LMFDB - Embedded newform 5445.2.a.cb.1.8

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			1.8
Root			$3.04904$ 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)

$f(q)$	$=$	$q+2.04904 q^{2} +2.19858 q^{4} +1.00000 q^{5} +0.589936 q^{7} +0.406903 q^{8} +O(q^{10})$	$q+2.04904 q^{2} +2.19858 q^{4} +1.00000 q^{5} +0.589936 q^{7} +0.406903 q^{8} +2.04904 q^{10} -1.50491 q^{13} +1.20880 q^{14} -3.56340 q^{16} -4.92000 q^{17} -6.82060 q^{19} +2.19858 q^{20} -0.0822934 q^{23} +1.00000 q^{25} -3.08363 q^{26} +1.29702 q^{28} -8.42443 q^{29} -1.97913 q^{31} -8.11537 q^{32} -10.0813 q^{34} +0.589936 q^{35} -3.57997 q^{37} -13.9757 q^{38} +0.406903 q^{40} -4.07321 q^{41} -4.86148 q^{43} -0.168623 q^{46} +5.77852 q^{47} -6.65198 q^{49} +2.04904 q^{50} -3.30868 q^{52} +6.77365 q^{53} +0.240047 q^{56} -17.2620 q^{58} -5.58403 q^{59} +0.960246 q^{61} -4.05533 q^{62} -9.50196 q^{64} -1.50491 q^{65} +13.6559 q^{67} -10.8170 q^{68} +1.20880 q^{70} -1.28379 q^{71} +15.6271 q^{73} -7.33551 q^{74} -14.9956 q^{76} +11.1522 q^{79} -3.56340 q^{80} -8.34619 q^{82} +4.01215 q^{83} -4.92000 q^{85} -9.96139 q^{86} +6.08177 q^{89} -0.887802 q^{91} -0.180929 q^{92} +11.8404 q^{94} -6.82060 q^{95} -6.70578 q^{97} -13.6302 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.04904	1.44889	0.724447	−	0.689331i	$-0.242095\pi$
$2$	2.04904	1.44889	0.724447	+	0.689331i	$0.242095\pi$
$3$	0	0
$3$	0	0
$4$	2.19858	1.09929
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.589936	0.222975	0.111487	−	0.993766i	$-0.464439\pi$
$7$	0.589936	0.222975	0.111487	+	0.993766i	$0.464439\pi$
$8$	0.406903	0.143862
$9$	0	0
$10$	2.04904	0.647965
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.50491	−0.417388	−0.208694	−	0.977981i	$-0.566921\pi$
$13$	−1.50491	−0.417388	−0.208694	+	0.977981i	$0.566921\pi$
$14$	1.20880	0.323067
$15$	0	0
$16$	−3.56340	−0.890850
$17$	−4.92000	−1.19327	−0.596637	−	0.802511i	$-0.703497\pi$
$17$	−4.92000	−1.19327	−0.596637	+	0.802511i	$0.703497\pi$
$18$	0	0
$19$	−6.82060	−1.56475	−0.782376	−	0.622806i	$-0.785993\pi$
$19$	−6.82060	−1.56475	−0.782376	+	0.622806i	$0.785993\pi$
$20$	2.19858	0.491618
$21$	0	0
$22$	0	0
$23$	−0.0822934	−0.0171594	−0.00857968	−	0.999963i	$-0.502731\pi$
$23$	−0.0822934	−0.0171594	−0.00857968	+	0.999963i	$0.502731\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.08363	−0.604751
$27$	0	0
$28$	1.29702	0.245114
$29$	−8.42443	−1.56438	−0.782188	−	0.623042i	$-0.785897\pi$
$29$	−8.42443	−1.56438	−0.782188	+	0.623042i	$0.785897\pi$
$30$	0	0
$31$	−1.97913	−0.355463	−0.177731	−	0.984079i	$-0.556876\pi$
$31$	−1.97913	−0.355463	−0.177731	+	0.984079i	$0.556876\pi$
$32$	−8.11537	−1.43461
$33$	0	0
$34$	−10.0813	−1.72893
$35$	0.589936	0.0997173
