LMFDB - Embedded newform 5445.2.a.ca.1.7

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 $ x^8 - 4x^7 - 3x^6 + 22x^5 - 3x^4 - 32x^3 + 9x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.13569$ 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)

$f(q)$	$=$	$q+1.13569 q^{2} -0.710206 q^{4} -1.00000 q^{5} +4.10132 q^{7} -3.07796 q^{8} +O(q^{10})$	$q+1.13569 q^{2} -0.710206 q^{4} -1.00000 q^{5} +4.10132 q^{7} -3.07796 q^{8} -1.13569 q^{10} -2.27388 q^{13} +4.65784 q^{14} -2.07520 q^{16} +5.01888 q^{17} -7.57794 q^{19} +0.710206 q^{20} +1.97494 q^{23} +1.00000 q^{25} -2.58242 q^{26} -2.91278 q^{28} -4.00136 q^{29} -3.81490 q^{31} +3.79913 q^{32} +5.69990 q^{34} -4.10132 q^{35} -6.73089 q^{37} -8.60619 q^{38} +3.07796 q^{40} -4.02336 q^{41} +9.57903 q^{43} +2.24292 q^{46} -3.18673 q^{47} +9.82086 q^{49} +1.13569 q^{50} +1.61492 q^{52} -11.2291 q^{53} -12.6237 q^{56} -4.54431 q^{58} +9.53769 q^{59} +3.61186 q^{61} -4.33255 q^{62} +8.46503 q^{64} +2.27388 q^{65} -6.31504 q^{67} -3.56444 q^{68} -4.65784 q^{70} -12.2950 q^{71} -3.89030 q^{73} -7.64422 q^{74} +5.38189 q^{76} +17.5747 q^{79} +2.07520 q^{80} -4.56930 q^{82} -13.8475 q^{83} -5.01888 q^{85} +10.8788 q^{86} -7.85399 q^{89} -9.32590 q^{91} -1.40261 q^{92} -3.61914 q^{94} +7.57794 q^{95} +5.17866 q^{97} +11.1535 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) $ 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8	$ 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} + 6 q^{13} - 14 q^{14} + 14 q^{16} - 8 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} + 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} + 20 q^{38} + 12 q^{40} - 22 q^{41} + 14 q^{43} - 2 q^{46} - 10 q^{47} - 4 q^{50} - 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} - 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} + 2 q^{71} + 16 q^{73} - 24 q^{74} - 22 q^{76} - 16 q^{79} - 14 q^{80} + 10 q^{82} - 46 q^{83} + 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} - 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8 + 4 * q^10 + 6 * q^13 - 14 * q^14 + 14 * q^16 - 8 * q^17 - 2 * q^19 - 6 * q^20 - 4 * q^23 + 8 * q^25 + 2 * q^26 + 24 * q^28 - 22 * q^29 + 10 * q^31 - 28 * q^32 - 2 * q^34 - 8 * q^35 - 14 * q^37 + 20 * q^38 + 12 * q^40 - 22 * q^41 + 14 * q^43 - 2 * q^46 - 10 * q^47 - 4 * q^50 - 10 * q^52 + 18 * q^53 - 34 * q^56 + 12 * q^58 - 2 * q^59 - 14 * q^61 - 30 * q^62 + 30 * q^64 - 6 * q^65 + 10 * q^67 - 6 * q^68 + 14 * q^70 + 2 * q^71 + 16 * q^73 - 24 * q^74 - 22 * q^76 - 16 * q^79 - 14 * q^80 + 10 * q^82 - 46 * q^83 + 8 * q^85 + 28 * q^86 - 38 * q^89 + 8 * q^91 + 24 * q^92 - 10 * q^94 + 2 * q^95 - 4 * q^97 - 4 * 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 $

$2$	1.13569	0.803055	0.401527	−	0.915847i	$-0.368479\pi$
$2$	1.13569	0.803055	0.401527	+	0.915847i	$0.368479\pi$
$3$	0	0
$3$	0	0
$4$	−0.710206	−0.355103
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.10132	1.55015	0.775077	−	0.631866i	$-0.217711\pi$
$7$	4.10132	1.55015	0.775077	+	0.631866i	$0.217711\pi$
$8$	−3.07796	−1.08822
$9$	0	0
$10$	−1.13569	−0.359137
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.27388	−0.630660	−0.315330	−	0.948982i	$-0.602115\pi$
$13$	−2.27388	−0.630660	−0.315330	+	0.948982i	$0.602115\pi$
$14$	4.65784	1.24486
$15$	0	0
$16$	−2.07520	−0.518799
$17$	5.01888	1.21726	0.608629	−	0.793455i	$-0.291720\pi$
$17$	5.01888	1.21726	0.608629	+	0.793455i	$0.291720\pi$
$18$	0	0
$19$	−7.57794	−1.73850	−0.869249	−	0.494375i	$-0.835397\pi$
$19$	−7.57794	−1.73850	−0.869249	+	0.494375i	$0.835397\pi$
$20$	0.710206	0.158807
$21$	0	0
$22$	0	0
$23$	1.97494	0.411803	0.205902	−	0.978573i	$-0.433987\pi$
$23$	1.97494	0.411803	0.205902	+	0.978573i	$0.433987\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.58242	−0.506455
$27$	0	0
$28$	−2.91278	−0.550464
$29$	−4.00136	−0.743033	−0.371517	−	0.928426i	$-0.621162\pi$
$29$	−4.00136	−0.743033	−0.371517	+	0.928426i	$0.621162\pi$
$30$	0	0
$31$	−3.81490	−0.685176	−0.342588	−	0.939486i	$-0.611304\pi$
$31$	−3.81490	−0.685176	−0.342588	+	0.939486i	$0.611304\pi$
$32$	3.79913	0.671598
$33$	0	0
$34$	5.69990	0.977525
$35$	−4.10132	−0.693250
$36$	0	0
$37$	−6.73089	−1.10655	−0.553276	−	0.832998i	$-0.686623\pi$
$37$	−6.73089	−1.10655	−0.553276	+	0.832998i	$0.686623\pi$
$38$	−8.60619	−1.39611
$39$	0	0
$40$	3.07796	0.486668
$41$	−4.02336	−0.628344	−0.314172	−	0.949366i	$-0.601727\pi$
$41$	−4.02336	−0.628344	−0.314172	+	0.949366i	$0.601727\pi$
$42$	0	0
$43$	9.57903	1.46079	0.730394	−	0.683026i	$-0.239336\pi$
$43$	9.57903	1.46079	0.730394	+	0.683026i	$0.239336\pi$
$44$	0	0
$45$	0	0
$46$	2.24292	0.330701
$47$	−3.18673	−0.464832	−0.232416	−	0.972616i	$-0.574663\pi$
$47$	−3.18673	−0.464832	−0.232416	+	0.972616i	$0.574663\pi$
$48$	0	0
$49$	9.82086	1.40298
$50$	1.13569	0.160611
$51$	0	0
$52$	1.61492	0.223949
$53$	−11.2291	−1.54244	−0.771219	−	0.636570i	$-0.780352\pi$
$53$	−11.2291	−1.54244	−0.771219	+	0.636570i	$0.780352\pi$
$54$	0	0
$55$	0	0
$56$	−12.6237	−1.68691
$57$	0	0
$58$	−4.54431	−0.596697
