LMFDB - Embedded newform 5445.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.09529$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09529 q^{2} +2.39026 q^{4} +1.00000 q^{5} +3.06719 q^{7} -0.817703 q^{8} +O(q^{10})$	$q-2.09529 q^{2} +2.39026 q^{4} +1.00000 q^{5} +3.06719 q^{7} -0.817703 q^{8} -2.09529 q^{10} -3.04981 q^{13} -6.42666 q^{14} -3.06719 q^{16} -0.463845 q^{17} +7.89563 q^{19} +2.39026 q^{20} +1.39026 q^{23} +1.00000 q^{25} +6.39026 q^{26} +7.33136 q^{28} +3.72162 q^{29} -10.4765 q^{31} +8.06206 q^{32} +0.971892 q^{34} +3.06719 q^{35} +1.84453 q^{37} -16.5437 q^{38} -0.817703 q^{40} -4.40763 q^{41} +1.31478 q^{43} -2.91300 q^{46} +2.98018 q^{47} +2.40763 q^{49} -2.09529 q^{50} -7.28984 q^{52} -4.18814 q^{53} -2.50805 q^{56} -7.79789 q^{58} -2.81502 q^{59} +2.01737 q^{61} +21.9513 q^{62} -10.7580 q^{64} -3.04981 q^{65} -6.75753 q^{67} -1.10871 q^{68} -6.42666 q^{70} +6.52195 q^{71} +9.87581 q^{73} -3.86484 q^{74} +18.8726 q^{76} +11.5579 q^{79} -3.06719 q^{80} +9.23528 q^{82} +8.91861 q^{83} -0.463845 q^{85} -2.75485 q^{86} +6.76978 q^{89} -9.35435 q^{91} +3.32307 q^{92} -6.24436 q^{94} +7.89563 q^{95} +15.3543 q^{97} -5.04469 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8	$ 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8} - q^{10} + q^{13} - 2 q^{14} - 3 q^{16} + q^{17} + 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} + 13 q^{28} - 12 q^{29} - 5 q^{31} + 8 q^{32} + 2 q^{34} + 3 q^{35} + 7 q^{37} - 20 q^{38} - 3 q^{40} - 11 q^{41} + 19 q^{43} - 4 q^{46} - 5 q^{47} + 3 q^{49} - q^{50} - 11 q^{52} + 11 q^{53} - 11 q^{56} - 14 q^{58} - 9 q^{59} + 12 q^{61} + 35 q^{62} - 3 q^{64} + q^{65} - 19 q^{67} + 3 q^{68} - 2 q^{70} - 5 q^{71} + 11 q^{73} + 34 q^{79} - 3 q^{80} - 6 q^{82} + 11 q^{83} + q^{85} - q^{86} + 8 q^{89} - 8 q^{91} + 12 q^{92} - q^{94} + 20 q^{95} + 32 q^{97} + 8 q^{98}+O(q^{100}) $ 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8 - q^10 + q^13 - 2 * q^14 - 3 * q^16 + q^17 + 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 + 13 * q^28 - 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^37 - 20 * q^38 - 3 * q^40 - 11 * q^41 + 19 * q^43 - 4 * q^46 - 5 * q^47 + 3 * q^49 - q^50 - 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 + 12 * q^61 + 35 * q^62 - 3 * q^64 + q^65 - 19 * q^67 + 3 * q^68 - 2 * q^70 - 5 * q^71 + 11 * q^73 + 34 * q^79 - 3 * q^80 - 6 * q^82 + 11 * q^83 + q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 - q^94 + 20 * q^95 + 32 * q^97 + 8 * 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 $

$2$	−2.09529	−1.48160	−0.740798	−	0.671728i	$-0.765553\pi$
$2$	−2.09529	−1.48160	−0.740798	+	0.671728i	$0.765553\pi$
$3$	0	0
$3$	0	0
$4$	2.39026	1.19513
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.06719	1.15929	0.579644	−	0.814870i	$-0.303192\pi$
$7$	3.06719	1.15929	0.579644	+	0.814870i	$0.303192\pi$
$8$	−0.817703	−0.289102
$9$	0	0
$10$	−2.09529	−0.662590
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.04981	−0.845866	−0.422933	−	0.906161i	$-0.638999\pi$
$13$	−3.04981	−0.845866	−0.422933	+	0.906161i	$0.638999\pi$
$14$	−6.42666	−1.71760
$15$	0	0
$16$	−3.06719	−0.766796
$17$	−0.463845	−0.112499	−0.0562495	−	0.998417i	$-0.517914\pi$
$17$	−0.463845	−0.112499	−0.0562495	+	0.998417i	$0.517914\pi$
$18$	0	0
$19$	7.89563	1.81138	0.905690	−	0.423940i	$-0.139353\pi$
$19$	7.89563	1.81138	0.905690	+	0.423940i	$0.139353\pi$
$20$	2.39026	0.534478
$21$	0	0
$22$	0	0
$23$	1.39026	0.289889	0.144944	−	0.989440i	$-0.453700\pi$
$23$	1.39026	0.289889	0.144944	+	0.989440i	$0.453700\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.39026	1.25323
$27$	0	0
$28$	7.33136	1.38550
$29$	3.72162	0.691087	0.345544	−	0.938403i	$-0.387695\pi$
$29$	3.72162	0.691087	0.345544	+	0.938403i	$0.387695\pi$
$30$	0	0
$31$	−10.4765	−1.88163	−0.940815	−	0.338921i	$-0.889938\pi$
$31$	−10.4765	−1.88163	−0.940815	+	0.338921i	$0.889938\pi$
$32$	8.06206	1.42518
$33$	0	0
$34$	0.971892	0.166678
$35$	3.06719	0.518449
$36$	0	0
$37$	1.84453	0.303239	0.151620	−	0.988439i	$-0.451551\pi$
$37$	1.84453	0.303239	0.151620	+	0.988439i	$0.451551\pi$
$38$	−16.5437	−2.68374
$39$	0	0
$40$	−0.817703	−0.129290
$41$	−4.40763	−0.688356	−0.344178	−	0.938904i	$-0.611842\pi$
$41$	−4.40763	−0.688356	−0.344178	+	0.938904i	$0.611842\pi$
$42$	0	0
$43$	1.31478	0.200502	0.100251	−	0.994962i	$-0.468035\pi$
$43$	1.31478	0.200502	0.100251	+	0.994962i	$0.468035\pi$
$44$	0	0
$45$	0	0
$46$	−2.91300	−0.429498
$47$	2.98018	0.434704	0.217352	−	0.976093i	$-0.430258\pi$
$47$	2.98018	0.434704	0.217352	+	0.976093i	$0.430258\pi$
$48$	0	0
$49$	2.40763	0.343947
$50$	−2.09529	−0.296319
$51$	0	0
$52$	−7.28984	−1.01092
$53$	−4.18814	−0.575286	−0.287643	−	0.957738i	$-0.592872\pi$
$53$	−4.18814	−0.575286	−0.287643	+	0.957738i	$0.592872\pi$
$54$	0	0
$55$	0	0
$56$	−2.50805	−0.335152
$57$	0	0
$58$	−7.79789	−1.02391
$59$	−2.81502	−0.366485	−0.183242	−	0.983068i	$-0.558659\pi$
$59$	−2.81502	−0.366485	−0.183242	+	0.983068i	$0.558659\pi$
$60$	0	0
$61$	2.01737	0.258298	0.129149	−	0.991625i	$-0.458775\pi$
$61$	2.01737	0.258298	0.129149	+	0.991625i	$0.458775\pi$
$62$	21.9513	2.78782
$63$	0	0
$64$	−10.7580	−1.34475
