LMFDB - Embedded newform 5445.2.a.ca.1.2

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			1.2
Root			$-1.39491$ 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)

$f(q)$	$=$	$q-2.39491 q^{2} +3.73560 q^{4} -1.00000 q^{5} +1.96984 q^{7} -4.15660 q^{8} +O(q^{10})$	$q-2.39491 q^{2} +3.73560 q^{4} -1.00000 q^{5} +1.96984 q^{7} -4.15660 q^{8} +2.39491 q^{10} +2.92022 q^{13} -4.71760 q^{14} +2.48350 q^{16} -6.62030 q^{17} -3.56013 q^{19} -3.73560 q^{20} +2.78918 q^{23} +1.00000 q^{25} -6.99366 q^{26} +7.35855 q^{28} -3.51753 q^{29} +8.11992 q^{31} +2.36545 q^{32} +15.8550 q^{34} -1.96984 q^{35} +4.42991 q^{37} +8.52619 q^{38} +4.15660 q^{40} +7.07811 q^{41} -10.8914 q^{43} -6.67984 q^{46} -0.0271524 q^{47} -3.11971 q^{49} -2.39491 q^{50} +10.9088 q^{52} +4.10948 q^{53} -8.18786 q^{56} +8.42417 q^{58} -9.51441 q^{59} +3.10231 q^{61} -19.4465 q^{62} -10.6320 q^{64} -2.92022 q^{65} +1.58295 q^{67} -24.7308 q^{68} +4.71760 q^{70} -6.05837 q^{71} -2.37292 q^{73} -10.6092 q^{74} -13.2992 q^{76} -12.5389 q^{79} -2.48350 q^{80} -16.9515 q^{82} -5.04333 q^{83} +6.62030 q^{85} +26.0840 q^{86} +12.0573 q^{89} +5.75238 q^{91} +10.4193 q^{92} +0.0650277 q^{94} +3.56013 q^{95} +10.4097 q^{97} +7.47144 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.39491	−1.69346	−0.846729	−	0.532025i	$-0.821431\pi$
$2$	−2.39491	−1.69346	−0.846729	+	0.532025i	$0.821431\pi$
$3$	0	0
$3$	0	0
$4$	3.73560	1.86780
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.96984	0.744531	0.372266	−	0.928126i	$-0.378581\pi$
$7$	1.96984	0.744531	0.372266	+	0.928126i	$0.378581\pi$
$8$	−4.15660	−1.46958
$9$	0	0
$10$	2.39491	0.757337
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.92022	0.809923	0.404961	−	0.914334i	$-0.367285\pi$
$13$	2.92022	0.809923	0.404961	+	0.914334i	$0.367285\pi$
$14$	−4.71760	−1.26083
$15$	0	0
$16$	2.48350	0.620875
$17$	−6.62030	−1.60566	−0.802829	−	0.596210i	$-0.796673\pi$
$17$	−6.62030	−1.60566	−0.802829	+	0.596210i	$0.796673\pi$
$18$	0	0
$19$	−3.56013	−0.816750	−0.408375	−	0.912814i	$-0.633904\pi$
$19$	−3.56013	−0.816750	−0.408375	+	0.912814i	$0.633904\pi$
$20$	−3.73560	−0.835305
$21$	0	0
$22$	0	0
$23$	2.78918	0.581585	0.290792	−	0.956786i	$-0.406081\pi$
$23$	2.78918	0.581585	0.290792	+	0.956786i	$0.406081\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.99366	−1.37157
$27$	0	0
$28$	7.35855	1.39063
$29$	−3.51753	−0.653188	−0.326594	−	0.945165i	$-0.605901\pi$
$29$	−3.51753	−0.653188	−0.326594	+	0.945165i	$0.605901\pi$
$30$	0	0
$31$	8.11992	1.45838	0.729191	−	0.684311i	$-0.239897\pi$
$31$	8.11992	1.45838	0.729191	+	0.684311i	$0.239897\pi$
$32$	2.36545	0.418156
$33$	0	0
$34$	15.8550	2.71911
$35$	−1.96984	−0.332964
$36$	0	0
$37$	4.42991	0.728273	0.364137	−	0.931346i	$-0.381364\pi$
$37$	4.42991	0.728273	0.364137	+	0.931346i	$0.381364\pi$
$38$	8.52619	1.38313
$39$	0	0
$40$	4.15660	0.657217
$41$	7.07811	1.10542	0.552708	−	0.833375i	$-0.313595\pi$
$41$	7.07811	1.10542	0.552708	+	0.833375i	$0.313595\pi$
$42$	0	0
$43$	−10.8914	−1.66092	−0.830462	−	0.557075i	$-0.811924\pi$
$43$	−10.8914	−1.66092	−0.830462	+	0.557075i	$0.811924\pi$
$44$	0	0
$45$	0	0
$46$	−6.67984	−0.984889
$47$	−0.0271524	−0.00396059	−0.00198029	−	0.999998i	$-0.500630\pi$
$47$	−0.0271524	−0.00396059	−0.00198029	+	0.999998i	$0.500630\pi$
$48$	0	0
$49$	−3.11971	−0.445673
$50$	−2.39491	−0.338692
$51$	0	0
$52$	10.9088	1.51277
$53$	4.10948	0.564481	0.282240	−	0.959344i	$-0.408922\pi$
$53$	4.10948	0.564481	0.282240	+	0.959344i	$0.408922\pi$
$54$	0	0
$55$	0	0
$56$	−8.18786	−1.09415
$57$	0	0
$58$	8.42417	1.10615
$59$	−9.51441	−1.23867	−0.619335	−	0.785127i	$-0.712598\pi$
$59$	−9.51441	−1.23867	−0.619335	+	0.785127i	$0.712598\pi$
$60$	0	0
$61$	3.10231	0.397210	0.198605	−	0.980080i	$-0.436359\pi$
$61$	3.10231	0.397210	0.198605	+	0.980080i	$0.436359\pi$
$62$	−19.4465	−2.46971
$63$	0	0
$64$	−10.6320	−1.32900
$65$	−2.92022	−0.362209
$66$	0	0
$67$	1.58295	0.193388	0.0966939	−	0.995314i	$-0.469173\pi$
$67$	1.58295	0.193388	0.0966939	+	0.995314i	$0.469173\pi$
$68$	−24.7308	−2.99905
$69$	0	0
$70$	4.71760	0.563861
$71$	−6.05837	−0.718997	−0.359498	−	0.933146i	$-0.617052\pi$
$71$	−6.05837	−0.718997	−0.359498	+	0.933146i	$0.617052\pi$
$72$	0	0
$73$	−2.37292	−0.277729	−0.138864	−	0.990311i	$-0.544345\pi$
$73$	−2.37292	−0.277729	−0.138864	+	0.990311i	$0.544345\pi$
$74$	−10.6092	−1.23330
$75$	0	0
$76$	−13.2992	−1.52552
$77$	0	0
$78$	0	0
$79$	−12.5389	−1.41074	−0.705369	−	0.708840i	$-0.749219\pi$
$79$	−12.5389	−1.41074	−0.705369	+	0.708840i	$0.749219\pi$
$80$	−2.48350	−0.277664
$81$	0	0
$82$	−16.9515	−1.87197
$83$	−5.04333	−0.553577	−0.276789	−	0.960931i	$-0.589270\pi$