$59$	9.53769	1.24170	0.620851	−	0.783929i	$-0.286787\pi$
$59$	9.53769	1.24170	0.620851	+	0.783929i	$0.286787\pi$
$60$	0	0
$61$	3.61186	0.462452	0.231226	−	0.972900i	$-0.425726\pi$
$61$	3.61186	0.462452	0.231226	+	0.972900i	$0.425726\pi$
$62$	−4.33255	−0.550234
$63$	0	0
$64$	8.46503	1.05813
$65$	2.27388	0.282040
$66$	0	0
$67$	−6.31504	−0.771505	−0.385752	−	0.922602i	$-0.626058\pi$
$67$	−6.31504	−0.771505	−0.385752	+	0.922602i	$0.626058\pi$
$68$	−3.56444	−0.432252
$69$	0	0
$70$	−4.65784	−0.556718
$71$	−12.2950	−1.45915	−0.729574	−	0.683901i	$-0.760282\pi$
$71$	−12.2950	−1.45915	−0.729574	+	0.683901i	$0.760282\pi$
$72$	0	0
$73$	−3.89030	−0.455325	−0.227663	−	0.973740i	$-0.573108\pi$
$73$	−3.89030	−0.455325	−0.227663	+	0.973740i	$0.573108\pi$
$74$	−7.64422	−0.888622
$75$	0	0
$76$	5.38189	0.617345
$77$	0	0
$78$	0	0
$79$	17.5747	1.97731	0.988656	−	0.150196i	$-0.0479905\pi$
$79$	17.5747	1.97731	0.988656	+	0.150196i	$0.0479905\pi$
$80$	2.07520	0.232014
$81$	0	0
$82$	−4.56930	−0.504594
$83$	−13.8475	−1.51996	−0.759982	−	0.649944i	$-0.774792\pi$
$83$	−13.8475	−1.51996	−0.759982	+	0.649944i	$0.774792\pi$
$84$	0	0
$85$	−5.01888	−0.544374
$86$	10.8788	1.17309
$87$	0	0
$88$	0	0
$89$	−7.85399	−0.832522	−0.416261	−	0.909245i	$-0.636660\pi$
$89$	−7.85399	−0.832522	−0.416261	+	0.909245i	$0.636660\pi$
$90$	0	0
$91$	−9.32590	−0.977620
$92$	−1.40261	−0.146232
$93$	0	0
$94$	−3.61914	−0.373286
$95$	7.57794	0.777480
$96$	0	0
$97$	5.17866	0.525813	0.262907	−	0.964821i	$-0.415319\pi$
$97$	5.17866	0.525813	0.262907	+	0.964821i	$0.415319\pi$
$98$	11.1535	1.12667
$99$	0	0
$100$	−0.710206	−0.0710206
$101$	3.99265	0.397284	0.198642	−	0.980072i	$-0.436347\pi$
$101$	3.99265	0.397284	0.198642	+	0.980072i	$0.436347\pi$
$102$	0	0
$103$	−0.940967	−0.0927162	−0.0463581	−	0.998925i	$-0.514762\pi$
$103$	−0.940967	−0.0927162	−0.0463581	+	0.998925i	$0.514762\pi$
$104$	6.99889	0.686298
$105$	0	0
$106$	−12.7528	−1.23866
$107$	−14.5295	−1.40462	−0.702308	−	0.711873i	$-0.747847\pi$
$107$	−14.5295	−1.40462	−0.702308	+	0.711873i	$0.747847\pi$
$108$	0	0
$109$	7.70193	0.737711	0.368855	−	0.929487i	$-0.379750\pi$
$109$	7.70193	0.737711	0.368855	+	0.929487i	$0.379750\pi$
$110$	0	0
$111$	0	0
$112$	−8.51105	−0.804219
$113$	0.675502	0.0635459	0.0317729	−	0.999495i	$-0.489885\pi$
$113$	0.675502	0.0635459	0.0317729	+	0.999495i	$0.489885\pi$
$114$	0	0
$115$	−1.97494	−0.184164
$116$	2.84179	0.263853
$117$	0	0
$118$	10.8319	0.997155
$119$	20.5841	1.88694
$120$	0	0
$121$	0	0
$122$	4.10196	0.371374
$123$	0	0
$124$	2.70936	0.243308
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.41105	−0.391418	−0.195709	−	0.980662i	$-0.562701\pi$
$127$	−4.41105	−0.391418	−0.195709	+	0.980662i	$0.562701\pi$
$128$	2.01540	0.178138
$129$	0	0
$130$	2.58242	0.226493
$131$	−21.4004	−1.86976	−0.934882	−	0.354959i	$-0.884495\pi$
$131$	−21.4004	−1.86976	−0.934882	+	0.354959i	$0.884495\pi$
$132$	0	0
$133$	−31.0796	−2.69494
$134$	−7.17193	−0.619561
$135$	0	0
$136$	−15.4479	−1.32465
$137$	3.82130	0.326475	0.163238	−	0.986587i	$-0.447806\pi$
$137$	3.82130	0.326475	0.163238	+	0.986587i	$0.447806\pi$
$138$	0	0
$139$	6.29721	0.534122	0.267061	−	0.963680i	$-0.413947\pi$
$139$	6.29721	0.534122	0.267061	+	0.963680i	$0.413947\pi$
$140$	2.91278	0.246175
$141$	0	0
$142$	−13.9633	−1.17178
$143$	0	0
$144$	0	0
$145$	4.00136	0.332295
$146$	−4.41818	−0.365651
$147$	0	0
$148$	4.78032	0.392940
$149$	−22.4959	−1.84293	−0.921467	−	0.388457i	$-0.873008\pi$
$149$	−22.4959	−1.84293	−0.921467	+	0.388457i	$0.873008\pi$
$150$	0	0
$151$	−2.99699	−0.243892	−0.121946	−	0.992537i	$-0.538913\pi$
$151$	−2.99699	−0.243892	−0.121946	+	0.992537i	$0.538913\pi$
$152$	23.3246	1.89187
$153$	0	0
$154$	0	0
$155$	3.81490	0.306420
$156$	0	0
$157$	−10.1899	−0.813244	−0.406622	−	0.913597i	$-0.633293\pi$
$157$	−10.1899	−0.813244	−0.406622	+	0.913597i	$0.633293\pi$
$158$	19.9595	1.58789
$159$	0	0
$160$	−3.79913	−0.300348
$161$	8.09986	0.638359
$162$	0	0
$163$	−19.8034	−1.55112	−0.775561	−	0.631273i	$-0.782533\pi$
$163$	−19.8034	−1.55112	−0.775561	+	0.631273i	$0.782533\pi$
$164$	2.85741	0.223127
$165$	0	0
$166$	−15.7265	−1.22061
$167$	−4.96739	−0.384388	−0.192194	−	0.981357i	$-0.561560\pi$
$167$	−4.96739	−0.384388	−0.192194	+	0.981357i	$0.561560\pi$
$168$	0	0
$169$	−7.82948	−0.602268
$170$	−5.69990	−0.437162
$171$	0	0
$172$	−6.80308	−0.518730
$173$	−16.2510	−1.23554	−0.617768	−	0.786360i	$-0.711963\pi$
$173$	−16.2510	−1.23554	−0.617768	+	0.786360i	$0.711963\pi$
$174$	0	0
$175$	4.10132	0.310031
$176$	0	0
$177$	0	0
$178$	−8.91971	−0.668561
$179$	−3.18448	−0.238019	−0.119010	−	0.992893i	$-0.537972\pi$
$179$	−3.18448	−0.238019	−0.119010	+	0.992893i	$0.537972\pi$
$180$	0	0
$181$	1.50099	0.111568	0.0557838	−	0.998443i	$-0.482234\pi$
$181$	1.50099	0.111568	0.0557838	+	0.998443i	$0.482234\pi$
$182$	−10.5913	−0.785083
$183$	0	0
$184$	−6.07877	−0.448133
$185$	6.73089	0.494865
$186$	0	0
$187$	0	0
$188$	2.26323	0.165063
$189$	0	0
$190$	8.60619	0.624359
$191$	−10.4428	−0.755613	−0.377807	−	0.925885i	$-0.623322\pi$
$191$	−10.4428	−0.755613	−0.377807	+	0.925885i	$0.623322\pi$