$36$	0	0
$37$	−3.57997	−0.588543	−0.294272	−	0.955722i	$-0.595077\pi$
$37$	−3.57997	−0.588543	−0.294272	+	0.955722i	$0.595077\pi$
$38$	−13.9757	−2.26716
$39$	0	0
$40$	0.406903	0.0643371
$41$	−4.07321	−0.636129	−0.318064	−	0.948069i	$-0.603033\pi$
$41$	−4.07321	−0.636129	−0.318064	+	0.948069i	$0.603033\pi$
$42$	0	0
$43$	−4.86148	−0.741369	−0.370685	−	0.928759i	$-0.620877\pi$
$43$	−4.86148	−0.741369	−0.370685	+	0.928759i	$0.620877\pi$
$44$	0	0
$45$	0	0
$46$	−0.168623	−0.0248621
$47$	5.77852	0.842884	0.421442	−	0.906855i	$-0.361524\pi$
$47$	5.77852	0.842884	0.421442	+	0.906855i	$0.361524\pi$
$48$	0	0
$49$	−6.65198	−0.950282
$50$	2.04904	0.289779
$51$	0	0
$52$	−3.30868	−0.458831
$53$	6.77365	0.930433	0.465217	−	0.885197i	$-0.345976\pi$
$53$	6.77365	0.930433	0.465217	+	0.885197i	$0.345976\pi$
$54$	0	0
$55$	0	0
$56$	0.240047	0.0320776
$57$	0	0
$58$	−17.2620	−2.26661
$59$	−5.58403	−0.726979	−0.363490	−	0.931598i	$-0.618415\pi$
$59$	−5.58403	−0.726979	−0.363490	+	0.931598i	$0.618415\pi$
$60$	0	0
$61$	0.960246	0.122947	0.0614735	−	0.998109i	$-0.480420\pi$
$61$	0.960246	0.122947	0.0614735	+	0.998109i	$0.480420\pi$
$62$	−4.05533	−0.515027
$63$	0	0
$64$	−9.50196	−1.18774
$65$	−1.50491	−0.186662
$66$	0	0
$67$	13.6559	1.66834	0.834168	−	0.551511i	$-0.185948\pi$
$67$	13.6559	1.66834	0.834168	+	0.551511i	$0.185948\pi$
$68$	−10.8170	−1.31176
$69$	0	0
$70$	1.20880	0.144480
$71$	−1.28379	−0.152358	−0.0761791	−	0.997094i	$-0.524272\pi$
$71$	−1.28379	−0.152358	−0.0761791	+	0.997094i	$0.524272\pi$
$72$	0	0
$73$	15.6271	1.82901	0.914507	−	0.404569i	$-0.132579\pi$
$73$	15.6271	1.82901	0.914507	+	0.404569i	$0.132579\pi$
$74$	−7.33551	−0.852736
$75$	0	0
$76$	−14.9956	−1.72012
$77$	0	0
$78$	0	0
$79$	11.1522	1.25472	0.627360	−	0.778729i	$-0.284135\pi$
$79$	11.1522	1.25472	0.627360	+	0.778729i	$0.284135\pi$
$80$	−3.56340	−0.398400
$81$	0	0
$82$	−8.34619	−0.921683
$83$	4.01215	0.440391	0.220196	−	0.975456i	$-0.429330\pi$
$83$	4.01215	0.440391	0.220196	+	0.975456i	$0.429330\pi$
$84$	0	0
$85$	−4.92000	−0.533648
$86$	−9.96139	−1.07416
$87$	0	0
$88$	0	0
$89$	6.08177	0.644667	0.322333	−	0.946626i	$-0.395533\pi$
$89$	6.08177	0.644667	0.322333	+	0.946626i	$0.395533\pi$
$90$	0	0
$91$	−0.887802	−0.0930670
$92$	−0.180929	−0.0188631
$93$	0	0
$94$	11.8404	1.22125
$95$	−6.82060	−0.699779
$96$	0	0
$97$	−6.70578	−0.680868	−0.340434	−	0.940268i	$-0.610574\pi$
$97$	−6.70578	−0.680868	−0.340434	+	0.940268i	$0.610574\pi$
$98$	−13.6302	−1.37686
$99$	0	0
$100$	2.19858	0.219858
$101$	5.94562	0.591611	0.295806	−	0.955248i	$-0.404412\pi$
$101$	5.94562	0.591611	0.295806	+	0.955248i	$0.404412\pi$
$102$	0	0
$103$	1.93558	0.190718	0.0953592	−	0.995443i	$-0.469600\pi$
$103$	1.93558	0.190718	0.0953592	+	0.995443i	$0.469600\pi$
