LMFDB - Embedded newform 6633.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6633,2,Mod(1,6633)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6633, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6633.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6633 = 3^{2} \cdot 11 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6633.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:	$52.9647716607$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 $ x^17 - x^16 - 26x^15 + 25x^14 + 272x^13 - 244x^12 - 1472x^11 + 1186x^10 + 4406x^9 - 3042x^8 - 7180x^7 + 4055x^6 + 5665x^5 - 2655x^4 - 1418x^3 + 810x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 737)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.72368$ of defining polynomial
Character	$\chi$	$=$	6633.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.72368 q^{2} +0.971084 q^{4} -1.06718 q^{5} +4.32688 q^{7} +1.77352 q^{8} +O(q^{10})$	\(q-1.72368 q^{2} +0.971084 q^{4} -1.06718 q^{5} +4.32688 q^{7} +1.77352 q^{8} +1.83948 q^{10} -1.00000 q^{11} +6.00251 q^{13} -7.45818 q^{14} -4.99916 q^{16} +2.14842 q^{17} +2.37519 q^{19} -1.03632 q^{20} +1.72368 q^{22} -1.08987 q^{23} -3.86112 q^{25} -10.3464 q^{26} +4.20177 q^{28} -7.27001 q^{29} -6.82333 q^{31} +5.06993 q^{32} -3.70320 q^{34} -4.61758 q^{35} +7.40982 q^{37} -4.09407 q^{38} -1.89267 q^{40} +4.40810 q^{41} +4.27535 q^{43} -0.971084 q^{44} +1.87859 q^{46} +6.39942 q^{47} +11.7219 q^{49} +6.65535 q^{50} +5.82895 q^{52} +11.4481 q^{53} +1.06718 q^{55} +7.67384 q^{56} +12.5312 q^{58} +10.1027 q^{59} +2.53640 q^{61} +11.7613 q^{62} +1.25938 q^{64} -6.40578 q^{65} -1.00000 q^{67} +2.08630 q^{68} +7.95924 q^{70} +12.8095 q^{71} -1.68587 q^{73} -12.7722 q^{74} +2.30651 q^{76} -4.32688 q^{77} +1.90386 q^{79} +5.33502 q^{80} -7.59816 q^{82} -6.54740 q^{83} -2.29276 q^{85} -7.36935 q^{86} -1.77352 q^{88} +4.54927 q^{89} +25.9722 q^{91} -1.05836 q^{92} -11.0306 q^{94} -2.53476 q^{95} -13.7516 q^{97} -20.2049 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * 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 $

$2$	−1.72368	−1.21883	−0.609414	−	0.792852i	$-0.708595\pi$
$2$	−1.72368	−1.21883	−0.609414	+	0.792852i	$0.708595\pi$
$3$	0	0
$3$	0	0
$4$	0.971084	0.485542
$5$	−1.06718	−0.477259	−0.238629	−	0.971111i	$-0.576698\pi$
$5$	−1.06718	−0.477259	−0.238629	+	0.971111i	$0.576698\pi$
$6$	0	0
$7$	4.32688	1.63541	0.817704	−	0.575638i	$-0.195246\pi$
$7$	4.32688	1.63541	0.817704	+	0.575638i	$0.195246\pi$
$8$	1.77352	0.627036
$9$	0	0
$10$	1.83948	0.581696
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.00251	1.66480	0.832399	−	0.554177i	$-0.186967\pi$
$13$	6.00251	1.66480	0.832399	+	0.554177i	$0.186967\pi$
$14$	−7.45818	−1.99328
$15$	0	0
$16$	−4.99916	−1.24979
$17$	2.14842	0.521068	0.260534	−	0.965465i	$-0.416101\pi$
$17$	2.14842	0.521068	0.260534	+	0.965465i	$0.416101\pi$
$18$	0	0
$19$	2.37519	0.544905	0.272453	−	0.962169i	$-0.412165\pi$
$19$	2.37519	0.544905	0.272453	+	0.962169i	$0.412165\pi$
$20$	−1.03632	−0.231729
$21$	0	0
$22$	1.72368	0.367491
$23$	−1.08987	−0.227254	−0.113627	−	0.993523i	$-0.536247\pi$
$23$	−1.08987	−0.227254	−0.113627	+	0.993523i	$0.536247\pi$
$24$	0	0
$25$	−3.86112	−0.772224
$26$	−10.3464	−2.02910
$27$	0	0
$28$	4.20177	0.794060
$29$	−7.27001	−1.35001	−0.675004	−	0.737815i	$-0.735858\pi$
$29$	−7.27001	−1.35001	−0.675004	+	0.737815i	$0.735858\pi$
$30$	0	0
$31$	−6.82333	−1.22551	−0.612753	−	0.790274i	$-0.709938\pi$
$31$	−6.82333	−1.22551	−0.612753	+	0.790274i	$0.709938\pi$
$32$	5.06993	0.896245
$33$	0	0
$34$	−3.70320	−0.635093
$35$	−4.61758	−0.780513
$36$	0	0
$37$	7.40982	1.21817	0.609084	−	0.793106i	$-0.291537\pi$
$37$	7.40982	1.21817	0.609084	+	0.793106i	$0.291537\pi$
$38$	−4.09407	−0.664146
$39$	0	0
$40$	−1.89267	−0.299258
$41$	4.40810	0.688429	0.344214	−	0.938891i	$-0.388145\pi$
$41$	4.40810	0.688429	0.344214	+	0.938891i	$0.388145\pi$
$42$	0	0
$43$	4.27535	0.651984	0.325992	−	0.945372i	$-0.394302\pi$
$43$	4.27535	0.651984	0.325992	+	0.945372i	$0.394302\pi$
$44$	−0.971084	−0.146396
$45$	0	0
$46$	1.87859	0.276983
$47$	6.39942	0.933452	0.466726	−	0.884402i	$-0.345433\pi$
$47$	6.39942	0.933452	0.466726	+	0.884402i	$0.345433\pi$
$48$	0	0
$49$	11.7219	1.67456
$50$	6.65535	0.941209
$51$	0	0
$52$	5.82895	0.808329
$53$	11.4481	1.57252	0.786262	−	0.617893i	$-0.212014\pi$
$53$	11.4481	1.57252	0.786262	+	0.617893i	$0.212014\pi$
$54$	0	0
$55$	1.06718	0.143899
$56$	7.67384	1.02546
$57$	0	0
$58$	12.5312	1.64543
$59$	10.1027	1.31526	0.657632	−	0.753339i	$-0.271558\pi$
$59$	10.1027	1.31526	0.657632	+	0.753339i	$0.271558\pi$
$60$	0	0
$61$	2.53640	0.324753	0.162376	−	0.986729i	$-0.448084\pi$
$61$	2.53640	0.324753	0.162376	+	0.986729i	$0.448084\pi$
$62$	11.7613	1.49368
$63$	0	0
$64$	1.25938	0.157422
$65$	−6.40578	−0.794539
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	2.08630	0.253001
$69$	0	0
