$583$	−71.5267	−2.96233
$584$	28.9395	1.19753
$585$	0	0
$586$	50.5561	2.08845
$587$	1.49454	0.0616861	0.0308431	−	0.999524i	$-0.490181\pi$
$587$	1.49454	0.0616861	0.0308431	+	0.999524i	$0.490181\pi$
$588$	0	0
$589$	7.33525	0.302244
$590$	11.3505	0.467291
$591$	0	0
$592$	10.0306	0.412257
$593$	−28.8126	−1.18319	−0.591596	−	0.806235i	$-0.701502\pi$
$593$	−28.8126	−1.18319	−0.591596	+	0.806235i	$0.701502\pi$
$594$	0	0
$595$	12.0136	0.492510
$596$	−3.62178	−0.148354
$597$	0	0
$598$	13.3642	0.546503
$599$	11.9948	0.490093	0.245046	−	0.969511i	$-0.421197\pi$
$599$	11.9948	0.490093	0.245046	+	0.969511i	$0.421197\pi$
$600$	0	0
$601$	2.73214	0.111446	0.0557231	−	0.998446i	$-0.482254\pi$
$601$	2.73214	0.111446	0.0557231	+	0.998446i	$0.482254\pi$
$602$	−22.8386	−0.930832
$603$	0	0
$604$	−14.5026	−0.590103
$605$	20.8938	0.849454
$606$	0	0
$607$	26.1655	1.06203	0.531013	−	0.847363i	$-0.321811\pi$
$607$	26.1655	1.06203	0.531013	+	0.847363i	$0.321811\pi$
$608$	11.1602	0.452608
$609$	0	0
$610$	4.92823	0.199538
$611$	−50.3436	−2.03668
$612$	0	0
$613$	−17.4491	−0.704761	−0.352381	−	0.935857i	$-0.614628\pi$
$613$	−17.4491	−0.704761	−0.352381	+	0.935857i	$0.614628\pi$
$614$	16.3048	0.658010
$615$	0	0
$616$	−51.6894	−2.08263
$617$	−17.9811	−0.723892	−0.361946	−	0.932199i	$-0.617887\pi$
$617$	−17.9811	−0.723892	−0.361946	+	0.932199i	$0.617887\pi$
$618$	0	0
$619$	7.61974	0.306263	0.153132	−	0.988206i	$-0.451064\pi$
$619$	7.61974	0.306263	0.153132	+	0.988206i	$0.451064\pi$
$620$	8.29016	0.332941
$621$	0	0
$622$	38.6859	1.55116
$623$	36.8228	1.47528
$624$	0	0
$625$	9.07606	0.363043
$626$	−6.05504	−0.242008
$627$	0	0
$628$	−69.5351	−2.77475
$629$	24.6364	0.982317
$630$	0	0
$631$	35.7129	1.42171	0.710854	−	0.703339i	$-0.248308\pi$
$631$	35.7129	1.42171	0.710854	+	0.703339i	$0.248308\pi$
$632$	8.54301	0.339823
$633$	0	0
$634$	−52.8994	−2.10090
$635$	3.72290	0.147739
$636$	0	0
$637$	−5.42473	−0.214936
$638$	−28.6751	−1.13526
$639$	0	0
$640$	22.1404	0.875176
$641$	38.5595	1.52301	0.761504	−	0.648160i	$-0.224461\pi$
$641$	38.5595	1.52301	0.761504	+	0.648160i	$0.224461\pi$
$642$	0	0
$643$	31.0438	1.22425	0.612125	−	0.790761i	$-0.290315\pi$
$643$	31.0438	1.22425	0.612125	+	0.790761i	$0.290315\pi$
$644$	−9.82909	−0.387320
$645$	0	0
$646$	−37.4940	−1.47518
$647$	32.1331	1.26328	0.631641	−	0.775261i	$-0.282382\pi$
$647$	32.1331	1.26328	0.631641	+	0.775261i	$0.282382\pi$
$648$	0	0
$649$	−24.6508	−0.967628
$650$	46.1601	1.81055
$651$	0	0
$652$	−76.7845	−3.00711
