LMFDB - Embedded newform 405.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51414 q^{2} +4.32088 q^{4} +1.00000 q^{5} -0.514137 q^{7} +5.83502 q^{8} +O(q^{10})$	\(q+2.51414 q^{2} +4.32088 q^{4} +1.00000 q^{5} -0.514137 q^{7} +5.83502 q^{8} +2.51414 q^{10} -3.32088 q^{11} -1.32088 q^{13} -1.29261 q^{14} +6.02827 q^{16} -3.32088 q^{17} -1.32088 q^{19} +4.32088 q^{20} -8.34916 q^{22} +4.12763 q^{23} +1.00000 q^{25} -3.32088 q^{26} -2.22153 q^{28} -1.38650 q^{29} +8.73566 q^{31} +3.48586 q^{32} -8.34916 q^{34} -0.514137 q^{35} +0.292611 q^{37} -3.32088 q^{38} +5.83502 q^{40} -11.3492 q^{41} +10.3492 q^{43} -14.3492 q^{44} +10.3774 q^{46} -4.86330 q^{47} -6.73566 q^{49} +2.51414 q^{50} -5.70739 q^{52} -5.02827 q^{53} -3.32088 q^{55} -3.00000 q^{56} -3.48586 q^{58} -5.02827 q^{59} +7.34916 q^{61} +21.9627 q^{62} -3.29261 q^{64} -1.32088 q^{65} +9.44852 q^{67} -14.3492 q^{68} -1.29261 q^{70} +8.99093 q^{71} +6.05655 q^{73} +0.735663 q^{74} -5.70739 q^{76} +1.70739 q^{77} -8.05655 q^{79} +6.02827 q^{80} -28.5333 q^{82} +1.54241 q^{83} -3.32088 q^{85} +26.0192 q^{86} -19.3774 q^{88} -3.00000 q^{89} +0.679116 q^{91} +17.8350 q^{92} -12.2270 q^{94} -1.32088 q^{95} -12.2553 q^{97} -16.9344 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8	$ 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 3 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} - 7 q^{29} + 8 q^{31} + 17 q^{32} - 4 q^{34} + 5 q^{35} + 6 q^{37} - 2 q^{38} + 3 q^{40} - 13 q^{41} + 10 q^{43} - 22 q^{44} - 3 q^{46} + 13 q^{47} - 2 q^{49} + q^{50} - 12 q^{52} - 2 q^{53} - 2 q^{55} - 9 q^{56} - 17 q^{58} - 2 q^{59} + q^{61} + 42 q^{62} - 15 q^{64} + 4 q^{65} + 11 q^{67} - 22 q^{68} - 9 q^{70} - 10 q^{71} - 8 q^{73} - 16 q^{74} - 12 q^{76} + 2 q^{79} + 5 q^{80} - 29 q^{82} - 15 q^{83} - 2 q^{85} + 28 q^{86} - 24 q^{88} - 9 q^{89} + 10 q^{91} + 39 q^{92} - 31 q^{94} + 4 q^{95} - 18 q^{97} - 40 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 + 5 * q^16 - 2 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 3 * q^23 + 3 * q^25 - 2 * q^26 + 5 * q^28 - 7 * q^29 + 8 * q^31 + 17 * q^32 - 4 * q^34 + 5 * q^35 + 6 * q^37 - 2 * q^38 + 3 * q^40 - 13 * q^41 + 10 * q^43 - 22 * q^44 - 3 * q^46 + 13 * q^47 - 2 * q^49 + q^50 - 12 * q^52 - 2 * q^53 - 2 * q^55 - 9 * q^56 - 17 * q^58 - 2 * q^59 + q^61 + 42 * q^62 - 15 * q^64 + 4 * q^65 + 11 * q^67 - 22 * q^68 - 9 * q^70 - 10 * q^71 - 8 * q^73 - 16 * q^74 - 12 * q^76 + 2 * q^79 + 5 * q^80 - 29 * q^82 - 15 * q^83 - 2 * q^85 + 28 * q^86 - 24 * q^88 - 9 * q^89 + 10 * q^91 + 39 * q^92 - 31 * q^94 + 4 * q^95 - 18 * q^97 - 40 * 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 $

$2$	2.51414	1.77776	0.888882	−	0.458137i	$-0.151483\pi$
$2$	2.51414	1.77776	0.888882	+	0.458137i	$0.151483\pi$
$3$	0	0
$3$	0	0
$4$	4.32088	2.16044
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.514137	−0.194325	−0.0971627	−	0.995269i	$-0.530977\pi$
$7$	−0.514137	−0.194325	−0.0971627	+	0.995269i	$0.530977\pi$
$8$	5.83502	2.06299
$9$	0	0
$10$	2.51414	0.795040
$11$	−3.32088	−1.00128	−0.500642	−	0.865654i	$-0.666903\pi$
$11$	−3.32088	−1.00128	−0.500642	+	0.865654i	$0.666903\pi$
$12$	0	0
$13$	−1.32088	−0.366347	−0.183174	−	0.983081i	$-0.558637\pi$
$13$	−1.32088	−0.366347	−0.183174	+	0.983081i	$0.558637\pi$
$14$	−1.29261	−0.345465
$15$	0	0
$16$	6.02827	1.50707
$17$	−3.32088	−0.805433	−0.402716	−	0.915325i	$-0.631934\pi$
$17$	−3.32088	−0.805433	−0.402716	+	0.915325i	$0.631934\pi$
$18$	0	0
$19$	−1.32088	−0.303032	−0.151516	−	0.988455i	$-0.548415\pi$
$19$	−1.32088	−0.303032	−0.151516	+	0.988455i	$0.548415\pi$
$20$	4.32088	0.966179
$21$	0	0
$22$	−8.34916	−1.78005
$23$	4.12763	0.860671	0.430335	−	0.902669i	$-0.358395\pi$
$23$	4.12763	0.860671	0.430335	+	0.902669i	$0.358395\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.32088	−0.651279
$27$	0	0
$28$	−2.22153	−0.419829
$29$	−1.38650	−0.257467	−0.128734	−	0.991679i	$-0.541091\pi$
$29$	−1.38650	−0.257467	−0.128734	+	0.991679i	$0.541091\pi$
$30$	0	0
$31$	8.73566	1.56897	0.784486	−	0.620147i	$-0.212927\pi$
$31$	8.73566	1.56897	0.784486	+	0.620147i	$0.212927\pi$
$32$	3.48586	0.616219
$33$	0	0
$34$	−8.34916	−1.43187
$35$	−0.514137	−0.0869050
$36$	0	0
$37$	0.292611	0.0481049	0.0240524	−	0.999711i	$-0.492343\pi$
$37$	0.292611	0.0481049	0.0240524	+	0.999711i	$0.492343\pi$
$38$	−3.32088	−0.538719
$39$	0	0
$40$	5.83502	0.922598
$41$	−11.3492	−1.77244	−0.886220	−	0.463264i	$-0.846678\pi$
$41$	−11.3492	−1.77244	−0.886220	+	0.463264i	$0.846678\pi$
$42$	0	0
$43$	10.3492	1.57823	0.789116	−	0.614244i	$-0.210539\pi$
$43$	10.3492	1.57823	0.789116	+	0.614244i	$0.210539\pi$
$44$	−14.3492	−2.16322
$45$	0	0
$46$	10.3774	1.53007
$47$	−4.86330	−0.709385	−0.354692	−	0.934983i	$-0.615414\pi$
$47$	−4.86330	−0.709385	−0.354692	+	0.934983i	$0.615414\pi$
$48$	0	0
$49$	−6.73566	−0.962238
$50$	2.51414	0.355553
$51$	0	0
$52$	−5.70739	−0.791472
$53$	−5.02827	−0.690687	−0.345343	−	0.938476i	$-0.612238\pi$
$53$	−5.02827	−0.690687	−0.345343	+	0.938476i	$0.612238\pi$
$54$	0	0
$55$	−3.32088	−0.447788
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−3.48586	−0.457716
$59$	−5.02827	−0.654625	−0.327313	−	0.944916i	$-0.606143\pi$
$59$	−5.02827	−0.654625	−0.327313	+	0.944916i	$0.606143\pi$
$60$	0	0
$61$	7.34916	0.940963	0.470482	−	0.882410i	$-0.344080\pi$