$192$	0	0
$193$	6.02750	0.433869	0.216935	−	0.976186i	$-0.430394\pi$
$193$	6.02750	0.433869	0.216935	+	0.976186i	$0.430394\pi$
$194$	5.88136	0.422257
$195$	0	0
$196$	−6.97483	−0.498202
$197$	9.07040	0.646239	0.323120	−	0.946358i	$-0.395268\pi$
$197$	9.07040	0.646239	0.323120	+	0.946358i	$0.395268\pi$
$198$	0	0
$199$	17.3143	1.22738	0.613689	−	0.789548i	$-0.289685\pi$
$199$	17.3143	1.22738	0.613689	+	0.789548i	$0.289685\pi$
$200$	−3.07796	−0.217644
$201$	0	0
$202$	4.53442	0.319041
$203$	−16.4109	−1.15182
$204$	0	0
$205$	4.02336	0.281004
$206$	−1.06865	−0.0744562
$207$	0	0
$208$	4.71874	0.327186
$209$	0	0
$210$	0	0
$211$	−15.7153	−1.08189	−0.540945	−	0.841058i	$-0.681933\pi$
$211$	−15.7153	−1.08189	−0.540945	+	0.841058i	$0.681933\pi$
$212$	7.97498	0.547724
$213$	0	0
$214$	−16.5010	−1.12798
$215$	−9.57903	−0.653285
$216$	0	0
$217$	−15.6461	−1.06213
$218$	8.74701	0.592422
$219$	0	0
$220$	0	0
$221$	−11.4123	−0.767676
$222$	0	0
$223$	1.08382	0.0725778	0.0362889	−	0.999341i	$-0.488446\pi$
$223$	1.08382	0.0725778	0.0362889	+	0.999341i	$0.488446\pi$
$224$	15.5815	1.04108
$225$	0	0
$226$	0.767162	0.0510308
$227$	1.99171	0.132195	0.0660974	−	0.997813i	$-0.478945\pi$
$227$	1.99171	0.132195	0.0660974	+	0.997813i	$0.478945\pi$
$228$	0	0
$229$	8.13974	0.537889	0.268945	−	0.963156i	$-0.413325\pi$
$229$	8.13974	0.537889	0.268945	+	0.963156i	$0.413325\pi$
$230$	−2.24292	−0.147894
$231$	0	0
$232$	12.3160	0.808585
$233$	−24.4296	−1.60043	−0.800217	−	0.599710i	$-0.795283\pi$
$233$	−24.4296	−1.60043	−0.800217	+	0.599710i	$0.795283\pi$
$234$	0	0
$235$	3.18673	0.207879
$236$	−6.77372	−0.440932
$237$	0	0
$238$	23.3771	1.51531
$239$	0.0709845	0.00459161	0.00229580	−	0.999997i	$-0.499269\pi$
$239$	0.0709845	0.00459161	0.00229580	+	0.999997i	$0.499269\pi$
$240$	0	0
$241$	−12.6653	−0.815842	−0.407921	−	0.913017i	$-0.633746\pi$
$241$	−12.6653	−0.815842	−0.407921	+	0.913017i	$0.633746\pi$
$242$	0	0
$243$	0	0
$244$	−2.56517	−0.164218
$245$	−9.82086	−0.627431
$246$	0	0
$247$	17.2313	1.09640
$248$	11.7421	0.745624
$249$	0	0
$250$	−1.13569	−0.0718274
$251$	7.65705	0.483309	0.241654	−	0.970362i	$-0.422310\pi$
$251$	7.65705	0.483309	0.241654	+	0.970362i	$0.422310\pi$
$252$	0	0
$253$	0	0
$254$	−5.00959	−0.314330
$255$	0	0
$256$	−14.6412	−0.915075
$257$	23.1101	1.44157	0.720784	−	0.693160i	$-0.243782\pi$
$257$	23.1101	1.44157	0.720784	+	0.693160i	$0.243782\pi$
$258$	0	0
$259$	−27.6056	−1.71533
$260$	−1.61492	−0.100153
$261$	0	0
$262$	−24.3043	−1.50152
$263$	9.67819	0.596783	0.298391	−	0.954444i	$-0.403550\pi$
$263$	9.67819	0.596783	0.298391	+	0.954444i	$0.403550\pi$
$264$	0	0
$265$	11.2291	0.689799
$266$	−35.2968	−2.16418
$267$	0	0
$268$	4.48498	0.273964
$269$	1.82359	0.111186	0.0555931	−	0.998454i	$-0.482295\pi$
$269$	1.82359	0.111186	0.0555931	+	0.998454i	$0.482295\pi$
$270$	0	0
$271$	−14.7934	−0.898635	−0.449318	−	0.893372i	$-0.648333\pi$
$271$	−14.7934	−0.898635	−0.449318	+	0.893372i	$0.648333\pi$
$272$	−10.4152	−0.631512
$273$	0	0
$274$	4.33981	0.262178
$275$	0	0
$276$	0	0
$277$	16.9801	1.02024	0.510118	−	0.860104i	$-0.329602\pi$
$277$	16.9801	1.02024	0.510118	+	0.860104i	$0.329602\pi$
$278$	7.15169	0.428930
$279$	0	0
$280$	12.6237	0.754410
$281$	−4.60152	−0.274504	−0.137252	−	0.990536i	$-0.543827\pi$
$281$	−4.60152	−0.274504	−0.137252	+	0.990536i	$0.543827\pi$
$282$	0	0
$283$	13.7869	0.819545	0.409772	−	0.912188i	$-0.365608\pi$
$283$	13.7869	0.819545	0.409772	+	0.912188i	$0.365608\pi$
$284$	8.73198	0.518148
$285$	0	0
$286$	0	0
$287$	−16.5011	−0.974030
$288$	0	0
$289$	8.18917	0.481716
$290$	4.54431	0.266851
$291$	0	0
$292$	2.76291	0.161687
$293$	−31.4245	−1.83584	−0.917919	−	0.396769i	$-0.870131\pi$
$293$	−31.4245	−1.83584	−0.917919	+	0.396769i	$0.870131\pi$
$294$	0	0
$295$	−9.53769	−0.555306
$296$	20.7174	1.20417
$297$	0	0
$298$	−25.5484	−1.47998
$299$	−4.49077	−0.259708
$300$	0	0
$301$	39.2867	2.26445
$302$	−3.40366	−0.195858
$303$	0	0
$304$	15.7257	0.901931
$305$	−3.61186	−0.206815
$306$	0	0
$307$	20.1733	1.15135	0.575676	−	0.817678i	$-0.304739\pi$
$307$	20.1733	1.15135	0.575676	+	0.817678i	$0.304739\pi$
$308$	0	0
$309$	0	0
$310$	4.33255	0.246072
$311$	18.5810	1.05363	0.526816	−	0.849979i	$-0.323386\pi$
$311$	18.5810	1.05363	0.526816	+	0.849979i	$0.323386\pi$
$312$	0	0
$313$	11.4086	0.644854	0.322427	−	0.946594i	$-0.395501\pi$
$313$	11.4086	0.644854	0.322427	+	0.946594i	$0.395501\pi$
$314$	−11.5726	−0.653079
$315$	0	0
$316$	−12.4817	−0.702149
$317$	−25.4611	−1.43004	−0.715018	−	0.699106i	$-0.753581\pi$
$317$	−25.4611	−1.43004	−0.715018	+	0.699106i	$0.753581\pi$
$318$	0	0
$319$	0	0
$320$	−8.46503	−0.473210
$321$	0	0
$322$	9.19894	0.512637
$323$	−38.0328	−2.11620
$324$	0	0
$325$	−2.27388	−0.126132
$326$	−22.4905	−1.24564
$327$	0	0
$328$	12.3837	0.683777
$329$	−13.0698	−0.720561
$330$	0	0
$331$	31.1055	1.70971	0.854857	−	0.518864i	$-0.173645\pi$
$331$	31.1055	1.70971	0.854857	+	0.518864i	$0.173645\pi$
$332$	9.83460	0.539744
$333$	0	0
$334$	−5.64142	−0.308684
$335$	6.31504	0.345027
$336$	0	0