$653$	19.3651	0.757813	0.378907	−	0.925435i	$-0.376300\pi$
$653$	19.3651	0.757813	0.378907	+	0.925435i	$0.376300\pi$
$654$	0	0
$655$	−2.64619	−0.103395
$656$	−0.995301	−0.0388600
$657$	0	0
$658$	57.4733	2.24054
$659$	−11.7647	−0.458289	−0.229145	−	0.973392i	$-0.573593\pi$
$659$	−11.7647	−0.458289	−0.229145	+	0.973392i	$0.573593\pi$
$660$	0	0
$661$	44.3806	1.72621	0.863103	−	0.505028i	$-0.168518\pi$
$661$	44.3806	1.72621	0.863103	+	0.505028i	$0.168518\pi$
$662$	−25.3639	−0.985794
$663$	0	0
$664$	50.0313	1.94159
$665$	8.96992	0.347839
$666$	0	0
$667$	−2.44167	−0.0945419
$668$	51.1091	1.97747
$669$	0	0
$670$	31.8401	1.23009
$671$	−10.7031	−0.413187
$672$	0	0
$673$	7.09514	0.273498	0.136749	−	0.990606i	$-0.456335\pi$
$673$	7.09514	0.273498	0.136749	+	0.990606i	$0.456335\pi$
$674$	15.6728	0.603692
$675$	0	0
$676$	45.5138	1.75053
$677$	−35.6343	−1.36954	−0.684769	−	0.728760i	$-0.740097\pi$
$677$	−35.6343	−1.36954	−0.684769	+	0.728760i	$0.740097\pi$
$678$	0	0
$679$	−46.1228	−1.77003
$680$	−18.9749	−0.727654
$681$	0	0
$682$	−27.9468	−1.07014
$683$	29.4656	1.12747	0.563734	−	0.825956i	$-0.309364\pi$
$683$	29.4656	1.12747	0.563734	+	0.825956i	$0.309364\pi$
$684$	0	0
$685$	4.06144	0.155180
$686$	46.6001	1.77920
$687$	0	0
$688$	7.41338	0.282633
$689$	−65.5022	−2.49543
$690$	0	0
$691$	3.05607	0.116258	0.0581292	−	0.998309i	$-0.481486\pi$
$691$	3.05607	0.116258	0.0581292	+	0.998309i	$0.481486\pi$
$692$	12.0140	0.456705
$693$	0	0
$694$	−55.1454	−2.09329
$695$	14.4791	0.549223
$696$	0	0
$697$	−2.44457	−0.0925948
$698$	7.98524	0.302246
$699$	0	0
$700$	−33.9498	−1.28318
$701$	25.5179	0.963796	0.481898	−	0.876227i	$-0.339948\pi$
$701$	25.5179	0.963796	0.481898	+	0.876227i	$0.339948\pi$
$702$	0	0
$703$	18.3947	0.693768
$704$	−63.2110	−2.38235
$705$	0	0
$706$	56.5963	2.13003
$707$	−7.05596	−0.265367
$708$	0	0
$709$	−24.5311	−0.921286	−0.460643	−	0.887586i	$-0.652381\pi$
$709$	−24.5311	−0.921286	−0.460643	+	0.887586i	$0.652381\pi$
$710$	4.27598	0.160475
$711$	0	0
$712$	−58.1599	−2.17963
$713$	−2.37965	−0.0891187
$714$	0	0
$715$	29.9353	1.11952
$716$	−45.2602	−1.69145
$717$	0	0
$718$	−38.4592	−1.43529
$719$	−24.5119	−0.914141	−0.457071	−	0.889430i	$-0.651101\pi$
$719$	−24.5119	−0.914141	−0.457071	+	0.889430i	$0.651101\pi$
$720$	0	0
$721$	36.0852	1.34388
$722$	17.0548	0.634713
$723$	0	0
$724$	−23.3573	−0.868066
$725$	−8.43356	−0.313215
$726$	0	0
$727$	1.72715	0.0640565	0.0320282	−	0.999487i	$-0.489803\pi$
$727$	1.72715	0.0640565	0.0320282	+	0.999487i	$0.489803\pi$