$61$	7.34916	0.940963	0.470482	+	0.882410i	$0.344080\pi$
$62$	21.9627	2.78926
$63$	0	0
$64$	−3.29261	−0.411576
$65$	−1.32088	−0.163836
$66$	0	0
$67$	9.44852	1.15432	0.577160	−	0.816631i	$-0.304161\pi$
$67$	9.44852	1.15432	0.577160	+	0.816631i	$0.304161\pi$
$68$	−14.3492	−1.74009
$69$	0	0
$70$	−1.29261	−0.154497
$71$	8.99093	1.06703	0.533513	−	0.845792i	$-0.320871\pi$
$71$	8.99093	1.06703	0.533513	+	0.845792i	$0.320871\pi$
$72$	0	0
$73$	6.05655	0.708865	0.354433	−	0.935082i	$-0.384674\pi$
$73$	6.05655	0.708865	0.354433	+	0.935082i	$0.384674\pi$
$74$	0.735663	0.0855191
$75$	0	0
$76$	−5.70739	−0.654682
$77$	1.70739	0.194575
$78$	0	0
$79$	−8.05655	−0.906432	−0.453216	−	0.891401i	$-0.649723\pi$
$79$	−8.05655	−0.906432	−0.453216	+	0.891401i	$0.649723\pi$
$80$	6.02827	0.673982
$81$	0	0
$82$	−28.5333	−3.15098
$83$	1.54241	0.169302	0.0846508	−	0.996411i	$-0.473023\pi$
$83$	1.54241	0.169302	0.0846508	+	0.996411i	$0.473023\pi$
$84$	0	0
$85$	−3.32088	−0.360200
$86$	26.0192	2.80572
$87$	0	0
$88$	−19.3774	−2.06564
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0.679116	0.0711906
$92$	17.8350	1.85943
$93$	0	0
$94$	−12.2270	−1.26112
$95$	−1.32088	−0.135520
$96$	0	0
$97$	−12.2553	−1.24433	−0.622167	−	0.782885i	$-0.713747\pi$
$97$	−12.2553	−1.24433	−0.622167	+	0.782885i	$0.713747\pi$
$98$	−16.9344	−1.71063
$99$	0	0
$100$	4.32088	0.432088
$101$	11.6700	1.16121	0.580606	−	0.814184i	$-0.302816\pi$
$101$	11.6700	1.16121	0.580606	+	0.814184i	$0.302816\pi$
$102$	0	0
$103$	0.292611	0.0288318	0.0144159	−	0.999896i	$-0.495411\pi$
$103$	0.292611	0.0288318	0.0144159	+	0.999896i	$0.495411\pi$
$104$	−7.70739	−0.755772
$105$	0	0
$106$	−12.6418	−1.22788
$107$	1.87237	0.181009	0.0905043	−	0.995896i	$-0.471152\pi$
$107$	1.87237	0.181009	0.0905043	+	0.995896i	$0.471152\pi$
$108$	0	0
$109$	5.54787	0.531390	0.265695	−	0.964057i	$-0.414399\pi$
$109$	5.54787	0.531390	0.265695	+	0.964057i	$0.414399\pi$
$110$	−8.34916	−0.796061
$111$	0	0
$112$	−3.09936	−0.292862
$113$	7.80128	0.733883	0.366942	−	0.930244i	$-0.380405\pi$
$113$	7.80128	0.733883	0.366942	+	0.930244i	$0.380405\pi$
$114$	0	0
$115$	4.12763	0.384904
$116$	−5.99093	−0.556244
$117$	0	0
$118$	−12.6418	−1.16377
$119$	1.70739	0.156516
$120$	0	0
$121$	0.0282739	0.00257035
$122$	18.4768	1.67281
$123$	0	0
$124$	37.7458	3.38967
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.8916	1.58762	0.793810	−	0.608166i	$-0.208094\pi$
$127$	17.8916	1.58762	0.793810	+	0.608166i	$0.208094\pi$
$128$	−15.2498	−1.34790
$129$	0	0
$130$	−3.32088	−0.291261
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	0.679116	0.0588868
$134$	23.7549	2.05211
$135$	0	0
$136$	−19.3774	−1.66160
$137$	5.67004	0.484424	0.242212	−	0.970223i	$-0.422127\pi$
$137$	5.67004	0.484424	0.242212	+	0.970223i	$0.422127\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−2.22153	−0.187753
$141$	0	0
$142$	22.6044	1.89692
$143$	4.38650	0.366818
$144$	0	0
$145$	−1.38650	−0.115143
$146$	15.2270	1.26019
$147$	0	0
$148$	1.26434	0.103928
$149$	17.6610	1.44684	0.723422	−	0.690407i	$-0.242568\pi$
$149$	17.6610	1.44684	0.723422	+	0.690407i	$0.242568\pi$
$150$	0	0
$151$	1.26434	0.102890	0.0514451	−	0.998676i	$-0.483617\pi$
$151$	1.26434	0.102890	0.0514451	+	0.998676i	$0.483617\pi$
$152$	−7.70739	−0.625152
$153$	0	0
$154$	4.29261	0.345908
$155$	8.73566	0.701665
$156$	0	0
$157$	−15.6700	−1.25061	−0.625303	−	0.780382i	$-0.715025\pi$
$157$	−15.6700	−1.25061	−0.625303	+	0.780382i	$0.715025\pi$
$158$	−20.2553	−1.61142
$159$	0	0
$160$	3.48586	0.275582
$161$	−2.12217	−0.167250
$162$	0	0
$163$	15.7074	1.23030	0.615149	−	0.788411i	$-0.289096\pi$
$163$	15.7074	1.23030	0.615149	+	0.788411i	$0.289096\pi$
$164$	−49.0384	−3.82926
$165$	0	0
$166$	3.87783	0.300978
$167$	6.16498	0.477060	0.238530	−	0.971135i	$-0.423334\pi$
$167$	6.16498	0.477060	0.238530	+	0.971135i	$0.423334\pi$
$168$	0	0
$169$	−11.2553	−0.865790
$170$	−8.34916	−0.640351
$171$	0	0
$172$	44.7175	3.40968
$173$	8.58522	0.652722	0.326361	−	0.945245i	$-0.394177\pi$
$173$	8.58522	0.652722	0.326361	+	0.945245i	$0.394177\pi$
$174$	0	0
$175$	−0.514137	−0.0388651
$176$	−20.0192	−1.50900
$177$	0	0
$178$	−7.54241	−0.565328
$179$	−1.06562	−0.0796482	−0.0398241	−	0.999207i	$-0.512680\pi$
$179$	−1.06562	−0.0796482	−0.0398241	+	0.999207i	$0.512680\pi$
$180$	0	0
$181$	−12.6700	−0.941757	−0.470878	−	0.882198i	$-0.656063\pi$
$181$	−12.6700	−0.941757	−0.470878	+	0.882198i	$0.656063\pi$
$182$	1.70739	0.126560
$183$	0	0
$184$	24.0848	1.77556
$185$	0.292611	0.0215132
$186$	0	0
$187$	11.0283	0.806467
$188$	−21.0137	−1.53258
$189$	0	0
$190$	−3.32088	−0.240922
$191$	−16.9344	−1.22533	−0.612664	−	0.790343i	$-0.709902\pi$
$191$	−16.9344	−1.22533	−0.612664	+	0.790343i	$0.709902\pi$
$192$	0	0
$193$	26.7175	1.92317	0.961585	−	0.274509i	$-0.0885153\pi$
$193$	26.7175	1.92317	0.961585	+	0.274509i	$0.0885153\pi$
$194$	−30.8114	−2.21213
$195$	0	0
$196$	−29.1040	−2.07886
$197$	−14.2553	−1.01565	−0.507823	−	0.861462i	$-0.669550\pi$
$197$	−14.2553	−1.01565	−0.507823	+	0.861462i	$0.669550\pi$
$198$	0	0
$199$	−24.6610	−1.74817	−0.874085	−	0.485773i	$-0.838538\pi$
$199$	−24.6610	−1.74817	−0.874085	+	0.485773i	$0.838538\pi$
$200$	5.83502	0.412598
$201$	0	0
$202$	29.3401	2.06436
$203$	0.712853	0.0500325
$204$	0	0
