LMFDB - Embedded newform 5445.2.a.cc.1.5

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:	$8$
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			1.5
Root			$-0.205878$ 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)

$f(q)$	$=$	$q+1.20588 q^{2} -0.545859 q^{4} -1.00000 q^{5} -4.31539 q^{7} -3.06999 q^{8} +O(q^{10})$	$q+1.20588 q^{2} -0.545859 q^{4} -1.00000 q^{5} -4.31539 q^{7} -3.06999 q^{8} -1.20588 q^{10} -6.18245 q^{13} -5.20383 q^{14} -2.61032 q^{16} -0.803988 q^{17} +1.87389 q^{19} +0.545859 q^{20} -8.08431 q^{23} +1.00000 q^{25} -7.45528 q^{26} +2.35559 q^{28} -1.05948 q^{29} -1.61509 q^{31} +2.99226 q^{32} -0.969511 q^{34} +4.31539 q^{35} -5.54134 q^{37} +2.25968 q^{38} +3.06999 q^{40} +5.05198 q^{41} +9.87101 q^{43} -9.74869 q^{46} -12.9857 q^{47} +11.6226 q^{49} +1.20588 q^{50} +3.37475 q^{52} +2.66437 q^{53} +13.2482 q^{56} -1.27760 q^{58} -6.60968 q^{59} +5.34760 q^{61} -1.94760 q^{62} +8.82894 q^{64} +6.18245 q^{65} -7.49018 q^{67} +0.438864 q^{68} +5.20383 q^{70} +5.82947 q^{71} -4.80148 q^{73} -6.68218 q^{74} -1.02288 q^{76} +13.9499 q^{79} +2.61032 q^{80} +6.09207 q^{82} +12.0568 q^{83} +0.803988 q^{85} +11.9032 q^{86} -12.7727 q^{89} +26.6797 q^{91} +4.41289 q^{92} -15.6592 q^{94} -1.87389 q^{95} +1.34761 q^{97} +14.0154 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.20588	0.852684	0.426342	−	0.904562i	$-0.359802\pi$
$2$	1.20588	0.852684	0.426342	+	0.904562i	$0.359802\pi$
$3$	0	0
$3$	0	0
$4$	−0.545859	−0.272929
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.31539	−1.63106	−0.815532	−	0.578712i	$-0.803555\pi$
$7$	−4.31539	−1.63106	−0.815532	+	0.578712i	$0.803555\pi$
$8$	−3.06999	−1.08541
$9$	0	0
$10$	−1.20588	−0.381332
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.18245	−1.71470	−0.857352	−	0.514731i	$-0.827892\pi$
$13$	−6.18245	−1.71470	−0.857352	+	0.514731i	$0.827892\pi$
$14$	−5.20383	−1.39078
$15$	0	0
$16$	−2.61032	−0.652580
$17$	−0.803988	−0.194996	−0.0974978	−	0.995236i	$-0.531084\pi$
$17$	−0.803988	−0.194996	−0.0974978	+	0.995236i	$0.531084\pi$
$18$	0	0
$19$	1.87389	0.429899	0.214949	−	0.976625i	$-0.431041\pi$
$19$	1.87389	0.429899	0.214949	+	0.976625i	$0.431041\pi$
$20$	0.545859	0.122058
$21$	0	0
$22$	0	0
$23$	−8.08431	−1.68570	−0.842848	−	0.538152i	$-0.819123\pi$
$23$	−8.08431	−1.68570	−0.842848	+	0.538152i	$0.819123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−7.45528	−1.46210
$27$	0	0
$28$	2.35559	0.445165
$29$	−1.05948	−0.196741	−0.0983703	−	0.995150i	$-0.531363\pi$
$29$	−1.05948	−0.196741	−0.0983703	+	0.995150i	$0.531363\pi$
$30$	0	0
$31$	−1.61509	−0.290078	−0.145039	−	0.989426i	$-0.546331\pi$
$31$	−1.61509	−0.290078	−0.145039	+	0.989426i	$0.546331\pi$
$32$	2.99226	0.528962
$33$	0	0
$34$	−0.969511	−0.166270
$35$	4.31539	0.729434
$36$	0	0
$37$	−5.54134	−0.910991	−0.455496	−	0.890238i	$-0.650538\pi$
$37$	−5.54134	−0.910991	−0.455496	+	0.890238i	$0.650538\pi$
$38$	2.25968	0.366568
$39$	0	0
$40$	3.06999	0.485409
$41$	5.05198	0.788987	0.394494	−	0.918899i	$-0.370920\pi$
$41$	5.05198	0.788987	0.394494	+	0.918899i	$0.370920\pi$
$42$	0	0
$43$	9.87101	1.50531	0.752657	−	0.658412i	$-0.228772\pi$
$43$	9.87101	1.50531	0.752657	+	0.658412i	$0.228772\pi$
$44$	0	0
$45$	0	0
$46$	−9.74869	−1.43737
$47$	−12.9857	−1.89416	−0.947079	−	0.321002i	$-0.895980\pi$
$47$	−12.9857	−1.89416	−0.947079	+	0.321002i	$0.895980\pi$
$48$	0	0
$49$	11.6226	1.66037
$50$	1.20588	0.170537
$51$	0	0
$52$	3.37475	0.467993
$53$	2.66437	0.365980	0.182990	−	0.983115i	$-0.441422\pi$
$53$	2.66437	0.365980	0.182990	+	0.983115i	$0.441422\pi$
$54$	0	0
$55$	0	0
$56$	13.2482	1.77037
$57$	0	0
$58$	−1.27760	−0.167758
$59$	−6.60968	−0.860507	−0.430254	−	0.902708i	$-0.641576\pi$
$59$	−6.60968	−0.860507	−0.430254	+	0.902708i	$0.641576\pi$
$60$	0	0
$61$	5.34760	0.684689	0.342345	−	0.939574i	$-0.388779\pi$
$61$	5.34760	0.684689	0.342345	+	0.939574i	$0.388779\pi$
$62$	−1.94760	−0.247345
$63$	0	0
$64$	8.82894	1.10362
$65$	6.18245	0.766839
$66$	0	0
$67$	−7.49018	−0.915071	−0.457536	−	0.889191i	$-0.651268\pi$
$67$	−7.49018	−0.915071	−0.457536	+	0.889191i	$0.651268\pi$
$68$	0.438864	0.0532201
$69$	0	0
$70$	5.20383	0.621977
$71$	5.82947	0.691830	0.345915	−	0.938266i	$-0.387568\pi$
$71$	5.82947	0.691830	0.345915	+	0.938266i	$0.387568\pi$
$72$	0	0
$73$	−4.80148	−0.561970	−0.280985	−	0.959712i	$-0.590661\pi$
$73$	−4.80148	−0.561970	−0.280985	+	0.959712i	$0.590661\pi$
$74$	−6.68218	−0.776788
$75$	0	0
$76$	−1.02288	−0.117332
$77$	0	0
$78$	0	0
$79$	13.9499	1.56949	0.784744	−	0.619820i	$-0.212794\pi$
$79$	13.9499	1.56949	0.784744	+	0.619820i	$0.212794\pi$
$80$	2.61032	0.291843
$81$	0	0
$82$	6.09207	0.672757
$83$	12.0568	1.32340	0.661701	−	0.749768i	$-0.269835\pi$
$83$	12.0568	1.32340	0.661701	+	0.749768i	$0.269835\pi$