$337$	−26.6950	−1.45417	−0.727084	−	0.686548i	$-0.759125\pi$
$337$	−26.6950	−1.45417	−0.727084	+	0.686548i	$0.759125\pi$
$338$	−8.89188	−0.483654
$339$	0	0
$340$	3.56444	0.193309
$341$	0	0
$342$	0	0
$343$	11.5692	0.624680
$344$	−29.4838	−1.58966
$345$	0	0
$346$	−18.4561	−0.992204
$347$	−6.79251	−0.364641	−0.182321	−	0.983239i	$-0.558361\pi$
$347$	−6.79251	−0.364641	−0.182321	+	0.983239i	$0.558361\pi$
$348$	0	0
$349$	−7.84626	−0.420001	−0.210000	−	0.977701i	$-0.567347\pi$
$349$	−7.84626	−0.420001	−0.210000	+	0.977701i	$0.567347\pi$
$350$	4.65784	0.248972
$351$	0	0
$352$	0	0
$353$	−18.4136	−0.980056	−0.490028	−	0.871707i	$-0.663014\pi$
$353$	−18.4136	−0.980056	−0.490028	+	0.871707i	$0.663014\pi$
$354$	0	0
$355$	12.2950	0.652551
$356$	5.57795	0.295631
$357$	0	0
$358$	−3.61659	−0.191142
$359$	25.4119	1.34119	0.670595	−	0.741823i	$-0.266039\pi$
$359$	25.4119	1.34119	0.670595	+	0.741823i	$0.266039\pi$
$360$	0	0
$361$	38.4251	2.02237
$362$	1.70466	0.0895949
$363$	0	0
$364$	6.62331	0.347156
$365$	3.89030	0.203628
$366$	0	0
$367$	−3.48293	−0.181808	−0.0909038	−	0.995860i	$-0.528976\pi$
$367$	−3.48293	−0.181808	−0.0909038	+	0.995860i	$0.528976\pi$
$368$	−4.09839	−0.213643
$369$	0	0
$370$	7.64422	0.397404
$371$	−46.0542	−2.39102
$372$	0	0
$373$	12.8348	0.664561	0.332281	−	0.943181i	$-0.392182\pi$
$373$	12.8348	0.664561	0.332281	+	0.943181i	$0.392182\pi$
$374$	0	0
$375$	0	0
$376$	9.80861	0.505840
$377$	9.09859	0.468601
$378$	0	0
$379$	−24.0638	−1.23607	−0.618037	−	0.786149i	$-0.712072\pi$
$379$	−24.0638	−1.23607	−0.618037	+	0.786149i	$0.712072\pi$
$380$	−5.38189	−0.276085
$381$	0	0
$382$	−11.8598	−0.606799
$383$	−3.17618	−0.162295	−0.0811477	−	0.996702i	$-0.525859\pi$
$383$	−3.17618	−0.162295	−0.0811477	+	0.996702i	$0.525859\pi$
$384$	0	0
$385$	0	0
$386$	6.84538	0.348421
$387$	0	0
$388$	−3.67791	−0.186718
$389$	−14.9001	−0.755463	−0.377731	−	0.925915i	$-0.623296\pi$
$389$	−14.9001	−0.755463	−0.377731	+	0.925915i	$0.623296\pi$
$390$	0	0
$391$	9.91198	0.501270
$392$	−30.2282	−1.52675
$393$	0	0
$394$	10.3012	0.518965
$395$	−17.5747	−0.884281
$396$	0	0
$397$	10.8837	0.546237	0.273119	−	0.961980i	$-0.411945\pi$
$397$	10.8837	0.546237	0.273119	+	0.961980i	$0.411945\pi$
$398$	19.6637	0.985651
$399$	0	0
$400$	−2.07520	−0.103760
$401$	−33.2847	−1.66216	−0.831080	−	0.556153i	$-0.812277\pi$
$401$	−33.2847	−1.66216	−0.831080	+	0.556153i	$0.812277\pi$
$402$	0	0
$403$	8.67461	0.432113
$404$	−2.83561	−0.141077
$405$	0	0
$406$	−18.6377	−0.924972
$407$	0	0
$408$	0	0
$409$	2.77694	0.137311	0.0686554	−	0.997640i	$-0.478129\pi$
$409$	2.77694	0.137311	0.0686554	+	0.997640i	$0.478129\pi$
$410$	4.56930	0.225661
$411$	0	0
$412$	0.668280	0.0329238
$413$	39.1172	1.92483
$414$	0	0
$415$	13.8475	0.679749
$416$	−8.63876	−0.423550
$417$	0	0
$418$	0	0
$419$	29.5565	1.44393	0.721965	−	0.691930i	$-0.243239\pi$
$419$	29.5565	1.44393	0.721965	+	0.691930i	$0.243239\pi$
$420$	0	0
$421$	0.291904	0.0142265	0.00711326	−	0.999975i	$-0.497736\pi$
$421$	0.291904	0.0142265	0.00711326	+	0.999975i	$0.497736\pi$
$422$	−17.8478	−0.868816
$423$	0	0
$424$	34.5627	1.67851
$425$	5.01888	0.243451
$426$	0	0
$427$	14.8134	0.716872
$428$	10.3189	0.498783
$429$	0	0
$430$	−10.8788	−0.524623
$431$	4.49580	0.216555	0.108277	−	0.994121i	$-0.465466\pi$
$431$	4.49580	0.216555	0.108277	+	0.994121i	$0.465466\pi$
$432$	0	0
$433$	−6.56823	−0.315649	−0.157824	−	0.987467i	$-0.550448\pi$
$433$	−6.56823	−0.315649	−0.157824	+	0.987467i	$0.550448\pi$
$434$	−17.7692	−0.852948
$435$	0	0
$436$	−5.46995	−0.261963
$437$	−14.9660	−0.715919
$438$	0	0
$439$	−26.7142	−1.27500	−0.637500	−	0.770451i	$-0.720031\pi$
$439$	−26.7142	−1.27500	−0.637500	+	0.770451i	$0.720031\pi$
$440$	0	0
$441$	0	0
$442$	−12.9609	−0.616486
$443$	23.1670	1.10070	0.550349	−	0.834935i	$-0.314495\pi$
$443$	23.1670	1.10070	0.550349	+	0.834935i	$0.314495\pi$
$444$	0	0
$445$	7.85399	0.372315
$446$	1.23088	0.0582839
$447$	0	0
$448$	34.7178	1.64026
$449$	29.0593	1.37139	0.685696	−	0.727888i	$-0.259498\pi$
$449$	29.0593	1.37139	0.685696	+	0.727888i	$0.259498\pi$
$450$	0	0
$451$	0	0
$452$	−0.479745	−0.0225653
$453$	0	0
$454$	2.26197	0.106160
$455$	9.32590	0.437205
$456$	0	0
$457$	7.62741	0.356795	0.178398	−	0.983958i	$-0.442909\pi$
$457$	7.62741	0.356795	0.178398	+	0.983958i	$0.442909\pi$
$458$	9.24423	0.431954
$459$	0	0
$460$	1.40261	0.0653971
$461$	34.0138	1.58418	0.792091	−	0.610404i	$-0.208993\pi$
$461$	34.0138	1.58418	0.792091	+	0.610404i	$0.208993\pi$
$462$	0	0
$463$	15.7564	0.732263	0.366131	−	0.930563i	$-0.380682\pi$
$463$	15.7564	0.732263	0.366131	+	0.930563i	$0.380682\pi$
$464$	8.30360	0.385485
$465$	0	0
$466$	−27.7445	−1.28524
$467$	−10.1508	−0.469724	−0.234862	−	0.972029i	$-0.575464\pi$
$467$	−10.1508	−0.469724	−0.234862	+	0.972029i	$0.575464\pi$
$468$	0	0
$469$	−25.9000	−1.19595
$470$	3.61914	0.166938
$471$	0	0
$472$	−29.3566	−1.35125
$473$	0	0
$474$	0	0
$475$	−7.57794	−0.347700
$476$	−14.6189	−0.670057
$477$	0	0
$478$	0.0806165	0.00368731
$479$	−4.04315	−0.184736	−0.0923681	−	0.995725i	$-0.529444\pi$
$479$	−4.04315	−0.184736	−0.0923681	+	0.995725i	$0.529444\pi$