$728$	−47.3357	−1.75438
$729$	0	0
$730$	19.1335	0.708164
$731$	18.2081	0.673451
$732$	0	0
$733$	−5.57305	−0.205845	−0.102923	−	0.994689i	$-0.532819\pi$
$733$	−5.57305	−0.205845	−0.102923	+	0.994689i	$0.532819\pi$
$734$	−16.6719	−0.615370
$735$	0	0
$736$	−3.62053	−0.133455
$737$	−69.1499	−2.54717
$738$	0	0
$739$	0.434174	0.0159713	0.00798567	−	0.999968i	$-0.497458\pi$
$739$	0.434174	0.0159713	0.00798567	+	0.999968i	$0.497458\pi$
$740$	20.7893	0.764230
$741$	0	0
$742$	74.7787	2.74521
$743$	−9.44909	−0.346653	−0.173327	−	0.984864i	$-0.555452\pi$
$743$	−9.44909	−0.346653	−0.173327	+	0.984864i	$0.555452\pi$
$744$	0	0
$745$	−1.07225	−0.0392842
$746$	−68.3948	−2.50411
$747$	0	0
$748$	92.0294	3.36493
$749$	−2.26568	−0.0827862
$750$	0	0
$751$	−14.7155	−0.536977	−0.268488	−	0.963283i	$-0.586524\pi$
$751$	−14.7155	−0.536977	−0.268488	+	0.963283i	$0.586524\pi$
$752$	−18.6558	−0.680306
$753$	0	0
$754$	−26.2599	−0.956329
$755$	−4.29358	−0.156260
$756$	0	0
$757$	−25.9373	−0.942708	−0.471354	−	0.881944i	$-0.656235\pi$
$757$	−25.9373	−0.942708	−0.471354	+	0.881944i	$0.656235\pi$
$758$	20.5941	0.748013
$759$	0	0
$760$	−14.1675	−0.513911
$761$	26.8838	0.974536	0.487268	−	0.873253i	$-0.337994\pi$
$761$	26.8838	0.974536	0.487268	+	0.873253i	$0.337994\pi$
$762$	0	0
$763$	−28.4420	−1.02967
$764$	87.7618	3.17511
$765$	0	0
$766$	−30.3641	−1.09710
$767$	−22.5745	−0.815118
$768$	0	0
$769$	−52.0352	−1.87644	−0.938218	−	0.346045i	$-0.887524\pi$
$769$	−52.0352	−1.87644	−0.938218	+	0.346045i	$0.887524\pi$
$770$	−34.1748	−1.23157
$771$	0	0
$772$	57.4591	2.06800
$773$	−25.8717	−0.930540	−0.465270	−	0.885169i	$-0.654043\pi$
$773$	−25.8717	−0.930540	−0.465270	+	0.885169i	$0.654043\pi$
$774$	0	0
$775$	−8.21935	−0.295248
$776$	72.8487	2.61512
$777$	0	0
$778$	84.5089	3.02979
$779$	−1.82523	−0.0653957
$780$	0	0
$781$	−9.28652	−0.332298
$782$	12.1636	0.434968
$783$	0	0
$784$	−2.01024	−0.0717942
$785$	−20.5863	−0.734757
$786$	0	0
$787$	48.0542	1.71295	0.856473	−	0.516192i	$-0.172651\pi$
$787$	48.0542	1.71295	0.856473	+	0.516192i	$0.172651\pi$
$788$	−47.9348	−1.70760
$789$	0	0
$790$	5.64826	0.200956
$791$	44.6454	1.58741
$792$	0	0
$793$	−9.80156	−0.348063
$794$	−9.16789	−0.325356
$795$	0	0
$796$	15.1384	0.536565
$797$	−20.5119	−0.726569	−0.363284	−	0.931678i	$-0.618345\pi$
$797$	−20.5119	−0.726569	−0.363284	+	0.931678i	$0.618345\pi$
$798$	0	0
$799$	−45.8207	−1.62102
$800$	−12.5054	−0.442131
$801$	0	0
$802$	14.1918	0.501129
$803$	−41.5539	−1.46641
$804$	0	0