$84$	0	0
$85$	0.803988	0.0872047
$86$	11.9032	1.28356
$87$	0	0
$88$	0	0
$89$	−12.7727	−1.35390	−0.676951	−	0.736028i	$-0.736699\pi$
$89$	−12.7727	−1.35390	−0.676951	+	0.736028i	$0.736699\pi$
$90$	0	0
$91$	26.6797	2.79679
$92$	4.41289	0.460076
$93$	0	0
$94$	−15.6592	−1.61512
$95$	−1.87389	−0.192257
$96$	0	0
$97$	1.34761	0.136829	0.0684145	−	0.997657i	$-0.478206\pi$
$97$	1.34761	0.136829	0.0684145	+	0.997657i	$0.478206\pi$
$98$	14.0154	1.41577
$99$	0	0
$100$	−0.545859	−0.0545859
$101$	−8.42949	−0.838766	−0.419383	−	0.907810i	$-0.637753\pi$
$101$	−8.42949	−0.838766	−0.419383	+	0.907810i	$0.637753\pi$
$102$	0	0
$103$	−9.90039	−0.975515	−0.487757	−	0.872979i	$-0.662185\pi$
$103$	−9.90039	−0.975515	−0.487757	+	0.872979i	$0.662185\pi$
$104$	18.9801	1.86115
$105$	0	0
$106$	3.21291	0.312065
$107$	8.16742	0.789574	0.394787	−	0.918773i	$-0.370818\pi$
$107$	8.16742	0.789574	0.394787	+	0.918773i	$0.370818\pi$
$108$	0	0
$109$	−13.4997	−1.29303	−0.646517	−	0.762900i	$-0.723775\pi$
$109$	−13.4997	−1.29303	−0.646517	+	0.762900i	$0.723775\pi$
$110$	0	0
$111$	0	0
$112$	11.2645	1.06440
$113$	13.5106	1.27097	0.635485	−	0.772114i	$-0.280800\pi$
$113$	13.5106	1.27097	0.635485	+	0.772114i	$0.280800\pi$
$114$	0	0
$115$	8.08431	0.753866
$116$	0.578327	0.0536963
$117$	0	0
$118$	−7.97047	−0.733741
$119$	3.46952	0.318050
$120$	0	0
$121$	0	0
$122$	6.44855	0.583824
$123$	0	0
$124$	0.881610	0.0791709
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.12012	0.188130	0.0940652	−	0.995566i	$-0.470014\pi$
$127$	2.12012	0.188130	0.0940652	+	0.995566i	$0.470014\pi$
$128$	4.66210	0.412075
$129$	0	0
$130$	7.45528	0.653871
$131$	8.72324	0.762153	0.381076	−	0.924544i	$-0.375553\pi$
$131$	8.72324	0.762153	0.381076	+	0.924544i	$0.375553\pi$
$132$	0	0
$133$	−8.08654	−0.701192
$134$	−9.03224	−0.780267
$135$	0	0
$136$	2.46824	0.211650
$137$	1.54382	0.131898	0.0659489	−	0.997823i	$-0.478993\pi$
$137$	1.54382	0.131898	0.0659489	+	0.997823i	$0.478993\pi$
$138$	0	0
$139$	−16.1982	−1.37391	−0.686957	−	0.726698i	$-0.741054\pi$
$139$	−16.1982	−1.37391	−0.686957	+	0.726698i	$0.741054\pi$
$140$	−2.35559	−0.199084
$141$	0	0
$142$	7.02962	0.589913
$143$	0	0
$144$	0	0
$145$	1.05948	0.0879851
$146$	−5.78999	−0.479183
$147$	0	0
$148$	3.02479	0.248636
$149$	14.0183	1.14842	0.574212	−	0.818707i	$-0.305308\pi$
$149$	14.0183	1.14842	0.574212	+	0.818707i	$0.305308\pi$
$150$	0	0
$151$	13.5852	1.10554	0.552772	−	0.833332i	$-0.313570\pi$
$151$	13.5852	1.10554	0.552772	+	0.833332i	$0.313570\pi$
$152$	−5.75282	−0.466615
$153$	0	0
$154$	0	0
$155$	1.61509	0.129727
$156$	0	0
$157$	−4.38074	−0.349621	−0.174810	−	0.984602i	$-0.555931\pi$
$157$	−4.38074	−0.349621	−0.174810	+	0.984602i	$0.555931\pi$
$158$	16.8219	1.33828
$159$	0	0
$160$	−2.99226	−0.236559
$161$	34.8869	2.74948
$162$	0	0
$163$	−4.68667	−0.367089	−0.183544	−	0.983011i	$-0.558757\pi$
$163$	−4.68667	−0.367089	−0.183544	+	0.983011i	$0.558757\pi$
$164$	−2.75767	−0.215338
$165$	0	0
$166$	14.5390	1.12844
$167$	1.64917	0.127617	0.0638084	−	0.997962i	$-0.479675\pi$
$167$	1.64917	0.127617	0.0638084	+	0.997962i	$0.479675\pi$
$168$	0	0
$169$	25.2227	1.94021
$170$	0.969511	0.0743581
$171$	0	0
$172$	−5.38818	−0.410845
$173$	−16.5722	−1.25996	−0.629981	−	0.776611i	$-0.716937\pi$
$173$	−16.5722	−1.25996	−0.629981	+	0.776611i	$0.716937\pi$
$174$	0	0
$175$	−4.31539	−0.326213
$176$	0	0
$177$	0	0
$178$	−15.4023	−1.15445
$179$	−17.5995	−1.31545	−0.657724	−	0.753259i	$-0.728481\pi$
$179$	−17.5995	−1.31545	−0.657724	+	0.753259i	$0.728481\pi$
$180$	0	0
$181$	8.40672	0.624867	0.312433	−	0.949940i	$-0.398856\pi$
$181$	8.40672	0.624867	0.312433	+	0.949940i	$0.398856\pi$
$182$	32.1724	2.38478
$183$	0	0
$184$	24.8188	1.82967
$185$	5.54134	0.407408
$186$	0	0
$187$	0	0
$188$	7.08835	0.516971
$189$	0	0
$190$	−2.25968	−0.163934
$191$	5.31771	0.384776	0.192388	−	0.981319i	$-0.438377\pi$
$191$	5.31771	0.384776	0.192388	+	0.981319i	$0.438377\pi$
$192$	0	0
$193$	−15.2872	−1.10040	−0.550199	−	0.835033i	$-0.685448\pi$
$193$	−15.2872	−1.10040	−0.550199	+	0.835033i	$0.685448\pi$
$194$	1.62505	0.116672
$195$	0	0
$196$	−6.34429	−0.453163
$197$	−1.69118	−0.120492	−0.0602458	−	0.998184i	$-0.519188\pi$
$197$	−1.69118	−0.120492	−0.0602458	+	0.998184i	$0.519188\pi$
$198$	0	0
$199$	3.44855	0.244461	0.122230	−	0.992502i	$-0.460995\pi$
$199$	3.44855	0.244461	0.122230	+	0.992502i	$0.460995\pi$
$200$	−3.06999	−0.217081
$201$	0	0
$202$	−10.1649	−0.715202
$203$	4.57207	0.320896
$204$	0	0
$205$	−5.05198	−0.352846
$206$	−11.9387	−0.831806
$207$	0	0
$208$	16.1382	1.11898
$209$	0	0
$210$	0	0
$211$	0.394446	0.0271548	0.0135774	−	0.999908i	$-0.495678\pi$
$211$	0.394446	0.0271548	0.0135774	+	0.999908i	$0.495678\pi$
$212$	−1.45437	−0.0998866