$480$	0	0
$481$	15.3052	0.697858
$482$	−14.3838	−0.655166
$483$	0	0
$484$	0	0
$485$	−5.17866	−0.235151
$486$	0	0
$487$	7.37591	0.334234	0.167117	−	0.985937i	$-0.446554\pi$
$487$	7.37591	0.334234	0.167117	+	0.985937i	$0.446554\pi$
$488$	−11.1172	−0.503250
$489$	0	0
$490$	−11.1535	−0.503862
$491$	2.19528	0.0990717	0.0495358	−	0.998772i	$-0.484226\pi$
$491$	2.19528	0.0990717	0.0495358	+	0.998772i	$0.484226\pi$
$492$	0	0
$493$	−20.0823	−0.904463
$494$	19.5694	0.880470
$495$	0	0
$496$	7.91667	0.355469
$497$	−50.4258	−2.26191
$498$	0	0
$499$	8.06180	0.360896	0.180448	−	0.983585i	$-0.442245\pi$
$499$	8.06180	0.360896	0.180448	+	0.983585i	$0.442245\pi$
$500$	0.710206	0.0317614
$501$	0	0
$502$	8.69605	0.388124
$503$	−33.3563	−1.48728	−0.743642	−	0.668579i	$-0.766903\pi$
$503$	−33.3563	−1.48728	−0.743642	+	0.668579i	$0.766903\pi$
$504$	0	0
$505$	−3.99265	−0.177671
$506$	0	0
$507$	0	0
$508$	3.13276	0.138994
$509$	16.7877	0.744100	0.372050	−	0.928213i	$-0.378655\pi$
$509$	16.7877	0.744100	0.372050	+	0.928213i	$0.378655\pi$
$510$	0	0
$511$	−15.9554	−0.705825
$512$	−20.6587	−0.912993
$513$	0	0
$514$	26.2459	1.15766
$515$	0.940967	0.0414639
$516$	0	0
$517$	0	0
$518$	−31.3514	−1.37750
$519$	0	0
$520$	−6.99889	−0.306922
$521$	14.7904	0.647981	0.323991	−	0.946060i	$-0.394975\pi$
$521$	14.7904	0.647981	0.323991	+	0.946060i	$0.394975\pi$
$522$	0	0
$523$	−1.41489	−0.0618688	−0.0309344	−	0.999521i	$-0.509848\pi$
$523$	−1.41489	−0.0618688	−0.0309344	+	0.999521i	$0.509848\pi$
$524$	15.1987	0.663958
$525$	0	0
$526$	10.9914	0.479249
$527$	−19.1465	−0.834036
$528$	0	0
$529$	−19.0996	−0.830418
$530$	12.7528	0.553946
$531$	0	0
$532$	22.0729	0.956981
$533$	9.14863	0.396271
$534$	0	0
$535$	14.5295	0.628163
$536$	19.4374	0.839568
$537$	0	0
$538$	2.07103	0.0892887
$539$	0	0
$540$	0	0
$541$	2.52676	0.108634	0.0543169	−	0.998524i	$-0.482702\pi$
$541$	2.52676	0.108634	0.0543169	+	0.998524i	$0.482702\pi$
$542$	−16.8007	−0.721653
$543$	0	0
$544$	19.0674	0.817507
$545$	−7.70193	−0.329914
$546$	0	0
$547$	0.519111	0.0221956	0.0110978	−	0.999938i	$-0.496467\pi$
$547$	0.519111	0.0221956	0.0110978	+	0.999938i	$0.496467\pi$
$548$	−2.71391	−0.115932
$549$	0	0
$550$	0	0
$551$	30.3220	1.29176
$552$	0	0
$553$	72.0797	3.06514
$554$	19.2842	0.819306
$555$	0	0
$556$	−4.47231	−0.189668
$557$	11.8026	0.500091	0.250045	−	0.968234i	$-0.419554\pi$
$557$	11.8026	0.500091	0.250045	+	0.968234i	$0.419554\pi$
$558$	0	0
$559$	−21.7815	−0.921261
$560$	8.51105	0.359658
$561$	0	0
$562$	−5.22591	−0.220442
$563$	6.73272	0.283750	0.141875	−	0.989885i	$-0.454687\pi$
$563$	6.73272	0.283750	0.141875	+	0.989885i	$0.454687\pi$
$564$	0	0
$565$	−0.675502	−0.0284186
$566$	15.6576	0.658140
$567$	0	0
$568$	37.8435	1.58788
$569$	31.4915	1.32019	0.660096	−	0.751181i	$-0.270516\pi$
$569$	31.4915	1.32019	0.660096	+	0.751181i	$0.270516\pi$
$570$	0	0
$571$	−24.1049	−1.00876	−0.504380	−	0.863482i	$-0.668279\pi$
$571$	−24.1049	−1.00876	−0.504380	+	0.863482i	$0.668279\pi$
$572$	0	0
$573$	0	0
$574$	−18.7402	−0.782199
$575$	1.97494	0.0823606
$576$	0	0
$577$	−34.2109	−1.42422	−0.712110	−	0.702068i	$-0.752260\pi$
$577$	−34.2109	−1.42422	−0.712110	+	0.702068i	$0.752260\pi$
$578$	9.30037	0.386844
$579$	0	0
$580$	−2.84179	−0.117999
$581$	−56.7932	−2.35618
$582$	0	0
$583$	0	0
$584$	11.9742	0.495495
$585$	0	0
$586$	−35.6885	−1.47428
$587$	17.6006	0.726457	0.363228	−	0.931700i	$-0.381675\pi$
$587$	17.6006	0.726457	0.363228	+	0.931700i	$0.381675\pi$
$588$	0	0
$589$	28.9091	1.19118
$590$	−10.8319	−0.445941
$591$	0	0
$592$	13.9679	0.574078
$593$	−33.9143	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$593$	−33.9143	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$594$	0	0
$595$	−20.5841	−0.843864
$596$	15.9767	0.654431
$597$	0	0
$598$	−5.10012	−0.208560
$599$	30.1345	1.23126	0.615630	−	0.788035i	$-0.288901\pi$
$599$	30.1345	1.23126	0.615630	+	0.788035i	$0.288901\pi$
$600$	0	0
$601$	−27.7612	−1.13240	−0.566202	−	0.824266i	$-0.691588\pi$
$601$	−27.7612	−1.13240	−0.566202	+	0.824266i	$0.691588\pi$
$602$	44.6176	1.81848
$603$	0	0
$604$	2.12848	0.0866067
$605$	0	0
$606$	0	0
$607$	−25.4949	−1.03481	−0.517404	−	0.855742i	$-0.673101\pi$
$607$	−25.4949	−1.03481	−0.517404	+	0.855742i	$0.673101\pi$
$608$	−28.7896	−1.16757
$609$	0	0
$610$	−4.10196	−0.166084
$611$	7.24622	0.293151
$612$	0	0
$613$	2.06728	0.0834967	0.0417484	−	0.999128i	$-0.486707\pi$
$613$	2.06728	0.0834967	0.0417484	+	0.999128i	$0.486707\pi$
$614$	22.9107	0.924599
$615$	0	0
$616$	0	0
$617$	−16.7249	−0.673321	−0.336660	−	0.941626i	$-0.609297\pi$
$617$	−16.7249	−0.673321	−0.336660	+	0.941626i	$0.609297\pi$
$618$	0	0
$619$	−42.2465	−1.69803	−0.849015	−	0.528369i	$-0.822804\pi$
$619$	−42.2465	−1.69803	−0.849015	+	0.528369i	$0.822804\pi$
$620$	−2.70936	−0.108811
$621$	0	0
$622$	21.1023	0.846125
$623$	−32.2118	−1.29054
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.9567	0.517853
$627$	0	0
$628$	7.23693	0.288785
$629$	−33.7816	−1.34696
$630$	0	0
$631$	38.8620	1.54707	0.773536	−	0.633753i	$-0.218486\pi$