$805$	−2.90996	−0.102563
$806$	−25.5929	−0.901471
$807$	0	0
$808$	11.1445	0.392064
$809$	−36.7380	−1.29164	−0.645819	−	0.763490i	$-0.723484\pi$
$809$	−36.7380	−1.29164	−0.645819	+	0.763490i	$0.723484\pi$
$810$	0	0
$811$	54.1313	1.90081	0.950403	−	0.311020i	$-0.100671\pi$
$811$	54.1313	1.90081	0.950403	+	0.311020i	$0.100671\pi$
$812$	19.3136	0.677775
$813$	0	0
$814$	−70.0824	−2.45639
$815$	−22.7325	−0.796285
$816$	0	0
$817$	13.5950	0.475629
$818$	53.0407	1.85453
$819$	0	0
$820$	−2.06284	−0.0720376
$821$	0.394710	0.0137755	0.00688774	−	0.999976i	$-0.497808\pi$
$821$	0.394710	0.0137755	0.00688774	+	0.999976i	$0.497808\pi$
$822$	0	0
$823$	30.1685	1.05161	0.525804	−	0.850606i	$-0.323765\pi$
$823$	30.1685	1.05161	0.525804	+	0.850606i	$0.323765\pi$
$824$	−56.9948	−1.98551
$825$	0	0
$826$	25.7716	0.896707
$827$	31.9318	1.11038	0.555189	−	0.831724i	$-0.312646\pi$
$827$	31.9318	1.11038	0.555189	+	0.831724i	$0.312646\pi$
$828$	0	0
$829$	12.3088	0.427504	0.213752	−	0.976888i	$-0.431432\pi$
$829$	12.3088	0.427504	0.213752	+	0.976888i	$0.431432\pi$
$830$	33.0785	1.14817
$831$	0	0
$832$	−57.8869	−2.00687
$833$	−4.93737	−0.171070
$834$	0	0
$835$	15.1312	0.523635
$836$	68.7135	2.37651
$837$	0	0
$838$	−54.2089	−1.87262
$839$	−2.81763	−0.0972754	−0.0486377	−	0.998816i	$-0.515488\pi$
$839$	−2.81763	−0.0972754	−0.0486377	+	0.998816i	$0.515488\pi$
$840$	0	0
$841$	−24.2023	−0.834560
$842$	−32.0440	−1.10431
$843$	0	0
$844$	86.7577	2.98632
$845$	13.4746	0.463542
$846$	0	0
$847$	47.4400	1.63006
$848$	−24.2731	−0.833541
$849$	0	0
$850$	42.0130	1.44104
$851$	−5.96748	−0.204563
$852$	0	0
$853$	1.47659	0.0505574	0.0252787	−	0.999680i	$-0.491953\pi$
$853$	1.47659	0.0505574	0.0252787	+	0.999680i	$0.491953\pi$
$854$	11.1897	0.382903
$855$	0	0
$856$	3.57853	0.122312
$857$	−6.96820	−0.238029	−0.119015	−	0.992893i	$-0.537974\pi$
$857$	−6.96820	−0.238029	−0.119015	+	0.992893i	$0.537974\pi$
$858$	0	0
$859$	14.3063	0.488123	0.244062	−	0.969760i	$-0.421520\pi$
$859$	14.3063	0.488123	0.244062	+	0.969760i	$0.421520\pi$
$860$	15.3648	0.523936
$861$	0	0
$862$	−92.8587	−3.16278
$863$	20.3236	0.691822	0.345911	−	0.938267i	$-0.387570\pi$
$863$	20.3236	0.691822	0.345911	+	0.938267i	$0.387570\pi$
$864$	0	0
$865$	3.55683	0.120936
$866$	75.0855	2.55151
$867$	0	0
$868$	18.8230	0.638895
$869$	−12.2668	−0.416123
$870$	0	0
$871$	−63.3255	−2.14570
$872$	44.9227	1.52128
$873$	0	0
$874$	9.08187	0.307199
$875$	−23.1034	−0.781037
$876$	0	0
$877$	0.830560	0.0280460	0.0140230	−	0.999902i	$-0.495536\pi$