$213$	0	0
$214$	9.84890	0.673257
$215$	−9.87101	−0.673197
$216$	0	0
$217$	6.96973	0.473136
$218$	−16.2790	−1.10255
$219$	0	0
$220$	0	0
$221$	4.97062	0.334360
$222$	0	0
$223$	−7.77146	−0.520416	−0.260208	−	0.965553i	$-0.583791\pi$
$223$	−7.77146	−0.520416	−0.260208	+	0.965553i	$0.583791\pi$
$224$	−12.9128	−0.862771
$225$	0	0
$226$	16.2921	1.08374
$227$	27.3162	1.81304	0.906519	−	0.422164i	$-0.138729\pi$
$227$	27.3162	1.81304	0.906519	+	0.422164i	$0.138729\pi$
$228$	0	0
$229$	15.5822	1.02970	0.514851	−	0.857280i	$-0.327847\pi$
$229$	15.5822	1.02970	0.514851	+	0.857280i	$0.327847\pi$
$230$	9.74869	0.642810
$231$	0	0
$232$	3.25260	0.213544
$233$	−27.8846	−1.82678	−0.913390	−	0.407086i	$-0.866545\pi$
$233$	−27.8846	−1.82678	−0.913390	+	0.407086i	$0.866545\pi$
$234$	0	0
$235$	12.9857	0.847093
$236$	3.60795	0.234858
$237$	0	0
$238$	4.18382	0.271197
$239$	18.7458	1.21257	0.606284	−	0.795248i	$-0.292660\pi$
$239$	18.7458	1.21257	0.606284	+	0.795248i	$0.292660\pi$
$240$	0	0
$241$	11.0049	0.708886	0.354443	−	0.935078i	$-0.384671\pi$
$241$	11.0049	0.708886	0.354443	+	0.935078i	$0.384671\pi$
$242$	0	0
$243$	0	0
$244$	−2.91903	−0.186872
$245$	−11.6226	−0.742539
$246$	0	0
$247$	−11.5852	−0.737149
$248$	4.95831	0.314853
$249$	0	0
$250$	−1.20588	−0.0762664
$251$	23.3136	1.47154	0.735769	−	0.677232i	$-0.236821\pi$
$251$	23.3136	1.47154	0.735769	+	0.677232i	$0.236821\pi$
$252$	0	0
$253$	0	0
$254$	2.55661	0.160416
$255$	0	0
$256$	−12.0360	−0.752248
$257$	−17.4730	−1.08994	−0.544969	−	0.838456i	$-0.683459\pi$
$257$	−17.4730	−1.08994	−0.544969	+	0.838456i	$0.683459\pi$
$258$	0	0
$259$	23.9130	1.48588
$260$	−3.37475	−0.209293
$261$	0	0
$262$	10.5192	0.649876
$263$	1.61187	0.0993921	0.0496961	−	0.998764i	$-0.484175\pi$
$263$	1.61187	0.0993921	0.0496961	+	0.998764i	$0.484175\pi$
$264$	0	0
$265$	−2.66437	−0.163671
$266$	−9.75138	−0.597896
$267$	0	0
$268$	4.08858	0.249750
$269$	−8.17266	−0.498296	−0.249148	−	0.968465i	$-0.580151\pi$
$269$	−8.17266	−0.498296	−0.249148	+	0.968465i	$0.580151\pi$
$270$	0	0
$271$	−10.4995	−0.637801	−0.318900	−	0.947788i	$-0.603314\pi$
$271$	−10.4995	−0.637801	−0.318900	+	0.947788i	$0.603314\pi$
$272$	2.09867	0.127250
$273$	0	0
$274$	1.86166	0.112467
$275$	0	0
$276$	0	0
$277$	−8.92222	−0.536084	−0.268042	−	0.963407i	$-0.586377\pi$
$277$	−8.92222	−0.536084	−0.268042	+	0.963407i	$0.586377\pi$
$278$	−19.5331	−1.17151
$279$	0	0
$280$	−13.2482	−0.791732
$281$	4.39358	0.262099	0.131049	−	0.991376i	$-0.458165\pi$
$281$	4.39358	0.262099	0.131049	+	0.991376i	$0.458165\pi$
$282$	0	0
$283$	−13.0048	−0.773056	−0.386528	−	0.922278i	$-0.626326\pi$
$283$	−13.0048	−0.773056	−0.386528	+	0.922278i	$0.626326\pi$
$284$	−3.18207	−0.188821
$285$	0	0
$286$	0	0
$287$	−21.8013	−1.28689
$288$	0	0
$289$	−16.3536	−0.961977
$290$	1.27760	0.0750235
$291$	0	0
$292$	2.62093	0.153378
$293$	15.5444	0.908115	0.454058	−	0.890972i	$-0.349976\pi$
$293$	15.5444	0.908115	0.454058	+	0.890972i	$0.349976\pi$
$294$	0	0
$295$	6.60968	0.384830
$296$	17.0119	0.988796
$297$	0	0
$298$	16.9044	0.979243
$299$	49.9809	2.89047
$300$	0	0
$301$	−42.5972	−2.45526
$302$	16.3820	0.942680
$303$	0	0
$304$	−4.89144	−0.280543
$305$	−5.34760	−0.306202
$306$	0	0
$307$	−8.76859	−0.500450	−0.250225	−	0.968188i	$-0.580505\pi$
$307$	−8.76859	−0.500450	−0.250225	+	0.968188i	$0.580505\pi$
$308$	0	0
$309$	0	0
$310$	1.94760	0.110616
$311$	1.75121	0.0993020	0.0496510	−	0.998767i	$-0.484189\pi$
$311$	1.75121	0.0993020	0.0496510	+	0.998767i	$0.484189\pi$
$312$	0	0
$313$	−17.5272	−0.990697	−0.495348	−	0.868694i	$-0.664960\pi$
$313$	−17.5272	−0.990697	−0.495348	+	0.868694i	$0.664960\pi$
$314$	−5.28263	−0.298116
$315$	0	0
$316$	−7.61468	−0.428359
$317$	1.99187	0.111874	0.0559372	−	0.998434i	$-0.482185\pi$
$317$	1.99187	0.111874	0.0559372	+	0.998434i	$0.482185\pi$
$318$	0	0
$319$	0	0
$320$	−8.82894	−0.493553
$321$	0	0
$322$	42.0694	2.34444
$323$	−1.50658	−0.0838284
$324$	0	0
$325$	−6.18245	−0.342941
$326$	−5.65156	−0.313011
$327$	0	0
$328$	−15.5096	−0.856372
$329$	56.0383	3.08949
$330$	0	0
$331$	−10.3437	−0.568541	−0.284271	−	0.958744i	$-0.591751\pi$
$331$	−10.3437	−0.568541	−0.284271	+	0.958744i	$0.591751\pi$
$332$	−6.58130	−0.361196
$333$	0	0
$334$	1.98870	0.108817
$335$	7.49018	0.409232
$336$	0	0
$337$	20.6187	1.12317	0.561587	−	0.827418i	$-0.310191\pi$
$337$	20.6187	1.12317	0.561587	+	0.827418i	$0.310191\pi$
$338$	30.4155	1.65439
$339$	0	0
$340$	−0.438864	−0.0238007
$341$	0	0
$342$	0	0
$343$	−19.9482	−1.07710
$344$	−30.3039	−1.63388
$345$	0	0
$346$	−19.9841	−1.07435
$347$	2.98375	0.160176	0.0800879	−	0.996788i	$-0.474480\pi$
$347$	2.98375	0.160176	0.0800879	+	0.996788i	$0.474480\pi$
$348$	0	0
$349$	−11.2897	−0.604323	−0.302161	−	0.953257i	$-0.597708\pi$