$205$	−11.3492	−0.792660
$206$	0.735663	0.0512561
$207$	0	0
$208$	−7.96265	−0.552111
$209$	4.38650	0.303421
$210$	0	0
$211$	−5.37743	−0.370198	−0.185099	−	0.982720i	$-0.559261\pi$
$211$	−5.37743	−0.370198	−0.185099	+	0.982720i	$0.559261\pi$
$212$	−21.7266	−1.49219
$213$	0	0
$214$	4.70739	0.321791
$215$	10.3492	0.705807
$216$	0	0
$217$	−4.49133	−0.304891
$218$	13.9481	0.944686
$219$	0	0
$220$	−14.3492	−0.967420
$221$	4.38650	0.295068
$222$	0	0
$223$	8.66458	0.580223	0.290112	−	0.956993i	$-0.406308\pi$
$223$	8.66458	0.580223	0.290112	+	0.956993i	$0.406308\pi$
$224$	−1.79221	−0.119747
$225$	0	0
$226$	19.6135	1.30467
$227$	3.32088	0.220415	0.110207	−	0.993909i	$-0.464848\pi$
$227$	3.32088	0.220415	0.110207	+	0.993909i	$0.464848\pi$
$228$	0	0
$229$	−25.3118	−1.67265	−0.836326	−	0.548233i	$-0.815301\pi$
$229$	−25.3118	−1.67265	−0.836326	+	0.548233i	$0.815301\pi$
$230$	10.3774	0.684268
$231$	0	0
$232$	−8.09029	−0.531153
$233$	27.6327	1.81028	0.905139	−	0.425116i	$-0.139767\pi$
$233$	27.6327	1.81028	0.905139	+	0.425116i	$0.139767\pi$
$234$	0	0
$235$	−4.86330	−0.317246
$236$	−21.7266	−1.41428
$237$	0	0
$238$	4.29261	0.278249
$239$	−4.19872	−0.271592	−0.135796	−	0.990737i	$-0.543359\pi$
$239$	−4.19872	−0.271592	−0.135796	+	0.990737i	$0.543359\pi$
$240$	0	0
$241$	3.60442	0.232181	0.116091	−	0.993239i	$-0.462964\pi$
$241$	3.60442	0.232181	0.116091	+	0.993239i	$0.462964\pi$
$242$	0.0710844	0.00456948
$243$	0	0
$244$	31.7549	2.03290
$245$	−6.73566	−0.430326
$246$	0	0
$247$	1.74474	0.111015
$248$	50.9728	3.23677
$249$	0	0
$250$	2.51414	0.159008
$251$	−6.87783	−0.434125	−0.217062	−	0.976158i	$-0.569648\pi$
$251$	−6.87783	−0.434125	−0.217062	+	0.976158i	$0.569648\pi$
$252$	0	0
$253$	−13.7074	−0.861776
$254$	44.9819	2.82241
$255$	0	0
$256$	−31.7549	−1.98468
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−0.150442	−0.00934801
$260$	−5.70739	−0.353957
$261$	0	0
$262$	−15.0848	−0.931943
$263$	6.23606	0.384532	0.192266	−	0.981343i	$-0.438416\pi$
$263$	6.23606	0.384532	0.192266	+	0.981343i	$0.438416\pi$
$264$	0	0
$265$	−5.02827	−0.308884
$266$	1.70739	0.104687
$267$	0	0
$268$	40.8259	2.49384
$269$	9.92345	0.605044	0.302522	−	0.953142i	$-0.402172\pi$
$269$	9.92345	0.605044	0.302522	+	0.953142i	$0.402172\pi$
$270$	0	0
$271$	6.60442	0.401190	0.200595	−	0.979674i	$-0.435712\pi$
$271$	6.60442	0.401190	0.200595	+	0.979674i	$0.435712\pi$
$272$	−20.0192	−1.21384
$273$	0	0
$274$	14.2553	0.861192
$275$	−3.32088	−0.200257
$276$	0	0
$277$	22.6610	1.36157	0.680783	−	0.732485i	$-0.261640\pi$
$277$	22.6610	1.36157	0.680783	+	0.732485i	$0.261640\pi$
$278$	20.1131	1.20630
$279$	0	0
$280$	−3.00000	−0.179284
$281$	−15.5479	−0.927508	−0.463754	−	0.885964i	$-0.653498\pi$
$281$	−15.5479	−0.927508	−0.463754	+	0.885964i	$0.653498\pi$
$282$	0	0
$283$	0.645378	0.0383637	0.0191819	−	0.999816i	$-0.493894\pi$
$283$	0.645378	0.0383637	0.0191819	+	0.999816i	$0.493894\pi$
$284$	38.8488	2.30525
$285$	0	0
$286$	11.0283	0.652116
$287$	5.83502	0.344430
$288$	0	0
$289$	−5.97173	−0.351278
$290$	−3.48586	−0.204697
$291$	0	0
$292$	26.1696	1.53146
$293$	−1.37743	−0.0804704	−0.0402352	−	0.999190i	$-0.512811\pi$
$293$	−1.37743	−0.0804704	−0.0402352	+	0.999190i	$0.512811\pi$
$294$	0	0
$295$	−5.02827	−0.292757
$296$	1.70739	0.0992400
$297$	0	0
$298$	44.4021	2.57214
$299$	−5.45213	−0.315305
$300$	0	0
$301$	−5.32088	−0.306691
$302$	3.17872	0.182915
$303$	0	0
$304$	−7.96265	−0.456689
$305$	7.34916	0.420812
$306$	0	0
$307$	−7.98546	−0.455754	−0.227877	−	0.973690i	$-0.573178\pi$
$307$	−7.98546	−0.455754	−0.227877	+	0.973690i	$0.573178\pi$
$308$	7.37743	0.420368
$309$	0	0
$310$	21.9627	1.24739
$311$	9.63270	0.546220	0.273110	−	0.961983i	$-0.411948\pi$
$311$	9.63270	0.546220	0.273110	+	0.961983i	$0.411948\pi$
$312$	0	0
$313$	−24.5369	−1.38691	−0.693455	−	0.720500i	$-0.743912\pi$
$313$	−24.5369	−1.38691	−0.693455	+	0.720500i	$0.743912\pi$
$314$	−39.3966	−2.22328
$315$	0	0
$316$	−34.8114	−1.95829
$317$	−20.3492	−1.14292	−0.571461	−	0.820629i	$-0.693623\pi$
$317$	−20.3492	−1.14292	−0.571461	+	0.820629i	$0.693623\pi$
$318$	0	0
$319$	4.60442	0.257798
$320$	−3.29261	−0.184063
$321$	0	0
$322$	−5.33542	−0.297331
$323$	4.38650	0.244072
$324$	0	0
$325$	−1.32088	−0.0732695
$326$	39.4905	2.18718
$327$	0	0
$328$	−66.2226	−3.65653
$329$	2.50040	0.137851
$330$	0	0
$331$	16.4431	0.903792	0.451896	−	0.892071i	$-0.350748\pi$
$331$	16.4431	0.903792	0.451896	+	0.892071i	$0.350748\pi$
$332$	6.66458	0.365766
$333$	0	0
$334$	15.4996	0.848100
$335$	9.44852	0.516228
$336$	0	0
$337$	4.89703	0.266758	0.133379	−	0.991065i	$-0.457417\pi$
$337$	4.89703	0.266758	0.133379	+	0.991065i	$0.457417\pi$
$338$	−28.2973	−1.53917
$339$	0	0
$340$	−14.3492	−0.778192
$341$	−29.0101	−1.57099
$342$	0	0
$343$	7.06201	0.381313
$344$	60.3876	3.25588
$345$	0	0
$346$	21.5844	1.16039
$347$	22.2745	1.19576	0.597878	−	0.801587i	$-0.296011\pi$
$347$	22.2745	1.19576	0.597878	+	0.801587i	$0.296011\pi$
$348$	0	0
$349$	−2.94345	−0.157559	−0.0787797	−	0.996892i	$-0.525102\pi$
$349$	−2.94345	−0.157559	−0.0787797	+	0.996892i	$0.525102\pi$
$350$	−1.29261	−0.0690929
$351$	0	0
$352$	−11.5761	−0.617011
$353$	18.8296	1.00220	0.501098	−	0.865390i	$-0.332930\pi$
$353$	18.8296	1.00220	0.501098	+	0.865390i	$0.332930\pi$
$354$	0	0
$355$	8.99093	0.477189
$356$	−12.9627	−0.687019
