LMFDB - Embedded newform 693.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.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:	$5.53363286007$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79129 q^{2} +1.20871 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.41742 q^{8} +O(q^{10})$	\(q-1.79129 q^{2} +1.20871 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.41742 q^{8} +5.37386 q^{10} +1.00000 q^{11} +1.00000 q^{13} -1.79129 q^{14} -4.95644 q^{16} -7.58258 q^{17} -6.58258 q^{19} -3.62614 q^{20} -1.79129 q^{22} +5.58258 q^{23} +4.00000 q^{25} -1.79129 q^{26} +1.20871 q^{28} +8.16515 q^{29} +3.58258 q^{31} +6.04356 q^{32} +13.5826 q^{34} -3.00000 q^{35} +1.00000 q^{37} +11.7913 q^{38} -4.25227 q^{40} +11.1652 q^{41} +1.58258 q^{43} +1.20871 q^{44} -10.0000 q^{46} -1.41742 q^{47} +1.00000 q^{49} -7.16515 q^{50} +1.20871 q^{52} +9.58258 q^{53} -3.00000 q^{55} +1.41742 q^{56} -14.6261 q^{58} -4.58258 q^{59} +10.0000 q^{61} -6.41742 q^{62} -0.912878 q^{64} -3.00000 q^{65} +8.58258 q^{67} -9.16515 q^{68} +5.37386 q^{70} -11.1652 q^{71} +7.00000 q^{73} -1.79129 q^{74} -7.95644 q^{76} +1.00000 q^{77} +7.16515 q^{79} +14.8693 q^{80} -20.0000 q^{82} +11.5826 q^{83} +22.7477 q^{85} -2.83485 q^{86} +1.41742 q^{88} -9.16515 q^{89} +1.00000 q^{91} +6.74773 q^{92} +2.53901 q^{94} +19.7477 q^{95} -2.41742 q^{97} -1.79129 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) $ 2 * q + q^2 + 7 * q^4 - 6 * q^5 + 2 * q^7 + 12 * q^8	$ 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8} - 3 q^{10} + 2 q^{11} + 2 q^{13} + q^{14} + 13 q^{16} - 6 q^{17} - 4 q^{19} - 21 q^{20} + q^{22} + 2 q^{23} + 8 q^{25} + q^{26} + 7 q^{28} - 2 q^{29} - 2 q^{31} + 35 q^{32} + 18 q^{34} - 6 q^{35} + 2 q^{37} + 19 q^{38} - 36 q^{40} + 4 q^{41} - 6 q^{43} + 7 q^{44} - 20 q^{46} - 12 q^{47} + 2 q^{49} + 4 q^{50} + 7 q^{52} + 10 q^{53} - 6 q^{55} + 12 q^{56} - 43 q^{58} + 20 q^{61} - 22 q^{62} + 44 q^{64} - 6 q^{65} + 8 q^{67} - 3 q^{70} - 4 q^{71} + 14 q^{73} + q^{74} + 7 q^{76} + 2 q^{77} - 4 q^{79} - 39 q^{80} - 40 q^{82} + 14 q^{83} + 18 q^{85} - 24 q^{86} + 12 q^{88} + 2 q^{91} - 14 q^{92} - 27 q^{94} + 12 q^{95} - 14 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 + 7 * q^4 - 6 * q^5 + 2 * q^7 + 12 * q^8 - 3 * q^10 + 2 * q^11 + 2 * q^13 + q^14 + 13 * q^16 - 6 * q^17 - 4 * q^19 - 21 * q^20 + q^22 + 2 * q^23 + 8 * q^25 + q^26 + 7 * q^28 - 2 * q^29 - 2 * q^31 + 35 * q^32 + 18 * q^34 - 6 * q^35 + 2 * q^37 + 19 * q^38 - 36 * q^40 + 4 * q^41 - 6 * q^43 + 7 * q^44 - 20 * q^46 - 12 * q^47 + 2 * q^49 + 4 * q^50 + 7 * q^52 + 10 * q^53 - 6 * q^55 + 12 * q^56 - 43 * q^58 + 20 * q^61 - 22 * q^62 + 44 * q^64 - 6 * q^65 + 8 * q^67 - 3 * q^70 - 4 * q^71 + 14 * q^73 + q^74 + 7 * q^76 + 2 * q^77 - 4 * q^79 - 39 * q^80 - 40 * q^82 + 14 * q^83 + 18 * q^85 - 24 * q^86 + 12 * q^88 + 2 * q^91 - 14 * q^92 - 27 * q^94 + 12 * q^95 - 14 * q^97 + 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 $

$2$	−1.79129	−1.26663	−0.633316	−	0.773893i	$-0.718307\pi$
$2$	−1.79129	−1.26663	−0.633316	+	0.773893i	$0.718307\pi$
$3$	0	0
$3$	0	0
$4$	1.20871	0.604356
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.41742	0.501135
$9$	0	0
$10$	5.37386	1.69936
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.79129	−0.478742
$15$	0	0
$16$	−4.95644	−1.23911
$17$	−7.58258	−1.83904	−0.919522	−	0.393038i	$-0.871424\pi$
$17$	−7.58258	−1.83904	−0.919522	+	0.393038i	$0.871424\pi$
$18$	0	0
$19$	−6.58258	−1.51015	−0.755073	−	0.655640i	$-0.772399\pi$
$19$	−6.58258	−1.51015	−0.755073	+	0.655640i	$0.772399\pi$
$20$	−3.62614	−0.810829
$21$	0	0
$22$	−1.79129	−0.381904
$23$	5.58258	1.16405	0.582024	−	0.813172i	$-0.302261\pi$
$23$	5.58258	1.16405	0.582024	+	0.813172i	$0.302261\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−1.79129	−0.351300
$27$	0	0
$28$	1.20871	0.228425
$29$	8.16515	1.51623	0.758115	−	0.652121i	$-0.226120\pi$
$29$	8.16515	1.51623	0.758115	+	0.652121i	$0.226120\pi$
$30$	0	0
$31$	3.58258	0.643450	0.321725	−	0.946833i	$-0.395737\pi$
$31$	3.58258	0.643450	0.321725	+	0.946833i	$0.395737\pi$
$32$	6.04356	1.06836
$33$	0	0
$34$	13.5826	2.32939
$35$	−3.00000	−0.507093
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	11.7913	1.91280
$39$	0	0
$40$	−4.25227	−0.672343
$41$	11.1652	1.74370	0.871852	−	0.489770i	$-0.162919\pi$
$41$	11.1652	1.74370	0.871852	+	0.489770i	$0.162919\pi$
$42$	0	0
$43$	1.58258	0.241341	0.120670	−	0.992693i	$-0.461496\pi$
$43$	1.58258	0.241341	0.120670	+	0.992693i	$0.461496\pi$
$44$	1.20871	0.182220
$45$	0	0
$46$	−10.0000	−1.47442
$47$	−1.41742	−0.206753	−0.103376	−	0.994642i	$-0.532965\pi$
$47$	−1.41742	−0.206753	−0.103376	+	0.994642i	$0.532965\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.16515	−1.01331
$51$	0	0
$52$	1.20871	0.167618
$53$	9.58258	1.31627	0.658134	−	0.752901i	$-0.271346\pi$
$53$	9.58258	1.31627	0.658134	+	0.752901i	$0.271346\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	1.41742	0.189411
$57$	0	0