$349$	−11.2897	−0.604323	−0.302161	+	0.953257i	$0.597708\pi$
$350$	−5.20383	−0.278156
$351$	0	0
$352$	0	0
$353$	13.4952	0.718275	0.359137	−	0.933285i	$-0.383071\pi$
$353$	13.4952	0.718275	0.359137	+	0.933285i	$0.383071\pi$
$354$	0	0
$355$	−5.82947	−0.309396
$356$	6.97209	0.369520
$357$	0	0
$358$	−21.2228	−1.12166
$359$	−24.6989	−1.30356	−0.651778	−	0.758410i	$-0.725977\pi$
$359$	−24.6989	−1.30356	−0.651778	+	0.758410i	$0.725977\pi$
$360$	0	0
$361$	−15.4886	−0.815187
$362$	10.1375	0.532814
$363$	0	0
$364$	−14.5633	−0.763327
$365$	4.80148	0.251321
$366$	0	0
$367$	6.34491	0.331201	0.165601	−	0.986193i	$-0.447044\pi$
$367$	6.34491	0.331201	0.165601	+	0.986193i	$0.447044\pi$
$368$	21.1026	1.10005
$369$	0	0
$370$	6.68218	0.347390
$371$	−11.4978	−0.596936
$372$	0	0
$373$	−13.1832	−0.682601	−0.341301	−	0.939954i	$-0.610867\pi$
$373$	−13.1832	−0.682601	−0.341301	+	0.939954i	$0.610867\pi$
$374$	0	0
$375$	0	0
$376$	39.8660	2.05593
$377$	6.55019	0.337352
$378$	0	0
$379$	−18.5363	−0.952144	−0.476072	−	0.879406i	$-0.657940\pi$
$379$	−18.5363	−0.952144	−0.476072	+	0.879406i	$0.657940\pi$
$380$	1.02288	0.0524725
$381$	0	0
$382$	6.41251	0.328093
$383$	28.3843	1.45037	0.725185	−	0.688554i	$-0.241754\pi$
$383$	28.3843	1.45037	0.725185	+	0.688554i	$0.241754\pi$
$384$	0	0
$385$	0	0
$386$	−18.4345	−0.938293
$387$	0	0
$388$	−0.735604	−0.0373446
$389$	−19.1714	−0.972029	−0.486014	−	0.873951i	$-0.661550\pi$
$389$	−19.1714	−0.972029	−0.486014	+	0.873951i	$0.661550\pi$
$390$	0	0
$391$	6.49969	0.328703
$392$	−35.6812	−1.80217
$393$	0	0
$394$	−2.03936	−0.102741
$395$	−13.9499	−0.701896
$396$	0	0
$397$	16.2455	0.815339	0.407670	−	0.913130i	$-0.366342\pi$
$397$	16.2455	0.815339	0.407670	+	0.913130i	$0.366342\pi$
$398$	4.15852	0.208448
$399$	0	0
$400$	−2.61032	−0.130516
$401$	12.3247	0.615465	0.307732	−	0.951473i	$-0.400430\pi$
$401$	12.3247	0.615465	0.307732	+	0.951473i	$0.400430\pi$
$402$	0	0
$403$	9.98520	0.497398
$404$	4.60131	0.228924
$405$	0	0
$406$	5.51336	0.273623
$407$	0	0
$408$	0	0
$409$	1.95751	0.0967926	0.0483963	−	0.998828i	$-0.484589\pi$
$409$	1.95751	0.0967926	0.0483963	+	0.998828i	$0.484589\pi$
$410$	−6.09207	−0.300866
$411$	0	0
$412$	5.40422	0.266247
$413$	28.5233	1.40354
$414$	0	0
$415$	−12.0568	−0.591843
$416$	−18.4995	−0.907013
$417$	0	0
$418$	0	0
$419$	−20.5776	−1.00528	−0.502640	−	0.864496i	$-0.667638\pi$
$419$	−20.5776	−1.00528	−0.502640	+	0.864496i	$0.667638\pi$
$420$	0	0
$421$	−17.1205	−0.834400	−0.417200	−	0.908815i	$-0.636989\pi$
$421$	−17.1205	−0.834400	−0.417200	+	0.908815i	$0.636989\pi$
$422$	0.475654	0.0231545
$423$	0	0
$424$	−8.17960	−0.397237
$425$	−0.803988	−0.0389991
$426$	0	0
$427$	−23.0769	−1.11677
$428$	−4.45826	−0.215498
$429$	0	0
$430$	−11.9032	−0.574025
$431$	10.6079	0.510963	0.255481	−	0.966814i	$-0.417766\pi$
$431$	10.6079	0.510963	0.255481	+	0.966814i	$0.417766\pi$
$432$	0	0
$433$	24.8262	1.19307	0.596536	−	0.802586i	$-0.296543\pi$
$433$	24.8262	1.19307	0.596536	+	0.802586i	$0.296543\pi$
$434$	8.40464	0.403436
$435$	0	0
$436$	7.36892	0.352907
$437$	−15.1491	−0.724678
$438$	0	0
$439$	−6.78781	−0.323965	−0.161982	−	0.986794i	$-0.551789\pi$
$439$	−6.78781	−0.323965	−0.161982	+	0.986794i	$0.551789\pi$
$440$	0	0
$441$	0	0
$442$	5.99395	0.285103
$443$	15.1500	0.719799	0.359899	−	0.932991i	$-0.382811\pi$
$443$	15.1500	0.719799	0.359899	+	0.932991i	$0.382811\pi$
$444$	0	0
$445$	12.7727	0.605484
$446$	−9.37143	−0.443750
$447$	0	0
$448$	−38.1003	−1.80007
$449$	17.9310	0.846217	0.423108	−	0.906079i	$-0.360939\pi$
$449$	17.9310	0.846217	0.423108	+	0.906079i	$0.360939\pi$
$450$	0	0
$451$	0	0
$452$	−7.37488	−0.346885
$453$	0	0
$454$	32.9400	1.54595
$455$	−26.6797	−1.25076
$456$	0	0
$457$	41.1984	1.92718	0.963590	−	0.267385i	$-0.0861594\pi$
$457$	41.1984	1.92718	0.963590	+	0.267385i	$0.0861594\pi$
$458$	18.7902	0.878010
$459$	0	0
$460$	−4.41289	−0.205752
$461$	16.8145	0.783128	0.391564	−	0.920151i	$-0.371934\pi$
$461$	16.8145	0.783128	0.391564	+	0.920151i	$0.371934\pi$
$462$	0	0
$463$	−35.7050	−1.65935	−0.829676	−	0.558245i	$-0.811475\pi$
$463$	−35.7050	−1.65935	−0.829676	+	0.558245i	$0.811475\pi$
$464$	2.76558	0.128389
$465$	0	0
$466$	−33.6254	−1.55767
$467$	35.9512	1.66362	0.831811	−	0.555059i	$-0.187304\pi$
$467$	35.9512	1.66362	0.831811	+	0.555059i	$0.187304\pi$
$468$	0	0
$469$	32.3230	1.49254
$470$	15.6592	0.722303
$471$	0	0
$472$	20.2917	0.934000
$473$	0	0
$474$	0	0
$475$	1.87389	0.0859798
$476$	−1.89387	−0.0868053
$477$	0	0
$478$	22.6052	1.03394
$479$	−13.8915	−0.634721	−0.317360	−	0.948305i	$-0.602796\pi$
$479$	−13.8915	−0.634721	−0.317360	+	0.948305i	$0.602796\pi$
$480$	0	0
$481$	34.2591	1.56208
$482$	13.2705	0.604456
$483$	0	0
$484$	0	0
$485$	−1.34761	−0.0611918
$486$	0	0