$631$	38.8620	1.54707	0.773536	+	0.633753i	$0.218486\pi$
$632$	−54.0943	−2.15175
$633$	0	0
$634$	−28.9159	−1.14840
$635$	4.41105	0.175047
$636$	0	0
$637$	−22.3314	−0.884803
$638$	0	0
$639$	0	0
$640$	−2.01540	−0.0796657
$641$	17.5446	0.692971	0.346486	−	0.938055i	$-0.387375\pi$
$641$	17.5446	0.692971	0.346486	+	0.938055i	$0.387375\pi$
$642$	0	0
$643$	36.6475	1.44524	0.722619	−	0.691246i	$-0.242938\pi$
$643$	36.6475	1.44524	0.722619	+	0.691246i	$0.242938\pi$
$644$	−5.75257	−0.226683
$645$	0	0
$646$	−43.1935	−1.69942
$647$	22.6182	0.889214	0.444607	−	0.895726i	$-0.353343\pi$
$647$	22.6182	0.889214	0.444607	+	0.895726i	$0.353343\pi$
$648$	0	0
$649$	0	0
$650$	−2.58242	−0.101291
$651$	0	0
$652$	14.0645	0.550808
$653$	−6.34671	−0.248366	−0.124183	−	0.992259i	$-0.539631\pi$
$653$	−6.34671	−0.248366	−0.124183	+	0.992259i	$0.539631\pi$
$654$	0	0
$655$	21.4004	0.836184
$656$	8.34927	0.325984
$657$	0	0
$658$	−14.8433	−0.578650
$659$	−45.8649	−1.78664	−0.893321	−	0.449420i	$-0.851631\pi$
$659$	−45.8649	−1.78664	−0.893321	+	0.449420i	$0.851631\pi$
$660$	0	0
$661$	46.4730	1.80759	0.903796	−	0.427964i	$-0.140769\pi$
$661$	46.4730	1.80759	0.903796	+	0.427964i	$0.140769\pi$
$662$	35.3263	1.37299
$663$	0	0
$664$	42.6221	1.65406
$665$	31.0796	1.20521
$666$	0	0
$667$	−7.90243	−0.305983
$668$	3.52787	0.136497
$669$	0	0
$670$	7.17193	0.277076
$671$	0	0
$672$	0	0
$673$	−11.1001	−0.427879	−0.213939	−	0.976847i	$-0.568629\pi$
$673$	−11.1001	−0.427879	−0.213939	+	0.976847i	$0.568629\pi$
$674$	−30.3173	−1.16778
$675$	0	0
$676$	5.56054	0.213867
$677$	15.9352	0.612441	0.306220	−	0.951961i	$-0.400936\pi$
$677$	15.9352	0.612441	0.306220	+	0.951961i	$0.400936\pi$
$678$	0	0
$679$	21.2394	0.815092
$680$	15.4479	0.592400
$681$	0	0
$682$	0	0
$683$	21.8344	0.835468	0.417734	−	0.908569i	$-0.362824\pi$
$683$	21.8344	0.835468	0.417734	+	0.908569i	$0.362824\pi$
$684$	0	0
$685$	−3.82130	−0.146004
$686$	13.1391	0.501653
$687$	0	0
$688$	−19.8784	−0.757856
$689$	25.5336	0.972753
$690$	0	0
$691$	−26.9932	−1.02687	−0.513435	−	0.858128i	$-0.671627\pi$
$691$	−26.9932	−1.02687	−0.513435	+	0.858128i	$0.671627\pi$
$692$	11.5415	0.438743
$693$	0	0
$694$	−7.71420	−0.292827
$695$	−6.29721	−0.238867
$696$	0	0
$697$	−20.1928	−0.764856
$698$	−8.91093	−0.337284
$699$	0	0
$700$	−2.91278	−0.110093
$701$	29.8366	1.12691	0.563455	−	0.826146i	$-0.309472\pi$
$701$	29.8366	1.12691	0.563455	+	0.826146i	$0.309472\pi$
$702$	0	0
$703$	51.0063	1.92374
$704$	0	0
$705$	0	0
$706$	−20.9121	−0.787039
$707$	16.3752	0.615851
$708$	0	0
$709$	33.4225	1.25521	0.627604	−	0.778533i	$-0.284036\pi$
$709$	33.4225	1.25521	0.627604	+	0.778533i	$0.284036\pi$
$710$	13.9633	0.524034
$711$	0	0
$712$	24.1743	0.905968
$713$	−7.53419	−0.282158
$714$	0	0
$715$	0	0
$716$	2.26164	0.0845213
$717$	0	0
$718$	28.8601	1.07705
$719$	11.2829	0.420782	0.210391	−	0.977617i	$-0.432526\pi$
$719$	11.2829	0.420782	0.210391	+	0.977617i	$0.432526\pi$
$720$	0	0
$721$	−3.85921	−0.143724
$722$	43.6391	1.62408
$723$	0	0
$724$	−1.06601	−0.0396180
$725$	−4.00136	−0.148607
$726$	0	0
$727$	43.3381	1.60732	0.803661	−	0.595087i	$-0.202882\pi$
$727$	43.3381	1.60732	0.803661	+	0.595087i	$0.202882\pi$
$728$	28.7047	1.06387
$729$	0	0
$730$	4.41818	0.163524
$731$	48.0760	1.77816
$732$	0	0
$733$	52.1642	1.92673	0.963365	−	0.268195i	$-0.0864273\pi$
$733$	52.1642	1.92673	0.963365	+	0.268195i	$0.0864273\pi$
$734$	−3.95554	−0.146002
$735$	0	0
$736$	7.50305	0.276566
$737$	0	0
$738$	0	0
$739$	−14.7098	−0.541109	−0.270555	−	0.962705i	$-0.587207\pi$
$739$	−14.7098	−0.541109	−0.270555	+	0.962705i	$0.587207\pi$
$740$	−4.78032	−0.175728
$741$	0	0
$742$	−52.3034	−1.92012
$743$	2.72748	0.100062	0.0500308	−	0.998748i	$-0.484068\pi$
$743$	2.72748	0.100062	0.0500308	+	0.998748i	$0.484068\pi$
$744$	0	0
$745$	22.4959	0.824185
$746$	14.5764	0.533679
$747$	0	0
$748$	0	0
$749$	−59.5900	−2.17737
$750$	0	0
$751$	16.7867	0.612555	0.306277	−	0.951942i	$-0.400916\pi$
$751$	16.7867	0.612555	0.306277	+	0.951942i	$0.400916\pi$
$752$	6.61309	0.241154
$753$	0	0
$754$	10.3332	0.376313
$755$	2.99699	0.109072
$756$	0	0
$757$	−22.9310	−0.833442	−0.416721	−	0.909034i	$-0.636821\pi$
$757$	−22.9310	−0.833442	−0.416721	+	0.909034i	$0.636821\pi$
$758$	−27.3291	−0.992636
$759$	0	0
$760$	−23.3246	−0.846071
$761$	7.81520	0.283301	0.141650	−	0.989917i	$-0.454759\pi$
$761$	7.81520	0.283301	0.141650	+	0.989917i	$0.454759\pi$
$762$	0	0
$763$	31.5881	1.14357
$764$	7.41652	0.268320
$765$	0	0
$766$	−3.60716	−0.130332
$767$	−21.6875	−0.783092
$768$	0	0
$769$	38.6765	1.39471	0.697355	−	0.716726i	$-0.254360\pi$
$769$	38.6765	1.39471	0.697355	+	0.716726i	$0.254360\pi$
$770$	0	0
$771$	0	0
$772$	−4.28077	−0.154068
$773$	16.8813	0.607178	0.303589	−	0.952803i	$-0.401815\pi$
$773$	16.8813	0.607178	0.303589	+	0.952803i	$0.401815\pi$
$774$	0	0
$775$	−3.81490	−0.137035
$776$	−15.9397	−0.572201
$777$	0	0
$778$	−16.9219	−0.606678
$779$	30.4888	1.09237
$780$	0	0
$781$	0	0
$782$	11.2569	0.402548
$783$	0	0
$784$	−20.3802	−0.727864
$785$	10.1899	0.363694
$786$	0	0