$357$	0	0
$358$	−2.67912	−0.141596
$359$	−31.8770	−1.68241	−0.841203	−	0.540720i	$-0.818152\pi$
$359$	−31.8770	−1.68241	−0.841203	+	0.540720i	$0.818152\pi$
$360$	0	0
$361$	−17.2553	−0.908172
$362$	−31.8542	−1.67422
$363$	0	0
$364$	2.93438	0.153803
$365$	6.05655	0.317014
$366$	0	0
$367$	−18.3492	−0.957818	−0.478909	−	0.877864i	$-0.658968\pi$
$367$	−18.3492	−0.957818	−0.478909	+	0.877864i	$0.658968\pi$
$368$	24.8825	1.29709
$369$	0	0
$370$	0.735663	0.0382453
$371$	2.58522	0.134218
$372$	0	0
$373$	−2.19872	−0.113845	−0.0569226	−	0.998379i	$-0.518129\pi$
$373$	−2.19872	−0.113845	−0.0569226	+	0.998379i	$0.518129\pi$
$374$	27.7266	1.43371
$375$	0	0
$376$	−28.3774	−1.46345
$377$	1.83141	0.0943226
$378$	0	0
$379$	15.4713	0.794709	0.397354	−	0.917665i	$-0.369928\pi$
$379$	15.4713	0.794709	0.397354	+	0.917665i	$0.369928\pi$
$380$	−5.70739	−0.292783
$381$	0	0
$382$	−42.5753	−2.17834
$383$	7.70739	0.393829	0.196915	−	0.980421i	$-0.436908\pi$
$383$	7.70739	0.393829	0.196915	+	0.980421i	$0.436908\pi$
$384$	0	0
$385$	1.70739	0.0870166
$386$	67.1715	3.41894
$387$	0	0
$388$	−52.9536	−2.68831
$389$	−24.6327	−1.24893	−0.624464	−	0.781054i	$-0.714682\pi$
$389$	−24.6327	−1.24893	−0.624464	+	0.781054i	$0.714682\pi$
$390$	0	0
$391$	−13.7074	−0.693212
$392$	−39.3027	−1.98509
$393$	0	0
$394$	−35.8397	−1.80558
$395$	−8.05655	−0.405369
$396$	0	0
$397$	−6.77301	−0.339928	−0.169964	−	0.985450i	$-0.554365\pi$
$397$	−6.77301	−0.339928	−0.169964	+	0.985450i	$0.554365\pi$
$398$	−62.0011	−3.10783
$399$	0	0
$400$	6.02827	0.301414
$401$	−18.4996	−0.923826	−0.461913	−	0.886925i	$-0.652837\pi$
$401$	−18.4996	−0.923826	−0.461913	+	0.886925i	$0.652837\pi$
$402$	0	0
$403$	−11.5388	−0.574789
$404$	50.4249	2.50873
$405$	0	0
$406$	1.79221	0.0889459
$407$	−0.971726	−0.0481667
$408$	0	0
$409$	−13.4148	−0.663318	−0.331659	−	0.943399i	$-0.607608\pi$
$409$	−13.4148	−0.663318	−0.331659	+	0.943399i	$0.607608\pi$
$410$	−28.5333	−1.40916
$411$	0	0
$412$	1.26434	0.0622894
$413$	2.58522	0.127210
$414$	0	0
$415$	1.54241	0.0757140
$416$	−4.60442	−0.225750
$417$	0	0
$418$	11.0283	0.539411
$419$	−33.1150	−1.61777	−0.808886	−	0.587966i	$-0.799929\pi$
$419$	−33.1150	−1.61777	−0.808886	+	0.587966i	$0.799929\pi$
$420$	0	0
$421$	−14.6983	−0.716352	−0.358176	−	0.933654i	$-0.616601\pi$
$421$	−14.6983	−0.716352	−0.358176	+	0.933654i	$0.616601\pi$
$422$	−13.5196	−0.658124
$423$	0	0
$424$	−29.3401	−1.42488
$425$	−3.32088	−0.161087
$426$	0	0
$427$	−3.77847	−0.182853
$428$	8.09029	0.391059
$429$	0	0
$430$	26.0192	1.25476
$431$	−32.7549	−1.57775	−0.788873	−	0.614556i	$-0.789335\pi$
$431$	−32.7549	−1.57775	−0.788873	+	0.614556i	$0.789335\pi$
$432$	0	0
$433$	−11.8314	−0.568581	−0.284291	−	0.958738i	$-0.591758\pi$
$433$	−11.8314	−0.568581	−0.284291	+	0.958738i	$0.591758\pi$
$434$	−11.2918	−0.542024
$435$	0	0
$436$	23.9717	1.14804
$437$	−5.45213	−0.260811
$438$	0	0
$439$	8.31181	0.396701	0.198351	−	0.980131i	$-0.436442\pi$
$439$	8.31181	0.396701	0.198351	+	0.980131i	$0.436442\pi$
$440$	−19.3774	−0.923783
$441$	0	0
$442$	11.0283	0.524561
$443$	−29.1751	−1.38615	−0.693076	−	0.720865i	$-0.743745\pi$
$443$	−29.1751	−1.38615	−0.693076	+	0.720865i	$0.743745\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	21.7839	1.03150
$447$	0	0
$448$	1.69285	0.0799798
$449$	18.9717	0.895331	0.447666	−	0.894201i	$-0.352256\pi$
$449$	18.9717	0.895331	0.447666	+	0.894201i	$0.352256\pi$
$450$	0	0
$451$	37.6892	1.77472
$452$	33.7084	1.58551
$453$	0	0
$454$	8.34916	0.391845
$455$	0.679116	0.0318374
$456$	0	0
$457$	−23.2353	−1.08690	−0.543450	−	0.839442i	$-0.682882\pi$
$457$	−23.2353	−1.08690	−0.543450	+	0.839442i	$0.682882\pi$
$458$	−63.6374	−2.97358
$459$	0	0
$460$	17.8350	0.831562
$461$	4.42571	0.206126	0.103063	−	0.994675i	$-0.467136\pi$
$461$	4.42571	0.206126	0.103063	+	0.994675i	$0.467136\pi$
$462$	0	0
$463$	−19.5087	−0.906645	−0.453322	−	0.891347i	$-0.649761\pi$
$463$	−19.5087	−0.906645	−0.453322	+	0.891347i	$0.649761\pi$
$464$	−8.35823	−0.388021
$465$	0	0
$466$	69.4724	3.21825
$467$	−24.5935	−1.13805	−0.569026	−	0.822320i	$-0.692679\pi$
$467$	−24.5935	−1.13805	−0.569026	+	0.822320i	$0.692679\pi$
$468$	0	0
$469$	−4.85783	−0.224314
$470$	−12.2270	−0.563989
$471$	0	0
$472$	−29.3401	−1.35049
$473$	−34.3684	−1.58026
$474$	0	0
$475$	−1.32088	−0.0606063
$476$	7.37743	0.338144
$477$	0	0
$478$	−10.5561	−0.482827
$479$	32.7549	1.49661	0.748304	−	0.663356i	$-0.230868\pi$
$479$	32.7549	1.49661	0.748304	+	0.663356i	$0.230868\pi$
$480$	0	0
$481$	−0.386505	−0.0176231
$482$	9.06201	0.412763
$483$	0	0
$484$	0.122168	0.00555309
$485$	−12.2553	−0.556483
$486$	0	0
$487$	−6.03735	−0.273578	−0.136789	−	0.990600i	$-0.543678\pi$
$487$	−6.03735	−0.273578	−0.136789	+	0.990600i	$0.543678\pi$
$488$	42.8825	1.94120
$489$	0	0
$490$	−16.9344	−0.765017
$491$	14.4431	0.651806	0.325903	−	0.945403i	$-0.394332\pi$
$491$	14.4431	0.651806	0.325903	+	0.945403i	$0.394332\pi$
$492$	0	0
$493$	4.60442	0.207373
$494$	4.38650	0.197358
$495$	0	0
$496$	52.6610	2.36455
$497$	−4.62257	−0.207351
$498$	0	0
$499$	20.9717	0.938823	0.469412	−	0.882979i	$-0.344466\pi$
$499$	20.9717	0.938823	0.469412	+	0.882979i	$0.344466\pi$
$500$	4.32088	0.193236
$501$	0	0
$502$	−17.2918	−0.771771
$503$	−5.31728	−0.237086	−0.118543	−	0.992949i	$-0.537822\pi$