$487$	−14.0003	−0.634413	−0.317207	−	0.948356i	$-0.602745\pi$
$487$	−14.0003	−0.634413	−0.317207	+	0.948356i	$0.602745\pi$
$488$	−16.4171	−0.743167
$489$	0	0
$490$	−14.0154	−0.633151
$491$	−25.2477	−1.13941	−0.569707	−	0.821848i	$-0.692943\pi$
$491$	−25.2477	−1.13941	−0.569707	+	0.821848i	$0.692943\pi$
$492$	0	0
$493$	0.851809	0.0383636
$494$	−13.9703	−0.628555
$495$	0	0
$496$	4.21590	0.189299
$497$	−25.1564	−1.12842
$498$	0	0
$499$	−8.86257	−0.396743	−0.198371	−	0.980127i	$-0.563565\pi$
$499$	−8.86257	−0.396743	−0.198371	+	0.980127i	$0.563565\pi$
$500$	0.545859	0.0244116
$501$	0	0
$502$	28.1133	1.25476
$503$	0.661184	0.0294807	0.0147404	−	0.999891i	$-0.495308\pi$
$503$	0.661184	0.0294807	0.0147404	+	0.999891i	$0.495308\pi$
$504$	0	0
$505$	8.42949	0.375107
$506$	0	0
$507$	0	0
$508$	−1.15729	−0.0513463
$509$	−38.7166	−1.71608	−0.858042	−	0.513580i	$-0.828319\pi$
$509$	−38.7166	−1.71608	−0.858042	+	0.513580i	$0.828319\pi$
$510$	0	0
$511$	20.7202	0.916609
$512$	−23.8381	−1.05351
$513$	0	0
$514$	−21.0704	−0.929373
$515$	9.90039	0.436263
$516$	0	0
$517$	0	0
$518$	28.8362	1.26699
$519$	0	0
$520$	−18.9801	−0.832332
$521$	24.5734	1.07658	0.538290	−	0.842760i	$-0.319071\pi$
$521$	24.5734	1.07658	0.538290	+	0.842760i	$0.319071\pi$
$522$	0	0
$523$	−33.6944	−1.47335	−0.736677	−	0.676245i	$-0.763606\pi$
$523$	−33.6944	−1.47335	−0.736677	+	0.676245i	$0.763606\pi$
$524$	−4.76166	−0.208014
$525$	0	0
$526$	1.94372	0.0847501
$527$	1.29851	0.0565640
$528$	0	0
$529$	42.3561	1.84157
$530$	−3.21291	−0.139560
$531$	0	0
$532$	4.41411	0.191376
$533$	−31.2336	−1.35288
$534$	0	0
$535$	−8.16742	−0.353108
$536$	22.9948	0.993225
$537$	0	0
$538$	−9.85523	−0.424889
$539$	0	0
$540$	0	0
$541$	8.20840	0.352906	0.176453	−	0.984309i	$-0.443538\pi$
$541$	8.20840	0.352906	0.176453	+	0.984309i	$0.443538\pi$
$542$	−12.6611	−0.543843
$543$	0	0
$544$	−2.40574	−0.103145
$545$	13.4997	0.578262
$546$	0	0
$547$	−35.0759	−1.49974	−0.749868	−	0.661587i	$-0.769883\pi$
$547$	−35.0759	−1.49974	−0.749868	+	0.661587i	$0.769883\pi$
$548$	−0.842711	−0.0359988
$549$	0	0
$550$	0	0
$551$	−1.98534	−0.0845785
$552$	0	0
$553$	−60.1993	−2.55993
$554$	−10.7591	−0.457111
$555$	0	0
$556$	8.84194	0.374982
$557$	39.6130	1.67846	0.839229	−	0.543779i	$-0.183007\pi$
$557$	39.6130	1.67846	0.839229	+	0.543779i	$0.183007\pi$
$558$	0	0
$559$	−61.0270	−2.58117
$560$	−11.2645	−0.476014
$561$	0	0
$562$	5.29812	0.223488
$563$	21.6803	0.913717	0.456858	−	0.889539i	$-0.348975\pi$
$563$	21.6803	0.913717	0.456858	+	0.889539i	$0.348975\pi$
$564$	0	0
$565$	−13.5106	−0.568395
$566$	−15.6822	−0.659173
$567$	0	0
$568$	−17.8964	−0.750918
$569$	−27.4638	−1.15134	−0.575672	−	0.817681i	$-0.695259\pi$
$569$	−27.4638	−1.15134	−0.575672	+	0.817681i	$0.695259\pi$
$570$	0	0
$571$	−23.8306	−0.997281	−0.498641	−	0.866809i	$-0.666167\pi$
$571$	−23.8306	−0.997281	−0.498641	+	0.866809i	$0.666167\pi$
$572$	0	0
$573$	0	0
$574$	−26.2897	−1.09731
$575$	−8.08431	−0.337139
$576$	0	0
$577$	−16.0731	−0.669130	−0.334565	−	0.942373i	$-0.608589\pi$
$577$	−16.0731	−0.669130	−0.334565	+	0.942373i	$0.608589\pi$
$578$	−19.7204	−0.820262
$579$	0	0
$580$	−0.578327	−0.0240137
$581$	−52.0296	−2.15855
$582$	0	0
$583$	0	0
$584$	14.7405	0.609966
$585$	0	0
$586$	18.7447	0.774336
$587$	17.7793	0.733829	0.366915	−	0.930255i	$-0.380414\pi$
$587$	17.7793	0.733829	0.366915	+	0.930255i	$0.380414\pi$
$588$	0	0
$589$	−3.02649	−0.124704
$590$	7.97047	0.328139
$591$	0	0
$592$	14.4647	0.594495
$593$	−20.2944	−0.833389	−0.416695	−	0.909046i	$-0.636812\pi$
$593$	−20.2944	−0.833389	−0.416695	+	0.909046i	$0.636812\pi$
$594$	0	0
$595$	−3.46952	−0.142236
$596$	−7.65202	−0.313439
$597$	0	0
$598$	60.2708	2.46466
$599$	−9.00707	−0.368019	−0.184009	−	0.982924i	$-0.558908\pi$
$599$	−9.00707	−0.368019	−0.184009	+	0.982924i	$0.558908\pi$
$600$	0	0
$601$	−41.1081	−1.67683	−0.838417	−	0.545029i	$-0.816519\pi$
$601$	−41.1081	−1.67683	−0.838417	+	0.545029i	$0.816519\pi$
$602$	−51.3671	−2.09356
$603$	0	0
$604$	−7.41558	−0.301736
$605$	0	0
$606$	0	0
$607$	−1.19057	−0.0483236	−0.0241618	−	0.999708i	$-0.507692\pi$
$607$	−1.19057	−0.0483236	−0.0241618	+	0.999708i	$0.507692\pi$
$608$	5.60716	0.227400
$609$	0	0
$610$	−6.44855	−0.261094
$611$	80.2834	3.24792
$612$	0	0
$613$	−0.136596	−0.00551706	−0.00275853	−	0.999996i	$-0.500878\pi$
$613$	−0.136596	−0.00551706	−0.00275853	+	0.999996i	$0.500878\pi$
$614$	−10.5738	−0.426726
$615$	0	0
$616$	0	0
$617$	17.7011	0.712620	0.356310	−	0.934368i	$-0.384035\pi$
$617$	17.7011	0.712620	0.356310	+	0.934368i	$0.384035\pi$
$618$	0	0
$619$	−7.08920	−0.284939	−0.142470	−	0.989799i	$-0.545504\pi$
$619$	−7.08920	−0.284939	−0.142470	+	0.989799i	$0.545504\pi$
$620$	−0.881610	−0.0354063
$621$	0	0
$622$	2.11174	0.0846733
$623$	55.1191	2.20830