$70$	7.95924	0.951311
$71$	12.8095	1.52021	0.760106	−	0.649799i	$-0.225147\pi$
$71$	12.8095	1.52021	0.760106	+	0.649799i	$0.225147\pi$
$72$	0	0
$73$	−1.68587	−0.197316	−0.0986581	−	0.995121i	$-0.531455\pi$
$73$	−1.68587	−0.197316	−0.0986581	+	0.995121i	$0.531455\pi$
$74$	−12.7722	−1.48474
$75$	0	0
$76$	2.30651	0.264575
$77$	−4.32688	−0.493094
$78$	0	0
$79$	1.90386	0.214201	0.107100	−	0.994248i	$-0.465843\pi$
$79$	1.90386	0.214201	0.107100	+	0.994248i	$0.465843\pi$
$80$	5.33502	0.596473
$81$	0	0
$82$	−7.59816	−0.839076
$83$	−6.54740	−0.718670	−0.359335	−	0.933209i	$-0.616997\pi$
$83$	−6.54740	−0.718670	−0.359335	+	0.933209i	$0.616997\pi$
$84$	0	0
$85$	−2.29276	−0.248684
$86$	−7.36935	−0.794657
$87$	0	0
$88$	−1.77352	−0.189058
$89$	4.54927	0.482222	0.241111	−	0.970498i	$-0.422488\pi$
$89$	4.54927	0.482222	0.241111	+	0.970498i	$0.422488\pi$
$90$	0	0
$91$	25.9722	2.72262
$92$	−1.05836	−0.110341
$93$	0	0
$94$	−11.0306	−1.13772
$95$	−2.53476	−0.260061
$96$	0	0
$97$	−13.7516	−1.39626	−0.698132	−	0.715970i	$-0.745985\pi$
$97$	−13.7516	−1.39626	−0.698132	+	0.715970i	$0.745985\pi$
$98$	−20.2049	−2.04100
$99$	0	0
$100$	−3.74947	−0.374947
$101$	0.414174	0.0412119	0.0206059	−	0.999788i	$-0.493440\pi$
$101$	0.414174	0.0412119	0.0206059	+	0.999788i	$0.493440\pi$
$102$	0	0
$103$	−8.46693	−0.834272	−0.417136	−	0.908844i	$-0.636966\pi$
$103$	−8.46693	−0.834272	−0.417136	+	0.908844i	$0.636966\pi$
$104$	10.6456	1.04389
$105$	0	0
$106$	−19.7330	−1.91664
$107$	−12.8109	−1.23848	−0.619240	−	0.785202i	$-0.712559\pi$
$107$	−12.8109	−1.23848	−0.619240	+	0.785202i	$0.712559\pi$
$108$	0	0
$109$	−13.3743	−1.28103	−0.640515	−	0.767946i	$-0.721279\pi$
$109$	−13.3743	−1.28103	−0.640515	+	0.767946i	$0.721279\pi$
$110$	−1.83948	−0.175388
$111$	0	0
$112$	−21.6308	−2.04392
$113$	−6.31968	−0.594505	−0.297253	−	0.954799i	$-0.596070\pi$
$113$	−6.31968	−0.594505	−0.297253	+	0.954799i	$0.596070\pi$
$114$	0	0
$115$	1.16309	0.108459
$116$	−7.05979	−0.655485
$117$	0	0
$118$	−17.4139	−1.60308
$119$	9.29597	0.852160
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.37195	−0.395818
$123$	0	0
$124$	−6.62603	−0.595035
$125$	9.45643	0.845809
$126$	0	0
$127$	−14.5338	−1.28967	−0.644835	−	0.764322i	$-0.723074\pi$
$127$	−14.5338	−1.28967	−0.644835	+	0.764322i	$0.723074\pi$
$128$	−12.3106	−1.08812
$129$	0	0
$130$	11.0415	0.968406
$131$	7.45295	0.651167	0.325583	−	0.945513i	$-0.394439\pi$
$131$	7.45295	0.651167	0.325583	+	0.945513i	$0.394439\pi$
$132$	0	0
$133$	10.2772	0.891143
$134$	1.72368	0.148904
$135$	0	0
$136$	3.81028	0.326729
$137$	17.9277	1.53167	0.765833	−	0.643039i	$-0.222327\pi$
$137$	17.9277	1.53167	0.765833	+	0.643039i	$0.222327\pi$
$138$	0	0
$139$	21.3775	1.81322	0.906609	−	0.421973i	$-0.138662\pi$
$139$	21.3775	1.81322	0.906609	+	0.421973i	$0.138662\pi$
$140$	−4.48406	−0.378972
$141$	0	0
$142$	−22.0796	−1.85288
$143$	−6.00251	−0.501955
$144$	0	0
$145$	7.75843	0.644302
$146$	2.90591	0.240495
$147$	0	0
$148$	7.19556	0.591472
$149$	−7.38417	−0.604935	−0.302468	−	0.953160i	$-0.597810\pi$
$149$	−7.38417	−0.604935	−0.302468	+	0.953160i	$0.597810\pi$
$150$	0	0
$151$	0.654488	0.0532615	0.0266307	−	0.999645i	$-0.491522\pi$
$151$	0.654488	0.0532615	0.0266307	+	0.999645i	$0.491522\pi$
$152$	4.21245	0.341675
$153$	0	0
$154$	7.45818	0.600997
$155$	7.28174	0.584883
$156$	0	0
$157$	−3.81535	−0.304498	−0.152249	−	0.988342i	$-0.548652\pi$
$157$	−3.81535	−0.304498	−0.152249	+	0.988342i	$0.548652\pi$
$158$	−3.28165	−0.261074
$159$	0	0
$160$	−5.41054	−0.427741
$161$	−4.71575	−0.371653
$162$	0	0
$163$	−8.28135	−0.648646	−0.324323	−	0.945946i	$-0.605136\pi$
$163$	−8.28135	−0.648646	−0.324323	+	0.945946i	$0.605136\pi$
$164$	4.28063	0.334261
$165$	0	0
$166$	11.2856	0.875936
$167$	2.11623	0.163759	0.0818794	−	0.996642i	$-0.473908\pi$
$167$	2.11623	0.163759	0.0818794	+	0.996642i	$0.473908\pi$
$168$	0	0
$169$	23.0302	1.77155
$170$	3.95199	0.303104
$171$	0	0
$172$	4.15172	0.316566
$173$	−20.4637	−1.55583	−0.777913	−	0.628372i	$-0.783721\pi$
$173$	−20.4637	−1.55583	−0.777913	+	0.628372i	$0.783721\pi$
$174$	0	0
$175$	−16.7066	−1.26290
$176$	4.99916	0.376826
$177$	0	0
$178$	−7.84151	−0.587746
$179$	−11.4912	−0.858893	−0.429447	−	0.903092i	$-0.641291\pi$
$179$	−11.4912	−0.858893	−0.429447	+	0.903092i	$0.641291\pi$
$180$	0	0
$181$	−1.07391	−0.0798228	−0.0399114	−	0.999203i	$-0.512708\pi$
$181$	−1.07391	−0.0798228	−0.0399114	+	0.999203i	$0.512708\pi$
$182$	−44.7678	−3.31841
$183$	0	0
$184$	−1.93291	−0.142496
$185$	−7.90763	−0.581381
$186$	0	0
$187$	−2.14842	−0.157108
$188$	6.21438	0.453230
$189$	0	0
$190$	4.36912	0.316969
$191$	−4.18845	−0.303066	−0.151533	−	0.988452i	$-0.548421\pi$
$191$	−4.18845	−0.303066	−0.151533	+	0.988452i	$0.548421\pi$
$192$	0	0
$193$	5.10660	0.367581	0.183790	−	0.982965i	$-0.441163\pi$
$193$	5.10660	0.367581	0.183790	+	0.982965i	$0.441163\pi$
$194$	23.7034	1.70180
$195$	0	0