$877$	0.830560	0.0280460	0.0140230	+	0.999902i	$0.495536\pi$
$878$	24.7911	0.836659
$879$	0	0
$880$	11.0931	0.373948
$881$	39.1178	1.31791	0.658955	−	0.752182i	$-0.270999\pi$
$881$	39.1178	1.31791	0.658955	+	0.752182i	$0.270999\pi$
$882$	0	0
$883$	17.0469	0.573674	0.286837	−	0.957979i	$-0.407396\pi$
$883$	17.0469	0.573674	0.286837	+	0.957979i	$0.407396\pi$
$884$	84.2780	2.83458
$885$	0	0
$886$	20.9714	0.704549
$887$	−17.7801	−0.596998	−0.298499	−	0.954410i	$-0.596486\pi$
$887$	−17.7801	−0.596998	−0.298499	+	0.954410i	$0.596486\pi$
$888$	0	0
$889$	8.45294	0.283502
$890$	−38.4528	−1.28894
$891$	0	0
$892$	−81.7916	−2.73859
$893$	−34.2119	−1.14486
$894$	0	0
$895$	−13.3996	−0.447898
$896$	50.2704	1.67942
$897$	0	0
$898$	21.1329	0.705215
$899$	4.67588	0.155949
$900$	0	0
$901$	−59.6174	−1.98614
$902$	6.95400	0.231543
$903$	0	0
$904$	−70.5152	−2.34530
$905$	−6.91506	−0.229864
$906$	0	0
$907$	−37.9831	−1.26121	−0.630604	−	0.776105i	$-0.717193\pi$
$907$	−37.9831	−1.26121	−0.630604	+	0.776105i	$0.717193\pi$
$908$	6.65692	0.220918
$909$	0	0
$910$	−31.2963	−1.03746
$911$	−23.7959	−0.788392	−0.394196	−	0.919026i	$-0.628977\pi$
$911$	−23.7959	−0.788392	−0.394196	+	0.919026i	$0.628977\pi$
$912$	0	0
$913$	−71.8394	−2.37754
$914$	21.1722	0.700313
$915$	0	0
$916$	71.7700	2.37135
$917$	−6.00825	−0.198410
$918$	0	0
$919$	−42.2443	−1.39351	−0.696756	−	0.717308i	$-0.745374\pi$
$919$	−42.2443	−1.39351	−0.696756	+	0.717308i	$0.745374\pi$
$920$	4.59614	0.151530
$921$	0	0
$922$	25.8671	0.851887
$923$	−8.50434	−0.279924
$924$	0	0
$925$	−20.6117	−0.677710
$926$	−89.9676	−2.95652
$927$	0	0
$928$	7.11414	0.233533
$929$	14.2178	0.466472	0.233236	−	0.972420i	$-0.425069\pi$
$929$	14.2178	0.466472	0.233236	+	0.972420i	$0.425069\pi$
$930$	0	0
$931$	−3.68647	−0.120819
$932$	−57.2766	−1.87616
$933$	0	0
$934$	−45.3772	−1.48479
$935$	27.2459	0.891035
$936$	0	0
$937$	−3.82472	−0.124948	−0.0624741	−	0.998047i	$-0.519899\pi$
$937$	−3.82472	−0.124948	−0.0624741	+	0.998047i	$0.519899\pi$
$938$	72.2938	2.36048
$939$	0	0
$940$	−38.6656	−1.26113
$941$	42.4235	1.38297	0.691483	−	0.722392i	$-0.256958\pi$
$941$	42.4235	1.38297	0.691483	+	0.722392i	$0.256958\pi$
$942$	0	0
$943$	0.592129	0.0192824
$944$	−8.36542	−0.272271
$945$	0	0
$946$	−51.7961	−1.68403
$947$	29.5450	0.960084	0.480042	−	0.877246i	$-0.340621\pi$
$947$	29.5450	0.960084	0.480042	+	0.877246i	$0.340621\pi$
$948$	0	0
$949$	−38.0539	−1.23528
$950$	31.3689	1.01774
$951$	0	0
$952$	−43.0830	−1.39633