$58$	−14.6261	−1.92051
$59$	−4.58258	−0.596601	−0.298300	−	0.954472i	$-0.596420\pi$
$59$	−4.58258	−0.596601	−0.298300	+	0.954472i	$0.596420\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−6.41742	−0.815014
$63$	0	0
$64$	−0.912878	−0.114110
$65$	−3.00000	−0.372104
$66$	0	0
$67$	8.58258	1.04853	0.524264	−	0.851556i	$-0.324340\pi$
$67$	8.58258	1.04853	0.524264	+	0.851556i	$0.324340\pi$
$68$	−9.16515	−1.11144
$69$	0	0
$70$	5.37386	0.642300
$71$	−11.1652	−1.32506	−0.662530	−	0.749036i	$-0.730517\pi$
$71$	−11.1652	−1.32506	−0.662530	+	0.749036i	$0.730517\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	−1.79129	−0.208233
$75$	0	0
$76$	−7.95644	−0.912666
$77$	1.00000	0.113961
$78$	0	0
$79$	7.16515	0.806143	0.403071	−	0.915169i	$-0.367943\pi$
$79$	7.16515	0.806143	0.403071	+	0.915169i	$0.367943\pi$
$80$	14.8693	1.66244
$81$	0	0
$82$	−20.0000	−2.20863
$83$	11.5826	1.27135	0.635676	−	0.771956i	$-0.280721\pi$
$83$	11.5826	1.27135	0.635676	+	0.771956i	$0.280721\pi$
$84$	0	0
$85$	22.7477	2.46734
$86$	−2.83485	−0.305690
$87$	0	0
$88$	1.41742	0.151098
$89$	−9.16515	−0.971504	−0.485752	−	0.874097i	$-0.661454\pi$
$89$	−9.16515	−0.971504	−0.485752	+	0.874097i	$0.661454\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	6.74773	0.703499
$93$	0	0
$94$	2.53901	0.261879
$95$	19.7477	2.02607
$96$	0	0
$97$	−2.41742	−0.245452	−0.122726	−	0.992441i	$-0.539164\pi$
$97$	−2.41742	−0.245452	−0.122726	+	0.992441i	$0.539164\pi$
$98$	−1.79129	−0.180947
$99$	0	0
$100$	4.83485	0.483485
$101$	−11.5826	−1.15251	−0.576255	−	0.817270i	$-0.695486\pi$
$101$	−11.5826	−1.15251	−0.576255	+	0.817270i	$0.695486\pi$
$102$	0	0
$103$	−1.16515	−0.114806	−0.0574029	−	0.998351i	$-0.518282\pi$
$103$	−1.16515	−0.114806	−0.0574029	+	0.998351i	$0.518282\pi$
$104$	1.41742	0.138990
$105$	0	0
$106$	−17.1652	−1.66723
$107$	12.5826	1.21640	0.608202	−	0.793782i	$-0.291891\pi$
$107$	12.5826	1.21640	0.608202	+	0.793782i	$0.291891\pi$
$108$	0	0
$109$	3.58258	0.343149	0.171574	−	0.985171i	$-0.445115\pi$
$109$	3.58258	0.343149	0.171574	+	0.985171i	$0.445115\pi$
$110$	5.37386	0.512378
$111$	0	0
$112$	−4.95644	−0.468339
$113$	−9.16515	−0.862185	−0.431092	−	0.902308i	$-0.641872\pi$
$113$	−9.16515	−0.862185	−0.431092	+	0.902308i	$0.641872\pi$
$114$	0	0
$115$	−16.7477	−1.56173
$116$	9.86932	0.916343
$117$	0	0
$118$	8.20871	0.755673
$119$	−7.58258	−0.695094
$120$	0	0
$121$	1.00000	0.0909091
$122$	−17.9129	−1.62176
$123$	0	0
$124$	4.33030	0.388873
$125$	3.00000	0.268328
$126$	0	0
$127$	−11.5826	−1.02779	−0.513894	−	0.857854i	$-0.671797\pi$
$127$	−11.5826	−1.02779	−0.513894	+	0.857854i	$0.671797\pi$
$128$	−10.4519	−0.923826
$129$	0	0
$130$	5.37386	0.471319
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	−6.58258	−0.570782
$134$	−15.3739	−1.32810
$135$	0	0
$136$	−10.7477	−0.921610
$137$	11.5826	0.989566	0.494783	−	0.869016i	$-0.335248\pi$
$137$	11.5826	0.989566	0.494783	+	0.869016i	$0.335248\pi$
$138$	0	0
$139$	11.1652	0.947016	0.473508	−	0.880790i	$-0.342988\pi$
$139$	11.1652	0.947016	0.473508	+	0.880790i	$0.342988\pi$
$140$	−3.62614	−0.306464
$141$	0	0
$142$	20.0000	1.67836
$143$	1.00000	0.0836242
$144$	0	0
$145$	−24.4955	−2.03424
$146$	−12.5390	−1.03774
$147$	0	0
$148$	1.20871	0.0993555
$149$	−6.16515	−0.505069	−0.252534	−	0.967588i	$-0.581264\pi$
$149$	−6.16515	−0.505069	−0.252534	+	0.967588i	$0.581264\pi$
$150$	0	0
$151$	3.58258	0.291546	0.145773	−	0.989318i	$-0.453433\pi$
$151$	3.58258	0.291546	0.145773	+	0.989318i	$0.453433\pi$
$152$	−9.33030	−0.756787
$153$	0	0
$154$	−1.79129	−0.144346
$155$	−10.7477	−0.863278
$156$	0	0
$157$	19.1652	1.52955	0.764773	−	0.644300i	$-0.222851\pi$
$157$	19.1652	1.52955	0.764773	+	0.644300i	$0.222851\pi$
$158$	−12.8348	−1.02109
$159$	0	0
$160$	−18.1307	−1.43336
$161$	5.58258	0.439969
$162$	0	0
$163$	8.58258	0.672239	0.336120	−	0.941819i	$-0.390885\pi$
$163$	8.58258	0.672239	0.336120	+	0.941819i	$0.390885\pi$
$164$	13.4955	1.05382
$165$	0	0
$166$	−20.7477	−1.61034
$167$	4.74773	0.367390	0.183695	−	0.982983i	$-0.441194\pi$
$167$	4.74773	0.367390	0.183695	+	0.982983i	$0.441194\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−40.7477	−3.12521
$171$	0	0
$172$	1.91288	0.145856
$173$	7.16515	0.544756	0.272378	−	0.962190i	$-0.412190\pi$
$173$	7.16515	0.544756	0.272378	+	0.962190i	$0.412190\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−4.95644	−0.373606
$177$	0	0
$178$	16.4174	1.23054
$179$	−14.3303	−1.07110	−0.535549	−	0.844504i	$-0.679895\pi$
$179$	−14.3303	−1.07110	−0.535549	+	0.844504i	$0.679895\pi$
$180$	0	0
$181$	−5.58258	−0.414950	−0.207475	−	0.978240i	$-0.566524\pi$
$181$	−5.58258	−0.414950	−0.207475	+	0.978240i	$0.566524\pi$
$182$	−1.79129	−0.132779
$183$	0	0
$184$	7.91288	0.583345
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−7.58258	−0.554493
$188$	−1.71326	−0.124952
$189$	0	0
$190$	−35.3739	−2.56629
$191$	−11.5826	−0.838086	−0.419043	−	0.907966i	$-0.637634\pi$
$191$	−11.5826	−0.838086	−0.419043	+	0.907966i	$0.637634\pi$