$196$	11.3830	0.813070
$197$	25.3064	1.80300	0.901502	−	0.432776i	$-0.142466\pi$
$197$	25.3064	1.80300	0.901502	+	0.432776i	$0.142466\pi$
$198$	0	0
$199$	13.0351	0.924037	0.462018	−	0.886870i	$-0.347125\pi$
$199$	13.0351	0.924037	0.462018	+	0.886870i	$0.347125\pi$
$200$	−6.84779	−0.484212
$201$	0	0
$202$	−0.713905	−0.0502302
$203$	−31.4565	−2.20781
$204$	0	0
$205$	−4.70424	−0.328559
$206$	14.5943	1.01683
$207$	0	0
$208$	−30.0075	−2.08065
$209$	−2.37519	−0.164295
$210$	0	0
$211$	14.3115	0.985245	0.492622	−	0.870243i	$-0.336038\pi$
$211$	14.3115	0.985245	0.492622	+	0.870243i	$0.336038\pi$
$212$	11.1171	0.763527
$213$	0	0
$214$	22.0820	1.50949
$215$	−4.56258	−0.311165
$216$	0	0
$217$	−29.5238	−2.00420
$218$	23.0531	1.56135
$219$	0	0
$220$	1.03632	0.0698690
$221$	12.8959	0.867473
$222$	0	0
$223$	11.3739	0.761654	0.380827	−	0.924646i	$-0.375639\pi$
$223$	11.3739	0.761654	0.380827	+	0.924646i	$0.375639\pi$
$224$	21.9370	1.46573
$225$	0	0
$226$	10.8931	0.724600
$227$	5.58500	0.370689	0.185345	−	0.982674i	$-0.440660\pi$
$227$	5.58500	0.370689	0.185345	+	0.982674i	$0.440660\pi$
$228$	0	0
$229$	10.4354	0.689591	0.344796	−	0.938678i	$-0.387948\pi$
$229$	10.4354	0.689591	0.344796	+	0.938678i	$0.387948\pi$
$230$	−2.00480	−0.132193
$231$	0	0
$232$	−12.8935	−0.846503
$233$	14.1258	0.925413	0.462706	−	0.886512i	$-0.346878\pi$
$233$	14.1258	0.925413	0.462706	+	0.886512i	$0.346878\pi$
$234$	0	0
$235$	−6.82935	−0.445498
$236$	9.81061	0.638617
$237$	0	0
$238$	−16.0233	−1.03864
$239$	4.25988	0.275549	0.137774	−	0.990464i	$-0.456005\pi$
$239$	4.25988	0.275549	0.137774	+	0.990464i	$0.456005\pi$
$240$	0	0
$241$	−4.10611	−0.264498	−0.132249	−	0.991217i	$-0.542220\pi$
$241$	−4.10611	−0.264498	−0.132249	+	0.991217i	$0.542220\pi$
$242$	−1.72368	−0.110803
$243$	0	0
$244$	2.46306	0.157681
$245$	−12.5094	−0.799199
$246$	0	0
$247$	14.2571	0.907157
$248$	−12.1013	−0.768436
$249$	0	0
$250$	−16.2999	−1.03090
$251$	−19.4224	−1.22593	−0.612966	−	0.790109i	$-0.710024\pi$
$251$	−19.4224	−1.22593	−0.612966	+	0.790109i	$0.710024\pi$
$252$	0	0
$253$	1.08987	0.0685196
$254$	25.0517	1.57188
$255$	0	0
$256$	18.7009	1.16880
$257$	−16.9024	−1.05434	−0.527172	−	0.849759i	$-0.676748\pi$
$257$	−16.9024	−1.05434	−0.527172	+	0.849759i	$0.676748\pi$
$258$	0	0
$259$	32.0614	1.99220
$260$	−6.22055	−0.385782
$261$	0	0
$262$	−12.8465	−0.793661
$263$	−12.1387	−0.748503	−0.374252	−	0.927327i	$-0.622100\pi$
$263$	−12.1387	−0.748503	−0.374252	+	0.927327i	$0.622100\pi$
$264$	0	0
$265$	−12.2173	−0.750501
$266$	−17.7146	−1.08615
$267$	0	0
$268$	−0.971084	−0.0593184
$269$	−29.8322	−1.81890	−0.909451	−	0.415810i	$-0.863498\pi$
$269$	−29.8322	−1.81890	−0.909451	+	0.415810i	$0.863498\pi$
$270$	0	0
$271$	27.0260	1.64171	0.820856	−	0.571135i	$-0.193497\pi$
$271$	27.0260	1.64171	0.820856	+	0.571135i	$0.193497\pi$
$272$	−10.7403	−0.651227
$273$	0	0
$274$	−30.9017	−1.86684
$275$	3.86112	0.232834
$276$	0	0
$277$	0.554577	0.0333213	0.0166607	−	0.999861i	$-0.494697\pi$
$277$	0.554577	0.0333213	0.0166607	+	0.999861i	$0.494697\pi$
$278$	−36.8481	−2.21000
$279$	0	0
$280$	−8.18938	−0.489409
$281$	−22.3830	−1.33526	−0.667629	−	0.744495i	$-0.732691\pi$
$281$	−22.3830	−1.33526	−0.667629	+	0.744495i	$0.732691\pi$
$282$	0	0
$283$	15.7779	0.937898	0.468949	−	0.883225i	$-0.344633\pi$
$283$	15.7779	0.937898	0.468949	+	0.883225i	$0.344633\pi$
$284$	12.4391	0.738127
$285$	0	0
$286$	10.3464	0.611797
$287$	19.0733	1.12586
$288$	0	0
$289$	−12.3843	−0.728488
$290$	−13.3731	−0.785294
$291$	0	0
$292$	−1.63712	−0.0958053
$293$	17.0868	0.998219	0.499110	−	0.866539i	$-0.333661\pi$
$293$	17.0868	0.998219	0.499110	+	0.866539i	$0.333661\pi$
$294$	0	0
$295$	−10.7815	−0.627721
$296$	13.1415	0.763834
$297$	0	0
$298$	12.7280	0.737312
$299$	−6.54197	−0.378332
$300$	0	0
$301$	18.4989	1.06626
$302$	−1.12813	−0.0649166
$303$	0	0
$304$	−11.8739	−0.681018
$305$	−2.70680	−0.154991
$306$	0	0
$307$	−14.5062	−0.827912	−0.413956	−	0.910297i	$-0.635853\pi$
$307$	−14.5062	−0.827912	−0.413956	+	0.910297i	$0.635853\pi$
$308$	−4.20177	−0.239418
$309$	0	0
$310$	−12.5514	−0.712872
$311$	6.64528	0.376819	0.188410	−	0.982091i	$-0.439667\pi$
$311$	6.64528	0.376819	0.188410	+	0.982091i	$0.439667\pi$
$312$	0	0
$313$	−23.3448	−1.31953	−0.659763	−	0.751474i	$-0.729343\pi$
$313$	−23.3448	−1.31953	−0.659763	+	0.751474i	$0.729343\pi$
$314$	6.57645	0.371131
$315$	0	0
$316$	1.84881	0.104003
$317$	26.2178	1.47254	0.736269	−	0.676689i	$-0.236586\pi$
$317$	26.2178	1.47254	0.736269	+	0.676689i	$0.236586\pi$
$318$	0	0
$319$	7.27001	0.407042
$320$	−1.34399	−0.0751312
$321$	0	0
$322$	8.12846	0.452981
$323$	5.10290	0.283933
$324$	0	0
$325$	−23.1764	−1.28560
$326$	14.2744	0.790588
$327$	0	0
$328$	7.81787	0.431669
$329$	27.6896	1.52657
$330$	0	0
$331$	7.13037	0.391920	0.195960	−	0.980612i	$-0.437218\pi$
$331$	7.13037	0.391920	0.195960	+	0.980612i	$0.437218\pi$
$332$	−6.35808	−0.348945