$83$	−5.04333	−0.553577	−0.276789	+	0.960931i	$0.589270\pi$
$84$	0	0
$85$	6.62030	0.718072
$86$	26.0840	2.81271
$87$	0	0
$88$	0	0
$89$	12.0573	1.27807	0.639035	−	0.769177i	$-0.279334\pi$
$89$	12.0573	1.27807	0.639035	+	0.769177i	$0.279334\pi$
$90$	0	0
$91$	5.75238	0.603013
$92$	10.4193	1.08628
$93$	0	0
$94$	0.0650277	0.00670709
$95$	3.56013	0.365262
$96$	0	0
$97$	10.4097	1.05694	0.528470	−	0.848952i	$-0.322766\pi$
$97$	10.4097	1.05694	0.528470	+	0.848952i	$0.322766\pi$
$98$	7.47144	0.754729
$99$	0	0
$100$	3.73560	0.373560
$101$	−17.2166	−1.71311	−0.856556	−	0.516054i	$-0.827400\pi$
$101$	−17.2166	−1.71311	−0.856556	+	0.516054i	$0.827400\pi$
$102$	0	0
$103$	6.66999	0.657213	0.328607	−	0.944467i	$-0.393421\pi$
$103$	6.66999	0.657213	0.328607	+	0.944467i	$0.393421\pi$
$104$	−12.1382	−1.19025
$105$	0	0
$106$	−9.84185	−0.955925
$107$	−9.61908	−0.929912	−0.464956	−	0.885334i	$-0.653930\pi$
$107$	−9.61908	−0.929912	−0.464956	+	0.885334i	$0.653930\pi$
$108$	0	0
$109$	−19.2602	−1.84480	−0.922398	−	0.386240i	$-0.873773\pi$
$109$	−19.2602	−1.84480	−0.922398	+	0.386240i	$0.873773\pi$
$110$	0	0
$111$	0	0
$112$	4.89211	0.462261
$113$	−3.56283	−0.335163	−0.167582	−	0.985858i	$-0.553596\pi$
$113$	−3.56283	−0.335163	−0.167582	+	0.985858i	$0.553596\pi$
$114$	0	0
$115$	−2.78918	−0.260093
$116$	−13.1401	−1.22003
$117$	0	0
$118$	22.7862	2.09764
$119$	−13.0410	−1.19546
$120$	0	0
$121$	0	0
$122$	−7.42975	−0.672658
$123$	0	0
$124$	30.3328	2.72396
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.3179	1.71419	0.857094	−	0.515159i	$-0.172267\pi$
$127$	19.3179	1.71419	0.857094	+	0.515159i	$0.172267\pi$
$128$	20.7319	1.83246
$129$	0	0
$130$	6.99366	0.613385
$131$	−13.9027	−1.21468	−0.607341	−	0.794441i	$-0.707764\pi$
$131$	−13.9027	−1.21468	−0.607341	+	0.794441i	$0.707764\pi$
$132$	0	0
$133$	−7.01290	−0.608096
$134$	−3.79102	−0.327494
$135$	0	0
$136$	27.5180	2.35964
$137$	22.7270	1.94169	0.970847	−	0.239699i	$-0.0770486\pi$
$137$	22.7270	1.94169	0.970847	+	0.239699i	$0.0770486\pi$
$138$	0	0
$139$	12.7100	1.07804	0.539022	−	0.842291i	$-0.318794\pi$
$139$	12.7100	1.07804	0.539022	+	0.842291i	$0.318794\pi$
$140$	−7.35855	−0.621911
$141$	0	0
$142$	14.5093	1.21759
$143$	0	0
$144$	0	0
$145$	3.51753	0.292115
$146$	5.68292	0.470322
$147$	0	0
$148$	16.5484	1.36027
$149$	4.30274	0.352494	0.176247	−	0.984346i	$-0.443604\pi$
$149$	4.30274	0.352494	0.176247	+	0.984346i	$0.443604\pi$
$150$	0	0
$151$	17.5844	1.43100	0.715499	−	0.698614i	$-0.246199\pi$
$151$	17.5844	1.43100	0.715499	+	0.698614i	$0.246199\pi$
$152$	14.7980	1.20028
$153$	0	0
$154$	0	0
$155$	−8.11992	−0.652208
$156$	0	0
$157$	−23.8112	−1.90034	−0.950172	−	0.311726i	$-0.899093\pi$
$157$	−23.8112	−1.90034	−0.950172	+	0.311726i	$0.899093\pi$
$158$	30.0296	2.38903
$159$	0	0
$160$	−2.36545	−0.187005
$161$	5.49425	0.433008
$162$	0	0
$163$	−19.8452	−1.55440	−0.777198	−	0.629256i	$-0.783360\pi$
$163$	−19.8452	−1.55440	−0.777198	+	0.629256i	$0.783360\pi$
$164$	26.4410	2.06469
$165$	0	0
$166$	12.0783	0.937460
$167$	14.4760	1.12019	0.560094	−	0.828429i	$-0.310765\pi$
$167$	14.4760	1.12019	0.560094	+	0.828429i	$0.310765\pi$
$168$	0	0
$169$	−4.47232	−0.344025
$170$	−15.8550	−1.21602
$171$	0	0
$172$	−40.6859	−3.10227
$173$	−4.31904	−0.328371	−0.164185	−	0.986429i	$-0.552500\pi$
$173$	−4.31904	−0.328371	−0.164185	+	0.986429i	$0.552500\pi$
$174$	0	0
$175$	1.96984	0.148906
$176$	0	0
$177$	0	0
$178$	−28.8761	−2.16436
$179$	7.45911	0.557520	0.278760	−	0.960361i	$-0.410077\pi$
$179$	7.45911	0.557520	0.278760	+	0.960361i	$0.410077\pi$
$180$	0	0
$181$	−20.4634	−1.52104	−0.760518	−	0.649317i	$-0.775055\pi$
$181$	−20.4634	−1.52104	−0.760518	+	0.649317i	$0.775055\pi$
$182$	−13.7764	−1.02118
$183$	0	0
$184$	−11.5935	−0.854686
$185$	−4.42991	−0.325694
$186$	0	0
$187$	0	0
$188$	−0.101431	−0.00739759
$189$	0	0
$190$	−8.52619	−0.618555
$191$	−6.55174	−0.474067	−0.237034	−	0.971501i	$-0.576175\pi$
$191$	−6.55174	−0.474067	−0.237034	+	0.971501i	$0.576175\pi$
$192$	0	0
$193$	−6.82579	−0.491331	−0.245665	−	0.969355i	$-0.579006\pi$
$193$	−6.82579	−0.491331	−0.245665	+	0.969355i	$0.579006\pi$
$194$	−24.9302	−1.78988
$195$	0	0
$196$	−11.6540	−0.832428
$197$	−3.45091	−0.245867	−0.122934	−	0.992415i	$-0.539230\pi$
$197$	−3.45091	−0.245867	−0.122934	+	0.992415i	$0.539230\pi$
$198$	0	0
$199$	−10.6742	−0.756671	−0.378336	−	0.925668i	$-0.623504\pi$
$199$	−10.6742	−0.756671	−0.378336	+	0.925668i	$0.623504\pi$
$200$	−4.15660	−0.293916
$201$	0	0
$202$	41.2321	2.90108
$203$	−6.92898	−0.486319
$204$	0	0
$205$	−7.07811	−0.494357
$206$	−15.9740	−1.11296
$207$	0	0
$208$	7.25237	0.502861
$209$	0	0
$210$	0	0