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.1357	−0.844752
$627$	0	0
$628$	2.39127	0.0954219
$629$	4.45517	0.177639
$630$	0	0
$631$	24.1403	0.961010	0.480505	−	0.876992i	$-0.340453\pi$
$631$	24.1403	0.961010	0.480505	+	0.876992i	$0.340453\pi$
$632$	−42.8262	−1.70353
$633$	0	0
$634$	2.40195	0.0953935
$635$	−2.12012	−0.0841344
$636$	0	0
$637$	−71.8560	−2.84704
$638$	0	0
$639$	0	0
$640$	−4.66210	−0.184286
$641$	42.8201	1.69129	0.845646	−	0.533745i	$-0.179216\pi$
$641$	42.8201	1.69129	0.845646	+	0.533745i	$0.179216\pi$
$642$	0	0
$643$	−34.2410	−1.35033	−0.675166	−	0.737666i	$-0.735928\pi$
$643$	−34.2410	−1.35033	−0.675166	+	0.737666i	$0.735928\pi$
$644$	−19.0433	−0.750413
$645$	0	0
$646$	−1.81675	−0.0714792
$647$	10.8930	0.428247	0.214124	−	0.976807i	$-0.431310\pi$
$647$	10.8930	0.428247	0.214124	+	0.976807i	$0.431310\pi$
$648$	0	0
$649$	0	0
$650$	−7.45528	−0.292420
$651$	0	0
$652$	2.55826	0.100189
$653$	42.0505	1.64556	0.822781	−	0.568359i	$-0.192421\pi$
$653$	42.0505	1.64556	0.822781	+	0.568359i	$0.192421\pi$
$654$	0	0
$655$	−8.72324	−0.340845
$656$	−13.1873	−0.514877
$657$	0	0
$658$	67.5753	2.63436
$659$	0.425249	0.0165654	0.00828268	−	0.999966i	$-0.497364\pi$
$659$	0.425249	0.0165654	0.00828268	+	0.999966i	$0.497364\pi$
$660$	0	0
$661$	−7.08477	−0.275565	−0.137783	−	0.990462i	$-0.543998\pi$
$661$	−7.08477	−0.275565	−0.137783	+	0.990462i	$0.543998\pi$
$662$	−12.4732	−0.484786
$663$	0	0
$664$	−37.0142	−1.43643
$665$	8.08654	0.313583
$666$	0	0
$667$	8.56517	0.331645
$668$	−0.900216	−0.0348304
$669$	0	0
$670$	9.03224	0.348946
$671$	0	0
$672$	0	0
$673$	12.2833	0.473487	0.236743	−	0.971572i	$-0.423920\pi$
$673$	12.2833	0.473487	0.236743	+	0.971572i	$0.423920\pi$
$674$	24.8637	0.957712
$675$	0	0
$676$	−13.7680	−0.529540
$677$	−17.2760	−0.663972	−0.331986	−	0.943284i	$-0.607719\pi$
$677$	−17.2760	−0.663972	−0.331986	+	0.943284i	$0.607719\pi$
$678$	0	0
$679$	−5.81545	−0.223177
$680$	−2.46824	−0.0946526
$681$	0	0
$682$	0	0
$683$	26.8858	1.02876	0.514379	−	0.857563i	$-0.328022\pi$
$683$	26.8858	1.02876	0.514379	+	0.857563i	$0.328022\pi$
$684$	0	0
$685$	−1.54382	−0.0589865
$686$	−24.0551	−0.918428
$687$	0	0
$688$	−25.7665	−0.982338
$689$	−16.4723	−0.627546
$690$	0	0
$691$	24.9330	0.948496	0.474248	−	0.880391i	$-0.342720\pi$
$691$	24.9330	0.948496	0.474248	+	0.880391i	$0.342720\pi$
$692$	9.04609	0.343881
$693$	0	0
$694$	3.59803	0.136579
$695$	16.1982	0.614433
$696$	0	0
$697$	−4.06173	−0.153849
$698$	−13.6140	−0.515297
$699$	0	0
$700$	2.35559	0.0890331
$701$	−20.0721	−0.758112	−0.379056	−	0.925374i	$-0.623751\pi$
$701$	−20.0721	−0.758112	−0.379056	+	0.925374i	$0.623751\pi$
$702$	0	0
$703$	−10.3838	−0.391634
$704$	0	0
$705$	0	0
$706$	16.2735	0.612461
$707$	36.3765	1.36808
$708$	0	0
$709$	−11.3356	−0.425717	−0.212858	−	0.977083i	$-0.568277\pi$
$709$	−11.3356	−0.425717	−0.212858	+	0.977083i	$0.568277\pi$
$710$	−7.02962	−0.263817
$711$	0	0
$712$	39.2121	1.46954
$713$	13.0569	0.488984
$714$	0	0
$715$	0	0
$716$	9.60684	0.359025
$717$	0	0
$718$	−29.7838	−1.11152
$719$	4.63537	0.172870	0.0864351	−	0.996257i	$-0.472452\pi$
$719$	4.63537	0.172870	0.0864351	+	0.996257i	$0.472452\pi$
$720$	0	0
$721$	42.7240	1.59113
$722$	−18.6773	−0.695097
$723$	0	0
$724$	−4.58889	−0.170545
$725$	−1.05948	−0.0393481
$726$	0	0
$727$	−2.22651	−0.0825766	−0.0412883	−	0.999147i	$-0.513146\pi$
$727$	−2.22651	−0.0825766	−0.0412883	+	0.999147i	$0.513146\pi$
$728$	−81.9065	−3.03566
$729$	0	0
$730$	5.78999	0.214297
$731$	−7.93617	−0.293530
$732$	0	0
$733$	30.4780	1.12573	0.562866	−	0.826548i	$-0.309699\pi$
$733$	30.4780	1.12573	0.562866	+	0.826548i	$0.309699\pi$
$734$	7.65118	0.282410
$735$	0	0
$736$	−24.1904	−0.891669
$737$	0	0
$738$	0	0
$739$	9.20178	0.338493	0.169246	−	0.985574i	$-0.445867\pi$
$739$	9.20178	0.338493	0.169246	+	0.985574i	$0.445867\pi$
$740$	−3.02479	−0.111194
$741$	0	0
$742$	−13.8649	−0.508998
$743$	−1.72902	−0.0634316	−0.0317158	−	0.999497i	$-0.510097\pi$
$743$	−1.72902	−0.0634316	−0.0317158	+	0.999497i	$0.510097\pi$
$744$	0	0
$745$	−14.0183	−0.513591
$746$	−15.8974	−0.582043
$747$	0	0
$748$	0	0
$749$	−35.2456	−1.28785
$750$	0	0
$751$	33.8778	1.23622	0.618109	−	0.786092i	$-0.287899\pi$
$751$	33.8778	1.23622	0.618109	+	0.786092i	$0.287899\pi$
$752$	33.8968	1.23609
$753$	0	0
$754$	7.89872	0.287655
$755$	−13.5852	−0.494415
$756$	0	0
$757$	−26.8677	−0.976522	−0.488261	−	0.872698i	$-0.662369\pi$
$757$	−26.8677	−0.976522	−0.488261	+	0.872698i	$0.662369\pi$
$758$	−22.3525	−0.811879
$759$	0	0
$760$	5.75282	0.208677
$761$	17.4413	0.632247	0.316124	−	0.948718i	$-0.397619\pi$
$761$	17.4413	0.632247	0.316124	+	0.948718i	$0.397619\pi$
$762$	0	0
$763$	58.2563	2.10902
$764$	−2.90272	−0.105017
$765$	0	0
$766$	34.2280	1.23671
$767$	40.8640	1.47551