$192$	0	0
$193$	−2.41742	−0.174010	−0.0870050	−	0.996208i	$-0.527730\pi$
$193$	−2.41742	−0.174010	−0.0870050	+	0.996208i	$0.527730\pi$
$194$	4.33030	0.310898
$195$	0	0
$196$	1.20871	0.0863366
$197$	−5.16515	−0.368002	−0.184001	−	0.982926i	$-0.558905\pi$
$197$	−5.16515	−0.368002	−0.184001	+	0.982926i	$0.558905\pi$
$198$	0	0
$199$	−9.58258	−0.679291	−0.339645	−	0.940554i	$-0.610307\pi$
$199$	−9.58258	−0.679291	−0.339645	+	0.940554i	$0.610307\pi$
$200$	5.66970	0.400908
$201$	0	0
$202$	20.7477	1.45980
$203$	8.16515	0.573081
$204$	0	0
$205$	−33.4955	−2.33942
$206$	2.08712	0.145417
$207$	0	0
$208$	−4.95644	−0.343667
$209$	−6.58258	−0.455326
$210$	0	0
$211$	−13.1652	−0.906326	−0.453163	−	0.891428i	$-0.649705\pi$
$211$	−13.1652	−0.906326	−0.453163	+	0.891428i	$0.649705\pi$
$212$	11.5826	0.795495
$213$	0	0
$214$	−22.5390	−1.54074
$215$	−4.74773	−0.323792
$216$	0	0
$217$	3.58258	0.243201
$218$	−6.41742	−0.434643
$219$	0	0
$220$	−3.62614	−0.244474
$221$	−7.58258	−0.510059
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	6.04356	0.403802
$225$	0	0
$226$	16.4174	1.09207
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	0.747727	0.0494112	0.0247056	−	0.999695i	$-0.492135\pi$
$229$	0.747727	0.0494112	0.0247056	+	0.999695i	$0.492135\pi$
$230$	30.0000	1.97814
$231$	0	0
$232$	11.5735	0.759836
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	4.25227	0.277388
$236$	−5.53901	−0.360559
$237$	0	0
$238$	13.5826	0.880428
$239$	−16.5826	−1.07264	−0.536319	−	0.844015i	$-0.680186\pi$
$239$	−16.5826	−1.07264	−0.536319	+	0.844015i	$0.680186\pi$
$240$	0	0
$241$	−10.1652	−0.654795	−0.327397	−	0.944887i	$-0.606172\pi$
$241$	−10.1652	−0.654795	−0.327397	+	0.944887i	$0.606172\pi$
$242$	−1.79129	−0.115148
$243$	0	0
$244$	12.0871	0.773799
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−6.58258	−0.418839
$248$	5.07803	0.322455
$249$	0	0
$250$	−5.37386	−0.339873
$251$	−7.41742	−0.468184	−0.234092	−	0.972214i	$-0.575212\pi$
$251$	−7.41742	−0.468184	−0.234092	+	0.972214i	$0.575212\pi$
$252$	0	0
$253$	5.58258	0.350974
$254$	20.7477	1.30183
$255$	0	0
$256$	20.5481	1.28426
$257$	−19.0000	−1.18519	−0.592594	−	0.805502i	$-0.701896\pi$
$257$	−19.0000	−1.18519	−0.592594	+	0.805502i	$0.701896\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	−3.62614	−0.224883
$261$	0	0
$262$	−28.6606	−1.77066
$263$	22.9129	1.41287	0.706434	−	0.707779i	$-0.250303\pi$
$263$	22.9129	1.41287	0.706434	+	0.707779i	$0.250303\pi$
$264$	0	0
$265$	−28.7477	−1.76596
$266$	11.7913	0.722970
$267$	0	0
$268$	10.3739	0.633685
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	5.41742	0.329085	0.164543	−	0.986370i	$-0.447385\pi$
$271$	5.41742	0.329085	0.164543	+	0.986370i	$0.447385\pi$
$272$	37.5826	2.27878
$273$	0	0
$274$	−20.7477	−1.25342
$275$	4.00000	0.241209
$276$	0	0
$277$	−19.1652	−1.15152	−0.575761	−	0.817618i	$-0.695294\pi$
$277$	−19.1652	−1.15152	−0.575761	+	0.817618i	$0.695294\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	−4.25227	−0.254122
$281$	27.3303	1.63039	0.815195	−	0.579187i	$-0.196630\pi$
$281$	27.3303	1.63039	0.815195	+	0.579187i	$0.196630\pi$
$282$	0	0
$283$	27.7477	1.64943	0.824716	−	0.565548i	$-0.191335\pi$
$283$	27.7477	1.64943	0.824716	+	0.565548i	$0.191335\pi$
$284$	−13.4955	−0.800808
$285$	0	0
$286$	−1.79129	−0.105921
$287$	11.1652	0.659058
$288$	0	0
$289$	40.4955	2.38209
$290$	43.8784	2.57663
$291$	0	0
$292$	8.46099	0.495142
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	13.7477	0.800424
$296$	1.41742	0.0823861
$297$	0	0
$298$	11.0436	0.639736
$299$	5.58258	0.322849
$300$	0	0
$301$	1.58258	0.0912181
$302$	−6.41742	−0.369281
$303$	0	0
$304$	32.6261	1.87124
$305$	−30.0000	−1.71780
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	1.20871	0.0688728
$309$	0	0
$310$	19.2523	1.09346
$311$	−14.3303	−0.812597	−0.406298	−	0.913740i	$-0.633181\pi$
$311$	−14.3303	−0.812597	−0.406298	+	0.913740i	$0.633181\pi$
$312$	0	0
$313$	19.5826	1.10687	0.553436	−	0.832891i	$-0.313316\pi$
$313$	19.5826	1.10687	0.553436	+	0.832891i	$0.313316\pi$
$314$	−34.3303	−1.93737
$315$	0	0
$316$	8.66061	0.487197
$317$	22.4174	1.25909	0.629544	−	0.776965i	$-0.283242\pi$
$317$	22.4174	1.25909	0.629544	+	0.776965i	$0.283242\pi$
$318$	0	0
$319$	8.16515	0.457161
$320$	2.73864	0.153094
$321$	0	0
$322$	−10.0000	−0.557278
$323$	49.9129	2.77723
$324$	0	0
$325$	4.00000	0.221880
$326$	−15.3739	−0.851480
$327$	0	0
$328$	15.8258	0.873831
$329$	−1.41742	−0.0781451
$330$	0	0
$331$	−3.16515	−0.173972	−0.0869862	−	0.996210i	$-0.527724\pi$
$331$	−3.16515	−0.173972	−0.0869862	+	0.996210i	$0.527724\pi$
$332$	14.0000	0.768350
$333$	0	0
$334$	−8.50455	−0.465348
$335$	−25.7477	−1.40675
$336$	0	0
$337$	17.5826	0.957784	0.478892	−	0.877874i	$-0.341039\pi$
$337$	17.5826	0.957784	0.478892	+	0.877874i	$0.341039\pi$
$338$	21.4955	1.16920
$339$	0	0
$340$	27.4955	1.49115
$341$	3.58258	0.194007
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.24318	0.120944
$345$	0	0
$346$	−12.8348	−0.690006