$211$	−1.37923	−0.0949503	−0.0474752	−	0.998872i	$-0.515117\pi$
$211$	−1.37923	−0.0949503	−0.0474752	+	0.998872i	$0.515117\pi$
$212$	15.3514	1.05434
$213$	0	0
$214$	23.0368	1.57477
$215$	10.8914	0.742788
$216$	0	0
$217$	15.9950	1.08581
$218$	46.1266	3.12408
$219$	0	0
$220$	0	0
$221$	−19.3327	−1.30046
$222$	0	0
$223$	12.1038	0.810531	0.405265	−	0.914199i	$-0.367179\pi$
$223$	12.1038	0.810531	0.405265	+	0.914199i	$0.367179\pi$
$224$	4.65956	0.311330
$225$	0	0
$226$	8.53267	0.567585
$227$	17.3320	1.15037	0.575183	−	0.818025i	$-0.304931\pi$
$227$	17.3320	1.15037	0.575183	+	0.818025i	$0.304931\pi$
$228$	0	0
$229$	−0.837178	−0.0553223	−0.0276611	−	0.999617i	$-0.508806\pi$
$229$	−0.837178	−0.0553223	−0.0276611	+	0.999617i	$0.508806\pi$
$230$	6.67984	0.440456
$231$	0	0
$232$	14.6210	0.959914
$233$	19.0481	1.24788	0.623941	−	0.781471i	$-0.285530\pi$
$233$	19.0481	1.24788	0.623941	+	0.781471i	$0.285530\pi$
$234$	0	0
$235$	0.0271524	0.00177123
$236$	−35.5420	−2.31359
$237$	0	0
$238$	31.2319	2.02446
$239$	−25.7936	−1.66845	−0.834223	−	0.551427i	$-0.814084\pi$
$239$	−25.7936	−1.66845	−0.834223	+	0.551427i	$0.814084\pi$
$240$	0	0
$241$	−17.0680	−1.09944	−0.549722	−	0.835348i	$-0.685266\pi$
$241$	−17.0680	−1.09944	−0.549722	+	0.835348i	$0.685266\pi$
$242$	0	0
$243$	0	0
$244$	11.5890	0.741908
$245$	3.11971	0.199311
$246$	0	0
$247$	−10.3964	−0.661504
$248$	−33.7513	−2.14321
$249$	0	0
$250$	2.39491	0.151467
$251$	10.3166	0.651177	0.325589	−	0.945512i	$-0.394438\pi$
$251$	10.3166	0.651177	0.325589	+	0.945512i	$0.394438\pi$
$252$	0	0
$253$	0	0
$254$	−46.2647	−2.90291
$255$	0	0
$256$	−28.3869	−1.77418
$257$	11.2636	0.702604	0.351302	−	0.936262i	$-0.385739\pi$
$257$	11.2636	0.702604	0.351302	+	0.936262i	$0.385739\pi$
$258$	0	0
$259$	8.72624	0.542222
$260$	−10.9088	−0.676533
$261$	0	0
$262$	33.2957	2.05701
$263$	−5.15082	−0.317613	−0.158806	−	0.987310i	$-0.550765\pi$
$263$	−5.15082	−0.317613	−0.158806	+	0.987310i	$0.550765\pi$
$264$	0	0
$265$	−4.10948	−0.252444
$266$	16.7953	1.02978
$267$	0	0
$268$	5.91326	0.361210
$269$	−23.1522	−1.41161	−0.705806	−	0.708405i	$-0.749415\pi$
$269$	−23.1522	−1.41161	−0.705806	+	0.708405i	$0.749415\pi$
$270$	0	0
$271$	13.1449	0.798498	0.399249	−	0.916843i	$-0.369271\pi$
$271$	13.1449	0.798498	0.399249	+	0.916843i	$0.369271\pi$
$272$	−16.4415	−0.996913
$273$	0	0
$274$	−54.4290	−3.28818
$275$	0	0
$276$	0	0
$277$	10.3489	0.621807	0.310903	−	0.950442i	$-0.399368\pi$
$277$	10.3489	0.621807	0.310903	+	0.950442i	$0.399368\pi$
$278$	−30.4392	−1.82562
$279$	0	0
$280$	8.18786	0.489319
$281$	−0.387020	−0.0230877	−0.0115438	−	0.999933i	$-0.503675\pi$
$281$	−0.387020	−0.0230877	−0.0115438	+	0.999933i	$0.503675\pi$
$282$	0	0
$283$	−14.5253	−0.863442	−0.431721	−	0.902007i	$-0.642094\pi$
$283$	−14.5253	−0.863442	−0.431721	+	0.902007i	$0.642094\pi$
$284$	−22.6317	−1.34294
$285$	0	0
$286$	0	0
$287$	13.9428	0.823016
$288$	0	0
$289$	26.8283	1.57814
$290$	−8.42417	−0.494684
$291$	0	0
$292$	−8.86426	−0.518742
$293$	14.8176	0.865652	0.432826	−	0.901478i	$-0.357517\pi$
$293$	14.8176	0.865652	0.432826	+	0.901478i	$0.357517\pi$
$294$	0	0
$295$	9.51441	0.553950
$296$	−18.4134	−1.07026
$297$	0	0
$298$	−10.3047	−0.596934
$299$	8.14502	0.471039
$300$	0	0
$301$	−21.4544	−1.23661
$302$	−42.1131	−2.42334
$303$	0	0
$304$	−8.84158	−0.507100
$305$	−3.10231	−0.177638
$306$	0	0
$307$	32.3539	1.84653	0.923266	−	0.384161i	$-0.125509\pi$
$307$	32.3539	1.84653	0.923266	+	0.384161i	$0.125509\pi$
$308$	0	0
$309$	0	0
$310$	19.4465	1.10449
$311$	−1.76355	−0.100002	−0.0500008	−	0.998749i	$-0.515922\pi$
$311$	−1.76355	−0.100002	−0.0500008	+	0.998749i	$0.515922\pi$
$312$	0	0
$313$	−26.1735	−1.47941	−0.739706	−	0.672930i	$-0.765036\pi$
$313$	−26.1735	−1.47941	−0.739706	+	0.672930i	$0.765036\pi$
$314$	57.0258	3.21815
$315$	0	0
$316$	−46.8404	−2.63498
$317$	−10.3755	−0.582744	−0.291372	−	0.956610i	$-0.594112\pi$
$317$	−10.3755	−0.582744	−0.291372	+	0.956610i	$0.594112\pi$
$318$	0	0
$319$	0	0
$320$	10.6320	0.594349
$321$	0	0
$322$	−13.1582	−0.733281
$323$	23.5691	1.31142
$324$	0	0
$325$	2.92022	0.161985
$326$	47.5275	2.63230
$327$	0	0
$328$	−29.4209	−1.62450
$329$	−0.0534861	−0.00294878
$330$	0	0
$331$	9.22380	0.506986	0.253493	−	0.967337i	$-0.418421\pi$
$331$	9.22380	0.506986	0.253493	+	0.967337i	$0.418421\pi$
$332$	−18.8399	−1.03397
$333$	0	0
$334$	−34.6688	−1.89699
$335$	−1.58295	−0.0864857
$336$	0	0
$337$	28.5372	1.55452	0.777261	−	0.629178i	$-0.216608\pi$
$337$	28.5372	1.55452	0.777261	+	0.629178i	$0.216608\pi$
$338$	10.7108	0.582592
$339$	0	0
$340$	24.7308	1.34121
$341$	0	0
$342$	0	0
$343$	−19.9343	−1.07635
$344$	45.2713	2.44086
$345$	0	0