$503$	−5.31728	−0.237086	−0.118543	+	0.992949i	$0.537822\pi$
$504$	0	0
$505$	11.6700	0.519310
$506$	−34.4623	−1.53203
$507$	0	0
$508$	77.3074	3.42996
$509$	18.2270	0.807897	0.403949	−	0.914782i	$-0.367637\pi$
$509$	18.2270	0.807897	0.403949	+	0.914782i	$0.367637\pi$
$510$	0	0
$511$	−3.11389	−0.137751
$512$	−49.3365	−2.18038
$513$	0	0
$514$	−45.2545	−1.99609
$515$	0.292611	0.0128940
$516$	0	0
$517$	16.1504	0.710296
$518$	−0.378232	−0.0166185
$519$	0	0
$520$	−7.70739	−0.337991
$521$	40.1232	1.75783	0.878915	−	0.476978i	$-0.158268\pi$
$521$	40.1232	1.75783	0.878915	+	0.476978i	$0.158268\pi$
$522$	0	0
$523$	18.9873	0.830257	0.415129	−	0.909763i	$-0.363737\pi$
$523$	18.9873	0.830257	0.415129	+	0.909763i	$0.363737\pi$
$524$	−25.9253	−1.13255
$525$	0	0
$526$	15.6783	0.683607
$527$	−29.0101	−1.26370
$528$	0	0
$529$	−5.96265	−0.259246
$530$	−12.6418	−0.549123
$531$	0	0
$532$	2.93438	0.127221
$533$	14.9909	0.649329
$534$	0	0
$535$	1.87237	0.0809495
$536$	55.1323	2.38135
$537$	0	0
$538$	24.9489	1.07562
$539$	22.3684	0.963473
$540$	0	0
$541$	16.5279	0.710589	0.355294	−	0.934754i	$-0.384381\pi$
$541$	16.5279	0.710589	0.355294	+	0.934754i	$0.384381\pi$
$542$	16.6044	0.713221
$543$	0	0
$544$	−11.5761	−0.496323
$545$	5.54787	0.237645
$546$	0	0
$547$	17.6737	0.755671	0.377835	−	0.925873i	$-0.376669\pi$
$547$	17.6737	0.755671	0.377835	+	0.925873i	$0.376669\pi$
$548$	24.4996	1.04657
$549$	0	0
$550$	−8.34916	−0.356009
$551$	1.83141	0.0780208
$552$	0	0
$553$	4.14217	0.176143
$554$	56.9728	2.42054
$555$	0	0
$556$	34.5671	1.46597
$557$	17.3401	0.734723	0.367362	−	0.930078i	$-0.380261\pi$
$557$	17.3401	0.734723	0.367362	+	0.930078i	$0.380261\pi$
$558$	0	0
$559$	−13.6700	−0.578181
$560$	−3.09936	−0.130972
$561$	0	0
$562$	−39.0895	−1.64889
$563$	12.9945	0.547654	0.273827	−	0.961779i	$-0.411710\pi$
$563$	12.9945	0.547654	0.273827	+	0.961779i	$0.411710\pi$
$564$	0	0
$565$	7.80128	0.328202
$566$	1.62257	0.0682016
$567$	0	0
$568$	52.4623	2.20127
$569$	−16.6802	−0.699269	−0.349635	−	0.936886i	$-0.613694\pi$
$569$	−16.6802	−0.699269	−0.349635	+	0.936886i	$0.613694\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	18.9536	0.792489
$573$	0	0
$574$	14.6700	0.612316
$575$	4.12763	0.172134
$576$	0	0
$577$	−23.5953	−0.982287	−0.491144	−	0.871079i	$-0.663421\pi$
$577$	−23.5953	−0.982287	−0.491144	+	0.871079i	$0.663421\pi$
$578$	−15.0137	−0.624489
$579$	0	0
$580$	−5.99093	−0.248760
$581$	−0.793010	−0.0328996
$582$	0	0
$583$	16.6983	0.691574
$584$	35.3401	1.46238
$585$	0	0
$586$	−3.46305	−0.143057
$587$	28.1276	1.16095	0.580476	−	0.814277i	$-0.302867\pi$
$587$	28.1276	1.16095	0.580476	+	0.814277i	$0.302867\pi$
$588$	0	0
$589$	−11.5388	−0.475448
$590$	−12.6418	−0.520453
$591$	0	0
$592$	1.76394	0.0724974
$593$	−9.17872	−0.376925	−0.188462	−	0.982080i	$-0.560350\pi$
$593$	−9.17872	−0.376925	−0.188462	+	0.982080i	$0.560350\pi$
$594$	0	0
$595$	1.70739	0.0699961
$596$	76.3110	3.12582
$597$	0	0
$598$	−13.7074	−0.560537
$599$	−31.4713	−1.28588	−0.642942	−	0.765915i	$-0.722286\pi$
$599$	−31.4713	−1.28588	−0.642942	+	0.765915i	$0.722286\pi$
$600$	0	0
$601$	−29.2654	−1.19376	−0.596880	−	0.802330i	$-0.703593\pi$
$601$	−29.2654	−1.19376	−0.596880	+	0.802330i	$0.703593\pi$
$602$	−13.3774	−0.545223
$603$	0	0
$604$	5.46305	0.222288
$605$	0.0282739	0.00114950
$606$	0	0
$607$	−44.2034	−1.79416	−0.897080	−	0.441868i	$-0.854316\pi$
$607$	−44.2034	−1.79416	−0.897080	+	0.441868i	$0.854316\pi$
$608$	−4.60442	−0.186734
$609$	0	0
$610$	18.4768	0.748103
$611$	6.42385	0.259881
$612$	0	0
$613$	−35.1715	−1.42056	−0.710282	−	0.703918i	$-0.751432\pi$
$613$	−35.1715	−1.42056	−0.710282	+	0.703918i	$0.751432\pi$
$614$	−20.0765	−0.810224
$615$	0	0
$616$	9.96265	0.401407
$617$	−7.42571	−0.298948	−0.149474	−	0.988766i	$-0.547758\pi$
$617$	−7.42571	−0.298948	−0.149474	+	0.988766i	$0.547758\pi$
$618$	0	0
$619$	8.54787	0.343568	0.171784	−	0.985135i	$-0.445047\pi$
$619$	8.54787	0.343568	0.171784	+	0.985135i	$0.445047\pi$
$620$	37.7458	1.51591
$621$	0	0
$622$	24.2179	0.971050
$623$	1.54241	0.0617954
$624$	0	0
$625$	1.00000	0.0400000
$626$	−61.6892	−2.46560
$627$	0	0
$628$	−67.7084	−2.70186
$629$	−0.971726	−0.0387453
$630$	0	0
$631$	−2.36836	−0.0942829	−0.0471415	−	0.998888i	$-0.515011\pi$
$631$	−2.36836	−0.0942829	−0.0471415	+	0.998888i	$0.515011\pi$
$632$	−47.0101	−1.86996
$633$	0	0
$634$	−51.1606	−2.03185
$635$	17.8916	0.710005
$636$	0	0
$637$	8.89703	0.352513
$638$	11.5761	0.458304
$639$	0	0
$640$	−15.2498	−0.602801
$641$	−0.133096	−0.00525698	−0.00262849	−	0.999997i	$-0.500837\pi$
$641$	−0.133096	−0.00525698	−0.00262849	+	0.999997i	$0.500837\pi$
$642$	0	0
$643$	−22.6464	−0.893088	−0.446544	−	0.894762i	$-0.647345\pi$
$643$	−22.6464	−0.893088	−0.446544	+	0.894762i	$0.647345\pi$
$644$	−9.16964	−0.361335
$645$	0	0
$646$	11.0283	0.433902
$647$	46.3912	1.82383	0.911913	−	0.410385i	$-0.134606\pi$
$647$	46.3912	1.82383	0.911913	+	0.410385i	$0.134606\pi$
$648$	0	0
$649$	16.6983	0.655466
$650$	−3.32088	−0.130256
$651$	0	0
$652$	67.8698	2.65799
$653$	36.4057	1.42467	0.712333	−	0.701842i	$-0.247639\pi$
$653$	36.4057	1.42467	0.712333	+	0.701842i	$0.247639\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	−68.4158	−2.67119
$657$	0	0
$658$	6.28635	0.245067
$659$	19.1414	0.745642	0.372821	−	0.927903i	$-0.378391\pi$