$347$	−26.3303	−1.41348	−0.706742	−	0.707471i	$-0.749836\pi$
$347$	−26.3303	−1.41348	−0.706742	+	0.707471i	$0.749836\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	−7.16515	−0.382993
$351$	0	0
$352$	6.04356	0.322123
$353$	−24.1652	−1.28618	−0.643091	−	0.765790i	$-0.722348\pi$
$353$	−24.1652	−1.28618	−0.643091	+	0.765790i	$0.722348\pi$
$354$	0	0
$355$	33.4955	1.77775
$356$	−11.0780	−0.587134
$357$	0	0
$358$	25.6697	1.35669
$359$	8.83485	0.466285	0.233143	−	0.972443i	$-0.425099\pi$
$359$	8.83485	0.466285	0.233143	+	0.972443i	$0.425099\pi$
$360$	0	0
$361$	24.3303	1.28054
$362$	10.0000	0.525588
$363$	0	0
$364$	1.20871	0.0633537
$365$	−21.0000	−1.09919
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	−27.6697	−1.44238
$369$	0	0
$370$	5.37386	0.279374
$371$	9.58258	0.497503
$372$	0	0
$373$	−34.7477	−1.79917	−0.899585	−	0.436747i	$-0.856131\pi$
$373$	−34.7477	−1.79917	−0.899585	+	0.436747i	$0.856131\pi$
$374$	13.5826	0.702338
$375$	0	0
$376$	−2.00909	−0.103611
$377$	8.16515	0.420527
$378$	0	0
$379$	−12.5826	−0.646323	−0.323162	−	0.946344i	$-0.604746\pi$
$379$	−12.5826	−0.646323	−0.323162	+	0.946344i	$0.604746\pi$
$380$	23.8693	1.22447
$381$	0	0
$382$	20.7477	1.06155
$383$	−10.3303	−0.527854	−0.263927	−	0.964543i	$-0.585018\pi$
$383$	−10.3303	−0.527854	−0.263927	+	0.964543i	$0.585018\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	4.33030	0.220407
$387$	0	0
$388$	−2.92197	−0.148341
$389$	26.3303	1.33500	0.667500	−	0.744610i	$-0.267365\pi$
$389$	26.3303	1.33500	0.667500	+	0.744610i	$0.267365\pi$
$390$	0	0
$391$	−42.3303	−2.14074
$392$	1.41742	0.0715907
$393$	0	0
$394$	9.25227	0.466123
$395$	−21.4955	−1.08155
$396$	0	0
$397$	−31.5826	−1.58508	−0.792542	−	0.609817i	$-0.791243\pi$
$397$	−31.5826	−1.58508	−0.792542	+	0.609817i	$0.791243\pi$
$398$	17.1652	0.860411
$399$	0	0
$400$	−19.8258	−0.991288
$401$	−31.9129	−1.59365	−0.796827	−	0.604208i	$-0.793490\pi$
$401$	−31.9129	−1.59365	−0.796827	+	0.604208i	$0.793490\pi$
$402$	0	0
$403$	3.58258	0.178461
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−14.6261	−0.725883
$407$	1.00000	0.0495682
$408$	0	0
$409$	8.33030	0.411907	0.205953	−	0.978562i	$-0.433970\pi$
$409$	8.33030	0.411907	0.205953	+	0.978562i	$0.433970\pi$
$410$	60.0000	2.96319
$411$	0	0
$412$	−1.40833	−0.0693836
$413$	−4.58258	−0.225494
$414$	0	0
$415$	−34.7477	−1.70570
$416$	6.04356	0.296310
$417$	0	0
$418$	11.7913	0.576731
$419$	−2.58258	−0.126167	−0.0630835	−	0.998008i	$-0.520093\pi$
$419$	−2.58258	−0.126167	−0.0630835	+	0.998008i	$0.520093\pi$
$420$	0	0
$421$	−33.6606	−1.64052	−0.820259	−	0.571993i	$-0.806171\pi$
$421$	−33.6606	−1.64052	−0.820259	+	0.571993i	$0.806171\pi$
$422$	23.5826	1.14798
$423$	0	0
$424$	13.5826	0.659628
$425$	−30.3303	−1.47124
$426$	0	0
$427$	10.0000	0.483934
$428$	15.2087	0.735141
$429$	0	0
$430$	8.50455	0.410126
$431$	−17.7477	−0.854878	−0.427439	−	0.904044i	$-0.640584\pi$
$431$	−17.7477	−0.854878	−0.427439	+	0.904044i	$0.640584\pi$
$432$	0	0
$433$	−11.1652	−0.536563	−0.268281	−	0.963341i	$-0.586456\pi$
$433$	−11.1652	−0.536563	−0.268281	+	0.963341i	$0.586456\pi$
$434$	−6.41742	−0.308046
$435$	0	0
$436$	4.33030	0.207384
$437$	−36.7477	−1.75788
$438$	0	0
$439$	17.4174	0.831288	0.415644	−	0.909527i	$-0.363556\pi$
$439$	17.4174	0.831288	0.415644	+	0.909527i	$0.363556\pi$
$440$	−4.25227	−0.202719
$441$	0	0
$442$	13.5826	0.646057
$443$	23.1652	1.10061	0.550305	−	0.834964i	$-0.314512\pi$
$443$	23.1652	1.10061	0.550305	+	0.834964i	$0.314512\pi$
$444$	0	0
$445$	27.4955	1.30341
$446$	−10.7477	−0.508920
$447$	0	0
$448$	−0.912878	−0.0431295
$449$	18.3303	0.865060	0.432530	−	0.901619i	$-0.357621\pi$
$449$	18.3303	0.865060	0.432530	+	0.901619i	$0.357621\pi$
$450$	0	0
$451$	11.1652	0.525746
$452$	−11.0780	−0.521067
$453$	0	0
$454$	−39.4083	−1.84952
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−19.9129	−0.931485	−0.465743	−	0.884920i	$-0.654213\pi$
$457$	−19.9129	−0.931485	−0.465743	+	0.884920i	$0.654213\pi$
$458$	−1.33939	−0.0625858
$459$	0	0
$460$	−20.2432	−0.943843
$461$	−18.3303	−0.853727	−0.426864	−	0.904316i	$-0.640382\pi$
$461$	−18.3303	−0.853727	−0.426864	+	0.904316i	$0.640382\pi$
$462$	0	0
$463$	8.58258	0.398866	0.199433	−	0.979911i	$-0.436090\pi$
$463$	8.58258	0.398866	0.199433	+	0.979911i	$0.436090\pi$
$464$	−40.4701	−1.87878
$465$	0	0
$466$	−25.0780	−1.16172
$467$	38.5826	1.78539	0.892694	−	0.450663i	$-0.148812\pi$
$467$	38.5826	1.78539	0.892694	+	0.450663i	$0.148812\pi$
$468$	0	0
$469$	8.58258	0.396307
$470$	−7.61704	−0.351348
$471$	0	0
$472$	−6.49545	−0.298978
$473$	1.58258	0.0727669
$474$	0	0
$475$	−26.3303	−1.20812
$476$	−9.16515	−0.420084
$477$	0	0
$478$	29.7042	1.35864
$479$	15.5826	0.711986	0.355993	−	0.934489i	$-0.384143\pi$
$479$	15.5826	0.711986	0.355993	+	0.934489i	$0.384143\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	18.2087	0.829384
$483$	0	0
$484$	1.20871	0.0549415
$485$	7.25227	0.329309
$486$	0	0
$487$	10.3303	0.468111	0.234055	−	0.972223i	$-0.424800\pi$