$333$	0	0
$334$	−3.64771	−0.199594
$335$	1.06718	0.0583064
$336$	0	0
$337$	26.6267	1.45045	0.725225	−	0.688512i	$-0.241736\pi$
$337$	26.6267	1.45045	0.725225	+	0.688512i	$0.241736\pi$
$338$	−39.6967	−2.15922
$339$	0	0
$340$	−2.22646	−0.120747
$341$	6.82333	0.369504
$342$	0	0
$343$	20.4312	1.10318
$344$	7.58244	0.408818
$345$	0	0
$346$	35.2729	1.89628
$347$	−27.6328	−1.48341	−0.741704	−	0.670728i	$-0.765982\pi$
$347$	−27.6328	−1.48341	−0.741704	+	0.670728i	$0.765982\pi$
$348$	0	0
$349$	−2.65151	−0.141932	−0.0709660	−	0.997479i	$-0.522608\pi$
$349$	−2.65151	−0.141932	−0.0709660	+	0.997479i	$0.522608\pi$
$350$	28.7969	1.53926
$351$	0	0
$352$	−5.06993	−0.270228
$353$	−4.26839	−0.227184	−0.113592	−	0.993528i	$-0.536236\pi$
$353$	−4.26839	−0.227184	−0.113592	+	0.993528i	$0.536236\pi$
$354$	0	0
$355$	−13.6701	−0.725534
$356$	4.41773	0.234139
$357$	0	0
$358$	19.8072	1.04684
$359$	0.579515	0.0305857	0.0152928	−	0.999883i	$-0.495132\pi$
$359$	0.579515	0.0305857	0.0152928	+	0.999883i	$0.495132\pi$
$360$	0	0
$361$	−13.3585	−0.703078
$362$	1.85107	0.0972903
$363$	0	0
$364$	25.2212	1.32195
$365$	1.79913	0.0941708
$366$	0	0
$367$	27.5414	1.43765	0.718824	−	0.695192i	$-0.244681\pi$
$367$	27.5414	1.43765	0.718824	+	0.695192i	$0.244681\pi$
$368$	5.44845	0.284020
$369$	0	0
$370$	13.6303	0.708603
$371$	49.5348	2.57172
$372$	0	0
$373$	32.9187	1.70447	0.852234	−	0.523161i	$-0.175247\pi$
$373$	32.9187	1.70447	0.852234	+	0.523161i	$0.175247\pi$
$374$	3.70320	0.191488
$375$	0	0
$376$	11.3495	0.585308
$377$	−43.6383	−2.24749
$378$	0	0
$379$	11.2104	0.575842	0.287921	−	0.957654i	$-0.407036\pi$
$379$	11.2104	0.575842	0.287921	+	0.957654i	$0.407036\pi$
$380$	−2.46146	−0.126270
$381$	0	0
$382$	7.21957	0.369385
$383$	−7.57877	−0.387257	−0.193629	−	0.981075i	$-0.562026\pi$
$383$	−7.57877	−0.387257	−0.193629	+	0.981075i	$0.562026\pi$
$384$	0	0
$385$	4.61758	0.235333
$386$	−8.80215	−0.448018
$387$	0	0
$388$	−13.3540	−0.677945
$389$	−2.05743	−0.104316	−0.0521580	−	0.998639i	$-0.516610\pi$
$389$	−2.05743	−0.104316	−0.0521580	+	0.998639i	$0.516610\pi$
$390$	0	0
$391$	−2.34150	−0.118415
$392$	20.7891	1.05001
$393$	0	0
$394$	−43.6201	−2.19755
$395$	−2.03176	−0.102229
$396$	0	0
$397$	−12.4304	−0.623862	−0.311931	−	0.950105i	$-0.600976\pi$
$397$	−12.4304	−0.623862	−0.311931	+	0.950105i	$0.600976\pi$
$398$	−22.4685	−1.12624
$399$	0	0
$400$	19.3024	0.965119
$401$	22.6526	1.13122	0.565608	−	0.824675i	$-0.308642\pi$
$401$	22.6526	1.13122	0.565608	+	0.824675i	$0.308642\pi$
$402$	0	0
$403$	−40.9571	−2.04022
$404$	0.402198	0.0200101
$405$	0	0
$406$	54.2210	2.69094
$407$	−7.40982	−0.367291
$408$	0	0
$409$	13.0276	0.644172	0.322086	−	0.946710i	$-0.395616\pi$
$409$	13.0276	0.644172	0.322086	+	0.946710i	$0.395616\pi$
$410$	8.10862	0.400456
$411$	0	0
$412$	−8.22211	−0.405074
$413$	43.7134	2.15100
$414$	0	0
$415$	6.98727	0.342992
$416$	30.4323	1.49207
$417$	0	0
$418$	4.09407	0.200248
$419$	17.2541	0.842919	0.421460	−	0.906847i	$-0.361518\pi$
$419$	17.2541	0.842919	0.421460	+	0.906847i	$0.361518\pi$
$420$	0	0
$421$	−12.6247	−0.615289	−0.307644	−	0.951501i	$-0.599541\pi$
$421$	−12.6247	−0.615289	−0.307644	+	0.951501i	$0.599541\pi$
$422$	−24.6685	−1.20084
$423$	0	0
$424$	20.3036	0.986029
$425$	−8.29531	−0.402382
$426$	0	0
$427$	10.9747	0.531103
$428$	−12.4405	−0.601334
$429$	0	0
$430$	7.86444	0.379257
$431$	8.52386	0.410580	0.205290	−	0.978701i	$-0.434186\pi$
$431$	8.52386	0.410580	0.205290	+	0.978701i	$0.434186\pi$
$432$	0	0
$433$	7.71656	0.370834	0.185417	−	0.982660i	$-0.440636\pi$
$433$	7.71656	0.370834	0.185417	+	0.982660i	$0.440636\pi$
$434$	50.8896	2.44278
$435$	0	0
$436$	−12.9876	−0.621994
$437$	−2.58865	−0.123832
$438$	0	0
$439$	−4.35494	−0.207850	−0.103925	−	0.994585i	$-0.533140\pi$
$439$	−4.35494	−0.207850	−0.103925	+	0.994585i	$0.533140\pi$
$440$	1.89267	0.0902297
$441$	0	0
$442$	−22.2285	−1.05730
$443$	7.68282	0.365022	0.182511	−	0.983204i	$-0.441578\pi$
$443$	7.68282	0.365022	0.182511	+	0.983204i	$0.441578\pi$
$444$	0	0
$445$	−4.85491	−0.230145
$446$	−19.6050	−0.928325
$447$	0	0
$448$	5.44919	0.257450
$449$	−11.7080	−0.552533	−0.276267	−	0.961081i	$-0.589097\pi$
$449$	−11.7080	−0.552533	−0.276267	+	0.961081i	$0.589097\pi$
$450$	0	0
$451$	−4.40810	−0.207569
$452$	−6.13694	−0.288657
$453$	0	0
$454$	−9.62677	−0.451807
$455$	−27.7171	−1.29940
$456$	0	0
$457$	−25.1779	−1.17777	−0.588886	−	0.808216i	$-0.700433\pi$
$457$	−25.1779	−1.17777	−0.588886	+	0.808216i	$0.700433\pi$
$458$	−17.9873	−0.840493
$459$	0	0
$460$	1.12946	0.0526614
$461$	2.65772	0.123782	0.0618911	−	0.998083i	$-0.480287\pi$
$461$	2.65772	0.123782	0.0618911	+	0.998083i	$0.480287\pi$
$462$	0	0
$463$	37.1695	1.72741	0.863706	−	0.503996i	$-0.168137\pi$
$463$	37.1695	1.72741	0.863706	+	0.503996i	$0.168137\pi$
$464$	36.3440	1.68723
$465$	0	0
$466$	−24.3484	−1.12792
$467$	−12.7434	−0.589696	−0.294848	−	0.955544i	$-0.595269\pi$