$65$	−3.04981	−0.378283
$66$	0	0
$67$	−6.75753	−0.825564	−0.412782	−	0.910830i	$-0.635443\pi$
$67$	−6.75753	−0.825564	−0.412782	+	0.910830i	$0.635443\pi$
$68$	−1.10871	−0.134451
$69$	0	0
$70$	−6.42666	−0.768132
$71$	6.52195	0.774013	0.387007	−	0.922077i	$-0.373509\pi$
$71$	6.52195	0.774013	0.387007	+	0.922077i	$0.373509\pi$
$72$	0	0
$73$	9.87581	1.15588	0.577938	−	0.816081i	$-0.303858\pi$
$73$	9.87581	1.15588	0.577938	+	0.816081i	$0.303858\pi$
$74$	−3.86484	−0.449278
$75$	0	0
$76$	18.8726	2.16483
$77$	0	0
$78$	0	0
$79$	11.5579	1.30036	0.650180	−	0.759780i	$-0.274693\pi$
$79$	11.5579	1.30036	0.650180	+	0.759780i	$0.274693\pi$
$80$	−3.06719	−0.342922
$81$	0	0
$82$	9.23528	1.01987
$83$	8.91861	0.978945	0.489472	−	0.872019i	$-0.337189\pi$
$83$	8.91861	0.978945	0.489472	+	0.872019i	$0.337189\pi$
$84$	0	0
$85$	−0.463845	−0.0503111
$86$	−2.75485	−0.297063
$87$	0	0
$88$	0	0
$89$	6.76978	0.717595	0.358797	−	0.933415i	$-0.383187\pi$
$89$	6.76978	0.717595	0.358797	+	0.933415i	$0.383187\pi$
$90$	0	0
$91$	−9.35435	−0.980602
$92$	3.32307	0.346454
$93$	0	0
$94$	−6.24436	−0.644056
$95$	7.89563	0.810074
$96$	0	0
$97$	15.3543	1.55900	0.779499	−	0.626404i	$-0.215474\pi$
$97$	15.3543	1.55900	0.779499	+	0.626404i	$0.215474\pi$
$98$	−5.04469	−0.509591
$99$	0	0
$100$	2.39026	0.239026
$101$	11.7326	1.16744	0.583718	−	0.811956i	$-0.301597\pi$
$101$	11.7326	1.16744	0.583718	+	0.811956i	$0.301597\pi$
$102$	0	0
$103$	13.8881	1.36843	0.684215	−	0.729280i	$-0.260145\pi$
$103$	13.8881	1.36843	0.684215	+	0.729280i	$0.260145\pi$
$104$	2.49384	0.244541
$105$	0	0
$106$	8.77539	0.852341
$107$	−7.32100	−0.707748	−0.353874	−	0.935293i	$-0.615136\pi$
$107$	−7.32100	−0.707748	−0.353874	+	0.935293i	$0.615136\pi$
$108$	0	0
$109$	7.43306	0.711958	0.355979	−	0.934494i	$-0.384147\pi$
$109$	7.43306	0.711958	0.355979	+	0.934494i	$0.384147\pi$
$110$	0	0
$111$	0	0
$112$	−9.40763	−0.888937
$113$	3.03640	0.285640	0.142820	−	0.989749i	$-0.454383\pi$
$113$	3.03640	0.285640	0.142820	+	0.989749i	$0.454383\pi$
$114$	0	0
$115$	1.39026	0.129642
$116$	8.89563	0.825938
$117$	0	0
$118$	5.89830	0.542983
$119$	−1.42270	−0.130419
$120$	0	0
$121$	0	0
$122$	−4.22699	−0.382693
$123$	0	0
$124$	−25.0415	−2.24879
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.451597	0.0400728	0.0200364	−	0.999799i	$-0.493622\pi$
$127$	0.451597	0.0400728	0.0200364	+	0.999799i	$0.493622\pi$
$128$	6.41709	0.567196
$129$	0	0
$130$	6.39026	0.560463
$131$	0.629003	0.0549563	0.0274781	−	0.999622i	$-0.491252\pi$
$131$	0.629003	0.0549563	0.0274781	+	0.999622i	$0.491252\pi$
$132$	0	0
$133$	24.2173	2.09991
$134$	14.1590	1.22315
$135$	0	0
$136$	0.379287	0.0325236
$137$	−11.1834	−0.955462	−0.477731	−	0.878506i	$-0.658541\pi$
$137$	−11.1834	−0.955462	−0.477731	+	0.878506i	$0.658541\pi$
$138$	0	0
$139$	2.37495	0.201441	0.100720	−	0.994915i	$-0.467885\pi$
$139$	2.37495	0.201441	0.100720	+	0.994915i	$0.467885\pi$
$140$	7.33136	0.619613
$141$	0	0
$142$	−13.6654	−1.14678
$143$	0	0
$144$	0	0
$145$	3.72162	0.309064
$146$	−20.6927	−1.71254
$147$	0	0
$148$	4.40891	0.362410
$149$	−8.72034	−0.714398	−0.357199	−	0.934028i	$-0.616268\pi$
$149$	−8.72034	−0.714398	−0.357199	+	0.934028i	$0.616268\pi$
$150$	0	0
$151$	−11.9354	−0.971291	−0.485646	−	0.874156i	$-0.661415\pi$
$151$	−11.9354	−0.971291	−0.485646	+	0.874156i	$0.661415\pi$
$152$	−6.45628	−0.523673
$153$	0	0
$154$	0	0
$155$	−10.4765	−0.841490
$156$	0	0
$157$	3.87532	0.309284	0.154642	−	0.987971i	$-0.450578\pi$
$157$	3.87532	0.309284	0.154642	+	0.987971i	$0.450578\pi$
$158$	−24.2171	−1.92661
$159$	0	0
$160$	8.06206	0.637362
$161$	4.26418	0.336064
$162$	0	0
$163$	−9.93621	−0.778264	−0.389132	−	0.921182i	$-0.627225\pi$
$163$	−9.93621	−0.778264	−0.389132	+	0.921182i	$0.627225\pi$
$164$	−10.5354	−0.822674
$165$	0	0
$166$	−18.6871	−1.45040
$167$	13.9200	1.07716	0.538581	−	0.842574i	$-0.318960\pi$
$167$	13.9200	1.07716	0.538581	+	0.842574i	$0.318960\pi$
$168$	0	0
$169$	−3.69863	−0.284510
$170$	0.971892	0.0745407
$171$	0	0
$172$	3.14266	0.239626
$173$	10.8311	0.823475	0.411737	−	0.911303i	$-0.364922\pi$
$173$	10.8311	0.823475	0.411737	+	0.911303i	$0.364922\pi$
$174$	0	0
$175$	3.06719	0.231857
$176$	0	0
$177$	0	0
$178$	−14.1847	−1.06319
$179$	−22.7335	−1.69918	−0.849589	−	0.527445i	$-0.823150\pi$
$179$	−22.7335	−1.69918	−0.849589	+	0.527445i	$0.823150\pi$
$180$	0	0
$181$	2.39831	0.178265	0.0891327	−	0.996020i	$-0.471590\pi$
$181$	2.39831	0.178265	0.0891327	+	0.996020i	$0.471590\pi$
$182$	19.6001	1.45286
$183$	0	0
$184$	−1.13682	−0.0838073
$185$	1.84453	0.135613
$186$	0	0
$187$	0	0
$188$	7.12340	0.519527
$189$	0	0
$190$	−16.5437	−1.20020
$191$	−17.2462	−1.24789	−0.623947	−	0.781466i	$-0.714472\pi$
$191$	−17.2462	−1.24789	−0.623947	+	0.781466i	$0.714472\pi$
$192$	0	0
$193$	2.58574	0.186125	0.0930627	−	0.995660i	$-0.470334\pi$
$193$	2.58574	0.186125	0.0930627	+	0.995660i	$0.470334\pi$
$194$	−32.1719	−2.30981
$195$	0	0
$196$	5.75485	0.411061
$197$	0.144731	0.0103116	0.00515582	−	0.999987i	$-0.498359\pi$
$197$	0.144731	0.0103116	0.00515582	+	0.999987i	$0.498359\pi$
$198$	0	0
$199$	−7.54177	−0.534622	−0.267311	−	0.963610i	$-0.586135\pi$
$199$	−7.54177	−0.534622	−0.267311	+	0.963610i	$0.586135\pi$