$487$	10.3303	0.468111	0.234055	+	0.972223i	$0.424800\pi$
$488$	14.1742	0.641638
$489$	0	0
$490$	5.37386	0.242766
$491$	22.9129	1.03404	0.517022	−	0.855972i	$-0.327041\pi$
$491$	22.9129	1.03404	0.517022	+	0.855972i	$0.327041\pi$
$492$	0	0
$493$	−61.9129	−2.78842
$494$	11.7913	0.530515
$495$	0	0
$496$	−17.7568	−0.797305
$497$	−11.1652	−0.500825
$498$	0	0
$499$	41.7477	1.86888	0.934442	−	0.356114i	$-0.115899\pi$
$499$	41.7477	1.86888	0.934442	+	0.356114i	$0.115899\pi$
$500$	3.62614	0.162166
$501$	0	0
$502$	13.2867	0.593016
$503$	0.747727	0.0333395	0.0166698	−	0.999861i	$-0.494694\pi$
$503$	0.747727	0.0333395	0.0166698	+	0.999861i	$0.494694\pi$
$504$	0	0
$505$	34.7477	1.54625
$506$	−10.0000	−0.444554
$507$	0	0
$508$	−14.0000	−0.621150
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	−15.9038	−0.702855
$513$	0	0
$514$	34.0345	1.50120
$515$	3.49545	0.154028
$516$	0	0
$517$	−1.41742	−0.0623382
$518$	−1.79129	−0.0787047
$519$	0	0
$520$	−4.25227	−0.186475
$521$	−15.8348	−0.693737	−0.346869	−	0.937914i	$-0.612755\pi$
$521$	−15.8348	−0.693737	−0.346869	+	0.937914i	$0.612755\pi$
$522$	0	0
$523$	15.4174	0.674157	0.337078	−	0.941477i	$-0.390561\pi$
$523$	15.4174	0.674157	0.337078	+	0.941477i	$0.390561\pi$
$524$	19.3394	0.844845
$525$	0	0
$526$	−41.0436	−1.78958
$527$	−27.1652	−1.18333
$528$	0	0
$529$	8.16515	0.355007
$530$	51.4955	2.23682
$531$	0	0
$532$	−7.95644	−0.344955
$533$	11.1652	0.483616
$534$	0	0
$535$	−37.7477	−1.63198
$536$	12.1652	0.525455
$537$	0	0
$538$	17.9129	0.772279
$539$	1.00000	0.0430730
$540$	0	0
$541$	18.3303	0.788081	0.394041	−	0.919093i	$-0.371077\pi$
$541$	18.3303	0.788081	0.394041	+	0.919093i	$0.371077\pi$
$542$	−9.70417	−0.416830
$543$	0	0
$544$	−45.8258	−1.96476
$545$	−10.7477	−0.460382
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	14.0000	0.598050
$549$	0	0
$550$	−7.16515	−0.305523
$551$	−53.7477	−2.28973
$552$	0	0
$553$	7.16515	0.304693
$554$	34.3303	1.45855
$555$	0	0
$556$	13.4955	0.572335
$557$	−9.33030	−0.395338	−0.197669	−	0.980269i	$-0.563337\pi$
$557$	−9.33030	−0.395338	−0.197669	+	0.980269i	$0.563337\pi$
$558$	0	0
$559$	1.58258	0.0669358
$560$	14.8693	0.628343
$561$	0	0
$562$	−48.9564	−2.06510
$563$	37.5826	1.58392	0.791958	−	0.610575i	$-0.209062\pi$
$563$	37.5826	1.58392	0.791958	+	0.610575i	$0.209062\pi$
$564$	0	0
$565$	27.4955	1.15674
$566$	−49.7042	−2.08922
$567$	0	0
$568$	−15.8258	−0.664034
$569$	26.6606	1.11767	0.558835	−	0.829279i	$-0.311248\pi$
$569$	26.6606	1.11767	0.558835	+	0.829279i	$0.311248\pi$
$570$	0	0
$571$	28.8348	1.20670	0.603350	−	0.797476i	$-0.293832\pi$
$571$	28.8348	1.20670	0.603350	+	0.797476i	$0.293832\pi$
$572$	1.20871	0.0505388
$573$	0	0
$574$	−20.0000	−0.834784
$575$	22.3303	0.931238
$576$	0	0
$577$	−21.9129	−0.912245	−0.456123	−	0.889917i	$-0.650762\pi$
$577$	−21.9129	−0.912245	−0.456123	+	0.889917i	$0.650762\pi$
$578$	−72.5390	−3.01723
$579$	0	0
$580$	−29.6080	−1.22940
$581$	11.5826	0.480526
$582$	0	0
$583$	9.58258	0.396870
$584$	9.92197	0.410574
$585$	0	0
$586$	0	0
$587$	−37.7477	−1.55802	−0.779008	−	0.627014i	$-0.784277\pi$
$587$	−37.7477	−1.55802	−0.779008	+	0.627014i	$0.784277\pi$
$588$	0	0
$589$	−23.5826	−0.971703
$590$	−24.6261	−1.01384
$591$	0	0
$592$	−4.95644	−0.203708
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	22.7477	0.932566
$596$	−7.45189	−0.305241
$597$	0	0
$598$	−10.0000	−0.408930
$599$	−7.16515	−0.292760	−0.146380	−	0.989228i	$-0.546762\pi$
$599$	−7.16515	−0.292760	−0.146380	+	0.989228i	$0.546762\pi$
$600$	0	0
$601$	24.4955	0.999190	0.499595	−	0.866259i	$-0.333482\pi$
$601$	24.4955	0.999190	0.499595	+	0.866259i	$0.333482\pi$
$602$	−2.83485	−0.115540
$603$	0	0
$604$	4.33030	0.176198
$605$	−3.00000	−0.121967
$606$	0	0
$607$	−21.7477	−0.882713	−0.441357	−	0.897332i	$-0.645503\pi$
$607$	−21.7477	−0.882713	−0.441357	+	0.897332i	$0.645503\pi$
$608$	−39.7822	−1.61338
$609$	0	0
$610$	53.7386	2.17581
$611$	−1.41742	−0.0573428
$612$	0	0
$613$	−26.7477	−1.08033	−0.540165	−	0.841559i	$-0.681638\pi$
$613$	−26.7477	−1.08033	−0.540165	+	0.841559i	$0.681638\pi$
$614$	0	0
$615$	0	0
$616$	1.41742	0.0571097
$617$	−2.83485	−0.114127	−0.0570634	−	0.998371i	$-0.518174\pi$
$617$	−2.83485	−0.114127	−0.0570634	+	0.998371i	$0.518174\pi$
$618$	0	0
$619$	−29.0780	−1.16874	−0.584372	−	0.811486i	$-0.698659\pi$
$619$	−29.0780	−1.16874	−0.584372	+	0.811486i	$0.698659\pi$
$620$	−12.9909	−0.521727
$621$	0	0
$622$	25.6697	1.02926
$623$	−9.16515	−0.367194
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−35.0780	−1.40200
$627$	0	0
$628$	23.1652	0.924390
$629$	−7.58258	−0.302337
$630$	0	0
$631$	23.1652	0.922190	0.461095	−	0.887351i	$-0.347457\pi$
$631$	23.1652	0.922190	0.461095	+	0.887351i	$0.347457\pi$
$632$	10.1561	0.403986
$633$	0	0
$634$	−40.1561	−1.59480
$635$	34.7477	1.37892
$636$	0	0
$637$	1.00000	0.0396214
$638$	−14.6261	−0.579054
$639$	0	0
$640$	31.3557	1.23944
$641$	−43.5826	−1.72141	−0.860704	−	0.509106i	$-0.829976\pi$