$659$	19.1414	0.745642	0.372821	+	0.927903i	$0.378391\pi$
$660$	0	0
$661$	39.9072	1.55221	0.776104	−	0.630605i	$-0.217193\pi$
$661$	39.9072	1.55221	0.776104	+	0.630605i	$0.217193\pi$
$662$	41.3401	1.60673
$663$	0	0
$664$	9.00000	0.349268
$665$	0.679116	0.0263350
$666$	0	0
$667$	−5.72298	−0.221595
$668$	26.6382	1.03066
$669$	0	0
$670$	23.7549	0.917730
$671$	−24.4057	−0.942172
$672$	0	0
$673$	23.6508	0.911673	0.455836	−	0.890064i	$-0.349340\pi$
$673$	23.6508	0.911673	0.455836	+	0.890064i	$0.349340\pi$
$674$	12.3118	0.474233
$675$	0	0
$676$	−48.6327	−1.87049
$677$	−14.8031	−0.568931	−0.284465	−	0.958686i	$-0.591816\pi$
$677$	−14.8031	−0.568931	−0.284465	+	0.958686i	$0.591816\pi$
$678$	0	0
$679$	6.30088	0.241806
$680$	−19.3774	−0.743091
$681$	0	0
$682$	−72.9354	−2.79284
$683$	−4.95252	−0.189503	−0.0947515	−	0.995501i	$-0.530206\pi$
$683$	−4.95252	−0.189503	−0.0947515	+	0.995501i	$0.530206\pi$
$684$	0	0
$685$	5.67004	0.216641
$686$	17.7549	0.677884
$687$	0	0
$688$	62.3876	2.37850
$689$	6.64177	0.253031
$690$	0	0
$691$	−19.2088	−0.730739	−0.365369	−	0.930863i	$-0.619057\pi$
$691$	−19.2088	−0.730739	−0.365369	+	0.930863i	$0.619057\pi$
$692$	37.0957	1.41017
$693$	0	0
$694$	56.0011	2.12577
$695$	8.00000	0.303457
$696$	0	0
$697$	37.6892	1.42758
$698$	−7.40024	−0.280103
$699$	0	0
$700$	−2.22153	−0.0839658
$701$	−29.3492	−1.10850	−0.554251	−	0.832349i	$-0.686995\pi$
$701$	−29.3492	−1.10850	−0.554251	+	0.832349i	$0.686995\pi$
$702$	0	0
$703$	−0.386505	−0.0145773
$704$	10.9344	0.412105
$705$	0	0
$706$	47.3401	1.78167
$707$	−6.00000	−0.225653
$708$	0	0
$709$	38.7266	1.45441	0.727204	−	0.686422i	$-0.240820\pi$
$709$	38.7266	1.45441	0.727204	+	0.686422i	$0.240820\pi$
$710$	22.6044	0.848329
$711$	0	0
$712$	−17.5051	−0.656030
$713$	36.0576	1.35037
$714$	0	0
$715$	4.38650	0.164046
$716$	−4.60442	−0.172075
$717$	0	0
$718$	−80.1432	−2.99092
$719$	−15.0848	−0.562569	−0.281284	−	0.959624i	$-0.590760\pi$
$719$	−15.0848	−0.562569	−0.281284	+	0.959624i	$0.590760\pi$
$720$	0	0
$721$	−0.150442	−0.00560275
$722$	−43.3821	−1.61451
$723$	0	0
$724$	−54.7458	−2.03461
$725$	−1.38650	−0.0514935
$726$	0	0
$727$	12.3455	0.457871	0.228936	−	0.973442i	$-0.426475\pi$
$727$	12.3455	0.457871	0.228936	+	0.973442i	$0.426475\pi$
$728$	3.96265	0.146866
$729$	0	0
$730$	15.2270	0.563576
$731$	−34.3684	−1.27116
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−46.1323	−1.70277
$735$	0	0
$736$	14.3884	0.530362
$737$	−31.3774	−1.15580
$738$	0	0
$739$	29.7266	1.09351	0.546755	−	0.837293i	$-0.315863\pi$
$739$	29.7266	1.09351	0.546755	+	0.837293i	$0.315863\pi$
$740$	1.26434	0.0464779
$741$	0	0
$742$	6.49960	0.238608
$743$	48.3648	1.77433	0.887165	−	0.461452i	$-0.152671\pi$
$743$	48.3648	1.77433	0.887165	+	0.461452i	$0.152671\pi$
$744$	0	0
$745$	17.6610	0.647048
$746$	−5.52787	−0.202390
$747$	0	0
$748$	47.6519	1.74233
$749$	−0.962653	−0.0351746
$750$	0	0
$751$	−31.8205	−1.16115	−0.580573	−	0.814208i	$-0.697171\pi$
$751$	−31.8205	−1.16115	−0.580573	+	0.814208i	$0.697171\pi$
$752$	−29.3173	−1.06909
$753$	0	0
$754$	4.60442	0.167683
$755$	1.26434	0.0460139
$756$	0	0
$757$	4.94531	0.179740	0.0898701	−	0.995953i	$-0.471355\pi$
$757$	4.94531	0.179740	0.0898701	+	0.995953i	$0.471355\pi$
$758$	38.8970	1.41280
$759$	0	0
$760$	−7.70739	−0.279576
$761$	35.4249	1.28415	0.642076	−	0.766641i	$-0.278073\pi$
$761$	35.4249	1.28415	0.642076	+	0.766641i	$0.278073\pi$
$762$	0	0
$763$	−2.85237	−0.103263
$764$	−73.1715	−2.64725
$765$	0	0
$766$	19.3774	0.700135
$767$	6.64177	0.239820
$768$	0	0
$769$	49.4249	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$769$	49.4249	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$770$	4.29261	0.154695
$771$	0	0
$772$	115.443	4.15490
$773$	12.6599	0.455345	0.227673	−	0.973738i	$-0.426888\pi$
$773$	12.6599	0.455345	0.227673	+	0.973738i	$0.426888\pi$
$774$	0	0
$775$	8.73566	0.313794
$776$	−71.5097	−2.56705
$777$	0	0
$778$	−61.9300	−2.22030
$779$	14.9909	0.537106
$780$	0	0
$781$	−29.8578	−1.06840
$782$	−34.4623	−1.23237
$783$	0	0
$784$	−40.6044	−1.45016
$785$	−15.6700	−0.559288
$786$	0	0
$787$	30.9344	1.10269	0.551346	−	0.834277i	$-0.314115\pi$
$787$	30.9344	1.10269	0.551346	+	0.834277i	$0.314115\pi$
$788$	−61.5953	−2.19424
$789$	0	0
$790$	−20.2553	−0.720650
$791$	−4.01093	−0.142612
$792$	0	0
$793$	−9.70739	−0.344720
$794$	−17.0283	−0.604311
$795$	0	0
$796$	−106.557	−3.77682
$797$	−30.5935	−1.08368	−0.541839	−	0.840483i	$-0.682272\pi$
$797$	−30.5935	−1.08368	−0.541839	+	0.840483i	$0.682272\pi$
$798$	0	0
$799$	16.1504	0.571362
$800$	3.48586	0.123244
$801$	0	0
$802$	−46.5105	−1.64234
$803$	−20.1131	−0.709776
$804$	0	0
$805$	−2.12217	−0.0747966
$806$	−29.0101	−1.02184
$807$	0	0
$808$	68.0950	2.39557
$809$	−2.89703	−0.101854	−0.0509271	−	0.998702i	$-0.516218\pi$
$809$	−2.89703	−0.101854	−0.0509271	+	0.998702i	$0.516218\pi$
$810$	0	0
$811$	−14.8861	−0.522722	−0.261361	−	0.965241i	$-0.584171\pi$
$811$	−14.8861	−0.522722	−0.261361	+	0.965241i	$0.584171\pi$
$812$	3.08016	0.108092
$813$	0	0
$814$	−2.44305	−0.0856289
$815$	15.7074	0.550206
$816$	0	0
$817$	−13.6700	−0.478254
$818$	−33.7266	−1.17922
$819$	0	0
$820$	−49.0384	−1.71250
$821$	−8.95173	−0.312417	−0.156209	−	0.987724i	$-0.549927\pi$
$821$	−8.95173	−0.312417	−0.156209	+	0.987724i	$0.549927\pi$
$822$	0	0