$346$	10.3437	0.556082
$347$	10.1698	0.545945	0.272972	−	0.962022i	$-0.411993\pi$
$347$	10.1698	0.545945	0.272972	+	0.962022i	$0.411993\pi$
$348$	0	0
$349$	−23.7174	−1.26957	−0.634783	−	0.772691i	$-0.718910\pi$
$349$	−23.7174	−1.26957	−0.634783	+	0.772691i	$0.718910\pi$
$350$	−4.71760	−0.252166
$351$	0	0
$352$	0	0
$353$	−14.1383	−0.752504	−0.376252	−	0.926517i	$-0.622787\pi$
$353$	−14.1383	−0.752504	−0.376252	+	0.926517i	$0.622787\pi$
$354$	0	0
$355$	6.05837	0.321545
$356$	45.0412	2.38718
$357$	0	0
$358$	−17.8639	−0.944137
$359$	−24.0330	−1.26841	−0.634207	−	0.773163i	$-0.718673\pi$
$359$	−24.0330	−1.26841	−0.634207	+	0.773163i	$0.718673\pi$
$360$	0	0
$361$	−6.32548	−0.332920
$362$	49.0081	2.57581
$363$	0	0
$364$	21.4886	1.12631
$365$	2.37292	0.124204
$366$	0	0
$367$	−1.79705	−0.0938055	−0.0469027	−	0.998899i	$-0.514935\pi$
$367$	−1.79705	−0.0938055	−0.0469027	+	0.998899i	$0.514935\pi$
$368$	6.92694	0.361091
$369$	0	0
$370$	10.6092	0.551549
$371$	8.09504	0.420274
$372$	0	0
$373$	−6.87802	−0.356131	−0.178065	−	0.984019i	$-0.556984\pi$
$373$	−6.87802	−0.356131	−0.178065	+	0.984019i	$0.556984\pi$
$374$	0	0
$375$	0	0
$376$	0.112862	0.00582041
$377$	−10.2719	−0.529032
$378$	0	0
$379$	−3.53329	−0.181493	−0.0907465	−	0.995874i	$-0.528925\pi$
$379$	−3.53329	−0.181493	−0.0907465	+	0.995874i	$0.528925\pi$
$380$	13.2992	0.682235
$381$	0	0
$382$	15.6908	0.802813
$383$	−1.90335	−0.0972564	−0.0486282	−	0.998817i	$-0.515485\pi$
$383$	−1.90335	−0.0972564	−0.0486282	+	0.998817i	$0.515485\pi$
$384$	0	0
$385$	0	0
$386$	16.3472	0.832048
$387$	0	0
$388$	38.8863	1.97415
$389$	−7.02219	−0.356039	−0.178020	−	0.984027i	$-0.556969\pi$
$389$	−7.02219	−0.356039	−0.178020	+	0.984027i	$0.556969\pi$
$390$	0	0
$391$	−18.4652	−0.933826
$392$	12.9674	0.654953
$393$	0	0
$394$	8.26463	0.416366
$395$	12.5389	0.630902
$396$	0	0
$397$	−12.5149	−0.628104	−0.314052	−	0.949406i	$-0.601687\pi$
$397$	−12.5149	−0.628104	−0.314052	+	0.949406i	$0.601687\pi$
$398$	25.5637	1.28139
$399$	0	0
$400$	2.48350	0.124175
$401$	27.3284	1.36472	0.682358	−	0.731018i	$-0.260955\pi$
$401$	27.3284	1.36472	0.682358	+	0.731018i	$0.260955\pi$
$402$	0	0
$403$	23.7120	1.18118
$404$	−64.3142	−3.19975
$405$	0	0
$406$	16.5943	0.823561
$407$	0	0
$408$	0	0
$409$	−11.5990	−0.573531	−0.286766	−	0.958001i	$-0.592580\pi$
$409$	−11.5990	−0.573531	−0.286766	+	0.958001i	$0.592580\pi$
$410$	16.9515	0.837172
$411$	0	0
$412$	24.9164	1.22754
$413$	−18.7419	−0.922229
$414$	0	0
$415$	5.04333	0.247567
$416$	6.90762	0.338674
$417$	0	0
$418$	0	0
$419$	9.07542	0.443363	0.221682	−	0.975119i	$-0.428845\pi$
$419$	9.07542	0.443363	0.221682	+	0.975119i	$0.428845\pi$
$420$	0	0
$421$	18.5151	0.902372	0.451186	−	0.892430i	$-0.351001\pi$
$421$	18.5151	0.902372	0.451186	+	0.892430i	$0.351001\pi$
$422$	3.30314	0.160794
$423$	0	0
$424$	−17.0815	−0.829551
$425$	−6.62030	−0.321132
$426$	0	0
$427$	6.11106	0.295735
$428$	−35.9330	−1.73689
$429$	0	0
$430$	−26.0840	−1.25788
$431$	−24.7913	−1.19415	−0.597077	−	0.802184i	$-0.703671\pi$
$431$	−24.7913	−1.19415	−0.597077	+	0.802184i	$0.703671\pi$
$432$	0	0
$433$	−35.3858	−1.70053	−0.850267	−	0.526352i	$-0.823559\pi$
$433$	−35.3858	−1.70053	−0.850267	+	0.526352i	$0.823559\pi$
$434$	−38.3066	−1.83877
$435$	0	0
$436$	−71.9485	−3.44571
$437$	−9.92985	−0.475009
$438$	0	0
$439$	−10.2916	−0.491192	−0.245596	−	0.969372i	$-0.578984\pi$
$439$	−10.2916	−0.491192	−0.245596	+	0.969372i	$0.578984\pi$
$440$	0	0
$441$	0	0
$442$	46.3001	2.20227
$443$	35.8502	1.70330	0.851648	−	0.524114i	$-0.175603\pi$
$443$	35.8502	1.70330	0.851648	+	0.524114i	$0.175603\pi$
$444$	0	0
$445$	−12.0573	−0.571571
$446$	−28.9875	−1.37260
$447$	0	0
$448$	−20.9435	−0.989485
$449$	−26.8436	−1.26683	−0.633414	−	0.773813i	$-0.718347\pi$
$449$	−26.8436	−1.26683	−0.633414	+	0.773813i	$0.718347\pi$
$450$	0	0
$451$	0	0
$452$	−13.3093	−0.626017
$453$	0	0
$454$	−41.5086	−1.94810
$455$	−5.75238	−0.269676
$456$	0	0
$457$	−19.6380	−0.918627	−0.459314	−	0.888274i	$-0.651905\pi$
$457$	−19.6380	−0.918627	−0.459314	+	0.888274i	$0.651905\pi$
$458$	2.00497	0.0936860
$459$	0	0
$460$	−10.4193	−0.485801
$461$	19.5626	0.911122	0.455561	−	0.890205i	$-0.349439\pi$
$461$	19.5626	0.911122	0.455561	+	0.890205i	$0.349439\pi$
$462$	0	0
$463$	−14.7474	−0.685371	−0.342686	−	0.939450i	$-0.611337\pi$
$463$	−14.7474	−0.685371	−0.342686	+	0.939450i	$0.611337\pi$
$464$	−8.73578	−0.405549
$465$	0	0
$466$	−45.6185	−2.11324
$467$	−10.3049	−0.476854	−0.238427	−	0.971160i	$-0.576632\pi$
$467$	−10.3049	−0.476854	−0.238427	+	0.971160i	$0.576632\pi$
$468$	0	0
$469$	3.11816	0.143983
$470$	−0.0650277	−0.00299950
$471$	0	0
$472$	39.5476	1.82033