$467$	−12.7434	−0.589696	−0.294848	+	0.955544i	$0.595269\pi$
$468$	0	0
$469$	−4.32688	−0.199797
$470$	11.7716	0.542985
$471$	0	0
$472$	17.9175	0.824718
$473$	−4.27535	−0.196581
$474$	0	0
$475$	−9.17089	−0.420789
$476$	9.02717	0.413760
$477$	0	0
$478$	−7.34268	−0.335846
$479$	−8.20889	−0.375074	−0.187537	−	0.982258i	$-0.560050\pi$
$479$	−8.20889	−0.375074	−0.187537	+	0.982258i	$0.560050\pi$
$480$	0	0
$481$	44.4776	2.02800
$482$	7.07763	0.322377
$483$	0	0
$484$	0.971084	0.0441402
$485$	14.6755	0.666378
$486$	0	0
$487$	−7.64496	−0.346426	−0.173213	−	0.984884i	$-0.555415\pi$
$487$	−7.64496	−0.346426	−0.173213	+	0.984884i	$0.555415\pi$
$488$	4.49837	0.203631
$489$	0	0
$490$	21.5623	0.974086
$491$	−20.8188	−0.939538	−0.469769	−	0.882789i	$-0.655663\pi$
$491$	−20.8188	−0.939538	−0.469769	+	0.882789i	$0.655663\pi$
$492$	0	0
$493$	−15.6190	−0.703446
$494$	−24.5747	−1.10567
$495$	0	0
$496$	34.1110	1.53163
$497$	55.4254	2.48617
$498$	0	0
$499$	19.5711	0.876123	0.438062	−	0.898945i	$-0.355665\pi$
$499$	19.5711	0.876123	0.438062	+	0.898945i	$0.355665\pi$
$500$	9.18300	0.410676
$501$	0	0
$502$	33.4781	1.49420
$503$	−6.28168	−0.280086	−0.140043	−	0.990145i	$-0.544724\pi$
$503$	−6.28168	−0.280086	−0.140043	+	0.990145i	$0.544724\pi$
$504$	0	0
$505$	−0.442000	−0.0196687
$506$	−1.87859	−0.0835137
$507$	0	0
$508$	−14.1136	−0.626189
$509$	−3.30016	−0.146277	−0.0731386	−	0.997322i	$-0.523302\pi$
$509$	−3.30016	−0.146277	−0.0731386	+	0.997322i	$0.523302\pi$
$510$	0	0
$511$	−7.29457	−0.322693
$512$	−7.61311	−0.336455
$513$	0	0
$514$	29.1344	1.28506
$515$	9.03576	0.398163
$516$	0	0
$517$	−6.39942	−0.281446
$518$	−55.2638	−2.42815
$519$	0	0
$520$	−11.3608	−0.498204
$521$	37.9469	1.66248	0.831242	−	0.555911i	$-0.187630\pi$
$521$	37.9469	1.66248	0.831242	+	0.555911i	$0.187630\pi$
$522$	0	0
$523$	35.2618	1.54189	0.770946	−	0.636901i	$-0.219784\pi$
$523$	35.2618	1.54189	0.770946	+	0.636901i	$0.219784\pi$
$524$	7.23744	0.316169
$525$	0	0
$526$	20.9232	0.912297
$527$	−14.6594	−0.638573
$528$	0	0
$529$	−21.8122	−0.948356
$530$	21.0587	0.914731
$531$	0	0
$532$	9.97999	0.432687
$533$	26.4596	1.14609
$534$	0	0
$535$	13.6716	0.591075
$536$	−1.77352	−0.0766046
$537$	0	0
$538$	51.4213	2.21693
$539$	−11.7219	−0.504899
$540$	0	0
$541$	−29.9010	−1.28555	−0.642773	−	0.766056i	$-0.722216\pi$
$541$	−29.9010	−1.28555	−0.642773	+	0.766056i	$0.722216\pi$
$542$	−46.5843	−2.00097
$543$	0	0
$544$	10.8923	0.467005
$545$	14.2729	0.611382
$546$	0	0
$547$	−37.5585	−1.60588	−0.802942	−	0.596057i	$-0.796733\pi$
$547$	−37.5585	−1.60588	−0.802942	+	0.596057i	$0.796733\pi$
$548$	17.4093	0.743689
$549$	0	0
$550$	−6.65535	−0.283785
$551$	−17.2676	−0.735626
$552$	0	0
$553$	8.23777	0.350306
$554$	−0.955916	−0.0406130
$555$	0	0
$556$	20.7594	0.880393
$557$	29.1666	1.23583	0.617913	−	0.786246i	$-0.287978\pi$
$557$	29.1666	1.23583	0.617913	+	0.786246i	$0.287978\pi$
$558$	0	0
$559$	25.6628	1.08542
$560$	23.0840	0.975478
$561$	0	0
$562$	38.5812	1.62745
$563$	7.63837	0.321919	0.160960	−	0.986961i	$-0.448541\pi$
$563$	7.63837	0.321919	0.160960	+	0.986961i	$0.448541\pi$
$564$	0	0
$565$	6.74425	0.283733
$566$	−27.1961	−1.14314
$567$	0	0
$568$	22.7180	0.953227
$569$	−33.7395	−1.41444	−0.707218	−	0.706996i	$-0.750050\pi$
$569$	−33.7395	−1.41444	−0.707218	+	0.706996i	$0.750050\pi$
$570$	0	0
$571$	−10.2457	−0.428768	−0.214384	−	0.976749i	$-0.568774\pi$
$571$	−10.2457	−0.428768	−0.214384	+	0.976749i	$0.568774\pi$
$572$	−5.82895	−0.243720
$573$	0	0
$574$	−32.8764	−1.37223
$575$	4.20813	0.175491
$576$	0	0
$577$	47.7090	1.98615	0.993077	−	0.117468i	$-0.0374777\pi$
$577$	47.7090	1.98615	0.993077	+	0.117468i	$0.0374777\pi$
$578$	21.3466	0.887901
$579$	0	0
$580$	7.53409	0.312836
$581$	−28.3298	−1.17532
$582$	0	0
$583$	−11.4481	−0.474134
$584$	−2.98993	−0.123724
$585$	0	0
$586$	−29.4522	−1.21666
$587$	14.1967	0.585961	0.292981	−	0.956118i	$-0.405353\pi$
$587$	14.1967	0.585961	0.292981	+	0.956118i	$0.405353\pi$
$588$	0	0
$589$	−16.2067	−0.667785
$590$	18.5838	0.765085
$591$	0	0
$592$	−37.0429	−1.52245
$593$	22.0584	0.905831	0.452916	−	0.891553i	$-0.350384\pi$
$593$	22.0584	0.905831	0.452916	+	0.891553i	$0.350384\pi$
$594$	0	0
$595$	−9.92049	−0.406701
$596$	−7.17066	−0.293721
$597$	0	0
$598$	11.2763	0.461121
$599$	−12.3729	−0.505541	−0.252771	−	0.967526i	$-0.581342\pi$
$599$	−12.3729	−0.505541	−0.252771	+	0.967526i	$0.581342\pi$
$600$	0	0
$601$	38.8194	1.58348	0.791739	−	0.610860i	$-0.209176\pi$
$601$	38.8194	1.58348	0.791739	+	0.610860i	$0.209176\pi$
$602$	−31.8863	−1.29959
$603$	0	0
$604$	0.635563	0.0258607
$605$	−1.06718	−0.0433871
$606$	0	0
$607$	−45.3595	−1.84109	−0.920543	−	0.390641i	$-0.872254\pi$
$607$	−45.3595	−1.84109	−0.920543	+	0.390641i	$0.872254\pi$
$608$	12.0420	0.488369
$609$	0	0
$610$	4.66567	0.188907
$611$	38.4126	1.55401
$612$	0	0
$613$	−11.3316	−0.457677	−0.228839	−	0.973464i	$-0.573493\pi$