$787$	27.1091	0.966335	0.483167	−	0.875528i	$-0.339486\pi$
$787$	27.1091	0.966335	0.483167	+	0.875528i	$0.339486\pi$
$788$	−6.44185	−0.229481
$789$	0	0
$790$	−19.9595	−0.710126
$791$	2.77045	0.0985060
$792$	0	0
$793$	−8.21293	−0.291650
$794$	12.3605	0.438658
$795$	0	0
$796$	−12.2967	−0.435845
$797$	23.5525	0.834272	0.417136	−	0.908844i	$-0.363034\pi$
$797$	23.5525	0.834272	0.417136	+	0.908844i	$0.363034\pi$
$798$	0	0
$799$	−15.9938	−0.565820
$800$	3.79913	0.134320
$801$	0	0
$802$	−37.8012	−1.33481
$803$	0	0
$804$	0	0
$805$	−8.09986	−0.285483
$806$	9.85168	0.347011
$807$	0	0
$808$	−12.2892	−0.432333
$809$	−18.0138	−0.633330	−0.316665	−	0.948537i	$-0.602563\pi$
$809$	−18.0138	−0.633330	−0.316665	+	0.948537i	$0.602563\pi$
$810$	0	0
$811$	6.25684	0.219707	0.109854	−	0.993948i	$-0.464962\pi$
$811$	6.25684	0.219707	0.109854	+	0.993948i	$0.464962\pi$
$812$	11.6551	0.409013
$813$	0	0
$814$	0	0
$815$	19.8034	0.693683
$816$	0	0
$817$	−72.5893	−2.53958
$818$	3.15374	0.110268
$819$	0	0
$820$	−2.85741	−0.0997853
$821$	−18.9344	−0.660815	−0.330408	−	0.943838i	$-0.607186\pi$
$821$	−18.9344	−0.660815	−0.330408	+	0.943838i	$0.607186\pi$
$822$	0	0
$823$	23.3683	0.814566	0.407283	−	0.913302i	$-0.366476\pi$
$823$	23.3683	0.814566	0.407283	+	0.913302i	$0.366476\pi$
$824$	2.89625	0.100896
$825$	0	0
$826$	44.4250	1.54574
$827$	5.01148	0.174266	0.0871330	−	0.996197i	$-0.472229\pi$
$827$	5.01148	0.174266	0.0871330	+	0.996197i	$0.472229\pi$
$828$	0	0
$829$	44.9519	1.56124	0.780622	−	0.625003i	$-0.214902\pi$
$829$	44.9519	1.56124	0.780622	+	0.625003i	$0.214902\pi$
$830$	15.7265	0.545876
$831$	0	0
$832$	−19.2484	−0.667320
$833$	49.2897	1.70779
$834$	0	0
$835$	4.96739	0.171903
$836$	0	0
$837$	0	0
$838$	33.5670	1.15955
$839$	52.4551	1.81095	0.905475	−	0.424399i	$-0.139514\pi$
$839$	52.4551	1.81095	0.905475	+	0.424399i	$0.139514\pi$
$840$	0	0
$841$	−12.9891	−0.447901
$842$	0.331512	0.0114247
$843$	0	0
$844$	11.1611	0.384182
$845$	7.82948	0.269342
$846$	0	0
$847$	0	0
$848$	23.3026	0.800215
$849$	0	0
$850$	5.69990	0.195505
$851$	−13.2931	−0.455682
$852$	0	0
$853$	−50.1056	−1.71558	−0.857791	−	0.513998i	$-0.828164\pi$
$853$	−50.1056	−1.71558	−0.857791	+	0.513998i	$0.828164\pi$
$854$	16.8235	0.575687
$855$	0	0
$856$	44.7211	1.52853
$857$	−13.2279	−0.451856	−0.225928	−	0.974144i	$-0.572541\pi$
$857$	−13.2279	−0.451856	−0.225928	+	0.974144i	$0.572541\pi$
$858$	0	0
$859$	−50.3765	−1.71883	−0.859413	−	0.511283i	$-0.829171\pi$
$859$	−50.3765	−1.71883	−0.859413	+	0.511283i	$0.829171\pi$
$860$	6.80308	0.231983
$861$	0	0
$862$	5.10584	0.173906
$863$	−13.6184	−0.463576	−0.231788	−	0.972766i	$-0.574458\pi$
$863$	−13.6184	−0.463576	−0.231788	+	0.972766i	$0.574458\pi$
$864$	0	0
$865$	16.2510	0.552549
$866$	−7.45948	−0.253483
$867$	0	0
$868$	11.1120	0.377165
$869$	0	0
$870$	0	0
$871$	14.3596	0.486557
$872$	−23.7062	−0.802793
$873$	0	0
$874$	−16.9967	−0.574922
$875$	−4.10132	−0.138650
$876$	0	0
$877$	−20.0237	−0.676152	−0.338076	−	0.941119i	$-0.609776\pi$
$877$	−20.0237	−0.676152	−0.338076	+	0.941119i	$0.609776\pi$
$878$	−30.3391	−1.02389
$879$	0	0
$880$	0	0
$881$	30.0543	1.01256	0.506278	−	0.862371i	$-0.331021\pi$
$881$	30.0543	1.01256	0.506278	+	0.862371i	$0.331021\pi$
$882$	0	0
$883$	29.0963	0.979168	0.489584	−	0.871956i	$-0.337149\pi$
$883$	29.0963	0.979168	0.489584	+	0.871956i	$0.337149\pi$
$884$	8.10509	0.272604
$885$	0	0
$886$	26.3106	0.883921
$887$	18.7486	0.629515	0.314758	−	0.949172i	$-0.398077\pi$
$887$	18.7486	0.629515	0.314758	+	0.949172i	$0.398077\pi$
$888$	0	0
$889$	−18.0912	−0.606758
$890$	8.91971	0.298989
$891$	0	0
$892$	−0.769733	−0.0257726
$893$	24.1488	0.808109
$894$	0	0
$895$	3.18448	0.106445
$896$	8.26581	0.276141
$897$	0	0
$898$	33.0024	1.10130
$899$	15.2648	0.509109
$900$	0	0
$901$	−56.3576	−1.87754
$902$	0	0
$903$	0	0
$904$	−2.07917	−0.0691520
$905$	−1.50099	−0.0498945
$906$	0	0
$907$	38.7590	1.28697	0.643486	−	0.765458i	$-0.277488\pi$
$907$	38.7590	1.28697	0.643486	+	0.765458i	$0.277488\pi$
$908$	−1.41453	−0.0469427
$909$	0	0
$910$	10.5913	0.351100
$911$	−34.8367	−1.15419	−0.577096	−	0.816677i	$-0.695814\pi$
$911$	−34.8367	−1.15419	−0.577096	+	0.816677i	$0.695814\pi$
$912$	0	0
$913$	0	0
$914$	8.66238	0.286526
$915$	0	0
$916$	−5.78089	−0.191006
$917$	−87.7701	−2.89842
$918$	0	0
$919$	−27.3492	−0.902167	−0.451083	−	0.892482i	$-0.648962\pi$
$919$	−27.3492	−0.902167	−0.451083	+	0.892482i	$0.648962\pi$
$920$	6.07877	0.200411
$921$	0	0
$922$	38.6292	1.27218
$923$	27.9573	0.920227
$924$	0	0
$925$	−6.73089	−0.221310
$926$	17.8944	0.588047
$927$	0	0
$928$	−15.2017	−0.499020
$929$	44.0757	1.44608	0.723039	−	0.690808i	$-0.242745\pi$
$929$	44.0757	1.44608	0.723039	+	0.690808i	$0.242745\pi$
$930$	0	0
$931$	−74.4218	−2.43908
$932$	17.3500	0.568319
$933$	0	0
$934$	−11.5282	−0.377214
$935$	0	0
$936$	0	0
$937$	20.3465	0.664690	0.332345	−	0.943158i	$-0.392160\pi$
$937$	20.3465	0.664690	0.332345	+	0.943158i	$0.392160\pi$
$938$	−29.4144	−0.960415
$939$	0	0
$940$	−2.26323	−0.0738185
$941$	−3.86942	−0.126140	−0.0630698	−	0.998009i	$-0.520089\pi$