$200$	−0.817703	−0.0578203
$201$	0	0
$202$	−24.5832	−1.72967
$203$	11.4149	0.801169
$204$	0	0
$205$	−4.40763	−0.307842
$206$	−29.0996	−2.02746
$207$	0	0
$208$	9.35435	0.648607
$209$	0	0
$210$	0	0
$211$	2.26881	0.156191	0.0780957	−	0.996946i	$-0.475116\pi$
$211$	2.26881	0.156191	0.0780957	+	0.996946i	$0.475116\pi$
$212$	−10.0107	−0.687540
$213$	0	0
$214$	15.3397	1.04860
$215$	1.31478	0.0896673
$216$	0	0
$217$	−32.1333	−2.18135
$218$	−15.5744	−1.05483
$219$	0	0
$220$	0	0
$221$	1.41464	0.0951591
$222$	0	0
$223$	8.57968	0.574538	0.287269	−	0.957850i	$-0.407253\pi$
$223$	8.57968	0.574538	0.287269	+	0.957850i	$0.407253\pi$
$224$	24.7278	1.65220
$225$	0	0
$226$	−6.36215	−0.423204
$227$	6.20039	0.411534	0.205767	−	0.978601i	$-0.434031\pi$
$227$	6.20039	0.411534	0.205767	+	0.978601i	$0.434031\pi$
$228$	0	0
$229$	23.1659	1.53084	0.765422	−	0.643528i	$-0.222530\pi$
$229$	23.1659	1.53084	0.765422	+	0.643528i	$0.222530\pi$
$230$	−2.91300	−0.192077
$231$	0	0
$232$	−3.04318	−0.199794
$233$	−27.8627	−1.82535	−0.912673	−	0.408690i	$-0.865986\pi$
$233$	−27.8627	−1.82535	−0.912673	+	0.408690i	$0.865986\pi$
$234$	0	0
$235$	2.98018	0.194406
$236$	−6.72863	−0.437997
$237$	0	0
$238$	2.98097	0.193228
$239$	−16.2862	−1.05347	−0.526734	−	0.850030i	$-0.676584\pi$
$239$	−16.2862	−1.05347	−0.526734	+	0.850030i	$0.676584\pi$
$240$	0	0
$241$	−4.39063	−0.282826	−0.141413	−	0.989951i	$-0.545164\pi$
$241$	−4.39063	−0.282826	−0.141413	+	0.989951i	$0.545164\pi$
$242$	0	0
$243$	0	0
$244$	4.82204	0.308699
$245$	2.40763	0.153818
$246$	0	0
$247$	−24.0802	−1.53219
$248$	8.56664	0.543982
$249$	0	0
$250$	−2.09529	−0.132518
$251$	2.66668	0.168319	0.0841597	−	0.996452i	$-0.473179\pi$
$251$	2.66668	0.168319	0.0841597	+	0.996452i	$0.473179\pi$
$252$	0	0
$253$	0	0
$254$	−0.946229	−0.0593717
$255$	0	0
$256$	8.07035	0.504397
$257$	21.6327	1.34941	0.674704	−	0.738088i	$-0.264271\pi$
$257$	21.6327	1.34941	0.674704	+	0.738088i	$0.264271\pi$
$258$	0	0
$259$	5.65752	0.351541
$260$	−7.28984	−0.452097
$261$	0	0
$262$	−1.31795	−0.0814230
$263$	22.1392	1.36516	0.682581	−	0.730810i	$-0.260858\pi$
$263$	22.1392	1.36516	0.682581	+	0.730810i	$0.260858\pi$
$264$	0	0
$265$	−4.18814	−0.257276
$266$	−50.7425	−3.11122
$267$	0	0
$268$	−16.1522	−0.986655
$269$	−20.7184	−1.26322	−0.631611	−	0.775285i	$-0.717606\pi$
$269$	−20.7184	−1.26322	−0.631611	+	0.775285i	$0.717606\pi$
$270$	0	0
$271$	0.423112	0.0257022	0.0128511	−	0.999917i	$-0.495909\pi$
$271$	0.423112	0.0257022	0.0128511	+	0.999917i	$0.495909\pi$
$272$	1.42270	0.0862638
$273$	0	0
$274$	23.4325	1.41561
$275$	0	0
$276$	0	0
$277$	8.57255	0.515075	0.257537	−	0.966268i	$-0.417089\pi$
$277$	8.57255	0.515075	0.257537	+	0.966268i	$0.417089\pi$
$278$	−4.97623	−0.298454
$279$	0	0
$280$	−2.50805	−0.149884
$281$	−7.01108	−0.418246	−0.209123	−	0.977889i	$-0.567061\pi$
$281$	−7.01108	−0.418246	−0.209123	+	0.977889i	$0.567061\pi$
$282$	0	0
$283$	−9.95317	−0.591655	−0.295827	−	0.955241i	$-0.595595\pi$
$283$	−9.95317	−0.591655	−0.295827	+	0.955241i	$0.595595\pi$
$284$	15.5891	0.925045
$285$	0	0
$286$	0	0
$287$	−13.5190	−0.798002
$288$	0	0
$289$	−16.7848	−0.987344
$290$	−7.79789	−0.457908
$291$	0	0
$292$	23.6057	1.38142
$293$	21.8209	1.27479	0.637394	−	0.770538i	$-0.280012\pi$
$293$	21.8209	1.27479	0.637394	+	0.770538i	$0.280012\pi$
$294$	0	0
$295$	−2.81502	−0.163897
$296$	−1.50828	−0.0876670
$297$	0	0
$298$	18.2717	1.05845
$299$	−4.24002	−0.245207
$300$	0	0
$301$	4.03268	0.232440
$302$	25.0082	1.43906
$303$	0	0
$304$	−24.2173	−1.38896
$305$	2.01737	0.115514
$306$	0	0
$307$	30.8674	1.76170	0.880849	−	0.473397i	$-0.156972\pi$
$307$	30.8674	1.76170	0.880849	+	0.473397i	$0.156972\pi$
$308$	0	0
$309$	0	0
$310$	21.9513	1.24675
$311$	19.4150	1.10092	0.550462	−	0.834860i	$-0.314451\pi$
$311$	19.4150	1.10092	0.550462	+	0.834860i	$0.314451\pi$
$312$	0	0
$313$	1.05147	0.0594326	0.0297163	−	0.999558i	$-0.490540\pi$
$313$	1.05147	0.0594326	0.0297163	+	0.999558i	$0.490540\pi$
$314$	−8.11993	−0.458234
$315$	0	0
$316$	27.6263	1.55410
$317$	23.8314	1.33851	0.669253	−	0.743034i	$-0.266614\pi$
$317$	23.8314	1.33851	0.669253	+	0.743034i	$0.266614\pi$
$318$	0	0
$319$	0	0
$320$	−10.7580	−0.601391
$321$	0	0
$322$	−8.93470	−0.497912
$323$	−3.66235	−0.203778
$324$	0	0
$325$	−3.04981	−0.169173
$326$	20.8193	1.15307
$327$	0	0
$328$	3.60413	0.199005
$329$	9.14077	0.503947
$330$	0	0
$331$	25.6693	1.41091	0.705457	−	0.708753i	$-0.250742\pi$
$331$	25.6693	1.41091	0.705457	+	0.708753i	$0.250742\pi$
$332$	21.3178	1.16996
$333$	0	0
$334$	−29.1665	−1.59592
$335$	−6.75753	−0.369203
$336$	0	0
$337$	23.8922	1.30149	0.650744	−	0.759297i	$-0.274457\pi$
$337$	23.8922	1.30149	0.650744	+	0.759297i	$0.274457\pi$
$338$	7.74973	0.421530
$339$	0	0
$340$	−1.10871	−0.0601282
$341$	0	0
$342$	0	0
$343$	−14.0857	−0.760554
$344$	−1.07510	−0.0579655
$345$	0	0
$346$	−22.6944	−1.22006
$347$	−0.332152	−0.0178309	−0.00891543	−	0.999960i	$-0.502838\pi$
$347$	−0.332152	−0.0178309	−0.00891543	+	0.999960i	$0.502838\pi$
$348$	0	0
$349$	−1.22149	−0.0653846	−0.0326923	−	0.999465i	$-0.510408\pi$
$349$	−1.22149	−0.0653846	−0.0326923	+	0.999465i	$0.510408\pi$
$350$	−6.42666	−0.343519