$473$	0	0
$474$	0	0
$475$	−3.56013	−0.163350
$476$	−48.7158	−2.23288
$477$	0	0
$478$	61.7733	2.82544
$479$	−11.3976	−0.520768	−0.260384	−	0.965505i	$-0.583849\pi$
$479$	−11.3976	−0.520768	−0.260384	+	0.965505i	$0.583849\pi$
$480$	0	0
$481$	12.9363	0.589845
$482$	40.8762	1.86186
$483$	0	0
$484$	0	0
$485$	−10.4097	−0.472678
$486$	0	0
$487$	−34.6435	−1.56984	−0.784922	−	0.619594i	$-0.787297\pi$
$487$	−34.6435	−1.56984	−0.784922	+	0.619594i	$0.787297\pi$
$488$	−12.8951	−0.583732
$489$	0	0
$490$	−7.47144	−0.337525
$491$	−36.6153	−1.65242	−0.826212	−	0.563360i	$-0.809509\pi$
$491$	−36.6153	−1.65242	−0.826212	+	0.563360i	$0.809509\pi$
$492$	0	0
$493$	23.2871	1.04880
$494$	24.8983	1.12023
$495$	0	0
$496$	20.1658	0.905473
$497$	−11.9341	−0.535315
$498$	0	0
$499$	−21.3515	−0.955824	−0.477912	−	0.878408i	$-0.658606\pi$
$499$	−21.3515	−0.955824	−0.477912	+	0.878408i	$0.658606\pi$
$500$	−3.73560	−0.167061
$501$	0	0
$502$	−24.7073	−1.10274
$503$	11.6575	0.519783	0.259891	−	0.965638i	$-0.416313\pi$
$503$	11.6575	0.519783	0.259891	+	0.965638i	$0.416313\pi$
$504$	0	0
$505$	17.2166	0.766127
$506$	0	0
$507$	0	0
$508$	72.1640	3.20176
$509$	29.3434	1.30062	0.650311	−	0.759668i	$-0.274639\pi$
$509$	29.3434	1.30062	0.650311	+	0.759668i	$0.274639\pi$
$510$	0	0
$511$	−4.67427	−0.206778
$512$	26.5205	1.17205
$513$	0	0
$514$	−26.9753	−1.18983
$515$	−6.66999	−0.293915
$516$	0	0
$517$	0	0
$518$	−20.8986	−0.918230
$519$	0	0
$520$	12.1382	0.532295
$521$	4.00057	0.175268	0.0876340	−	0.996153i	$-0.472069\pi$
$521$	4.00057	0.175268	0.0876340	+	0.996153i	$0.472069\pi$
$522$	0	0
$523$	−4.44571	−0.194397	−0.0971986	−	0.995265i	$-0.530988\pi$
$523$	−4.44571	−0.194397	−0.0971986	+	0.995265i	$0.530988\pi$
$524$	−51.9348	−2.26878
$525$	0	0
$526$	12.3357	0.537864
$527$	−53.7563	−2.34166
$528$	0	0
$529$	−15.2205	−0.661759
$530$	9.84185	0.427502
$531$	0	0
$532$	−26.1974	−1.13580
$533$	20.6696	0.895301
$534$	0	0
$535$	9.61908	0.415869
$536$	−6.57969	−0.284199
$537$	0	0
$538$	55.4474	2.39051
$539$	0	0
$540$	0	0
$541$	−13.1859	−0.566905	−0.283452	−	0.958986i	$-0.591480\pi$
$541$	−13.1859	−0.566905	−0.283452	+	0.958986i	$0.591480\pi$
$542$	−31.4810	−1.35222
$543$	0	0
$544$	−15.6599	−0.671415
$545$	19.2602	0.825018
$546$	0	0
$547$	1.86825	0.0798806	0.0399403	−	0.999202i	$-0.487283\pi$
$547$	1.86825	0.0798806	0.0399403	+	0.999202i	$0.487283\pi$
$548$	84.8988	3.62670
$549$	0	0
$550$	0	0
$551$	12.5229	0.533491
$552$	0	0
$553$	−24.6997	−1.05034
$554$	−24.7848	−1.05300
$555$	0	0
$556$	47.4793	2.01357
$557$	13.5209	0.572900	0.286450	−	0.958095i	$-0.407525\pi$
$557$	13.5209	0.572900	0.286450	+	0.958095i	$0.407525\pi$
$558$	0	0
$559$	−31.8053	−1.34522
$560$	−4.89211	−0.206729
$561$	0	0
$562$	0.926879	0.0390980
$563$	−30.5720	−1.28846	−0.644228	−	0.764833i	$-0.722821\pi$
$563$	−30.5720	−1.28846	−0.644228	+	0.764833i	$0.722821\pi$
$564$	0	0
$565$	3.56283	0.149890
$566$	34.7869	1.46220
$567$	0	0
$568$	25.1823	1.05662
$569$	36.4840	1.52949	0.764745	−	0.644333i	$-0.222865\pi$
$569$	36.4840	1.52949	0.764745	+	0.644333i	$0.222865\pi$
$570$	0	0
$571$	−19.9929	−0.836679	−0.418339	−	0.908291i	$-0.637388\pi$
$571$	−19.9929	−0.836679	−0.418339	+	0.908291i	$0.637388\pi$
$572$	0	0
$573$	0	0
$574$	−33.3917	−1.39374
$575$	2.78918	0.116317
$576$	0	0
$577$	−31.1211	−1.29559	−0.647794	−	0.761816i	$-0.724308\pi$
$577$	−31.1211	−1.29559	−0.647794	+	0.761816i	$0.724308\pi$
$578$	−64.2514	−2.67251
$579$	0	0
$580$	13.1401	0.545612
$581$	−9.93458	−0.412156
$582$	0	0
$583$	0	0
$584$	9.86327	0.408145
$585$	0	0
$586$	−35.4868	−1.46594
$587$	−31.9068	−1.31693	−0.658467	−	0.752610i	$-0.728795\pi$
$587$	−31.9068	−1.31693	−0.658467	+	0.752610i	$0.728795\pi$
$588$	0	0
$589$	−28.9080	−1.19113
$590$	−22.7862	−0.938091
$591$	0	0
$592$	11.0017	0.452167
$593$	−34.5829	−1.42015	−0.710074	−	0.704127i	$-0.751338\pi$
$593$	−34.5829	−1.42015	−0.710074	+	0.704127i	$0.751338\pi$
$594$	0	0
$595$	13.0410	0.534627
$596$	16.0733	0.658388
$597$	0	0
$598$	−19.5066	−0.797684
$599$	−17.7818	−0.726543	−0.363271	−	0.931683i	$-0.618340\pi$
$599$	−17.7818	−0.726543	−0.363271	+	0.931683i	$0.618340\pi$
$600$	0	0
$601$	−14.8208	−0.604554	−0.302277	−	0.953220i	$-0.597747\pi$
$601$	−14.8208	−0.604554	−0.302277	+	0.953220i	$0.597747\pi$
$602$	51.3813	2.09415
$603$	0	0
$604$	65.6883	2.67282
$605$	0	0
$606$	0	0
$607$	−13.8034	−0.560264	−0.280132	−	0.959962i	$-0.590378\pi$
$607$	−13.8034	−0.560264	−0.280132	+	0.959962i	$0.590378\pi$
$608$	−8.42129	−0.341528
$609$	0	0
$610$	7.42975	0.300822
$611$	−0.0792910	−0.00320777
$612$	0	0
$613$	22.4951	0.908570	0.454285	−	0.890856i	$-0.349895\pi$