$823$	2.99454	0.104383	0.0521915	−	0.998637i	$-0.483379\pi$
$823$	2.99454	0.104383	0.0521915	+	0.998637i	$0.483379\pi$
$824$	1.70739	0.0594797
$825$	0	0
$826$	6.49960	0.226150
$827$	31.9663	1.11158	0.555788	−	0.831324i	$-0.312417\pi$
$827$	31.9663	1.11158	0.555788	+	0.831324i	$0.312417\pi$
$828$	0	0
$829$	22.7458	0.789994	0.394997	−	0.918682i	$-0.370746\pi$
$829$	22.7458	0.789994	0.394997	+	0.918682i	$0.370746\pi$
$830$	3.87783	0.134602
$831$	0	0
$832$	4.34916	0.150780
$833$	22.3684	0.775018
$834$	0	0
$835$	6.16498	0.213348
$836$	18.9536	0.655523
$837$	0	0
$838$	−83.2555	−2.87601
$839$	−23.2643	−0.803174	−0.401587	−	0.915821i	$-0.631541\pi$
$839$	−23.2643	−0.803174	−0.401587	+	0.915821i	$0.631541\pi$
$840$	0	0
$841$	−27.0776	−0.933710
$842$	−36.9536	−1.27350
$843$	0	0
$844$	−23.2353	−0.799791
$845$	−11.2553	−0.387193
$846$	0	0
$847$	−0.0145366	−0.000499485 0
$848$	−30.3118	−1.04091
$849$	0	0
$850$	−8.34916	−0.286374
$851$	1.20779	0.0414025
$852$	0	0
$853$	10.9909	0.376322	0.188161	−	0.982138i	$-0.439747\pi$
$853$	10.9909	0.376322	0.188161	+	0.982138i	$0.439747\pi$
$854$	−9.49960	−0.325070
$855$	0	0
$856$	10.9253	0.373419
$857$	−16.1504	−0.551689	−0.275844	−	0.961202i	$-0.588957\pi$
$857$	−16.1504	−0.551689	−0.275844	+	0.961202i	$0.588957\pi$
$858$	0	0
$859$	28.5188	0.973049	0.486524	−	0.873667i	$-0.338264\pi$
$859$	28.5188	0.973049	0.486524	+	0.873667i	$0.338264\pi$
$860$	44.7175	1.52485
$861$	0	0
$862$	−82.3502	−2.80486
$863$	12.2890	0.418322	0.209161	−	0.977881i	$-0.432927\pi$
$863$	12.2890	0.418322	0.209161	+	0.977881i	$0.432927\pi$
$864$	0	0
$865$	8.58522	0.291906
$866$	−29.7458	−1.01080
$867$	0	0
$868$	−19.4065	−0.658700
$869$	26.7549	0.907597
$870$	0	0
$871$	−12.4804	−0.422882
$872$	32.3720	1.09625
$873$	0	0
$874$	−13.7074	−0.463659
$875$	−0.514137	−0.0173810
$876$	0	0
$877$	−39.7002	−1.34058	−0.670290	−	0.742099i	$-0.733830\pi$
$877$	−39.7002	−1.34058	−0.670290	+	0.742099i	$0.733830\pi$
$878$	20.8970	0.705241
$879$	0	0
$880$	−20.0192	−0.674847
$881$	32.1040	1.08161	0.540806	−	0.841147i	$-0.318119\pi$
$881$	32.1040	1.08161	0.540806	+	0.841147i	$0.318119\pi$
$882$	0	0
$883$	13.5051	0.454482	0.227241	−	0.973839i	$-0.427030\pi$
$883$	13.5051	0.454482	0.227241	+	0.973839i	$0.427030\pi$
$884$	18.9536	0.637478
$885$	0	0
$886$	−73.3502	−2.46425
$887$	35.1222	1.17929	0.589643	−	0.807664i	$-0.299268\pi$
$887$	35.1222	1.17929	0.589643	+	0.807664i	$0.299268\pi$
$888$	0	0
$889$	−9.19872	−0.308515
$890$	−7.54241	−0.252822
$891$	0	0
$892$	37.4386	1.25354
$893$	6.42385	0.214966
$894$	0	0
$895$	−1.06562	−0.0356198
$896$	7.84049	0.261932
$897$	0	0
$898$	47.6975	1.59169
$899$	−12.1120	−0.403959
$900$	0	0
$901$	16.6983	0.556302
$902$	94.7559	3.15503
$903$	0	0
$904$	45.5207	1.51399
$905$	−12.6700	−0.421166
$906$	0	0
$907$	15.1186	0.502004	0.251002	−	0.967987i	$-0.419240\pi$
$907$	15.1186	0.502004	0.251002	+	0.967987i	$0.419240\pi$
$908$	14.3492	0.476194
$909$	0	0
$910$	1.70739	0.0565994
$911$	52.5561	1.74126	0.870631	−	0.491936i	$-0.163711\pi$
$911$	52.5561	1.74126	0.870631	+	0.491936i	$0.163711\pi$
$912$	0	0
$913$	−5.12217	−0.169519
$914$	−58.4166	−1.93225
$915$	0	0
$916$	−109.369	−3.61367
$917$	3.08482	0.101870
$918$	0	0
$919$	−54.5489	−1.79940	−0.899702	−	0.436505i	$-0.856216\pi$
$919$	−54.5489	−1.79940	−0.899702	+	0.436505i	$0.856216\pi$
$920$	24.0848	0.794053
$921$	0	0
$922$	11.1268	0.366443
$923$	−11.8760	−0.390903
$924$	0	0
$925$	0.292611	0.00962098
$926$	−49.0475	−1.61180
$927$	0	0
$928$	−4.83317	−0.158656
$929$	20.3793	0.668623	0.334311	−	0.942463i	$-0.391496\pi$
$929$	20.3793	0.668623	0.334311	+	0.942463i	$0.391496\pi$
$930$	0	0
$931$	8.89703	0.291588
$932$	119.398	3.91100
$933$	0	0
$934$	−61.8314	−2.02319
$935$	11.0283	0.360663
$936$	0	0
$937$	49.1979	1.60723	0.803613	−	0.595152i	$-0.202908\pi$
$937$	49.1979	1.60723	0.803613	+	0.595152i	$0.202908\pi$
$938$	−12.2133	−0.398777
$939$	0	0
$940$	−21.0137	−0.685393
$941$	−23.2371	−0.757508	−0.378754	−	0.925497i	$-0.623647\pi$
$941$	−23.2371	−0.757508	−0.378754	+	0.925497i	$0.623647\pi$
$942$	0	0
$943$	−46.8452	−1.52549
$944$	−30.3118	−0.986565
$945$	0	0
$946$	−86.4068	−2.80933
$947$	37.1642	1.20767	0.603837	−	0.797108i	$-0.293638\pi$
$947$	37.1642	1.20767	0.603837	+	0.797108i	$0.293638\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	−3.32088	−0.107744
$951$	0	0
$952$	9.96265	0.322891
$953$	−23.5761	−0.763706	−0.381853	−	0.924223i	$-0.624714\pi$
$953$	−23.5761	−0.763706	−0.381853	+	0.924223i	$0.624714\pi$
$954$	0	0
$955$	−16.9344	−0.547984
$956$	−18.1422	−0.586760
$957$	0	0
$958$	82.3502	2.66061
$959$	−2.91518	−0.0941360
$960$	0	0
$961$	45.3118	1.46167
$962$	−0.971726	−0.0313297
$963$	0	0
$964$	15.5743	0.501614
$965$	26.7175	0.860067
$966$	0	0
$967$	8.38290	0.269576	0.134788	−	0.990874i	$-0.456965\pi$
$967$	8.38290	0.269576	0.134788	+	0.990874i	$0.456965\pi$
$968$	0.164979	0.00530261
$969$	0	0
$970$	−30.8114	−0.989295
$971$	13.2078	0.423858	0.211929	−	0.977285i	$-0.432025\pi$
$971$	13.2078	0.423858	0.211929	+	0.977285i	$0.432025\pi$
$972$	0	0
$973$	−4.11310	−0.131860
$974$	−15.1787	−0.486357
$975$	0	0
$976$	44.3027	1.41810
$977$	−14.3310	−0.458490	−0.229245	−	0.973369i	$-0.573626\pi$
$977$	−14.3310	−0.458490	−0.229245	+	0.973369i	$0.573626\pi$
$978$	0	0
$979$	9.96265	0.318408