$351$	0	0
$352$	0	0
$353$	25.7038	1.36808	0.684039	−	0.729446i	$-0.260222\pi$
$353$	25.7038	1.36808	0.684039	+	0.729446i	$0.260222\pi$
$354$	0	0
$355$	6.52195	0.346149
$356$	16.1815	0.857618
$357$	0	0
$358$	47.6333	2.51750
$359$	−17.9315	−0.946387	−0.473193	−	0.880959i	$-0.656899\pi$
$359$	−17.9315	−0.946387	−0.473193	+	0.880959i	$0.656899\pi$
$360$	0	0
$361$	43.3409	2.28110
$362$	−5.02517	−0.264117
$363$	0	0
$364$	−22.3593	−1.17195
$365$	9.87581	0.516923
$366$	0	0
$367$	−8.49091	−0.443222	−0.221611	−	0.975135i	$-0.571131\pi$
$367$	−8.49091	−0.443222	−0.221611	+	0.975135i	$0.571131\pi$
$368$	−4.26418	−0.222286
$369$	0	0
$370$	−3.86484	−0.200923
$371$	−12.8458	−0.666921
$372$	0	0
$373$	−35.8450	−1.85598	−0.927991	−	0.372604i	$-0.878465\pi$
$373$	−35.8450	−1.85598	−0.927991	+	0.372604i	$0.878465\pi$
$374$	0	0
$375$	0	0
$376$	−2.43690	−0.125674
$377$	−11.3502	−0.584567
$378$	0	0
$379$	17.5516	0.901564	0.450782	−	0.892634i	$-0.351145\pi$
$379$	17.5516	0.901564	0.450782	+	0.892634i	$0.351145\pi$
$380$	18.8726	0.968143
$381$	0	0
$382$	36.1360	1.84888
$383$	21.6250	1.10499	0.552494	−	0.833517i	$-0.313676\pi$
$383$	21.6250	1.10499	0.552494	+	0.833517i	$0.313676\pi$
$384$	0	0
$385$	0	0
$386$	−5.41788	−0.275763
$387$	0	0
$388$	36.7008	1.86320
$389$	−35.7416	−1.81217	−0.906086	−	0.423093i	$-0.860944\pi$
$389$	−35.7416	−1.81217	−0.906086	+	0.423093i	$0.860944\pi$
$390$	0	0
$391$	−0.644864	−0.0326122
$392$	−1.96872	−0.0994356
$393$	0	0
$394$	−0.303254	−0.0152777
$395$	11.5579	0.581539
$396$	0	0
$397$	−20.0447	−1.00601	−0.503007	−	0.864282i	$-0.667773\pi$
$397$	−20.0447	−1.00601	−0.503007	+	0.864282i	$0.667773\pi$
$398$	15.8022	0.792094
$399$	0	0
$400$	−3.06719	−0.153359
$401$	20.8987	1.04363	0.521815	−	0.853059i	$-0.325255\pi$
$401$	20.8987	1.04363	0.521815	+	0.853059i	$0.325255\pi$
$402$	0	0
$403$	31.9513	1.59161
$404$	28.0439	1.39524
$405$	0	0
$406$	−23.9176	−1.18701
$407$	0	0
$408$	0	0
$409$	−23.4982	−1.16191	−0.580955	−	0.813936i	$-0.697321\pi$
$409$	−23.4982	−1.16191	−0.580955	+	0.813936i	$0.697321\pi$
$410$	9.23528	0.456098
$411$	0	0
$412$	33.1960	1.63545
$413$	−8.63420	−0.424861
$414$	0	0
$415$	8.91861	0.437797
$416$	−24.5878	−1.20552
$417$	0	0
$418$	0	0
$419$	10.1128	0.494043	0.247022	−	0.969010i	$-0.420548\pi$
$419$	10.1128	0.494043	0.247022	+	0.969010i	$0.420548\pi$
$420$	0	0
$421$	8.92283	0.434872	0.217436	−	0.976075i	$-0.430231\pi$
$421$	8.92283	0.434872	0.217436	+	0.976075i	$0.430231\pi$
$422$	−4.75383	−0.231413
$423$	0	0
$424$	3.42466	0.166316
$425$	−0.463845	−0.0224998
$426$	0	0
$427$	6.18765	0.299442
$428$	−17.4991	−0.845850
$429$	0	0
$430$	−2.75485	−0.132851
$431$	−17.6122	−0.848352	−0.424176	−	0.905580i	$-0.639436\pi$
$431$	−17.6122	−0.848352	−0.424176	+	0.905580i	$0.639436\pi$
$432$	0	0
$433$	9.14397	0.439431	0.219716	−	0.975564i	$-0.429487\pi$
$433$	9.14397	0.439431	0.219716	+	0.975564i	$0.429487\pi$
$434$	67.3287	3.23188
$435$	0	0
$436$	17.7669	0.850881
$437$	10.9769	0.525099
$438$	0	0
$439$	−6.46946	−0.308770	−0.154385	−	0.988011i	$-0.549340\pi$
$439$	−6.46946	−0.308770	−0.154385	+	0.988011i	$0.549340\pi$
$440$	0	0
$441$	0	0
$442$	−2.96409	−0.140987
$443$	40.8842	1.94247	0.971233	−	0.238132i	$-0.0765350\pi$
$443$	40.8842	1.94247	0.971233	+	0.238132i	$0.0765350\pi$
$444$	0	0
$445$	6.76978	0.320918
$446$	−17.9769	−0.851233
$447$	0	0
$448$	−32.9968	−1.55895
$449$	12.1608	0.573902	0.286951	−	0.957945i	$-0.407358\pi$
$449$	12.1608	0.573902	0.286951	+	0.957945i	$0.407358\pi$
$450$	0	0
$451$	0	0
$452$	7.25777	0.341377
$453$	0	0
$454$	−12.9916	−0.609728
$455$	−9.35435	−0.438539
$456$	0	0
$457$	9.90240	0.463215	0.231607	−	0.972809i	$-0.425602\pi$
$457$	9.90240	0.463215	0.231607	+	0.972809i	$0.425602\pi$
$458$	−48.5393	−2.26809
$459$	0	0
$460$	3.32307	0.154939
$461$	−3.12529	−0.145559	−0.0727796	−	0.997348i	$-0.523187\pi$
$461$	−3.12529	−0.145559	−0.0727796	+	0.997348i	$0.523187\pi$
$462$	0	0
$463$	−24.3518	−1.13173	−0.565863	−	0.824499i	$-0.691457\pi$
$463$	−24.3518	−1.13173	−0.565863	+	0.824499i	$0.691457\pi$
$464$	−11.4149	−0.529923
$465$	0	0
$466$	58.3806	2.70443
$467$	−33.1737	−1.53510	−0.767548	−	0.640991i	$-0.778523\pi$
$467$	−33.1737	−1.53510	−0.767548	+	0.640991i	$0.778523\pi$
$468$	0	0
$469$	−20.7266	−0.957065
$470$	−6.24436	−0.288031
$471$	0	0
$472$	2.30185	0.105951
$473$	0	0
$474$	0	0
$475$	7.89563	0.362276
$476$	−3.40062	−0.155867
$477$	0	0
$478$	34.1244	1.56082
$479$	−17.2977	−0.790353	−0.395176	−	0.918605i	$-0.629317\pi$
$479$	−17.2977	−0.790353	−0.395176	+	0.918605i	$0.629317\pi$
$480$	0	0
$481$	−5.62548	−0.256500
$482$	9.19967	0.419033
$483$	0	0
$484$	0	0
$485$	15.3543	0.697205
$486$	0	0
$487$	7.64061	0.346229	0.173114	−	0.984902i	$-0.444617\pi$
$487$	7.64061	0.346229	0.173114	+	0.984902i	$0.444617\pi$
$488$	−1.64961	−0.0746744
$489$	0	0
$490$	−5.04469	−0.227896
$491$	−30.6563	−1.38350	−0.691750	−	0.722137i	$-0.743160\pi$
$491$	−30.6563	−1.38350	−0.691750	+	0.722137i	$0.743160\pi$
$492$	0	0
$493$	−1.72625	−0.0777466
$494$	50.4551	2.27008
$495$	0	0
$496$	32.1333	1.44283
$497$	20.0040	0.897303
$498$	0	0
$499$	−37.4783	−1.67776	−0.838879	−	0.544317i	$-0.816789\pi$