$953$	−24.1244	−0.781466	−0.390733	−	0.920504i	$-0.627778\pi$
$953$	−24.1244	−0.781466	−0.390733	+	0.920504i	$0.627778\pi$
$954$	0	0
$955$	25.9824	0.840771
$956$	−56.7106	−1.83415
$957$	0	0
$958$	85.3739	2.75831
$959$	9.22161	0.297781
$960$	0	0
$961$	−26.4429	−0.852996
$962$	−64.1795	−2.06923
$963$	0	0
$964$	80.1656	2.58196
$965$	17.0111	0.547607
$966$	0	0
$967$	19.9475	0.641470	0.320735	−	0.947169i	$-0.396070\pi$
$967$	19.9475	0.641470	0.320735	+	0.947169i	$0.396070\pi$
$968$	−74.9291	−2.40831
$969$	0	0
$970$	48.1644	1.54647
$971$	10.7773	0.345862	0.172931	−	0.984934i	$-0.444676\pi$
$971$	10.7773	0.345862	0.172931	+	0.984934i	$0.444676\pi$
$972$	0	0
$973$	32.8752	1.05393
$974$	19.9558	0.639424
$975$	0	0
$976$	−3.63216	−0.116262
$977$	−19.5515	−0.625508	−0.312754	−	0.949834i	$-0.601252\pi$
$977$	−19.5515	−0.625508	−0.312754	+	0.949834i	$0.601252\pi$
$978$	0	0
$979$	83.5112	2.66903
$980$	−4.16638	−0.133090
$981$	0	0
$982$	−8.23388	−0.262754
$983$	−22.7909	−0.726917	−0.363459	−	0.931610i	$-0.618404\pi$
$983$	−22.7909	−0.726917	−0.363459	+	0.931610i	$0.618404\pi$
$984$	0	0
$985$	−14.1914	−0.452175
$986$	−23.9007	−0.761153
$987$	0	0
$988$	62.9259	2.00194
$989$	−4.41040	−0.140243
$990$	0	0
$991$	40.1534	1.27552	0.637758	−	0.770237i	$-0.279862\pi$
$991$	40.1534	1.27552	0.637758	+	0.770237i	$0.279862\pi$
$992$	6.93344	0.220137
$993$	0	0
$994$	9.70874	0.307943
$995$	4.48180	0.142083
$996$	0	0
$997$	34.0796	1.07931	0.539657	−	0.841885i	$-0.318554\pi$
$997$	34.0796	1.07931	0.539657	+	0.841885i	$0.318554\pi$
$998$	−73.7377	−2.33413
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.e.1.1		9
3.2	odd	2		149.2.a.b.1.9	✓	9
12.11	even	2		2384.2.a.j.1.7		9
15.14	odd	2		3725.2.a.c.1.1		9
21.20	even	2		7301.2.a.j.1.9		9
24.5	odd	2		9536.2.a.v.1.7		9
24.11	even	2		9536.2.a.w.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.a.b.1.9	✓	9	3.2	odd	2
1341.2.a.e.1.1		9	1.1	even	1	trivial
2384.2.a.j.1.7		9	12.11	even	2
3725.2.a.c.1.1		9	15.14	odd	2
7301.2.a.j.1.9		9	21.20	even	2
9536.2.a.v.1.7		9	24.5	odd	2
9536.2.a.w.1.3		9	24.11	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:	\( N \)	\(=\)	\( 1341 = 3^{2} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1341.