$641$	−43.5826	−1.72141	−0.860704	+	0.509106i	$0.829976\pi$
$642$	0	0
$643$	38.2432	1.50816	0.754082	−	0.656780i	$-0.228082\pi$
$643$	38.2432	1.50816	0.754082	+	0.656780i	$0.228082\pi$
$644$	6.74773	0.265898
$645$	0	0
$646$	−89.4083	−3.51772
$647$	10.9129	0.429030	0.214515	−	0.976721i	$-0.431183\pi$
$647$	10.9129	0.429030	0.214515	+	0.976721i	$0.431183\pi$
$648$	0	0
$649$	−4.58258	−0.179882
$650$	−7.16515	−0.281040
$651$	0	0
$652$	10.3739	0.406272
$653$	30.3303	1.18692	0.593458	−	0.804865i	$-0.297762\pi$
$653$	30.3303	1.18692	0.593458	+	0.804865i	$0.297762\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	−55.3394	−2.16064
$657$	0	0
$658$	2.53901	0.0989811
$659$	28.5826	1.11342	0.556710	−	0.830707i	$-0.312064\pi$
$659$	28.5826	1.11342	0.556710	+	0.830707i	$0.312064\pi$
$660$	0	0
$661$	−39.0780	−1.51996	−0.759980	−	0.649947i	$-0.774791\pi$
$661$	−39.0780	−1.51996	−0.759980	+	0.649947i	$0.774791\pi$
$662$	5.66970	0.220359
$663$	0	0
$664$	16.4174	0.637120
$665$	19.7477	0.765784
$666$	0	0
$667$	45.5826	1.76496
$668$	5.73864	0.222034
$669$	0	0
$670$	46.1216	1.78183
$671$	10.0000	0.386046
$672$	0	0
$673$	11.2523	0.433743	0.216872	−	0.976200i	$-0.430415\pi$
$673$	11.2523	0.433743	0.216872	+	0.976200i	$0.430415\pi$
$674$	−31.4955	−1.21316
$675$	0	0
$676$	−14.5045	−0.557867
$677$	45.1652	1.73584	0.867919	−	0.496706i	$-0.165457\pi$
$677$	45.1652	1.73584	0.867919	+	0.496706i	$0.165457\pi$
$678$	0	0
$679$	−2.41742	−0.0927722
$680$	32.2432	1.23647
$681$	0	0
$682$	−6.41742	−0.245736
$683$	33.0780	1.26570	0.632848	−	0.774276i	$-0.281886\pi$
$683$	33.0780	1.26570	0.632848	+	0.774276i	$0.281886\pi$
$684$	0	0
$685$	−34.7477	−1.32764
$686$	−1.79129	−0.0683917
$687$	0	0
$688$	−7.84394	−0.299047
$689$	9.58258	0.365067
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	8.66061	0.329227
$693$	0	0
$694$	47.1652	1.79036
$695$	−33.4955	−1.27055
$696$	0	0
$697$	−84.6606	−3.20675
$698$	26.8693	1.01702
$699$	0	0
$700$	4.83485	0.182740
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	−6.58258	−0.248267
$704$	−0.912878	−0.0344054
$705$	0	0
$706$	43.2867	1.62912
$707$	−11.5826	−0.435608
$708$	0	0
$709$	27.6606	1.03882	0.519408	−	0.854526i	$-0.326153\pi$
$709$	27.6606	1.03882	0.519408	+	0.854526i	$0.326153\pi$
$710$	−60.0000	−2.25176
$711$	0	0
$712$	−12.9909	−0.486855
$713$	20.0000	0.749006
$714$	0	0
$715$	−3.00000	−0.112194
$716$	−17.3212	−0.647324
$717$	0	0
$718$	−15.8258	−0.590612
$719$	14.0780	0.525022	0.262511	−	0.964929i	$-0.415449\pi$
$719$	14.0780	0.525022	0.262511	+	0.964929i	$0.415449\pi$
$720$	0	0
$721$	−1.16515	−0.0433925
$722$	−43.5826	−1.62198
$723$	0	0
$724$	−6.74773	−0.250777
$725$	32.6606	1.21298
$726$	0	0
$727$	−15.9129	−0.590176	−0.295088	−	0.955470i	$-0.595349\pi$
$727$	−15.9129	−0.590176	−0.295088	+	0.955470i	$0.595349\pi$
$728$	1.41742	0.0525332
$729$	0	0
$730$	37.6170	1.39227
$731$	−12.0000	−0.443836
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−39.4083	−1.45459
$735$	0	0
$736$	33.7386	1.24362
$737$	8.58258	0.316143
$738$	0	0
$739$	−31.9129	−1.17393	−0.586967	−	0.809611i	$-0.699678\pi$
$739$	−31.9129	−1.17393	−0.586967	+	0.809611i	$0.699678\pi$
$740$	−3.62614	−0.133299
$741$	0	0
$742$	−17.1652	−0.630153
$743$	53.2432	1.95330	0.976651	−	0.214830i	$-0.0689198\pi$
$743$	53.2432	1.95330	0.976651	+	0.214830i	$0.0689198\pi$
$744$	0	0
$745$	18.4955	0.677621
$746$	62.2432	2.27888
$747$	0	0
$748$	−9.16515	−0.335111
$749$	12.5826	0.459757
$750$	0	0
$751$	−8.91288	−0.325236	−0.162618	−	0.986689i	$-0.551994\pi$
$751$	−8.91288	−0.325236	−0.162618	+	0.986689i	$0.551994\pi$
$752$	7.02538	0.256189
$753$	0	0
$754$	−14.6261	−0.532652
$755$	−10.7477	−0.391150
$756$	0	0
$757$	27.3303	0.993337	0.496668	−	0.867940i	$-0.334557\pi$
$757$	27.3303	0.993337	0.496668	+	0.867940i	$0.334557\pi$
$758$	22.5390	0.818654
$759$	0	0
$760$	27.9909	1.01534
$761$	−42.3303	−1.53447	−0.767236	−	0.641365i	$-0.778369\pi$
$761$	−42.3303	−1.53447	−0.767236	+	0.641365i	$0.778369\pi$
$762$	0	0
$763$	3.58258	0.129698
$764$	−14.0000	−0.506502
$765$	0	0
$766$	18.5045	0.668596
$767$	−4.58258	−0.165467
$768$	0	0
$769$	6.49545	0.234232	0.117116	−	0.993118i	$-0.462635\pi$
$769$	6.49545	0.234232	0.117116	+	0.993118i	$0.462635\pi$
$770$	5.37386	0.193661
$771$	0	0
$772$	−2.92197	−0.105164
$773$	−6.16515	−0.221745	−0.110873	−	0.993835i	$-0.535365\pi$
$773$	−6.16515	−0.221745	−0.110873	+	0.993835i	$0.535365\pi$
$774$	0	0
$775$	14.3303	0.514760
$776$	−3.42652	−0.123005
$777$	0	0
$778$	−47.1652	−1.69095
$779$	−73.4955	−2.63325
$780$	0	0
$781$	−11.1652	−0.399521
$782$	75.8258	2.71152
$783$	0	0
$784$	−4.95644	−0.177016
$785$	−57.4955	−2.05210
$786$	0	0
$787$	38.5826	1.37532	0.687660	−	0.726033i	$-0.258638\pi$
$787$	38.5826	1.37532	0.687660	+	0.726033i	$0.258638\pi$
$788$	−6.24318	−0.222404
$789$	0	0
$790$	38.5045	1.36993
$791$	−9.16515	−0.325875
$792$	0	0
$793$	10.0000	0.355110
$794$	56.5735	2.00772
$795$	0	0
$796$	−11.5826	−0.410534
$797$	52.4955	1.85948	0.929742	−	0.368211i	$-0.120030\pi$