$499$	−37.4783	−1.67776	−0.838879	+	0.544317i	$0.816789\pi$
$500$	2.39026	0.106896
$501$	0	0
$502$	−5.58748	−0.249381
$503$	8.47695	0.377969	0.188984	−	0.981980i	$-0.439480\pi$
$503$	8.47695	0.377969	0.188984	+	0.981980i	$0.439480\pi$
$504$	0	0
$505$	11.7326	0.522093
$506$	0	0
$507$	0	0
$508$	1.07943	0.0478921
$509$	1.69723	0.0752286	0.0376143	−	0.999292i	$-0.488024\pi$
$509$	1.69723	0.0752286	0.0376143	+	0.999292i	$0.488024\pi$
$510$	0	0
$511$	30.2909	1.33999
$512$	−29.7439	−1.31451
$513$	0	0
$514$	−45.3268	−1.99928
$515$	13.8881	0.611981
$516$	0	0
$517$	0	0
$518$	−11.8542	−0.520843
$519$	0	0
$520$	2.49384	0.109362
$521$	37.0929	1.62507	0.812535	−	0.582912i	$-0.198087\pi$
$521$	37.0929	1.62507	0.812535	+	0.582912i	$0.198087\pi$
$522$	0	0
$523$	18.0818	0.790662	0.395331	−	0.918539i	$-0.370630\pi$
$523$	18.0818	0.790662	0.395331	+	0.918539i	$0.370630\pi$
$524$	1.50348	0.0656798
$525$	0	0
$526$	−46.3881	−2.02262
$527$	4.85946	0.211681
$528$	0	0
$529$	−21.0672	−0.915965
$530$	8.77539	0.381179
$531$	0	0
$532$	57.8857	2.50966
$533$	13.4424	0.582257
$534$	0	0
$535$	−7.32100	−0.316515
$536$	5.52565	0.238672
$537$	0	0
$538$	43.4111	1.87159
$539$	0	0
$540$	0	0
$541$	−11.7524	−0.505275	−0.252638	−	0.967561i	$-0.581298\pi$
$541$	−11.7524	−0.505275	−0.252638	+	0.967561i	$0.581298\pi$
$542$	−0.886544	−0.0380803
$543$	0	0
$544$	−3.73955	−0.160332
$545$	7.43306	0.318397
$546$	0	0
$547$	21.7569	0.930256	0.465128	−	0.885243i	$-0.346008\pi$
$547$	21.7569	0.930256	0.465128	+	0.885243i	$0.346008\pi$
$548$	−26.7312	−1.14190
$549$	0	0
$550$	0	0
$551$	29.3845	1.25182
$552$	0	0
$553$	35.4501	1.50749
$554$	−17.9620	−0.763133
$555$	0	0
$556$	5.67675	0.240748
$557$	4.83432	0.204837	0.102418	−	0.994741i	$-0.467342\pi$
$557$	4.83432	0.204837	0.102418	+	0.994741i	$0.467342\pi$
$558$	0	0
$559$	−4.00984	−0.169598
$560$	−9.40763	−0.397545
$561$	0	0
$562$	14.6903	0.619672
$563$	4.77199	0.201115	0.100558	−	0.994931i	$-0.467937\pi$
$563$	4.77199	0.201115	0.100558	+	0.994931i	$0.467937\pi$
$564$	0	0
$565$	3.03640	0.127742
$566$	20.8548	0.876594
$567$	0	0
$568$	−5.33302	−0.223768
$569$	35.7187	1.49741	0.748703	−	0.662905i	$-0.230677\pi$
$569$	35.7187	1.49741	0.748703	+	0.662905i	$0.230677\pi$
$570$	0	0
$571$	33.9838	1.42218	0.711090	−	0.703101i	$-0.248202\pi$
$571$	33.9838	1.42218	0.711090	+	0.703101i	$0.248202\pi$
$572$	0	0
$573$	0	0
$574$	28.3263	1.18232
$575$	1.39026	0.0579777
$576$	0	0
$577$	−20.6579	−0.860000	−0.430000	−	0.902829i	$-0.641486\pi$
$577$	−20.6579	−0.860000	−0.430000	+	0.902829i	$0.641486\pi$
$578$	35.1692	1.46285
$579$	0	0
$580$	8.89563	0.369371
$581$	27.3550	1.13488
$582$	0	0
$583$	0	0
$584$	−8.07548	−0.334166
$585$	0	0
$586$	−45.7211	−1.88872
$587$	−13.3600	−0.551428	−0.275714	−	0.961240i	$-0.588914\pi$
$587$	−13.3600	−0.551428	−0.275714	+	0.961240i	$0.588914\pi$
$588$	0	0
$589$	−82.7183	−3.40835
$590$	5.89830	0.242829
$591$	0	0
$592$	−5.65752	−0.232523
$593$	20.8062	0.854410	0.427205	−	0.904155i	$-0.359498\pi$
$593$	20.8062	0.854410	0.427205	+	0.904155i	$0.359498\pi$
$594$	0	0
$595$	−1.42270	−0.0583250
$596$	−20.8439	−0.853798
$597$	0	0
$598$	8.88410	0.363298
$599$	14.2456	0.582061	0.291030	−	0.956714i	$-0.406002\pi$
$599$	14.2456	0.582061	0.291030	+	0.956714i	$0.406002\pi$
$600$	0	0
$601$	20.7462	0.846255	0.423127	−	0.906070i	$-0.360932\pi$
$601$	20.7462	0.846255	0.423127	+	0.906070i	$0.360932\pi$
$602$	−8.44964	−0.344382
$603$	0	0
$604$	−28.5287	−1.16082
$605$	0	0
$606$	0	0
$607$	−17.9219	−0.727428	−0.363714	−	0.931511i	$-0.618492\pi$
$607$	−17.9219	−0.727428	−0.363714	+	0.931511i	$0.618492\pi$
$608$	63.6550	2.58155
$609$	0	0
$610$	−4.22699	−0.171146
$611$	−9.08900	−0.367702
$612$	0	0
$613$	28.2019	1.13906	0.569531	−	0.821970i	$-0.307125\pi$
$613$	28.2019	1.13906	0.569531	+	0.821970i	$0.307125\pi$
$614$	−64.6764	−2.61013
$615$	0	0
$616$	0	0
$617$	4.72930	0.190394	0.0951972	−	0.995458i	$-0.469652\pi$
$617$	4.72930	0.190394	0.0951972	+	0.995458i	$0.469652\pi$
$618$	0	0
$619$	−30.0575	−1.20811	−0.604056	−	0.796942i	$-0.706450\pi$
$619$	−30.0575	−1.20811	−0.604056	+	0.796942i	$0.706450\pi$
$620$	−25.0415	−1.00569
$621$	0	0
$622$	−40.6802	−1.63113
$623$	20.7642	0.831899
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.20314	−0.0880551
$627$	0	0
$628$	9.26301	0.369634
$629$	−0.855577	−0.0341141
$630$	0	0
$631$	−31.7036	−1.26210	−0.631050	−	0.775742i	$-0.717376\pi$
$631$	−31.7036	−1.26210	−0.631050	+	0.775742i	$0.717376\pi$
$632$	−9.45090	−0.375936
$633$	0	0
$634$	−49.9338	−1.98313
$635$	0.451597	0.0179211
$636$	0	0
$637$	−7.34282	−0.290933
$638$	0	0
$639$	0	0
$640$	6.41709	0.253658
$641$	18.9573	0.748770	0.374385	−	0.927273i	$-0.377854\pi$
$641$	18.9573	0.748770	0.374385	+	0.927273i	$0.377854\pi$
$642$	0	0
$643$	−36.2489	−1.42952	−0.714758	−	0.699372i	$-0.753463\pi$
$643$	−36.2489	−1.42952	−0.714758	+	0.699372i	$0.753463\pi$
$644$	10.1925	0.401640
$645$	0	0
$646$	7.67369	0.301917
$647$	34.9519	1.37410	0.687050	−	0.726610i	$-0.258905\pi$
$647$	34.9519	1.37410	0.687050	+	0.726610i	$0.258905\pi$
$648$	0	0
$649$	0	0
$650$	6.39026	0.250646
$651$	0	0
$652$	−23.7501	−0.930126
$653$	29.7893	1.16574	0.582872	−	0.812564i	$-0.301929\pi$