$613$	22.4951	0.908570	0.454285	+	0.890856i	$0.349895\pi$
$614$	−77.4846	−3.12702
$615$	0	0
$616$	0	0
$617$	−10.6195	−0.427525	−0.213762	−	0.976886i	$-0.568572\pi$
$617$	−10.6195	−0.427525	−0.213762	+	0.976886i	$0.568572\pi$
$618$	0	0
$619$	−2.63250	−0.105809	−0.0529046	−	0.998600i	$-0.516848\pi$
$619$	−2.63250	−0.105809	−0.0529046	+	0.998600i	$0.516848\pi$
$620$	−30.3328	−1.21819
$621$	0	0
$622$	4.22354	0.169349
$623$	23.7510	0.951563
$624$	0	0
$625$	1.00000	0.0400000
$626$	62.6832	2.50532
$627$	0	0
$628$	−88.9493	−3.54946
$629$	−29.3273	−1.16936
$630$	0	0
$631$	24.1315	0.960660	0.480330	−	0.877088i	$-0.340517\pi$
$631$	24.1315	0.960660	0.480330	+	0.877088i	$0.340517\pi$
$632$	52.1193	2.07320
$633$	0	0
$634$	24.8483	0.986852
$635$	−19.3179	−0.766608
$636$	0	0
$637$	−9.11024	−0.360961
$638$	0	0
$639$	0	0
$640$	−20.7319	−0.819500
$641$	45.7507	1.80705	0.903523	−	0.428540i	$-0.140972\pi$
$641$	45.7507	1.80705	0.903523	+	0.428540i	$0.140972\pi$
$642$	0	0
$643$	−6.60925	−0.260643	−0.130322	−	0.991472i	$-0.541601\pi$
$643$	−6.60925	−0.260643	−0.130322	+	0.991472i	$0.541601\pi$
$644$	20.5243	0.808772
$645$	0	0
$646$	−56.4459	−2.22083
$647$	1.86148	0.0731824	0.0365912	−	0.999330i	$-0.488350\pi$
$647$	1.86148	0.0731824	0.0365912	+	0.999330i	$0.488350\pi$
$648$	0	0
$649$	0	0
$650$	−6.99366	−0.274314
$651$	0	0
$652$	−74.1337	−2.90330
$653$	−31.9277	−1.24943	−0.624713	−	0.780854i	$-0.714784\pi$
$653$	−31.9277	−1.24943	−0.624713	+	0.780854i	$0.714784\pi$
$654$	0	0
$655$	13.9027	0.543223
$656$	17.5785	0.686325
$657$	0	0
$658$	0.128094	0.00499364
$659$	8.36924	0.326019	0.163010	−	0.986624i	$-0.447880\pi$
$659$	8.36924	0.326019	0.163010	+	0.986624i	$0.447880\pi$
$660$	0	0
$661$	−34.0773	−1.32545	−0.662726	−	0.748862i	$-0.730601\pi$
$661$	−34.0773	−1.32545	−0.662726	+	0.748862i	$0.730601\pi$
$662$	−22.0902	−0.858559
$663$	0	0
$664$	20.9631	0.813527
$665$	7.01290	0.271949
$666$	0	0
$667$	−9.81102	−0.379884
$668$	54.0766	2.09228
$669$	0	0
$670$	3.79102	0.146460
$671$	0	0
$672$	0	0
$673$	−2.06786	−0.0797103	−0.0398551	−	0.999205i	$-0.512690\pi$
$673$	−2.06786	−0.0797103	−0.0398551	+	0.999205i	$0.512690\pi$
$674$	−68.3441	−2.63252
$675$	0	0
$676$	−16.7068	−0.642569
$677$	−18.0905	−0.695275	−0.347637	−	0.937629i	$-0.613016\pi$
$677$	−18.0905	−0.695275	−0.347637	+	0.937629i	$0.613016\pi$
$678$	0	0
$679$	20.5054	0.786925
$680$	−27.5180	−1.05527
$681$	0	0
$682$	0	0
$683$	47.3828	1.81305	0.906526	−	0.422149i	$-0.138724\pi$
$683$	47.3828	1.81305	0.906526	+	0.422149i	$0.138724\pi$
$684$	0	0
$685$	−22.7270	−0.868352
$686$	47.7408	1.82275
$687$	0	0
$688$	−27.0488	−1.03123
$689$	12.0006	0.457186
$690$	0	0
$691$	15.8475	0.602869	0.301435	−	0.953487i	$-0.402535\pi$
$691$	15.8475	0.602869	0.301435	+	0.953487i	$0.402535\pi$
$692$	−16.1342	−0.613331
$693$	0	0
$694$	−24.3558	−0.924534
$695$	−12.7100	−0.482116
$696$	0	0
$697$	−46.8592	−1.77492
$698$	56.8012	2.14996
$699$	0	0
$700$	7.35855	0.278127
$701$	−26.3059	−0.993560	−0.496780	−	0.867876i	$-0.665484\pi$
$701$	−26.3059	−0.993560	−0.496780	+	0.867876i	$0.665484\pi$
$702$	0	0
$703$	−15.7711	−0.594817
$704$	0	0
$705$	0	0
$706$	33.8599	1.27433
$707$	−33.9139	−1.27547
$708$	0	0
$709$	30.8329	1.15795	0.578977	−	0.815344i	$-0.303452\pi$
$709$	30.8329	1.15795	0.578977	+	0.815344i	$0.303452\pi$
$710$	−14.5093	−0.544523
$711$	0	0
$712$	−50.1174	−1.87823
$713$	22.6479	0.848172
$714$	0	0
$715$	0	0
$716$	27.8643	1.04134
$717$	0	0
$718$	57.5569	2.14801
$719$	−47.2434	−1.76188	−0.880940	−	0.473228i	$-0.843089\pi$
$719$	−47.2434	−1.76188	−0.880940	+	0.473228i	$0.843089\pi$
$720$	0	0
$721$	13.1388	0.489316
$722$	15.1490	0.563786
$723$	0	0
$724$	−76.4432	−2.84099
$725$	−3.51753	−0.130638
$726$	0	0
$727$	−24.4299	−0.906054	−0.453027	−	0.891497i	$-0.649656\pi$
$727$	−24.4299	−0.906054	−0.453027	+	0.891497i	$0.649656\pi$
$728$	−23.9104	−0.886177
$729$	0	0
$730$	−5.68292	−0.210334
$731$	72.1043	2.66688
$732$	0	0
$733$	27.9875	1.03374	0.516870	−	0.856064i	$-0.327097\pi$
$733$	27.9875	1.03374	0.516870	+	0.856064i	$0.327097\pi$
$734$	4.30379	0.158856
$735$	0	0
$736$	6.59766	0.243193
$737$	0	0
$738$	0	0
$739$	−2.70533	−0.0995172	−0.0497586	−	0.998761i	$-0.515845\pi$
$739$	−2.70533	−0.0995172	−0.0497586	+	0.998761i	$0.515845\pi$
$740$	−16.5484	−0.608331
$741$	0	0
$742$	−19.3869	−0.711716
$743$	−4.14414	−0.152034	−0.0760168	−	0.997107i	$-0.524220\pi$
$743$	−4.14414	−0.152034	−0.0760168	+	0.997107i	$0.524220\pi$
$744$	0	0
$745$	−4.30274	−0.157640
$746$	16.4723	0.603092
$747$	0	0
$748$	0	0
$749$	−18.9481	−0.692348
$750$	0	0
$751$	−44.4316	−1.62133	−0.810666	−	0.585508i	$-0.800895\pi$
$751$	−44.4316	−1.62133	−0.810666	+	0.585508i	$0.800895\pi$