$797$	52.4955	1.85948	0.929742	+	0.368211i	$0.120030\pi$
$798$	0	0
$799$	10.7477	0.380227
$800$	24.1742	0.854689
$801$	0	0
$802$	57.1652	2.01857
$803$	7.00000	0.247025
$804$	0	0
$805$	−16.7477	−0.590280
$806$	−6.41742	−0.226044
$807$	0	0
$808$	−16.4174	−0.577563
$809$	−9.33030	−0.328036	−0.164018	−	0.986457i	$-0.552446\pi$
$809$	−9.33030	−0.328036	−0.164018	+	0.986457i	$0.552446\pi$
$810$	0	0
$811$	−2.25227	−0.0790880	−0.0395440	−	0.999218i	$-0.512591\pi$
$811$	−2.25227	−0.0790880	−0.0395440	+	0.999218i	$0.512591\pi$
$812$	9.86932	0.346345
$813$	0	0
$814$	−1.79129	−0.0627846
$815$	−25.7477	−0.901904
$816$	0	0
$817$	−10.4174	−0.364460
$818$	−14.9220	−0.521734
$819$	0	0
$820$	−40.4864	−1.41385
$821$	47.0000	1.64031	0.820156	−	0.572140i	$-0.193887\pi$
$821$	47.0000	1.64031	0.820156	+	0.572140i	$0.193887\pi$
$822$	0	0
$823$	−30.5826	−1.06604	−0.533021	−	0.846102i	$-0.678943\pi$
$823$	−30.5826	−1.06604	−0.533021	+	0.846102i	$0.678943\pi$
$824$	−1.65151	−0.0575332
$825$	0	0
$826$	8.20871	0.285618
$827$	−8.91288	−0.309931	−0.154966	−	0.987920i	$-0.549527\pi$
$827$	−8.91288	−0.309931	−0.154966	+	0.987920i	$0.549527\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	62.2432	2.16049
$831$	0	0
$832$	−0.912878	−0.0316484
$833$	−7.58258	−0.262721
$834$	0	0
$835$	−14.2432	−0.492906
$836$	−7.95644	−0.275179
$837$	0	0
$838$	4.62614	0.159807
$839$	−7.08712	−0.244675	−0.122337	−	0.992489i	$-0.539039\pi$
$839$	−7.08712	−0.244675	−0.122337	+	0.992489i	$0.539039\pi$
$840$	0	0
$841$	37.6697	1.29896
$842$	60.2958	2.07793
$843$	0	0
$844$	−15.9129	−0.547744
$845$	36.0000	1.23844
$846$	0	0
$847$	1.00000	0.0343604
$848$	−47.4955	−1.63100
$849$	0	0
$850$	54.3303	1.86351
$851$	5.58258	0.191368
$852$	0	0
$853$	17.1652	0.587724	0.293862	−	0.955848i	$-0.405059\pi$
$853$	17.1652	0.587724	0.293862	+	0.955848i	$0.405059\pi$
$854$	−17.9129	−0.612966
$855$	0	0
$856$	17.8348	0.609583
$857$	7.66970	0.261992	0.130996	−	0.991383i	$-0.458183\pi$
$857$	7.66970	0.261992	0.130996	+	0.991383i	$0.458183\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	−5.73864	−0.195686
$861$	0	0
$862$	31.7913	1.08282
$863$	−14.4174	−0.490775	−0.245387	−	0.969425i	$-0.578915\pi$
$863$	−14.4174	−0.490775	−0.245387	+	0.969425i	$0.578915\pi$
$864$	0	0
$865$	−21.4955	−0.730867
$866$	20.0000	0.679628
$867$	0	0
$868$	4.33030	0.146980
$869$	7.16515	0.243061
$870$	0	0
$871$	8.58258	0.290809
$872$	5.07803	0.171964
$873$	0	0
$874$	65.8258	2.22659
$875$	3.00000	0.101419
$876$	0	0
$877$	9.49545	0.320639	0.160319	−	0.987065i	$-0.448748\pi$
$877$	9.49545	0.320639	0.160319	+	0.987065i	$0.448748\pi$
$878$	−31.1996	−1.05294
$879$	0	0
$880$	14.8693	0.501245
$881$	37.6606	1.26882	0.634409	−	0.772998i	$-0.281244\pi$
$881$	37.6606	1.26882	0.634409	+	0.772998i	$0.281244\pi$
$882$	0	0
$883$	−55.7477	−1.87606	−0.938030	−	0.346554i	$-0.887352\pi$
$883$	−55.7477	−1.87606	−0.938030	+	0.346554i	$0.887352\pi$
$884$	−9.16515	−0.308257
$885$	0	0
$886$	−41.4955	−1.39407
$887$	−38.7477	−1.30102	−0.650511	−	0.759497i	$-0.725445\pi$
$887$	−38.7477	−1.30102	−0.650511	+	0.759497i	$0.725445\pi$
$888$	0	0
$889$	−11.5826	−0.388467
$890$	−49.2523	−1.65094
$891$	0	0
$892$	7.25227	0.242824
$893$	9.33030	0.312227
$894$	0	0
$895$	42.9909	1.43703
$896$	−10.4519	−0.349173
$897$	0	0
$898$	−32.8348	−1.09571
$899$	29.2523	0.975618
$900$	0	0
$901$	−72.6606	−2.42068
$902$	−20.0000	−0.665927
$903$	0	0
$904$	−12.9909	−0.432071
$905$	16.7477	0.556713
$906$	0	0
$907$	42.3303	1.40555	0.702777	−	0.711410i	$-0.251943\pi$
$907$	42.3303	1.40555	0.702777	+	0.711410i	$0.251943\pi$
$908$	26.5917	0.882475
$909$	0	0
$910$	5.37386	0.178142
$911$	3.49545	0.115810	0.0579048	−	0.998322i	$-0.481558\pi$
$911$	3.49545	0.115810	0.0579048	+	0.998322i	$0.481558\pi$
$912$	0	0
$913$	11.5826	0.383327
$914$	35.6697	1.17985
$915$	0	0
$916$	0.903787	0.0298620
$917$	16.0000	0.528367
$918$	0	0
$919$	−8.08712	−0.266770	−0.133385	−	0.991064i	$-0.542585\pi$
$919$	−8.08712	−0.266770	−0.133385	+	0.991064i	$0.542585\pi$
$920$	−23.7386	−0.782640
$921$	0	0
$922$	32.8348	1.08136
$923$	−11.1652	−0.367505
$924$	0	0
$925$	4.00000	0.131519
$926$	−15.3739	−0.505217
$927$	0	0
$928$	49.3466	1.61988
$929$	21.3303	0.699825	0.349912	−	0.936782i	$-0.386211\pi$
$929$	21.3303	0.699825	0.349912	+	0.936782i	$0.386211\pi$
$930$	0	0
$931$	−6.58258	−0.215735
$932$	16.9220	0.554298
$933$	0	0
$934$	−69.1125	−2.26143
$935$	22.7477	0.743930
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	−15.3739	−0.501974
$939$	0	0
$940$	5.13977	0.167641
$941$	−35.1652	−1.14635	−0.573176	−	0.819433i	$-0.694289\pi$
$941$	−35.1652	−1.14635	−0.573176	+	0.819433i	$0.694289\pi$
$942$	0	0
$943$	62.3303	2.02975
$944$	22.7133	0.739254
$945$	0	0
$946$	−2.83485	−0.0921689
$947$	−45.1652	−1.46767	−0.733835	−	0.679328i	$-0.762272\pi$
$947$	−45.1652	−1.46767	−0.733835	+	0.679328i	$0.762272\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	47.1652	1.53024
$951$	0	0
$952$	−10.7477	−0.348336