$653$	29.7893	1.16574	0.582872	+	0.812564i	$0.301929\pi$
$654$	0	0
$655$	0.629003	0.0245772
$656$	13.5190	0.527829
$657$	0	0
$658$	−19.1526	−0.746646
$659$	−7.30532	−0.284575	−0.142287	−	0.989825i	$-0.545446\pi$
$659$	−7.30532	−0.284575	−0.142287	+	0.989825i	$0.545446\pi$
$660$	0	0
$661$	−22.7352	−0.884296	−0.442148	−	0.896942i	$-0.645783\pi$
$661$	−22.7352	−0.884296	−0.442148	+	0.896942i	$0.645783\pi$
$662$	−53.7848	−2.09040
$663$	0	0
$664$	−7.29277	−0.283014
$665$	24.2173	0.939109
$666$	0	0
$667$	5.17401	0.200338
$668$	33.2724	1.28735
$669$	0	0
$670$	14.1590	0.547010
$671$	0	0
$672$	0	0
$673$	17.7451	0.684024	0.342012	−	0.939696i	$-0.388892\pi$
$673$	17.7451	0.684024	0.342012	+	0.939696i	$0.388892\pi$
$674$	−50.0611	−1.92828
$675$	0	0
$676$	−8.84069	−0.340026
$677$	8.16216	0.313697	0.156849	−	0.987623i	$-0.449867\pi$
$677$	8.16216	0.313697	0.156849	+	0.987623i	$0.449867\pi$
$678$	0	0
$679$	47.0946	1.80733
$680$	0.379287	0.0145450
$681$	0	0
$682$	0	0
$683$	6.19100	0.236892	0.118446	−	0.992960i	$-0.462209\pi$
$683$	6.19100	0.236892	0.118446	+	0.992960i	$0.462209\pi$
$684$	0	0
$685$	−11.1834	−0.427296
$686$	29.5136	1.12683
$687$	0	0
$688$	−4.03268	−0.153744
$689$	12.7731	0.486615
$690$	0	0
$691$	23.6051	0.897981	0.448990	−	0.893537i	$-0.351784\pi$
$691$	23.6051	0.897981	0.448990	+	0.893537i	$0.351784\pi$
$692$	25.8892	0.984158
$693$	0	0
$694$	0.695956	0.0264181
$695$	2.37495	0.0900871
$696$	0	0
$697$	2.04446	0.0774393
$698$	2.55937	0.0968737
$699$	0	0
$700$	7.33136	0.277099
$701$	37.2284	1.40610	0.703049	−	0.711142i	$-0.251822\pi$
$701$	37.2284	1.40610	0.703049	+	0.711142i	$0.251822\pi$
$702$	0	0
$703$	14.5637	0.549282
$704$	0	0
$705$	0	0
$706$	−53.8571	−2.02694
$707$	35.9860	1.35339
$708$	0	0
$709$	18.2537	0.685534	0.342767	−	0.939420i	$-0.388636\pi$
$709$	18.2537	0.685534	0.342767	+	0.939420i	$0.388636\pi$
$710$	−13.6654	−0.512853
$711$	0	0
$712$	−5.53567	−0.207458
$713$	−14.5650	−0.545463
$714$	0	0
$715$	0	0
$716$	−54.3388	−2.03074
$717$	0	0
$718$	37.5717	1.40216
$719$	−9.57389	−0.357046	−0.178523	−	0.983936i	$-0.557132\pi$
$719$	−9.57389	−0.357046	−0.178523	+	0.983936i	$0.557132\pi$
$720$	0	0
$721$	42.5972	1.58640
$722$	−90.8119	−3.37967
$723$	0	0
$724$	5.73259	0.213050
$725$	3.72162	0.138217
$726$	0	0
$727$	−14.0175	−0.519882	−0.259941	−	0.965625i	$-0.583703\pi$
$727$	−14.0175	−0.519882	−0.259941	+	0.965625i	$0.583703\pi$
$728$	7.64908	0.283494
$729$	0	0
$730$	−20.6927	−0.765872
$731$	−0.609854	−0.0225563
$732$	0	0
$733$	−9.28772	−0.343050	−0.171525	−	0.985180i	$-0.554869\pi$
$733$	−9.28772	−0.343050	−0.171525	+	0.985180i	$0.554869\pi$
$734$	17.7909	0.656676
$735$	0	0
$736$	11.2083	0.413145
$737$	0	0
$738$	0	0
$739$	38.9586	1.43312	0.716558	−	0.697527i	$-0.245716\pi$
$739$	38.9586	1.43312	0.716558	+	0.697527i	$0.245716\pi$
$740$	4.40891	0.162075
$741$	0	0
$742$	26.9158	0.988108
$743$	45.9296	1.68499	0.842497	−	0.538701i	$-0.181085\pi$
$743$	45.9296	1.68499	0.842497	+	0.538701i	$0.181085\pi$
$744$	0	0
$745$	−8.72034	−0.319489
$746$	75.1057	2.74982
$747$	0	0
$748$	0	0
$749$	−22.4549	−0.820483
$750$	0	0
$751$	23.1928	0.846318	0.423159	−	0.906055i	$-0.360921\pi$
$751$	23.1928	0.846318	0.423159	+	0.906055i	$0.360921\pi$
$752$	−9.14077	−0.333330
$753$	0	0
$754$	23.7821	0.866093
$755$	−11.9354	−0.434375
$756$	0	0
$757$	−6.52202	−0.237047	−0.118523	−	0.992951i	$-0.537816\pi$
$757$	−6.52202	−0.237047	−0.118523	+	0.992951i	$0.537816\pi$
$758$	−36.7757	−1.33575
$759$	0	0
$760$	−6.45628	−0.234194
$761$	−6.56682	−0.238047	−0.119024	−	0.992891i	$-0.537976\pi$
$761$	−6.56682	−0.238047	−0.119024	+	0.992891i	$0.537976\pi$
$762$	0	0
$763$	22.7986	0.825364
$764$	−41.2230	−1.49139
$765$	0	0
$766$	−45.3108	−1.63715
$767$	8.58530	0.309997
$768$	0	0
$769$	−12.5950	−0.454188	−0.227094	−	0.973873i	$-0.572922\pi$
$769$	−12.5950	−0.454188	−0.227094	+	0.973873i	$0.572922\pi$
$770$	0	0
$771$	0	0
$772$	6.18057	0.222444
$773$	−21.9448	−0.789299	−0.394650	−	0.918832i	$-0.629134\pi$
$773$	−21.9448	−0.789299	−0.394650	+	0.918832i	$0.629134\pi$
$774$	0	0
$775$	−10.4765	−0.376326
$776$	−12.5553	−0.450709
$777$	0	0
$778$	74.8892	2.68491
$779$	−34.8010	−1.24687
$780$	0	0
$781$	0	0
$782$	1.35118	0.0483181
$783$	0	0
$784$	−7.38464	−0.263737
$785$	3.87532	0.138316
$786$	0	0
$787$	−16.7298	−0.596354	−0.298177	−	0.954511i	$-0.596379\pi$
$787$	−16.7298	−0.596354	−0.298177	+	0.954511i	$0.596379\pi$
$788$	0.345944	0.0123237
$789$	0	0
$790$	−24.2171	−0.861606
$791$	9.31320	0.331139
$792$	0	0
$793$	−6.15261	−0.218486
$794$	41.9995	1.49051
$795$	0	0
$796$	−18.0268	−0.638942
$797$	11.7707	0.416940	0.208470	−	0.978029i	$-0.433152\pi$
$797$	11.7707	0.416940	0.208470	+	0.978029i	$0.433152\pi$
$798$	0	0
$799$	−1.38234	−0.0489038
$800$	8.06206	0.285037
$801$	0	0
$802$	−43.7889	−1.54624
$803$	0	0
$804$	0	0
$805$	4.26418	0.150292
$806$	−66.9473	−2.35812
$807$	0	0
$808$	−9.59377	−0.337508
$809$	1.19595	0.0420472	0.0210236	−	0.999779i	$-0.493307\pi$
$809$	1.19595	0.0420472	0.0210236	+	0.999779i	$0.493307\pi$
$810$	0	0
$811$	−35.8253	−1.25800	−0.628998	−	0.777407i	$-0.716535\pi$
$811$	−35.8253	−1.25800	−0.628998	+	0.777407i	$0.716535\pi$
$812$	27.2845	0.957499
$813$	0	0
$814$	0	0