$941$	−3.86942	−0.126140	−0.0630698	+	0.998009i	$0.520089\pi$
$942$	0	0
$943$	−7.94589	−0.258754
$944$	−19.7926	−0.644194
$945$	0	0
$946$	0	0
$947$	54.4898	1.77068	0.885340	−	0.464943i	$-0.153925\pi$
$947$	54.4898	1.77068	0.885340	+	0.464943i	$0.153925\pi$
$948$	0	0
$949$	8.84607	0.287155
$950$	−8.60619	−0.279222
$951$	0	0
$952$	−63.3568	−2.05341
$953$	40.5032	1.31203	0.656014	−	0.754748i	$-0.272241\pi$
$953$	40.5032	1.31203	0.656014	+	0.754748i	$0.272241\pi$
$954$	0	0
$955$	10.4428	0.337921
$956$	−0.0504136	−0.00163049
$957$	0	0
$958$	−4.59177	−0.148353
$959$	15.6724	0.506087
$960$	0	0
$961$	−16.4465	−0.530533
$962$	17.3820	0.560418
$963$	0	0
$964$	8.99495	0.289708
$965$	−6.02750	−0.194032
$966$	0	0
$967$	22.9360	0.737572	0.368786	−	0.929514i	$-0.379774\pi$
$967$	22.9360	0.737572	0.368786	+	0.929514i	$0.379774\pi$
$968$	0	0
$969$	0	0
$970$	−5.88136	−0.188839
$971$	−46.3553	−1.48761	−0.743806	−	0.668396i	$-0.766981\pi$
$971$	−46.3553	−1.48761	−0.743806	+	0.668396i	$0.766981\pi$
$972$	0	0
$973$	25.8269	0.827972
$974$	8.37675	0.268408
$975$	0	0
$976$	−7.49533	−0.239920
$977$	−26.4057	−0.844794	−0.422397	−	0.906411i	$-0.638811\pi$
$977$	−26.4057	−0.844794	−0.422397	+	0.906411i	$0.638811\pi$
$978$	0	0
$979$	0	0
$980$	6.97483	0.222803
$981$	0	0
$982$	2.49316	0.0795600
$983$	−15.7849	−0.503461	−0.251730	−	0.967797i	$-0.581000\pi$
$983$	−15.7849	−0.503461	−0.251730	+	0.967797i	$0.581000\pi$
$984$	0	0
$985$	−9.07040	−0.289007
$986$	−22.8073	−0.726333
$987$	0	0
$988$	−12.2378	−0.389335
$989$	18.9180	0.601557
$990$	0	0
$991$	27.9885	0.889085	0.444542	−	0.895758i	$-0.353366\pi$
$991$	27.9885	0.889085	0.444542	+	0.895758i	$0.353366\pi$
$992$	−14.4933	−0.460163
$993$	0	0
$994$	−57.2681	−1.81644
$995$	−17.3143	−0.548900
$996$	0	0
$997$	39.5825	1.25359	0.626796	−	0.779184i	$-0.284366\pi$
$997$	39.5825	1.25359	0.626796	+	0.779184i	$0.284366\pi$
$998$	9.15572	0.289819
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.ca.1.7		8
3.2	odd	2		5445.2.a.cd.1.2		8
11.5	even	5		495.2.n.h.91.1	yes	16
11.9	even	5		495.2.n.h.136.1	yes	16
11.10	odd	2		5445.2.a.cc.1.2		8
33.5	odd	10		495.2.n.g.91.4	✓	16
33.20	odd	10		495.2.n.g.136.4	yes	16
33.32	even	2		5445.2.a.cb.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.91.4	✓	16	33.5	odd	10
495.2.n.g.136.4	yes	16	33.20	odd	10
495.2.n.h.91.1	yes	16	11.5	even	5
495.2.n.h.136.1	yes	16	11.9	even	5
5445.2.a.ca.1.7		8	1.1	even	1	trivial
5445.2.a.cb.1.7		8	33.32	even	2
5445.2.a.cc.1.2		8	11.10	odd	2
5445.2.a.cd.1.2		8	3.2	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)

\(f(q)\)	\(=\)	\(q+1.13569 q^{2} -0.710206 q^{4} -1.00000 q^{5} +4.10132 q^{7} -3.07796 q^{8} +O(q^{10})\)	\(q+1.13569 q^{2} -0.710206 q^{4} -1.00000 q^{5} +4.10132 q^{7} -3.07796 q^{8} -1.13569 q^{10} -2.27388 q^{13} +4.65784 q^{14} -2.07520 q^{16} +5.01888 q^{17} -7.57794 q^{19} +0.710206 q^{20} +1.97494 q^{23} +1.00000 q^{25} -2.58242 q^{26} -2.91278 q^{28} -4.00136 q^{29} -3.81490 q^{31} +3.79913 q^{32} +5.69990 q^{34} -4.10132 q^{35} -6.73089 q^{37} -8.60619 q^{38} +3.07796 q^{40} -4.02336 q^{41} +9.57903 q^{43} +2.24292 q^{46} -3.18673 q^{47} +9.82086 q^{49} +1.13569 q^{50} +1.61492 q^{52} -11.2291 q^{53} -12.6237 q^{56} -4.54431 q^{58} +9.53769 q^{59} +3.61186 q^{61} -4.33255 q^{62} +8.46503 q^{64} +2.27388 q^{65} -6.31504 q^{67} -3.56444 q^{68} -4.65784 q^{70} -12.2950 q^{71} -3.89030 q^{73} -7.64422 q^{74} +5.38189 q^{76} +17.5747 q^{79} +2.07520 q^{80} -4.56930 q^{82} -13.8475 q^{83} -5.01888 q^{85} +10.8788 q^{86} -7.85399 q^{89} -9.32590 q^{91} -1.40261 q^{92} -3.61914 q^{94} +7.57794 q^{95} +5.17866 q^{97} +11.1535 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8	\( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 4 q^{10} + 6 q^{13} - 14 q^{14} + 14 q^{16} - 8 q^{17} - 2 q^{19} - 6 q^{20} - 4 q^{23} + 8 q^{25} + 2 q^{26} + 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} + 20 q^{38} + 12 q^{40} - 22 q^{41} + 14 q^{43} - 2 q^{46} - 10 q^{47} - 4 q^{50} - 10 q^{52} + 18 q^{53} - 34 q^{56} + 12 q^{58} - 2 q^{59} - 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} + 2 q^{71} + 16 q^{73} - 24 q^{74} - 22 q^{76} - 16 q^{79} - 14 q^{80} + 10 q^{82} - 46 q^{83} + 8 q^{85} + 28 q^{86} - 38 q^{89} + 8 q^{91} + 24 q^{92} - 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 6 * q^4 - 8 * q^5 + 8 * q^7 - 12 * q^8 + 4 * q^10 + 6 * q^13 - 14 * q^14 + 14 * q^16 - 8 * q^17 - 2 * q^19 - 6 * q^20 - 4 * q^23 + 8 * q^25 + 2 * q^26 + 24 * q^28 - 22 * q^29 + 10 * q^31 - 28 * q^32 - 2 * q^34 - 8 * q^35 - 14 * q^37 + 20 * q^38 + 12 * q^40 - 22 * q^41 + 14 * q^43 - 2 * q^46 - 10 * q^47 - 4 * q^50 - 10 * q^52 + 18 * q^53 - 34 * q^56 + 12 * q^58 - 2 * q^59 - 14 * q^61 - 30 * q^62 + 30 * q^64 - 6 * q^65 + 10 * q^67 - 6 * q^68 + 14 * q^70 + 2 * q^71 + 16 * q^73 - 24 * q^74 - 22 * q^76 - 16 * q^79 - 14 * q^80 + 10 * q^82 - 46 * q^83 + 8 * q^85 + 28 * q^86 - 38 * q^89 + 8 * q^91 + 24 * q^92 - 10 * q^94 + 2 * q^95 - 4 * q^97 - 4 * q^98

Introduction

L-functions

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