$613$	−11.3316	−0.457677	−0.228839	+	0.973464i	$0.573493\pi$
$614$	25.0041	1.00908
$615$	0	0
$616$	−7.67384	−0.309188
$617$	37.9175	1.52650	0.763251	−	0.646103i	$-0.223602\pi$
$617$	37.9175	1.52650	0.763251	+	0.646103i	$0.223602\pi$
$618$	0	0
$619$	−42.7598	−1.71866	−0.859330	−	0.511421i	$-0.829119\pi$
$619$	−42.7598	−1.71866	−0.859330	+	0.511421i	$0.829119\pi$
$620$	7.07118	0.283986
$621$	0	0
$622$	−11.4544	−0.459278
$623$	19.6842	0.788630
$624$	0	0
$625$	9.21387	0.368555
$626$	40.2391	1.60828
$627$	0	0
$628$	−3.70503	−0.147847
$629$	15.9194	0.634749
$630$	0	0
$631$	−40.3326	−1.60562	−0.802808	−	0.596238i	$-0.796661\pi$
$631$	−40.3326	−1.60562	−0.802808	+	0.596238i	$0.796661\pi$
$632$	3.37654	0.134311
$633$	0	0
$634$	−45.1912	−1.79477
$635$	15.5103	0.615506
$636$	0	0
$637$	70.3610	2.78781
$638$	−12.5312	−0.496115
$639$	0	0
$640$	13.1377	0.519313
$641$	17.6341	0.696503	0.348252	−	0.937401i	$-0.386775\pi$
$641$	17.6341	0.696503	0.348252	+	0.937401i	$0.386775\pi$
$642$	0	0
$643$	−11.5496	−0.455470	−0.227735	−	0.973723i	$-0.573132\pi$
$643$	−11.5496	−0.455470	−0.227735	+	0.973723i	$0.573132\pi$
$644$	−4.57939	−0.180453
$645$	0	0
$646$	−8.79578	−0.346066
$647$	−44.4834	−1.74882	−0.874411	−	0.485185i	$-0.838752\pi$
$647$	−44.4834	−1.74882	−0.874411	+	0.485185i	$0.838752\pi$
$648$	0	0
$649$	−10.1027	−0.396567
$650$	39.9488	1.56692
$651$	0	0
$652$	−8.04189	−0.314945
$653$	−43.0579	−1.68499	−0.842493	−	0.538708i	$-0.818913\pi$
$653$	−43.0579	−1.68499	−0.842493	+	0.538708i	$0.818913\pi$
$654$	0	0
$655$	−7.95365	−0.310775
$656$	−22.0368	−0.860392
$657$	0	0
$658$	−47.7280	−1.86063
$659$	3.65097	0.142222	0.0711108	−	0.997468i	$-0.477346\pi$
$659$	3.65097	0.142222	0.0711108	+	0.997468i	$0.477346\pi$
$660$	0	0
$661$	−43.5318	−1.69319	−0.846595	−	0.532237i	$-0.821351\pi$
$661$	−43.5318	−1.69319	−0.846595	+	0.532237i	$0.821351\pi$
$662$	−12.2905	−0.477684
$663$	0	0
$664$	−11.6120	−0.450632
$665$	−10.9676	−0.425306
$666$	0	0
$667$	7.92338	0.306794
$668$	2.05504	0.0795118
$669$	0	0
$670$	−1.83948	−0.0710655
$671$	−2.53640	−0.0979166
$672$	0	0
$673$	−16.1418	−0.622220	−0.311110	−	0.950374i	$-0.600701\pi$
$673$	−16.1418	−0.622220	−0.311110	+	0.950374i	$0.600701\pi$
$674$	−45.8960	−1.76785
$675$	0	0
$676$	22.3642	0.860162
$677$	−20.5329	−0.789145	−0.394573	−	0.918865i	$-0.629107\pi$
$677$	−20.5329	−0.789145	−0.394573	+	0.918865i	$0.629107\pi$
$678$	0	0
$679$	−59.5016	−2.28346
$680$	−4.06626	−0.155934
$681$	0	0
$682$	−11.7613	−0.450362
$683$	40.1877	1.53774	0.768870	−	0.639405i	$-0.220819\pi$
$683$	40.1877	1.53774	0.768870	+	0.639405i	$0.220819\pi$
$684$	0	0
$685$	−19.1321	−0.731001
$686$	−35.2170	−1.34459
$687$	0	0
$688$	−21.3732	−0.814844
$689$	68.7176	2.61793
$690$	0	0
$691$	44.6781	1.69964	0.849818	−	0.527076i	$-0.176712\pi$
$691$	44.6781	1.69964	0.849818	+	0.527076i	$0.176712\pi$
$692$	−19.8720	−0.755419
$693$	0	0
$694$	47.6302	1.80802
$695$	−22.8137	−0.865373
$696$	0	0
$697$	9.47044	0.358719
$698$	4.57036	0.172991
$699$	0	0
$700$	−16.2235	−0.613192
$701$	31.0370	1.17225	0.586126	−	0.810220i	$-0.300652\pi$
$701$	31.0370	1.17225	0.586126	+	0.810220i	$0.300652\pi$
$702$	0	0
$703$	17.5997	0.663786
$704$	−1.25938	−0.0474647
$705$	0	0
$706$	7.35736	0.276898
$707$	1.79208	0.0673983
$708$	0	0
$709$	46.0804	1.73058	0.865292	−	0.501267i	$-0.167133\pi$
$709$	46.0804	1.73058	0.865292	+	0.501267i	$0.167133\pi$
$710$	23.5629	0.884302
$711$	0	0
$712$	8.06825	0.302370
$713$	7.43655	0.278501
$714$	0	0
$715$	6.40578	0.239562
$716$	−11.1589	−0.417029
$717$	0	0
$718$	−0.998901	−0.0372787
$719$	23.6458	0.881839	0.440920	−	0.897547i	$-0.354652\pi$
$719$	23.6458	0.881839	0.440920	+	0.897547i	$0.354652\pi$
$720$	0	0
$721$	−36.6354	−1.36438
$722$	23.0258	0.856932
$723$	0	0
$724$	−1.04285	−0.0387573
$725$	28.0704	1.04251
$726$	0	0
$727$	8.57687	0.318098	0.159049	−	0.987271i	$-0.449157\pi$
$727$	8.57687	0.318098	0.159049	+	0.987271i	$0.449157\pi$
$728$	46.0623	1.70718
$729$	0	0
$730$	−3.10113	−0.114778
$731$	9.18524	0.339729
$732$	0	0
$733$	−2.32323	−0.0858103	−0.0429051	−	0.999079i	$-0.513661\pi$
$733$	−2.32323	−0.0858103	−0.0429051	+	0.999079i	$0.513661\pi$
$734$	−47.4726	−1.75225
$735$	0	0
$736$	−5.52557	−0.203675
$737$	1.00000	0.0368355
$738$	0	0
$739$	46.2492	1.70130	0.850652	−	0.525730i	$-0.176208\pi$
$739$	46.2492	1.70130	0.850652	+	0.525730i	$0.176208\pi$
$740$	−7.67898	−0.282285
$741$	0	0
$742$	−85.3823	−3.13448
$743$	−13.8339	−0.507518	−0.253759	−	0.967267i	$-0.581667\pi$
$743$	−13.8339	−0.507518	−0.253759	+	0.967267i	$0.581667\pi$
$744$	0	0
$745$	7.88026	0.288710
$746$	−56.7415	−2.07745
$747$	0	0
$748$	−2.08630	−0.0762826
$749$	−55.4314	−2.02542
$750$	0	0
$751$	−25.9583	−0.947231	−0.473615	−	0.880732i	$-0.657051\pi$
$751$	−25.9583	−0.947231	−0.473615	+	0.880732i	$0.657051\pi$
$752$	−31.9918	−1.16662
$753$	0	0
$754$	75.2187	2.73930
$755$	−0.698458	−0.0254195