$768$	0	0
$769$	41.6485	1.50188	0.750942	−	0.660368i	$-0.229600\pi$
$769$	41.6485	1.50188	0.750942	+	0.660368i	$0.229600\pi$
$770$	0	0
$771$	0	0
$772$	8.34467	0.300331
$773$	22.1515	0.796733	0.398366	−	0.917226i	$-0.369577\pi$
$773$	22.1515	0.796733	0.398366	+	0.917226i	$0.369577\pi$
$774$	0	0
$775$	−1.61509	−0.0580157
$776$	−4.13715	−0.148515
$777$	0	0
$778$	−23.1184	−0.828834
$779$	9.46684	0.339185
$780$	0	0
$781$	0	0
$782$	7.83783	0.280280
$783$	0	0
$784$	−30.3386	−1.08352
$785$	4.38074	0.156355
$786$	0	0
$787$	8.41235	0.299868	0.149934	−	0.988696i	$-0.452094\pi$
$787$	8.41235	0.299868	0.149934	+	0.988696i	$0.452094\pi$
$788$	0.923145	0.0328857
$789$	0	0
$790$	−16.8219	−0.598496
$791$	−58.3034	−2.07303
$792$	0	0
$793$	−33.0612	−1.17404
$794$	19.5901	0.695227
$795$	0	0
$796$	−1.88242	−0.0667206
$797$	−40.9234	−1.44958	−0.724791	−	0.688969i	$-0.758063\pi$
$797$	−40.9234	−1.44958	−0.724791	+	0.688969i	$0.758063\pi$
$798$	0	0
$799$	10.4403	0.369352
$800$	2.99226	0.105792
$801$	0	0
$802$	14.8621	0.524797
$803$	0	0
$804$	0	0
$805$	−34.8869	−1.22960
$806$	12.0409	0.424124
$807$	0	0
$808$	25.8785	0.910402
$809$	−21.6716	−0.761931	−0.380966	−	0.924589i	$-0.624408\pi$
$809$	−21.6716	−0.761931	−0.380966	+	0.924589i	$0.624408\pi$
$810$	0	0
$811$	54.5031	1.91386	0.956931	−	0.290314i	$-0.0937599\pi$
$811$	54.5031	1.91386	0.956931	+	0.290314i	$0.0937599\pi$
$812$	−2.49571	−0.0875821
$813$	0	0
$814$	0	0
$815$	4.68667	0.164167
$816$	0	0
$817$	18.4971	0.647133
$818$	2.36052	0.0825335
$819$	0	0
$820$	2.75767	0.0963020
$821$	17.9510	0.626495	0.313247	−	0.949672i	$-0.398583\pi$
$821$	17.9510	0.626495	0.313247	+	0.949672i	$0.398583\pi$
$822$	0	0
$823$	35.2352	1.22822	0.614110	−	0.789220i	$-0.289515\pi$
$823$	35.2352	1.22822	0.614110	+	0.789220i	$0.289515\pi$
$824$	30.3942	1.05883
$825$	0	0
$826$	34.3957	1.19678
$827$	39.9373	1.38876	0.694378	−	0.719610i	$-0.255680\pi$
$827$	39.9373	1.38876	0.694378	+	0.719610i	$0.255680\pi$
$828$	0	0
$829$	−45.5317	−1.58138	−0.790690	−	0.612216i	$-0.790278\pi$
$829$	−45.5317	−1.58138	−0.790690	+	0.612216i	$0.790278\pi$
$830$	−14.5390	−0.504656
$831$	0	0
$832$	−54.5845	−1.89238
$833$	−9.34441	−0.323765
$834$	0	0
$835$	−1.64917	−0.0570720
$836$	0	0
$837$	0	0
$838$	−24.8140	−0.857186
$839$	18.0373	0.622718	0.311359	−	0.950292i	$-0.399216\pi$
$839$	18.0373	0.622718	0.311359	+	0.950292i	$0.399216\pi$
$840$	0	0
$841$	−27.8775	−0.961293
$842$	−20.6452	−0.711480
$843$	0	0
$844$	−0.215312	−0.00741134
$845$	−25.2227	−0.867688
$846$	0	0
$847$	0	0
$848$	−6.95486	−0.238831
$849$	0	0
$850$	−0.969511	−0.0332540
$851$	44.7979	1.53565
$852$	0	0
$853$	1.18704	0.0406434	0.0203217	−	0.999793i	$-0.493531\pi$
$853$	1.18704	0.0406434	0.0203217	+	0.999793i	$0.493531\pi$
$854$	−27.8280	−0.952254
$855$	0	0
$856$	−25.0739	−0.857009
$857$	−5.59493	−0.191119	−0.0955597	−	0.995424i	$-0.530464\pi$
$857$	−5.59493	−0.191119	−0.0955597	+	0.995424i	$0.530464\pi$
$858$	0	0
$859$	−18.5783	−0.633882	−0.316941	−	0.948445i	$-0.602656\pi$
$859$	−18.5783	−0.633882	−0.316941	+	0.948445i	$0.602656\pi$
$860$	5.38818	0.183735
$861$	0	0
$862$	12.7918	0.435690
$863$	14.2249	0.484220	0.242110	−	0.970249i	$-0.422161\pi$
$863$	14.2249	0.484220	0.242110	+	0.970249i	$0.422161\pi$
$864$	0	0
$865$	16.5722	0.563472
$866$	29.9374	1.01731
$867$	0	0
$868$	−3.80449	−0.129133
$869$	0	0
$870$	0	0
$871$	46.3077	1.56908
$872$	41.4439	1.40347
$873$	0	0
$874$	−18.2679	−0.617922
$875$	4.31539	0.145887
$876$	0	0
$877$	18.6019	0.628141	0.314070	−	0.949400i	$-0.398307\pi$
$877$	18.6019	0.628141	0.314070	+	0.949400i	$0.398307\pi$
$878$	−8.18527	−0.276240
$879$	0	0
$880$	0	0
$881$	−30.5201	−1.02825	−0.514124	−	0.857716i	$-0.671883\pi$
$881$	−30.5201	−1.02825	−0.514124	+	0.857716i	$0.671883\pi$
$882$	0	0
$883$	−16.2877	−0.548125	−0.274062	−	0.961712i	$-0.588367\pi$
$883$	−16.2877	−0.548125	−0.274062	+	0.961712i	$0.588367\pi$
$884$	−2.71326	−0.0912566
$885$	0	0
$886$	18.2691	0.613761
$887$	5.29250	0.177705	0.0888524	−	0.996045i	$-0.471680\pi$
$887$	5.29250	0.177705	0.0888524	+	0.996045i	$0.471680\pi$
$888$	0	0
$889$	−9.14915	−0.306853
$890$	15.4023	0.516286
$891$	0	0
$892$	4.24212	0.142037
$893$	−24.3337	−0.814296
$894$	0	0
$895$	17.5995	0.588286
$896$	−20.1188	−0.672121
$897$	0	0
$898$	21.6226	0.721556
$899$	1.71115	0.0570702
$900$	0	0
$901$	−2.14212	−0.0713644
$902$	0	0
$903$	0	0
$904$	−41.4774	−1.37952
$905$	−8.40672	−0.279449
$906$	0	0
$907$	1.15084	0.0382132	0.0191066	−	0.999817i	$-0.493918\pi$
$907$	1.15084	0.0382132	0.0191066	+	0.999817i	$0.493918\pi$
$908$	−14.9108	−0.494832
$909$	0	0
$910$	−32.1724	−1.06651
$911$	24.4697	0.810717	0.405359	−	0.914158i	$-0.367147\pi$
$911$	24.4697	0.810717	0.405359	+	0.914158i	$0.367147\pi$
$912$	0	0
$913$	0	0