$980$	−29.1040	−0.929694
$981$	0	0
$982$	36.3118	1.15876
$983$	−32.3082	−1.03047	−0.515236	−	0.857048i	$-0.672296\pi$
$983$	−32.3082	−1.03047	−0.515236	+	0.857048i	$0.672296\pi$
$984$	0	0
$985$	−14.2553	−0.454210
$986$	11.5761	0.368660
$987$	0	0
$988$	7.53880	0.239841
$989$	42.7175	1.35834
$990$	0	0
$991$	−39.6700	−1.26016	−0.630080	−	0.776530i	$-0.716978\pi$
$991$	−39.6700	−1.26016	−0.630080	+	0.776530i	$0.716978\pi$
$992$	30.4513	0.966831
$993$	0	0
$994$	−11.6218	−0.368620
$995$	−24.6610	−0.781805
$996$	0	0
$997$	38.6874	1.22524	0.612621	−	0.790377i	$-0.290115\pi$
$997$	38.6874	1.22524	0.612621	+	0.790377i	$0.290115\pi$
$998$	52.7258	1.66901
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.a.j.1.3		3
3.2	odd	2		405.2.a.i.1.1		3
4.3	odd	2		6480.2.a.bv.1.3		3
5.2	odd	4		2025.2.b.l.649.6		6
5.3	odd	4		2025.2.b.l.649.1		6
5.4	even	2		2025.2.a.n.1.1		3
9.2	odd	6		135.2.e.b.91.3		6
9.4	even	3		45.2.e.b.16.1	✓	6
9.5	odd	6		135.2.e.b.46.3		6
9.7	even	3		45.2.e.b.31.1	yes	6
12.11	even	2		6480.2.a.bs.1.3		3
15.2	even	4		2025.2.b.m.649.1		6
15.8	even	4		2025.2.b.m.649.6		6
15.14	odd	2		2025.2.a.o.1.3		3
36.7	odd	6		720.2.q.i.481.1		6
36.11	even	6		2160.2.q.k.1441.1		6
36.23	even	6		2160.2.q.k.721.1		6
36.31	odd	6		720.2.q.i.241.1		6
45.2	even	12		675.2.k.b.199.1		12
45.4	even	6		225.2.e.b.151.3		6
45.7	odd	12		225.2.k.b.49.6		12
45.13	odd	12		225.2.k.b.124.6		12
45.14	odd	6		675.2.e.b.451.1		6
45.22	odd	12		225.2.k.b.124.1		12
45.23	even	12		675.2.k.b.424.1		12
45.29	odd	6		675.2.e.b.226.1		6
45.32	even	12		675.2.k.b.424.6		12
45.34	even	6		225.2.e.b.76.3		6
45.38	even	12		675.2.k.b.199.6		12
45.43	odd	12		225.2.k.b.49.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.1	✓	6	9.4	even	3
45.2.e.b.31.1	yes	6	9.7	even	3
135.2.e.b.46.3		6	9.5	odd	6
135.2.e.b.91.3		6	9.2	odd	6
225.2.e.b.76.3		6	45.34	even	6
225.2.e.b.151.3		6	45.4	even	6
225.2.k.b.49.1		12	45.43	odd	12
225.2.k.b.49.6		12	45.7	odd	12
225.2.k.b.124.1		12	45.22	odd	12
225.2.k.b.124.6		12	45.13	odd	12
405.2.a.i.1.1		3	3.2	odd	2
405.2.a.j.1.3		3	1.1	even	1	trivial
675.2.e.b.226.1		6	45.29	odd	6
675.2.e.b.451.1		6	45.14	odd	6
675.2.k.b.199.1		12	45.2	even	12
675.2.k.b.199.6		12	45.38	even	12
675.2.k.b.424.1		12	45.23	even	12
675.2.k.b.424.6		12	45.32	even	12
720.2.q.i.241.1		6	36.31	odd	6
720.2.q.i.481.1		6	36.7	odd	6
2025.2.a.n.1.1		3	5.4	even	2
2025.2.a.o.1.3		3	15.14	odd	2
2025.2.b.l.649.1		6	5.3	odd	4
2025.2.b.l.649.6		6	5.2	odd	4
2025.2.b.m.649.1		6	15.2	even	4
2025.2.b.m.649.6		6	15.8	even	4
2160.2.q.k.721.1		6	36.23	even	6
2160.2.q.k.1441.1		6	36.11	even	6
6480.2.a.bs.1.3		3	12.11	even	2
6480.2.a.bv.1.3		3	4.3	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51414 q^{2} +4.32088 q^{4} +1.00000 q^{5} -0.514137 q^{7} +5.83502 q^{8} +O(q^{10})\)	\(q+2.51414 q^{2} +4.32088 q^{4} +1.00000 q^{5} -0.514137 q^{7} +5.83502 q^{8} +2.51414 q^{10} -3.32088 q^{11} -1.32088 q^{13} -1.29261 q^{14} +6.02827 q^{16} -3.32088 q^{17} -1.32088 q^{19} +4.32088 q^{20} -8.34916 q^{22} +4.12763 q^{23} +1.00000 q^{25} -3.32088 q^{26} -2.22153 q^{28} -1.38650 q^{29} +8.73566 q^{31} +3.48586 q^{32} -8.34916 q^{34} -0.514137 q^{35} +0.292611 q^{37} -3.32088 q^{38} +5.83502 q^{40} -11.3492 q^{41} +10.3492 q^{43} -14.3492 q^{44} +10.3774 q^{46} -4.86330 q^{47} -6.73566 q^{49} +2.51414 q^{50} -5.70739 q^{52} -5.02827 q^{53} -3.32088 q^{55} -3.00000 q^{56} -3.48586 q^{58} -5.02827 q^{59} +7.34916 q^{61} +21.9627 q^{62} -3.29261 q^{64} -1.32088 q^{65} +9.44852 q^{67} -14.3492 q^{68} -1.29261 q^{70} +8.99093 q^{71} +6.05655 q^{73} +0.735663 q^{74} -5.70739 q^{76} +1.70739 q^{77} -8.05655 q^{79} +6.02827 q^{80} -28.5333 q^{82} +1.54241 q^{83} -3.32088 q^{85} +26.0192 q^{86} -19.3774 q^{88} -3.00000 q^{89} +0.679116 q^{91} +17.8350 q^{92} -12.2270 q^{94} -1.32088 q^{95} -12.2553 q^{97} -16.9344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8	\( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 3 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} - 7 q^{29} + 8 q^{31} + 17 q^{32} - 4 q^{34} + 5 q^{35} + 6 q^{37} - 2 q^{38} + 3 q^{40} - 13 q^{41} + 10 q^{43} - 22 q^{44} - 3 q^{46} + 13 q^{47} - 2 q^{49} + q^{50} - 12 q^{52} - 2 q^{53} - 2 q^{55} - 9 q^{56} - 17 q^{58} - 2 q^{59} + q^{61} + 42 q^{62} - 15 q^{64} + 4 q^{65} + 11 q^{67} - 22 q^{68} - 9 q^{70} - 10 q^{71} - 8 q^{73} - 16 q^{74} - 12 q^{76} + 2 q^{79} + 5 q^{80} - 29 q^{82} - 15 q^{83} - 2 q^{85} + 28 q^{86} - 24 q^{88} - 9 q^{89} + 10 q^{91} + 39 q^{92} - 31 q^{94} + 4 q^{95} - 18 q^{97} - 40 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 + 5 * q^16 - 2 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 3 * q^23 + 3 * q^25 - 2 * q^26 + 5 * q^28 - 7 * q^29 + 8 * q^31 + 17 * q^32 - 4 * q^34 + 5 * q^35 + 6 * q^37 - 2 * q^38 + 3 * q^40 - 13 * q^41 + 10 * q^43 - 22 * q^44 - 3 * q^46 + 13 * q^47 - 2 * q^49 + q^50 - 12 * q^52 - 2 * q^53 - 2 * q^55 - 9 * q^56 - 17 * q^58 - 2 * q^59 + q^61 + 42 * q^62 - 15 * q^64 + 4 * q^65 + 11 * q^67 - 22 * q^68 - 9 * q^70 - 10 * q^71 - 8 * q^73 - 16 * q^74 - 12 * q^76 + 2 * q^79 + 5 * q^80 - 29 * q^82 - 15 * q^83 - 2 * q^85 + 28 * q^86 - 24 * q^88 - 9 * q^89 + 10 * q^91 + 39 * q^92 - 31 * q^94 + 4 * q^95 - 18 * q^97 - 40 * q^98

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