$756$	0	0
$757$	25.8728	0.940361	0.470181	−	0.882570i	$-0.344189\pi$
$757$	25.8728	0.940361	0.470181	+	0.882570i	$0.344189\pi$
$758$	−19.3233	−0.701853
$759$	0	0
$760$	−4.49546	−0.163067
$761$	−42.5231	−1.54146	−0.770730	−	0.637162i	$-0.780108\pi$
$761$	−42.5231	−1.54146	−0.770730	+	0.637162i	$0.780108\pi$
$762$	0	0
$763$	−57.8692	−2.09501
$764$	−4.06734	−0.147151
$765$	0	0
$766$	13.0634	0.472000
$767$	60.6418	2.18965
$768$	0	0
$769$	6.35590	0.229200	0.114600	−	0.993412i	$-0.463441\pi$
$769$	6.35590	0.229200	0.114600	+	0.993412i	$0.463441\pi$
$770$	−7.95924	−0.286831
$771$	0	0
$772$	4.95894	0.178476
$773$	−27.0040	−0.971266	−0.485633	−	0.874163i	$-0.661411\pi$
$773$	−27.0040	−0.971266	−0.485633	+	0.874163i	$0.661411\pi$
$774$	0	0
$775$	26.3457	0.946366
$776$	−24.3888	−0.875507
$777$	0	0
$778$	3.54636	0.127143
$779$	10.4701	0.375129
$780$	0	0
$781$	−12.8095	−0.458361
$782$	4.03601	0.144327
$783$	0	0
$784$	−58.5998	−2.09285
$785$	4.07167	0.145324
$786$	0	0
$787$	−14.3224	−0.510539	−0.255269	−	0.966870i	$-0.582164\pi$
$787$	−14.3224	−0.510539	−0.255269	+	0.966870i	$0.582164\pi$
$788$	24.5746	0.875434
$789$	0	0
$790$	3.50212	0.124600
$791$	−27.3445	−0.972259
$792$	0	0
$793$	15.2248	0.540647
$794$	21.4260	0.760381
$795$	0	0
$796$	12.6582	0.448659
$797$	53.2038	1.88457	0.942287	−	0.334806i	$-0.108671\pi$
$797$	53.2038	1.88457	0.942287	+	0.334806i	$0.108671\pi$
$798$	0	0
$799$	13.7486	0.486392
$800$	−19.5756	−0.692102
$801$	0	0
$802$	−39.0458	−1.37876
$803$	1.68587	0.0594931
$804$	0	0
$805$	5.03256	0.177375
$806$	70.5971	2.48668
$807$	0	0
$808$	0.734548	0.0258413
$809$	29.3289	1.03115	0.515574	−	0.856845i	$-0.327579\pi$
$809$	29.3289	1.03115	0.515574	+	0.856845i	$0.327579\pi$
$810$	0	0
$811$	−5.72445	−0.201012	−0.100506	−	0.994936i	$-0.532046\pi$
$811$	−5.72445	−0.201012	−0.100506	+	0.994936i	$0.532046\pi$
$812$	−30.5469	−1.07199
$813$	0	0
$814$	12.7722	0.447665
$815$	8.83771	0.309572
$816$	0	0
$817$	10.1548	0.355270
$818$	−22.4554	−0.785135
$819$	0	0
$820$	−4.56822	−0.159529
$821$	18.9410	0.661045	0.330523	−	0.943798i	$-0.392775\pi$
$821$	18.9410	0.661045	0.330523	+	0.943798i	$0.392775\pi$
$822$	0	0
$823$	−37.9440	−1.32264	−0.661322	−	0.750102i	$-0.730004\pi$
$823$	−37.9440	−1.32264	−0.661322	+	0.750102i	$0.730004\pi$
$824$	−15.0163	−0.523118
$825$	0	0
$826$	−75.3481	−2.62169
$827$	−30.3265	−1.05456	−0.527278	−	0.849693i	$-0.676787\pi$
$827$	−30.3265	−1.05456	−0.527278	+	0.849693i	$0.676787\pi$
$828$	0	0
$829$	25.4799	0.884953	0.442476	−	0.896780i	$-0.354100\pi$
$829$	25.4799	0.884953	0.442476	+	0.896780i	$0.354100\pi$
$830$	−12.0438	−0.418048
$831$	0	0
$832$	7.55944	0.262077
$833$	25.1836	0.872561
$834$	0	0
$835$	−2.25840	−0.0781553
$836$	−2.30651	−0.0797722
$837$	0	0
$838$	−29.7406	−1.02737
$839$	51.8704	1.79077	0.895383	−	0.445297i	$-0.146902\pi$
$839$	51.8704	1.79077	0.895383	+	0.445297i	$0.146902\pi$
$840$	0	0
$841$	23.8531	0.822519
$842$	21.7609	0.749931
$843$	0	0
$844$	13.8977	0.478378
$845$	−24.5774	−0.845488
$846$	0	0
$847$	4.32688	0.148674
$848$	−57.2312	−1.96533
$849$	0	0
$850$	14.2985	0.490434
$851$	−8.07575	−0.276833
$852$	0	0
$853$	−9.44150	−0.323271	−0.161635	−	0.986851i	$-0.551677\pi$
$853$	−9.44150	−0.323271	−0.161635	+	0.986851i	$0.551677\pi$
$854$	−18.9169	−0.647324
$855$	0	0
$856$	−22.7205	−0.776571
$857$	15.8360	0.540949	0.270474	−	0.962727i	$-0.412819\pi$
$857$	15.8360	0.540949	0.270474	+	0.962727i	$0.412819\pi$
$858$	0	0
$859$	−19.1747	−0.654233	−0.327117	−	0.944984i	$-0.606077\pi$
$859$	−19.1747	−0.654233	−0.327117	+	0.944984i	$0.606077\pi$
$860$	−4.43065	−0.151084
$861$	0	0
$862$	−14.6924	−0.500427
$863$	36.1098	1.22919	0.614596	−	0.788842i	$-0.289319\pi$
$863$	36.1098	1.22919	0.614596	+	0.788842i	$0.289319\pi$
$864$	0	0
$865$	21.8385	0.742531
$866$	−13.3009	−0.451983
$867$	0	0
$868$	−28.6701	−0.973126
$869$	−1.90386	−0.0645839
$870$	0	0
$871$	−6.00251	−0.203387
$872$	−23.7197	−0.803251
$873$	0	0
$874$	4.46201	0.150930
$875$	40.9169	1.38324
$876$	0	0
$877$	−37.7502	−1.27473	−0.637367	−	0.770561i	$-0.719976\pi$
$877$	−37.7502	−1.27473	−0.637367	+	0.770561i	$0.719976\pi$
$878$	7.50654	0.253334
$879$	0	0
$880$	−5.33502	−0.179844
$881$	10.4464	0.351949	0.175975	−	0.984395i	$-0.443692\pi$
$881$	10.4464	0.351949	0.175975	+	0.984395i	$0.443692\pi$
$882$	0	0
$883$	53.5209	1.80112	0.900560	−	0.434731i	$-0.143157\pi$
$883$	53.5209	1.80112	0.900560	+	0.434731i	$0.143157\pi$
$884$	12.5230	0.421195
$885$	0	0
$886$	−13.2427	−0.444899
$887$	3.64469	0.122377	0.0611883	−	0.998126i	$-0.480511\pi$
$887$	3.64469	0.122377	0.0611883	+	0.998126i	$0.480511\pi$
$888$	0	0
$889$	−62.8862	−2.10914
$890$	8.36832	0.280507
$891$	0	0
$892$	11.0450	0.369815
$893$	15.1998	0.508643
$894$	0	0
$895$	12.2632	0.409914
$896$	−53.2667	−1.77951
$897$	0	0
$898$	20.1808	0.673443
$899$	49.6057	1.65444
$900$	0	0
$901$	24.5954	0.819393
$902$	7.59816	0.252991