$914$	49.6802	1.64328
$915$	0	0
$916$	−8.50569	−0.281036
$917$	−37.6442	−1.24312
$918$	0	0
$919$	27.1167	0.894499	0.447249	−	0.894409i	$-0.352404\pi$
$919$	27.1167	0.894499	0.447249	+	0.894409i	$0.352404\pi$
$920$	−24.8188	−0.818251
$921$	0	0
$922$	20.2762	0.667761
$923$	−36.0404	−1.18628
$924$	0	0
$925$	−5.54134	−0.182198
$926$	−43.0559	−1.41490
$927$	0	0
$928$	−3.17024	−0.104068
$929$	29.0648	0.953587	0.476793	−	0.879015i	$-0.341799\pi$
$929$	29.0648	0.953587	0.476793	+	0.879015i	$0.341799\pi$
$930$	0	0
$931$	21.7794	0.713790
$932$	15.2211	0.498582
$933$	0	0
$934$	43.3527	1.41854
$935$	0	0
$936$	0	0
$937$	59.8539	1.95534	0.977671	−	0.210143i	$-0.0673931\pi$
$937$	59.8539	1.95534	0.977671	+	0.210143i	$0.0673931\pi$
$938$	38.9776	1.27266
$939$	0	0
$940$	−7.08835	−0.231197
$941$	−30.2191	−0.985115	−0.492558	−	0.870280i	$-0.663938\pi$
$941$	−30.2191	−0.985115	−0.492558	+	0.870280i	$0.663938\pi$
$942$	0	0
$943$	−40.8418	−1.32999
$944$	17.2534	0.561550
$945$	0	0
$946$	0	0
$947$	38.6489	1.25592	0.627960	−	0.778245i	$-0.283890\pi$
$947$	38.6489	1.25592	0.627960	+	0.778245i	$0.283890\pi$
$948$	0	0
$949$	29.6849	0.963612
$950$	2.25968	0.0733136
$951$	0	0
$952$	−10.6514	−0.345214
$953$	−44.7169	−1.44852	−0.724261	−	0.689526i	$-0.757819\pi$
$953$	−44.7169	−1.44852	−0.724261	+	0.689526i	$0.757819\pi$
$954$	0	0
$955$	−5.31771	−0.172077
$956$	−10.2326	−0.330945
$957$	0	0
$958$	−16.7515	−0.541216
$959$	−6.66220	−0.215134
$960$	0	0
$961$	−28.3915	−0.915855
$962$	41.3123	1.33196
$963$	0	0
$964$	−6.00711	−0.193476
$965$	15.2872	0.492113
$966$	0	0
$967$	34.5876	1.11226	0.556131	−	0.831095i	$-0.312285\pi$
$967$	34.5876	1.11226	0.556131	+	0.831095i	$0.312285\pi$
$968$	0	0
$969$	0	0
$970$	−1.62505	−0.0521773
$971$	−6.91193	−0.221814	−0.110907	−	0.993831i	$-0.535376\pi$
$971$	−6.91193	−0.221814	−0.110907	+	0.993831i	$0.535376\pi$
$972$	0	0
$973$	69.9016	2.24094
$974$	−16.8826	−0.540954
$975$	0	0
$976$	−13.9589	−0.446815
$977$	12.2661	0.392427	0.196214	−	0.980561i	$-0.437135\pi$
$977$	12.2661	0.392427	0.196214	+	0.980561i	$0.437135\pi$
$978$	0	0
$979$	0	0
$980$	6.34429	0.202661
$981$	0	0
$982$	−30.4457	−0.971561
$983$	−0.161755	−0.00515918	−0.00257959	−	0.999997i	$-0.500821\pi$
$983$	−0.161755	−0.00515918	−0.00257959	+	0.999997i	$0.500821\pi$
$984$	0	0
$985$	1.69118	0.0538855
$986$	1.02718	0.0327120
$987$	0	0
$988$	6.32389	0.201190
$989$	−79.8003	−2.53750
$990$	0	0
$991$	43.1252	1.36992	0.684958	−	0.728582i	$-0.259820\pi$
$991$	43.1252	1.36992	0.684958	+	0.728582i	$0.259820\pi$
$992$	−4.83277	−0.153440
$993$	0	0
$994$	−30.3356	−0.962185
$995$	−3.44855	−0.109326
$996$	0	0
$997$	1.47228	0.0466276	0.0233138	−	0.999728i	$-0.492578\pi$
$997$	1.47228	0.0466276	0.0233138	+	0.999728i	$0.492578\pi$
$998$	−10.6872	−0.338296
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.cc.1.5		8
3.2	odd	2		5445.2.a.cb.1.4		8
11.7	odd	10		495.2.n.h.181.2	yes	16
11.8	odd	10		495.2.n.h.361.2	yes	16
11.10	odd	2		5445.2.a.ca.1.4		8
33.8	even	10		495.2.n.g.361.3	yes	16
33.29	even	10		495.2.n.g.181.3	✓	16
33.32	even	2		5445.2.a.cd.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.181.3	✓	16	33.29	even	10
495.2.n.g.361.3	yes	16	33.8	even	10
495.2.n.h.181.2	yes	16	11.7	odd	10
495.2.n.h.361.2	yes	16	11.8	odd	10
5445.2.a.ca.1.4		8	11.10	odd	2
5445.2.a.cb.1.4		8	3.2	odd	2
5445.2.a.cc.1.5		8	1.1	even	1	trivial
5445.2.a.cd.1.5		8	33.32	even	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)

\(f(q)\)	\(=\)	\(q+1.20588 q^{2} -0.545859 q^{4} -1.00000 q^{5} -4.31539 q^{7} -3.06999 q^{8} +O(q^{10})\)	\(q+1.20588 q^{2} -0.545859 q^{4} -1.00000 q^{5} -4.31539 q^{7} -3.06999 q^{8} -1.20588 q^{10} -6.18245 q^{13} -5.20383 q^{14} -2.61032 q^{16} -0.803988 q^{17} +1.87389 q^{19} +0.545859 q^{20} -8.08431 q^{23} +1.00000 q^{25} -7.45528 q^{26} +2.35559 q^{28} -1.05948 q^{29} -1.61509 q^{31} +2.99226 q^{32} -0.969511 q^{34} +4.31539 q^{35} -5.54134 q^{37} +2.25968 q^{38} +3.06999 q^{40} +5.05198 q^{41} +9.87101 q^{43} -9.74869 q^{46} -12.9857 q^{47} +11.6226 q^{49} +1.20588 q^{50} +3.37475 q^{52} +2.66437 q^{53} +13.2482 q^{56} -1.27760 q^{58} -6.60968 q^{59} +5.34760 q^{61} -1.94760 q^{62} +8.82894 q^{64} +6.18245 q^{65} -7.49018 q^{67} +0.438864 q^{68} +5.20383 q^{70} +5.82947 q^{71} -4.80148 q^{73} -6.68218 q^{74} -1.02288 q^{76} +13.9499 q^{79} +2.61032 q^{80} +6.09207 q^{82} +12.0568 q^{83} +0.803988 q^{85} +11.9032 q^{86} -12.7727 q^{89} +26.6797 q^{91} +4.41289 q^{92} -15.6592 q^{94} -1.87389 q^{95} +1.34761 q^{97} +14.0154 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 6 q^{4} - 8 q^{5} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) 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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