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37103 q^{2} +3.62178 q^{4} +1.07225 q^{5} +2.43457 q^{7} -3.84529 q^{8} +O(q^{10})\)	\(q-2.37103 q^{2} +3.62178 q^{4} +1.07225 q^{5} +2.43457 q^{7} -3.84529 q^{8} -2.54234 q^{10} +5.52141 q^{11} +5.05635 q^{13} -5.77244 q^{14} +1.87373 q^{16} +4.60209 q^{17} +3.43613 q^{19} +3.88345 q^{20} -13.0914 q^{22} -1.11473 q^{23} -3.85028 q^{25} -11.9888 q^{26} +8.81749 q^{28} +2.19038 q^{29} +2.13474 q^{31} +3.24791 q^{32} -10.9117 q^{34} +2.61047 q^{35} +5.35331 q^{37} -8.14717 q^{38} -4.12311 q^{40} -0.531188 q^{41} +3.95649 q^{43} +19.9973 q^{44} +2.64305 q^{46} -9.95650 q^{47} -1.07285 q^{49} +9.12913 q^{50} +18.3130 q^{52} -12.9544 q^{53} +5.92033 q^{55} -9.36163 q^{56} -5.19345 q^{58} -4.46458 q^{59} -1.93846 q^{61} -5.06153 q^{62} -11.4483 q^{64} +5.42167 q^{65} -12.5240 q^{67} +16.6677 q^{68} -6.18950 q^{70} -1.68191 q^{71} -7.52597 q^{73} -12.6929 q^{74} +12.4449 q^{76} +13.4423 q^{77} -2.22168 q^{79} +2.00910 q^{80} +1.25946 q^{82} -13.0111 q^{83} +4.93459 q^{85} -9.38095 q^{86} -21.2314 q^{88} +15.1250 q^{89} +12.3101 q^{91} -4.03730 q^{92} +23.6072 q^{94} +3.68439 q^{95} -18.9449 q^{97} +2.54377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8	\( 9 q + q^{2} + 13 q^{4} + q^{5} + 3 q^{7} + 6 q^{8} - 6 q^{10} - 5 q^{11} + 7 q^{13} + 8 q^{14} + 13 q^{16} + 5 q^{17} + 30 q^{19} + 10 q^{20} - 7 q^{22} + 4 q^{23} + 6 q^{25} + 15 q^{26} - 12 q^{28} + 16 q^{29} + 22 q^{31} + 38 q^{32} + 9 q^{34} + 11 q^{35} - 7 q^{37} + 18 q^{38} - 7 q^{40} - 6 q^{41} + 4 q^{43} - 6 q^{44} + q^{46} + 6 q^{47} + 14 q^{49} + 16 q^{50} + 50 q^{52} + 2 q^{53} - 2 q^{55} - 7 q^{56} + 2 q^{58} - 43 q^{59} + q^{61} - 33 q^{62} + 18 q^{64} + 20 q^{65} + 33 q^{67} + 16 q^{68} - 3 q^{70} - 15 q^{71} - 11 q^{73} - 33 q^{74} + 59 q^{76} + 30 q^{77} + q^{79} - 65 q^{80} + 5 q^{82} + 4 q^{83} - 34 q^{85} + 7 q^{86} - 37 q^{88} + 19 q^{89} + 62 q^{91} - 17 q^{92} + 17 q^{94} + 21 q^{95} - q^{97} - 36 q^{98}+O(q^{100}) \) 9 * q + q^2 + 13 * q^4 + q^5 + 3 * q^7 + 6 * q^8 - 6 * q^10 - 5 * q^11 + 7 * q^13 + 8 * q^14 + 13 * q^16 + 5 * q^17 + 30 * q^19 + 10 * q^20 - 7 * q^22 + 4 * q^23 + 6 * q^25 + 15 * q^26 - 12 * q^28 + 16 * q^29 + 22 * q^31 + 38 * q^32 + 9 * q^34 + 11 * q^35 - 7 * q^37 + 18 * q^38 - 7 * q^40 - 6 * q^41 + 4 * q^43 - 6 * q^44 + q^46 + 6 * q^47 + 14 * q^49 + 16 * q^50 + 50 * q^52 + 2 * q^53 - 2 * q^55 - 7 * q^56 + 2 * q^58 - 43 * q^59 + q^61 - 33 * q^62 + 18 * q^64 + 20 * q^65 + 33 * q^67 + 16 * q^68 - 3 * q^70 - 15 * q^71 - 11 * q^73 - 33 * q^74 + 59 * q^76 + 30 * q^77 + q^79 - 65 * q^80 + 5 * q^82 + 4 * q^83 - 34 * q^85 + 7 * q^86 - 37 * q^88 + 19 * q^89 + 62 * q^91 - 17 * q^92 + 17 * q^94 + 21 * q^95 - q^97 - 36 * q^98

Introduction

L-functions

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