$903$	0	0
$904$	−11.2081	−0.372776
$905$	1.14605	0.0380961
$906$	0	0
$907$	40.1654	1.33367	0.666835	−	0.745205i	$-0.267649\pi$
$907$	40.1654	1.33367	0.666835	+	0.745205i	$0.267649\pi$
$908$	5.42351	0.179985
$909$	0	0
$910$	47.7754	1.58374
$911$	−16.4344	−0.544496	−0.272248	−	0.962227i	$-0.587767\pi$
$911$	−16.4344	−0.544496	−0.272248	+	0.962227i	$0.587767\pi$
$912$	0	0
$913$	6.54740	0.216687
$914$	43.3987	1.43550
$915$	0	0
$916$	10.1337	0.334826
$917$	32.2480	1.06492
$918$	0	0
$919$	−21.5574	−0.711112	−0.355556	−	0.934655i	$-0.615708\pi$
$919$	−21.5574	−0.711112	−0.355556	+	0.934655i	$0.615708\pi$
$920$	2.06277	0.0680076
$921$	0	0
$922$	−4.58106	−0.150869
$923$	76.8894	2.53085
$924$	0	0
$925$	−28.6102	−0.940698
$926$	−64.0684	−2.10542
$927$	0	0
$928$	−36.8584	−1.20994
$929$	−6.07005	−0.199152	−0.0995759	−	0.995030i	$-0.531749\pi$
$929$	−6.07005	−0.199152	−0.0995759	+	0.995030i	$0.531749\pi$
$930$	0	0
$931$	27.8418	0.912477
$932$	13.7174	0.449327
$933$	0	0
$934$	21.9656	0.718738
$935$	2.29276	0.0749812
$936$	0	0
$937$	0.143334	0.00468250	0.00234125	−	0.999997i	$-0.499255\pi$
$937$	0.143334	0.00468250	0.00234125	+	0.999997i	$0.499255\pi$
$938$	7.45818	0.243518
$939$	0	0
$940$	−6.63188	−0.216308
$941$	20.4624	0.667055	0.333528	−	0.942740i	$-0.391761\pi$
$941$	20.4624	0.667055	0.333528	+	0.942740i	$0.391761\pi$
$942$	0	0
$943$	−4.80426	−0.156448
$944$	−50.5053	−1.64381
$945$	0	0
$946$	7.36935	0.239598
$947$	−46.7586	−1.51945	−0.759725	−	0.650244i	$-0.774667\pi$
$947$	−46.7586	−1.51945	−0.759725	+	0.650244i	$0.774667\pi$
$948$	0	0
$949$	−10.1195	−0.328491
$950$	15.8077	0.512870
$951$	0	0
$952$	16.4866	0.534335
$953$	32.3660	1.04844	0.524219	−	0.851583i	$-0.324357\pi$
$953$	32.3660	1.04844	0.524219	+	0.851583i	$0.324357\pi$
$954$	0	0
$955$	4.46985	0.144641
$956$	4.13670	0.133790
$957$	0	0
$958$	14.1495	0.457150
$959$	77.5711	2.50490
$960$	0	0
$961$	15.5579	0.501866
$962$	−76.6652	−2.47179
$963$	0	0
$964$	−3.98738	−0.128425
$965$	−5.44967	−0.175431
$966$	0	0
$967$	19.7965	0.636614	0.318307	−	0.947988i	$-0.396886\pi$
$967$	19.7965	0.636614	0.318307	+	0.947988i	$0.396886\pi$
$968$	1.77352	0.0570032
$969$	0	0
$970$	−25.2959	−0.812201
$971$	3.58830	0.115154	0.0575770	−	0.998341i	$-0.481663\pi$
$971$	3.58830	0.115154	0.0575770	+	0.998341i	$0.481663\pi$
$972$	0	0
$973$	92.4980	2.96535
$974$	13.1775	0.422234
$975$	0	0
$976$	−12.6799	−0.405873
$977$	17.0384	0.545106	0.272553	−	0.962141i	$-0.412132\pi$
$977$	17.0384	0.545106	0.272553	+	0.962141i	$0.412132\pi$
$978$	0	0
$979$	−4.54927	−0.145395
$980$	−12.1477	−0.388045
$981$	0	0
$982$	35.8850	1.14514
$983$	−12.2217	−0.389813	−0.194906	−	0.980822i	$-0.562440\pi$
$983$	−12.2217	−0.389813	−0.194906	+	0.980822i	$0.562440\pi$
$984$	0	0
$985$	−27.0065	−0.860499
$986$	26.9223	0.857380
$987$	0	0
$988$	13.8448	0.440463
$989$	−4.65958	−0.148166
$990$	0	0
$991$	43.4757	1.38105	0.690526	−	0.723308i	$-0.257379\pi$
$991$	43.4757	1.38105	0.690526	+	0.723308i	$0.257379\pi$
$992$	−34.5938	−1.09835
$993$	0	0
$994$	−95.5358	−3.03021
$995$	−13.9109	−0.441004
$996$	0	0
$997$	−52.6247	−1.66664	−0.833321	−	0.552790i	$-0.813563\pi$
$997$	−52.6247	−1.66664	−0.833321	+	0.552790i	$0.813563\pi$
$998$	−33.7344	−1.06784
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6633.2.a.w.1.5		17
3.2	odd	2		737.2.a.f.1.13	✓	17
33.32	even	2		8107.2.a.o.1.5		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
737.2.a.f.1.13	✓	17	3.2	odd	2
6633.2.a.w.1.5		17	1.1	even	1	trivial
8107.2.a.o.1.5		17	33.32	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:	\( N \)	\(=\)	\( 6633 = 3^{2} \cdot 11 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6633.a (trivial)

\(f(q)\)	\(=\)	\(q-1.72368 q^{2} +0.971084 q^{4} -1.06718 q^{5} +4.32688 q^{7} +1.77352 q^{8} +O(q^{10})\)	\(q-1.72368 q^{2} +0.971084 q^{4} -1.06718 q^{5} +4.32688 q^{7} +1.77352 q^{8} +1.83948 q^{10} -1.00000 q^{11} +6.00251 q^{13} -7.45818 q^{14} -4.99916 q^{16} +2.14842 q^{17} +2.37519 q^{19} -1.03632 q^{20} +1.72368 q^{22} -1.08987 q^{23} -3.86112 q^{25} -10.3464 q^{26} +4.20177 q^{28} -7.27001 q^{29} -6.82333 q^{31} +5.06993 q^{32} -3.70320 q^{34} -4.61758 q^{35} +7.40982 q^{37} -4.09407 q^{38} -1.89267 q^{40} +4.40810 q^{41} +4.27535 q^{43} -0.971084 q^{44} +1.87859 q^{46} +6.39942 q^{47} +11.7219 q^{49} +6.65535 q^{50} +5.82895 q^{52} +11.4481 q^{53} +1.06718 q^{55} +7.67384 q^{56} +12.5312 q^{58} +10.1027 q^{59} +2.53640 q^{61} +11.7613 q^{62} +1.25938 q^{64} -6.40578 q^{65} -1.00000 q^{67} +2.08630 q^{68} +7.95924 q^{70} +12.8095 q^{71} -1.68587 q^{73} -12.7722 q^{74} +2.30651 q^{76} -4.32688 q^{77} +1.90386 q^{79} +5.33502 q^{80} -7.59816 q^{82} -6.54740 q^{83} -2.29276 q^{85} -7.36935 q^{86} -1.77352 q^{88} +4.54927 q^{89} +25.9722 q^{91} -1.05836 q^{92} -11.0306 q^{94} -2.53476 q^{95} -13.7516 q^{97} -20.2049 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more