$815$	−9.93621	−0.348050
$816$	0	0
$817$	10.3810	0.363186
$818$	49.2356	1.72148
$819$	0	0
$820$	−10.5354	−0.367911
$821$	5.49200	0.191672	0.0958360	−	0.995397i	$-0.469448\pi$
$821$	5.49200	0.191672	0.0958360	+	0.995397i	$0.469448\pi$
$822$	0	0
$823$	42.7252	1.48931	0.744654	−	0.667451i	$-0.232614\pi$
$823$	42.7252	1.48931	0.744654	+	0.667451i	$0.232614\pi$
$824$	−11.3563	−0.395616
$825$	0	0
$826$	18.0912	0.629473
$827$	32.4208	1.12738	0.563691	−	0.825986i	$-0.309381\pi$
$827$	32.4208	1.12738	0.563691	+	0.825986i	$0.309381\pi$
$828$	0	0
$829$	−2.91149	−0.101120	−0.0505600	−	0.998721i	$-0.516101\pi$
$829$	−2.91149	−0.101120	−0.0505600	+	0.998721i	$0.516101\pi$
$830$	−18.6871	−0.648639
$831$	0	0
$832$	32.8100	1.13748
$833$	−1.11677	−0.0386937
$834$	0	0
$835$	13.9200	0.481722
$836$	0	0
$837$	0	0
$838$	−21.1893	−0.731973
$839$	20.8465	0.719701	0.359850	−	0.933010i	$-0.382828\pi$
$839$	20.8465	0.719701	0.359850	+	0.933010i	$0.382828\pi$
$840$	0	0
$841$	−15.1496	−0.522398
$842$	−18.6960	−0.644305
$843$	0	0
$844$	5.42304	0.186669
$845$	−3.69863	−0.127237
$846$	0	0
$847$	0	0
$848$	12.8458	0.441127
$849$	0	0
$850$	0.971892	0.0333356
$851$	2.56437	0.0879056
$852$	0	0
$853$	−34.5509	−1.18300	−0.591500	−	0.806305i	$-0.701464\pi$
$853$	−34.5509	−1.18300	−0.591500	+	0.806305i	$0.701464\pi$
$854$	−12.9650	−0.443652
$855$	0	0
$856$	5.98641	0.204611
$857$	33.2969	1.13740	0.568699	−	0.822545i	$-0.307447\pi$
$857$	33.2969	1.13740	0.568699	+	0.822545i	$0.307447\pi$
$858$	0	0
$859$	−16.7665	−0.572067	−0.286034	−	0.958220i	$-0.592337\pi$
$859$	−16.7665	−0.572067	−0.286034	+	0.958220i	$0.592337\pi$
$860$	3.14266	0.107164
$861$	0	0
$862$	36.9028	1.25692
$863$	1.48415	0.0505211	0.0252605	−	0.999681i	$-0.491958\pi$
$863$	1.48415	0.0505211	0.0252605	+	0.999681i	$0.491958\pi$
$864$	0	0
$865$	10.8311	0.368269
$866$	−19.1593	−0.651060
$867$	0	0
$868$	−76.8068	−2.60699
$869$	0	0
$870$	0	0
$871$	20.6092	0.698316
$872$	−6.07803	−0.205828
$873$	0	0
$874$	−22.9999	−0.777984
$875$	3.06719	0.103690
$876$	0	0
$877$	24.8615	0.839513	0.419756	−	0.907637i	$-0.362115\pi$
$877$	24.8615	0.839513	0.419756	+	0.907637i	$0.362115\pi$
$878$	13.5554	0.457473
$879$	0	0
$880$	0	0
$881$	−32.6968	−1.10158	−0.550792	−	0.834643i	$-0.685674\pi$
$881$	−32.6968	−1.10158	−0.550792	+	0.834643i	$0.685674\pi$
$882$	0	0
$883$	47.6218	1.60260	0.801300	−	0.598263i	$-0.204142\pi$
$883$	47.6218	1.60260	0.801300	+	0.598263i	$0.204142\pi$
$884$	3.38136	0.113727
$885$	0	0
$886$	−85.6644	−2.87795
$887$	59.0960	1.98425	0.992125	−	0.125251i	$-0.0399736\pi$
$887$	59.0960	1.98425	0.992125	+	0.125251i	$0.0399736\pi$
$888$	0	0
$889$	1.38513	0.0464559
$890$	−14.1847	−0.475471
$891$	0	0
$892$	20.5076	0.686646
$893$	23.5304	0.787415
$894$	0	0
$895$	−22.7335	−0.759896
$896$	19.6824	0.657543
$897$	0	0
$898$	−25.4804	−0.850291
$899$	−38.9894	−1.30037
$900$	0	0
$901$	1.94265	0.0647190
$902$	0	0
$903$	0	0
$904$	−2.48287	−0.0825791
$905$	2.39831	0.0797227
$906$	0	0
$907$	34.6576	1.15079	0.575393	−	0.817877i	$-0.304849\pi$
$907$	34.6576	1.15079	0.575393	+	0.817877i	$0.304849\pi$
$908$	14.8205	0.491836
$909$	0	0
$910$	19.6001	0.649737
$911$	10.1883	0.337552	0.168776	−	0.985654i	$-0.446019\pi$
$911$	10.1883	0.337552	0.168776	+	0.985654i	$0.446019\pi$
$912$	0	0
$913$	0	0
$914$	−20.7484	−0.686298
$915$	0	0
$916$	55.3724	1.82956
$917$	1.92927	0.0637101
$918$	0	0
$919$	36.1289	1.19178	0.595891	−	0.803065i	$-0.296799\pi$
$919$	36.1289	1.19178	0.595891	+	0.803065i	$0.296799\pi$
$920$	−1.13682	−0.0374797
$921$	0	0
$922$	6.54840	0.215660
$923$	−19.8907	−0.654711
$924$	0	0
$925$	1.84453	0.0606479
$926$	51.0242	1.67676
$927$	0	0
$928$	30.0039	0.984927
$929$	−28.1240	−0.922717	−0.461359	−	0.887214i	$-0.652638\pi$
$929$	−28.1240	−0.922717	−0.461359	+	0.887214i	$0.652638\pi$
$930$	0	0
$931$	19.0097	0.623019
$932$	−66.5990	−2.18152
$933$	0	0
$934$	69.5087	2.27439
$935$	0	0
$936$	0	0
$937$	−0.0851677	−0.00278231	−0.00139115	−	0.999999i	$-0.500443\pi$
$937$	−0.0851677	−0.00278231	−0.00139115	+	0.999999i	$0.500443\pi$
$938$	43.4283	1.41798
$939$	0	0
$940$	7.12340	0.232340
$941$	−41.8154	−1.36314	−0.681572	−	0.731751i	$-0.738703\pi$
$941$	−41.8154	−1.36314	−0.681572	+	0.731751i	$0.738703\pi$
$942$	0	0
$943$	−6.12774	−0.199547
$944$	8.63420	0.281019
$945$	0	0
$946$	0	0
$947$	−8.92463	−0.290012	−0.145006	−	0.989431i	$-0.546320\pi$
$947$	−8.92463	−0.290012	−0.145006	+	0.989431i	$0.546320\pi$
$948$	0	0
$949$	−30.1194	−0.977716
$950$	−16.5437	−0.536747
$951$	0	0
$952$	1.16335	0.0377042
$953$	5.26383	0.170512	0.0852561	−	0.996359i	$-0.472829\pi$
$953$	5.26383	0.170512	0.0852561	+	0.996359i	$0.472829\pi$
$954$	0	0
$955$	−17.2462	−0.558075
$956$	−38.9283	−1.25903
$957$	0	0
$958$	36.2438	1.17098
$959$	−34.3016	−1.10765
$960$	0	0
$961$	78.7564	2.54053
$962$	11.7870	0.380029
$963$	0	0
$964$	−10.4947	−0.338013
$965$	2.58574	0.0832378
$966$	0	0
$967$	18.5421	0.596275	0.298138	−	0.954523i	$-0.403635\pi$
$967$	18.5421	0.596275	0.298138	+	0.954523i	$0.403635\pi$
$968$	0	0
$969$	0	0
$970$	−32.1719	−1.03298
$971$	24.2230	0.777354	0.388677	−	0.921374i	$-0.372932\pi$
$971$	24.2230	0.777354	0.388677	+	0.921374i	$0.372932\pi$
$972$	0	0