$752$	−0.0674331	−0.00245903
$753$	0	0
$754$	24.6004	0.895894
$755$	−17.5844	−0.639962
$756$	0	0
$757$	29.9901	1.09001	0.545004	−	0.838433i	$-0.316528\pi$
$757$	29.9901	1.09001	0.545004	+	0.838433i	$0.316528\pi$
$758$	8.46192	0.307351
$759$	0	0
$760$	−14.7980	−0.536782
$761$	−39.8420	−1.44427	−0.722135	−	0.691752i	$-0.756839\pi$
$761$	−39.8420	−1.44427	−0.722135	+	0.691752i	$0.756839\pi$
$762$	0	0
$763$	−37.9397	−1.37351
$764$	−24.4747	−0.885462
$765$	0	0
$766$	4.55834	0.164700
$767$	−27.7841	−1.00323
$768$	0	0
$769$	20.5304	0.740346	0.370173	−	0.928963i	$-0.379298\pi$
$769$	20.5304	0.740346	0.370173	+	0.928963i	$0.379298\pi$
$770$	0	0
$771$	0	0
$772$	−25.4984	−0.917708
$773$	−16.3315	−0.587401	−0.293701	−	0.955897i	$-0.594887\pi$
$773$	−16.3315	−0.587401	−0.293701	+	0.955897i	$0.594887\pi$
$774$	0	0
$775$	8.11992	0.291676
$776$	−43.2688	−1.55326
$777$	0	0
$778$	16.8175	0.602938
$779$	−25.1990	−0.902848
$780$	0	0
$781$	0	0
$782$	44.2225	1.58139
$783$	0	0
$784$	−7.74781	−0.276708
$785$	23.8112	0.849860
$786$	0	0
$787$	−7.38766	−0.263342	−0.131671	−	0.991293i	$-0.542034\pi$
$787$	−7.38766	−0.263342	−0.131671	+	0.991293i	$0.542034\pi$
$788$	−12.8912	−0.459231
$789$	0	0
$790$	−30.0296	−1.06841
$791$	−7.01823	−0.249539
$792$	0	0
$793$	9.05941	0.321709
$794$	29.9720	1.06367
$795$	0	0
$796$	−39.8744	−1.41331
$797$	−23.1554	−0.820207	−0.410104	−	0.912039i	$-0.634508\pi$
$797$	−23.1554	−0.820207	−0.410104	+	0.912039i	$0.634508\pi$
$798$	0	0
$799$	0.179757	0.00635935
$800$	2.36545	0.0836311
$801$	0	0
$802$	−65.4491	−2.31109
$803$	0	0
$804$	0	0
$805$	−5.49425	−0.193647
$806$	−56.7880	−2.00027
$807$	0	0
$808$	71.5624	2.51756
$809$	14.9484	0.525559	0.262779	−	0.964856i	$-0.415361\pi$
$809$	14.9484	0.525559	0.262779	+	0.964856i	$0.415361\pi$
$810$	0	0
$811$	−25.1135	−0.881855	−0.440927	−	0.897543i	$-0.645350\pi$
$811$	−25.1135	−0.881855	−0.440927	+	0.897543i	$0.645350\pi$
$812$	−25.8839	−0.908347
$813$	0	0
$814$	0	0
$815$	19.8452	0.695147
$816$	0	0
$817$	38.7748	1.35656
$818$	27.7785	0.971251
$819$	0	0
$820$	−26.4410	−0.923359
$821$	13.1035	0.457315	0.228657	−	0.973507i	$-0.426566\pi$
$821$	13.1035	0.457315	0.228657	+	0.973507i	$0.426566\pi$
$822$	0	0
$823$	−20.5741	−0.717169	−0.358584	−	0.933497i	$-0.616741\pi$
$823$	−20.5741	−0.717169	−0.358584	+	0.933497i	$0.616741\pi$
$824$	−27.7245	−0.965828
$825$	0	0
$826$	44.8852	1.56176
$827$	−12.6343	−0.439338	−0.219669	−	0.975574i	$-0.570498\pi$
$827$	−12.6343	−0.439338	−0.219669	+	0.975574i	$0.570498\pi$
$828$	0	0
$829$	7.78876	0.270515	0.135257	−	0.990810i	$-0.456814\pi$
$829$	7.78876	0.270515	0.135257	+	0.990810i	$0.456814\pi$
$830$	−12.0783	−0.419245
$831$	0	0
$832$	−31.0479	−1.07639
$833$	20.6534	0.715599
$834$	0	0
$835$	−14.4760	−0.500963
$836$	0	0
$837$	0	0
$838$	−21.7348	−0.750817
$839$	−30.5522	−1.05478	−0.527389	−	0.849624i	$-0.676829\pi$
$839$	−30.5522	−1.05478	−0.527389	+	0.849624i	$0.676829\pi$
$840$	0	0
$841$	−16.6270	−0.573345
$842$	−44.3421	−1.52813
$843$	0	0
$844$	−5.15226	−0.177348
$845$	4.47232	0.153853
$846$	0	0
$847$	0	0
$848$	10.2059	0.350472
$849$	0	0
$850$	15.8550	0.543823
$851$	12.3558	0.423553
$852$	0	0
$853$	25.6737	0.879051	0.439525	−	0.898230i	$-0.355147\pi$
$853$	25.6737	0.879051	0.439525	+	0.898230i	$0.355147\pi$
$854$	−14.6354	−0.500815
$855$	0	0
$856$	39.9827	1.36658
$857$	−14.7370	−0.503407	−0.251703	−	0.967804i	$-0.580991\pi$
$857$	−14.7370	−0.503407	−0.251703	+	0.967804i	$0.580991\pi$
$858$	0	0
$859$	−1.68741	−0.0575738	−0.0287869	−	0.999586i	$-0.509164\pi$
$859$	−1.68741	−0.0575738	−0.0287869	+	0.999586i	$0.509164\pi$
$860$	40.6859	1.38738
$861$	0	0
$862$	59.3729	2.02225
$863$	13.1473	0.447538	0.223769	−	0.974642i	$-0.428164\pi$
$863$	13.1473	0.447538	0.223769	+	0.974642i	$0.428164\pi$
$864$	0	0
$865$	4.31904	0.146852
$866$	84.7458	2.87978
$867$	0	0
$868$	59.7509	2.02808
$869$	0	0
$870$	0	0
$871$	4.62255	0.156629
$872$	80.0572	2.71108
$873$	0	0
$874$	23.7811	0.804408
$875$	−1.96984	−0.0665929
$876$	0	0
$877$	−46.9492	−1.58536	−0.792681	−	0.609637i	$-0.791315\pi$
$877$	−46.9492	−1.58536	−0.792681	+	0.609637i	$0.791315\pi$
$878$	24.6475	0.831813
$879$	0	0
$880$	0	0
$881$	14.1843	0.477880	0.238940	−	0.971034i	$-0.423200\pi$
$881$	14.1843	0.477880	0.238940	+	0.971034i	$0.423200\pi$
$882$	0	0
$883$	−10.3899	−0.349647	−0.174824	−	0.984600i	$-0.555936\pi$
$883$	−10.3899	−0.349647	−0.174824	+	0.984600i	$0.555936\pi$
$884$	−72.2193	−2.42900
$885$	0	0
$886$	−85.8581	−2.88446
$887$	−3.84764	−0.129191	−0.0645955	−	0.997912i	$-0.520576\pi$
$887$	−3.84764	−0.129191	−0.0645955	+	0.997912i	$0.520576\pi$
$888$	0	0
$889$	38.0533	1.27627