$953$	28.1652	0.912359	0.456179	−	0.889888i	$-0.349218\pi$
$953$	28.1652	0.912359	0.456179	+	0.889888i	$0.349218\pi$
$954$	0	0
$955$	34.7477	1.12441
$956$	−20.0436	−0.648255
$957$	0	0
$958$	−27.9129	−0.901824
$959$	11.5826	0.374021
$960$	0	0
$961$	−18.1652	−0.585973
$962$	−1.79129	−0.0577534
$963$	0	0
$964$	−12.2867	−0.395729
$965$	7.25227	0.233459
$966$	0	0
$967$	−13.1652	−0.423363	−0.211681	−	0.977339i	$-0.567894\pi$
$967$	−13.1652	−0.423363	−0.211681	+	0.977339i	$0.567894\pi$
$968$	1.41742	0.0455577
$969$	0	0
$970$	−12.9909	−0.417113
$971$	12.5826	0.403794	0.201897	−	0.979407i	$-0.435289\pi$
$971$	12.5826	0.403794	0.201897	+	0.979407i	$0.435289\pi$
$972$	0	0
$973$	11.1652	0.357938
$974$	−18.5045	−0.592924
$975$	0	0
$976$	−49.5644	−1.58652
$977$	−23.2523	−0.743906	−0.371953	−	0.928252i	$-0.621312\pi$
$977$	−23.2523	−0.743906	−0.371953	+	0.928252i	$0.621312\pi$
$978$	0	0
$979$	−9.16515	−0.292920
$980$	−3.62614	−0.115833
$981$	0	0
$982$	−41.0436	−1.30975
$983$	4.83485	0.154208	0.0771039	−	0.997023i	$-0.475433\pi$
$983$	4.83485	0.154208	0.0771039	+	0.997023i	$0.475433\pi$
$984$	0	0
$985$	15.4955	0.493726
$986$	110.904	3.53190
$987$	0	0
$988$	−7.95644	−0.253128
$989$	8.83485	0.280932
$990$	0	0
$991$	−20.2523	−0.643335	−0.321667	−	0.946853i	$-0.604243\pi$
$991$	−20.2523	−0.643335	−0.321667	+	0.946853i	$0.604243\pi$
$992$	21.6515	0.687436
$993$	0	0
$994$	20.0000	0.634361
$995$	28.7477	0.911364
$996$	0	0
$997$	10.6606	0.337625	0.168812	−	0.985648i	$-0.446007\pi$
$997$	10.6606	0.337625	0.168812	+	0.985648i	$0.446007\pi$
$998$	−74.7822	−2.36719
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.j.1.1		2
3.2	odd	2		231.2.a.b.1.2	✓	2
7.6	odd	2		4851.2.a.ba.1.1		2
11.10	odd	2		7623.2.a.bf.1.2		2
12.11	even	2		3696.2.a.bl.1.2		2
15.14	odd	2		5775.2.a.bn.1.1		2
21.20	even	2		1617.2.a.o.1.2		2
33.32	even	2		2541.2.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.b.1.2	✓	2	3.2	odd	2
693.2.a.j.1.1		2	1.1	even	1	trivial
1617.2.a.o.1.2		2	21.20	even	2
2541.2.a.z.1.1		2	33.32	even	2
3696.2.a.bl.1.2		2	12.11	even	2
4851.2.a.ba.1.1		2	7.6	odd	2
5775.2.a.bn.1.1		2	15.14	odd	2
7623.2.a.bf.1.2		2	11.10	odd	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{2} +1.20871 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.41742 q^{8} +O(q^{10})\)	\(q-1.79129 q^{2} +1.20871 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.41742 q^{8} +5.37386 q^{10} +1.00000 q^{11} +1.00000 q^{13} -1.79129 q^{14} -4.95644 q^{16} -7.58258 q^{17} -6.58258 q^{19} -3.62614 q^{20} -1.79129 q^{22} +5.58258 q^{23} +4.00000 q^{25} -1.79129 q^{26} +1.20871 q^{28} +8.16515 q^{29} +3.58258 q^{31} +6.04356 q^{32} +13.5826 q^{34} -3.00000 q^{35} +1.00000 q^{37} +11.7913 q^{38} -4.25227 q^{40} +11.1652 q^{41} +1.58258 q^{43} +1.20871 q^{44} -10.0000 q^{46} -1.41742 q^{47} +1.00000 q^{49} -7.16515 q^{50} +1.20871 q^{52} +9.58258 q^{53} -3.00000 q^{55} +1.41742 q^{56} -14.6261 q^{58} -4.58258 q^{59} +10.0000 q^{61} -6.41742 q^{62} -0.912878 q^{64} -3.00000 q^{65} +8.58258 q^{67} -9.16515 q^{68} +5.37386 q^{70} -11.1652 q^{71} +7.00000 q^{73} -1.79129 q^{74} -7.95644 q^{76} +1.00000 q^{77} +7.16515 q^{79} +14.8693 q^{80} -20.0000 q^{82} +11.5826 q^{83} +22.7477 q^{85} -2.83485 q^{86} +1.41742 q^{88} -9.16515 q^{89} +1.00000 q^{91} +6.74773 q^{92} +2.53901 q^{94} +19.7477 q^{95} -2.41742 q^{97} -1.79129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) 2 * q + q^2 + 7 * q^4 - 6 * q^5 + 2 * q^7 + 12 * q^8	\( 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8} - 3 q^{10} + 2 q^{11} + 2 q^{13} + q^{14} + 13 q^{16} - 6 q^{17} - 4 q^{19} - 21 q^{20} + q^{22} + 2 q^{23} + 8 q^{25} + q^{26} + 7 q^{28} - 2 q^{29} - 2 q^{31} + 35 q^{32} + 18 q^{34} - 6 q^{35} + 2 q^{37} + 19 q^{38} - 36 q^{40} + 4 q^{41} - 6 q^{43} + 7 q^{44} - 20 q^{46} - 12 q^{47} + 2 q^{49} + 4 q^{50} + 7 q^{52} + 10 q^{53} - 6 q^{55} + 12 q^{56} - 43 q^{58} + 20 q^{61} - 22 q^{62} + 44 q^{64} - 6 q^{65} + 8 q^{67} - 3 q^{70} - 4 q^{71} + 14 q^{73} + q^{74} + 7 q^{76} + 2 q^{77} - 4 q^{79} - 39 q^{80} - 40 q^{82} + 14 q^{83} + 18 q^{85} - 24 q^{86} + 12 q^{88} + 2 q^{91} - 14 q^{92} - 27 q^{94} + 12 q^{95} - 14 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 + 7 * q^4 - 6 * q^5 + 2 * q^7 + 12 * q^8 - 3 * q^10 + 2 * q^11 + 2 * q^13 + q^14 + 13 * q^16 - 6 * q^17 - 4 * q^19 - 21 * q^20 + q^22 + 2 * q^23 + 8 * q^25 + q^26 + 7 * q^28 - 2 * q^29 - 2 * q^31 + 35 * q^32 + 18 * q^34 - 6 * q^35 + 2 * q^37 + 19 * q^38 - 36 * q^40 + 4 * q^41 - 6 * q^43 + 7 * q^44 - 20 * q^46 - 12 * q^47 + 2 * q^49 + 4 * q^50 + 7 * q^52 + 10 * q^53 - 6 * q^55 + 12 * q^56 - 43 * q^58 + 20 * q^61 - 22 * q^62 + 44 * q^64 - 6 * q^65 + 8 * q^67 - 3 * q^70 - 4 * q^71 + 14 * q^73 + q^74 + 7 * q^76 + 2 * q^77 - 4 * q^79 - 39 * q^80 - 40 * q^82 + 14 * q^83 + 18 * q^85 - 24 * q^86 + 12 * q^88 + 2 * q^91 - 14 * q^92 - 27 * q^94 + 12 * q^95 - 14 * q^97 + q^98

Introduction

L-functions

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