$973$	7.28442	0.233528
$974$	−16.0093	−0.512972
$975$	0	0
$976$	−6.18765	−0.198062
$977$	−49.0618	−1.56963	−0.784813	−	0.619732i	$-0.787241\pi$
$977$	−49.0618	−1.56963	−0.784813	+	0.619732i	$0.787241\pi$
$978$	0	0
$979$	0	0
$980$	5.75485	0.183832
$981$	0	0
$982$	64.2340	2.04979
$983$	−49.1394	−1.56731	−0.783653	−	0.621199i	$-0.786646\pi$
$983$	−49.1394	−1.56731	−0.783653	+	0.621199i	$0.786646\pi$
$984$	0	0
$985$	0.144731	0.00461151
$986$	3.61701	0.115189
$987$	0	0
$988$	−57.5578	−1.83116
$989$	1.82788	0.0581233
$990$	0	0
$991$	30.9620	0.983541	0.491771	−	0.870725i	$-0.336350\pi$
$991$	30.9620	0.983541	0.491771	+	0.870725i	$0.336350\pi$
$992$	−84.4619	−2.68167
$993$	0	0
$994$	−41.9143	−1.32944
$995$	−7.54177	−0.239090
$996$	0	0
$997$	−22.8939	−0.725058	−0.362529	−	0.931972i	$-0.618087\pi$
$997$	−22.8939	−0.725058	−0.362529	+	0.931972i	$0.618087\pi$
$998$	78.5280	2.48576
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bi.1.1		4
3.2	odd	2		605.2.a.k.1.4		4
11.7	odd	10		495.2.n.e.181.2		8
11.8	odd	10		495.2.n.e.361.2		8
11.10	odd	2		5445.2.a.bp.1.4		4
12.11	even	2		9680.2.a.cm.1.1		4
15.14	odd	2		3025.2.a.w.1.1		4
33.2	even	10		605.2.g.m.81.2		8
33.5	odd	10		605.2.g.e.366.1		8
33.8	even	10		55.2.g.b.31.1	yes	8
33.14	odd	10		605.2.g.k.251.2		8
33.17	even	10		605.2.g.m.366.2		8
33.20	odd	10		605.2.g.e.81.1		8
33.26	odd	10		605.2.g.k.511.2		8
33.29	even	10		55.2.g.b.16.1	✓	8
33.32	even	2		605.2.a.j.1.1		4
132.95	odd	10		880.2.bo.h.401.2		8
132.107	odd	10		880.2.bo.h.801.2		8
132.131	odd	2		9680.2.a.cn.1.1		4
165.8	odd	20		275.2.z.a.174.4		16
165.29	even	10		275.2.h.a.126.2		8
165.62	odd	20		275.2.z.a.49.4		16
165.74	even	10		275.2.h.a.251.2		8
165.107	odd	20		275.2.z.a.174.1		16
165.128	odd	20		275.2.z.a.49.1		16
165.164	even	2		3025.2.a.bd.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.16.1	✓	8	33.29	even	10
55.2.g.b.31.1	yes	8	33.8	even	10
275.2.h.a.126.2		8	165.29	even	10
275.2.h.a.251.2		8	165.74	even	10
275.2.z.a.49.1		16	165.128	odd	20
275.2.z.a.49.4		16	165.62	odd	20
275.2.z.a.174.1		16	165.107	odd	20
275.2.z.a.174.4		16	165.8	odd	20
495.2.n.e.181.2		8	11.7	odd	10
495.2.n.e.361.2		8	11.8	odd	10
605.2.a.j.1.1		4	33.32	even	2
605.2.a.k.1.4		4	3.2	odd	2
605.2.g.e.81.1		8	33.20	odd	10
605.2.g.e.366.1		8	33.5	odd	10
605.2.g.k.251.2		8	33.14	odd	10
605.2.g.k.511.2		8	33.26	odd	10
605.2.g.m.81.2		8	33.2	even	10
605.2.g.m.366.2		8	33.17	even	10
880.2.bo.h.401.2		8	132.95	odd	10
880.2.bo.h.801.2		8	132.107	odd	10
3025.2.a.w.1.1		4	15.14	odd	2
3025.2.a.bd.1.4		4	165.164	even	2
5445.2.a.bi.1.1		4	1.1	even	1	trivial
5445.2.a.bp.1.4		4	11.10	odd	2
9680.2.a.cm.1.1		4	12.11	even	2
9680.2.a.cn.1.1		4	132.131	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09529 q^{2} +2.39026 q^{4} +1.00000 q^{5} +3.06719 q^{7} -0.817703 q^{8} +O(q^{10})\)	\(q-2.09529 q^{2} +2.39026 q^{4} +1.00000 q^{5} +3.06719 q^{7} -0.817703 q^{8} -2.09529 q^{10} -3.04981 q^{13} -6.42666 q^{14} -3.06719 q^{16} -0.463845 q^{17} +7.89563 q^{19} +2.39026 q^{20} +1.39026 q^{23} +1.00000 q^{25} +6.39026 q^{26} +7.33136 q^{28} +3.72162 q^{29} -10.4765 q^{31} +8.06206 q^{32} +0.971892 q^{34} +3.06719 q^{35} +1.84453 q^{37} -16.5437 q^{38} -0.817703 q^{40} -4.40763 q^{41} +1.31478 q^{43} -2.91300 q^{46} +2.98018 q^{47} +2.40763 q^{49} -2.09529 q^{50} -7.28984 q^{52} -4.18814 q^{53} -2.50805 q^{56} -7.79789 q^{58} -2.81502 q^{59} +2.01737 q^{61} +21.9513 q^{62} -10.7580 q^{64} -3.04981 q^{65} -6.75753 q^{67} -1.10871 q^{68} -6.42666 q^{70} +6.52195 q^{71} +9.87581 q^{73} -3.86484 q^{74} +18.8726 q^{76} +11.5579 q^{79} -3.06719 q^{80} +9.23528 q^{82} +8.91861 q^{83} -0.463845 q^{85} -2.75485 q^{86} +6.76978 q^{89} -9.35435 q^{91} +3.32307 q^{92} -6.24436 q^{94} +7.89563 q^{95} +15.3543 q^{97} -5.04469 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8	\( 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8} - q^{10} + q^{13} - 2 q^{14} - 3 q^{16} + q^{17} + 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} + 13 q^{28} - 12 q^{29} - 5 q^{31} + 8 q^{32} + 2 q^{34} + 3 q^{35} + 7 q^{37} - 20 q^{38} - 3 q^{40} - 11 q^{41} + 19 q^{43} - 4 q^{46} - 5 q^{47} + 3 q^{49} - q^{50} - 11 q^{52} + 11 q^{53} - 11 q^{56} - 14 q^{58} - 9 q^{59} + 12 q^{61} + 35 q^{62} - 3 q^{64} + q^{65} - 19 q^{67} + 3 q^{68} - 2 q^{70} - 5 q^{71} + 11 q^{73} + 34 q^{79} - 3 q^{80} - 6 q^{82} + 11 q^{83} + q^{85} - q^{86} + 8 q^{89} - 8 q^{91} + 12 q^{92} - q^{94} + 20 q^{95} + 32 q^{97} + 8 q^{98}+O(q^{100}) \) 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8 - q^10 + q^13 - 2 * q^14 - 3 * q^16 + q^17 + 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 + 13 * q^28 - 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^37 - 20 * q^38 - 3 * q^40 - 11 * q^41 + 19 * q^43 - 4 * q^46 - 5 * q^47 + 3 * q^49 - q^50 - 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 + 12 * q^61 + 35 * q^62 - 3 * q^64 + q^65 - 19 * q^67 + 3 * q^68 - 2 * q^70 - 5 * q^71 + 11 * q^73 + 34 * q^79 - 3 * q^80 - 6 * q^82 + 11 * q^83 + q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 - q^94 + 20 * q^95 + 32 * q^97 + 8 * q^98

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