$890$	28.8761	0.967931
$891$	0	0
$892$	45.2150	1.51391
$893$	0.0966662	0.00323481
$894$	0	0
$895$	−7.45911	−0.249331
$896$	40.8386	1.36432
$897$	0	0
$898$	64.2881	2.14532
$899$	−28.5621	−0.952598
$900$	0	0
$901$	−27.2060	−0.906363
$902$	0	0
$903$	0	0
$904$	14.8093	0.492550
$905$	20.4634	0.680228
$906$	0	0
$907$	1.20437	0.0399904	0.0199952	−	0.999800i	$-0.493635\pi$
$907$	1.20437	0.0399904	0.0199952	+	0.999800i	$0.493635\pi$
$908$	64.7455	2.14865
$909$	0	0
$910$	13.7764	0.456684
$911$	38.9307	1.28983	0.644915	−	0.764254i	$-0.276893\pi$
$911$	38.9307	1.28983	0.644915	+	0.764254i	$0.276893\pi$
$912$	0	0
$913$	0	0
$914$	47.0313	1.55566
$915$	0	0
$916$	−3.12736	−0.103331
$917$	−27.3861	−0.904369
$918$	0	0
$919$	−9.56757	−0.315605	−0.157802	−	0.987471i	$-0.550441\pi$
$919$	−9.56757	−0.315605	−0.157802	+	0.987471i	$0.550441\pi$
$920$	11.5935	0.382227
$921$	0	0
$922$	−46.8507	−1.54295
$923$	−17.6918	−0.582332
$924$	0	0
$925$	4.42991	0.145655
$926$	35.3188	1.16065
$927$	0	0
$928$	−8.32052	−0.273134
$929$	−6.45603	−0.211816	−0.105908	−	0.994376i	$-0.533775\pi$
$929$	−6.45603	−0.211816	−0.105908	+	0.994376i	$0.533775\pi$
$930$	0	0
$931$	11.1066	0.364003
$932$	71.1561	2.33079
$933$	0	0
$934$	24.6793	0.807532
$935$	0	0
$936$	0	0
$937$	0.478554	0.0156337	0.00781684	−	0.999969i	$-0.497512\pi$
$937$	0.478554	0.0156337	0.00781684	+	0.999969i	$0.497512\pi$
$938$	−7.46772	−0.243830
$939$	0	0
$940$	0.101431	0.00330830
$941$	18.3345	0.597689	0.298845	−	0.954302i	$-0.403399\pi$
$941$	18.3345	0.597689	0.298845	+	0.954302i	$0.403399\pi$
$942$	0	0
$943$	19.7421	0.642893
$944$	−23.6290	−0.769060
$945$	0	0
$946$	0	0
$947$	−23.6619	−0.768908	−0.384454	−	0.923144i	$-0.625610\pi$
$947$	−23.6619	−0.768908	−0.384454	+	0.923144i	$0.625610\pi$
$948$	0	0
$949$	−6.92943	−0.224939
$950$	8.52619	0.276626
$951$	0	0
$952$	54.2061	1.75683
$953$	19.5111	0.632026	0.316013	−	0.948755i	$-0.397656\pi$
$953$	19.5111	0.632026	0.316013	+	0.948755i	$0.397656\pi$
$954$	0	0
$955$	6.55174	0.212009
$956$	−96.3544	−3.11632
$957$	0	0
$958$	27.2962	0.881899
$959$	44.7686	1.44565
$960$	0	0
$961$	34.9332	1.12688
$962$	−30.9813	−0.998878
$963$	0	0
$964$	−63.7591	−2.05354
$965$	6.82579	0.219730
$966$	0	0
$967$	−13.3686	−0.429905	−0.214953	−	0.976624i	$-0.568960\pi$
$967$	−13.3686	−0.429905	−0.214953	+	0.976624i	$0.568960\pi$
$968$	0	0
$969$	0	0
$970$	24.9302	0.800461
$971$	55.6001	1.78429	0.892146	−	0.451747i	$-0.149199\pi$
$971$	55.6001	1.78429	0.892146	+	0.451747i	$0.149199\pi$
$972$	0	0
$973$	25.0366	0.802638
$974$	82.9680	2.65847
$975$	0	0
$976$	7.70458	0.246618
$977$	−42.9068	−1.37271	−0.686355	−	0.727267i	$-0.740790\pi$
$977$	−42.9068	−1.37271	−0.686355	+	0.727267i	$0.740790\pi$
$978$	0	0
$979$	0	0
$980$	11.6540	0.372273
$981$	0	0
$982$	87.6903	2.79831
$983$	34.5788	1.10289	0.551446	−	0.834211i	$-0.314076\pi$
$983$	34.5788	1.10289	0.551446	+	0.834211i	$0.314076\pi$
$984$	0	0
$985$	3.45091	0.109955
$986$	−55.7705	−1.77609
$987$	0	0
$988$	−38.8366	−1.23556
$989$	−30.3781	−0.965968
$990$	0	0
$991$	15.5839	0.495038	0.247519	−	0.968883i	$-0.420385\pi$
$991$	15.5839	0.495038	0.247519	+	0.968883i	$0.420385\pi$
$992$	19.2072	0.609830
$993$	0	0
$994$	28.5810	0.906534
$995$	10.6742	0.338394
$996$	0	0
$997$	4.04074	0.127972	0.0639858	−	0.997951i	$-0.479619\pi$
$997$	4.04074	0.127972	0.0639858	+	0.997951i	$0.479619\pi$
$998$	51.1349	1.61865
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.39491 q^{2} +3.73560 q^{4} -1.00000 q^{5} +1.96984 q^{7} -4.15660 q^{8} +O(q^{10})\)	\(q-2.39491 q^{2} +3.73560 q^{4} -1.00000 q^{5} +1.96984 q^{7} -4.15660 q^{8} +2.39491 q^{10} +2.92022 q^{13} -4.71760 q^{14} +2.48350 q^{16} -6.62030 q^{17} -3.56013 q^{19} -3.73560 q^{20} +2.78918 q^{23} +1.00000 q^{25} -6.99366 q^{26} +7.35855 q^{28} -3.51753 q^{29} +8.11992 q^{31} +2.36545 q^{32} +15.8550 q^{34} -1.96984 q^{35} +4.42991 q^{37} +8.52619 q^{38} +4.15660 q^{40} +7.07811 q^{41} -10.8914 q^{43} -6.67984 q^{46} -0.0271524 q^{47} -3.11971 q^{49} -2.39491 q^{50} +10.9088 q^{52} +4.10948 q^{53} -8.18786 q^{56} +8.42417 q^{58} -9.51441 q^{59} +3.10231 q^{61} -19.4465 q^{62} -10.6320 q^{64} -2.92022 q^{65} +1.58295 q^{67} -24.7308 q^{68} +4.71760 q^{70} -6.05837 q^{71} -2.37292 q^{73} -10.6092 q^{74} -13.2992 q^{76} -12.5389 q^{79} -2.48350 q^{80} -16.9515 q^{82} -5.04333 q^{83} +6.62030 q^{85} +26.0840 q^{86} +12.0573 q^{89} +5.75238 q^{91} +10.4193 q^{92} +0.0650277 q^{94} +3.56013 q^{95} +10.4097 q^{97} +7.47144 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 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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