$104$	−0.612355	−0.0600463
$105$	0	0
$106$	13.8795	1.34810
$107$	−8.28721	−0.801154	−0.400577	−	0.916263i	$-0.631190\pi$
$107$	−8.28721	−0.801154	−0.400577	+	0.916263i	$0.631190\pi$
$108$	0	0
$109$	−3.18782	−0.305338	−0.152669	−	0.988277i	$-0.548787\pi$
$109$	−3.18782	−0.305338	−0.152669	+	0.988277i	$0.548787\pi$
$110$	0	0
$111$	0	0
$112$	−2.10218	−0.198637
$113$	11.4906	1.08094	0.540470	−	0.841363i	$-0.318246\pi$
$113$	11.4906	1.08094	0.540470	+	0.841363i	$0.318246\pi$
$114$	0	0
$115$	−0.0822934	−0.00767390
$116$	−18.5218	−1.71971
$117$	0	0
$118$	−11.4419	−1.05331
$119$	−2.90248	−0.266070
$120$	0	0
$121$	0	0
$122$	1.96759	0.178137
$123$	0	0
$124$	−4.35128	−0.390757
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.1914	−1.61423	−0.807115	−	0.590395i	$-0.798972\pi$
$127$	−18.1914	−1.61423	−0.807115	+	0.590395i	$0.798972\pi$
$128$	−3.23918	−0.286306
$129$	0	0
$130$	−3.08363	−0.270453
$131$	−18.7758	−1.64045	−0.820225	−	0.572041i	$-0.806152\pi$
$131$	−18.7758	−1.64045	−0.820225	+	0.572041i	$0.806152\pi$
$132$	0	0
$133$	−4.02371	−0.348900
$134$	27.9816	2.41724
$135$	0	0
$136$	−2.00196	−0.171667
$137$	−4.72703	−0.403857	−0.201929	−	0.979400i	$-0.564721\pi$
$137$	−4.72703	−0.403857	−0.201929	+	0.979400i	$0.564721\pi$
$138$	0	0
$139$	18.4737	1.56692	0.783460	−	0.621442i	$-0.213453\pi$
$139$	18.4737	1.56692	0.783460	+	0.621442i	$0.213453\pi$
$140$	1.29702	0.109618
$141$	0	0
$142$	−2.63055	−0.220751
$143$	0	0
$144$	0	0
$145$	−8.42443	−0.699611
$146$	32.0206	2.65005
$147$	0	0
$148$	−7.87085	−0.646980
$149$	−15.5011	−1.26990	−0.634949	−	0.772554i	$-0.718979\pi$
$149$	−15.5011	−1.26990	−0.634949	+	0.772554i	$0.718979\pi$
$150$	0	0
$151$	−5.55440	−0.452011	−0.226005	−	0.974126i	$-0.572567\pi$
$151$	−5.55440	−0.452011	−0.226005	+	0.974126i	$0.572567\pi$
$152$	−2.77532	−0.225109
$153$	0	0
$154$	0	0
$155$	−1.97913	−0.158968
$156$	0	0
$157$	−18.5115	−1.47738	−0.738689	−	0.674046i	$-0.764555\pi$
$157$	−18.5115	−1.47738	−0.738689	+	0.674046i	$0.764555\pi$
$158$	22.8513	1.81796
$159$	0	0
$160$	−8.11537	−0.641577
$161$	−0.0485478	−0.00382610
$162$	0	0
$163$	7.81457	0.612085	0.306042	−	0.952018i	$-0.400995\pi$
$163$	7.81457	0.612085	0.306042	+	0.952018i	$0.400995\pi$
$164$	−8.95529	−0.699291
$165$	0	0
$166$	8.22108	0.638080
$167$	−5.96190	−0.461345	−0.230673	−	0.973031i	$-0.574093\pi$
$167$	−5.96190	−0.461345	−0.230673	+	0.973031i	$0.574093\pi$
$168$	0	0
$169$	−10.7352	−0.825787
$170$	−10.0813	−0.773200
$171$	0	0
$172$	−10.6884	−0.814981
$173$	16.5738	1.26009	0.630043	−	0.776560i	$-0.283037\pi$
$173$	16.5738	1.26009	0.630043	+	0.776560i	$0.283037\pi$
$174$	0	0
$175$	0.589936	0.0445949
$176$	0	0
$177$	0	0
$178$	12.4618	0.934053
$179$	−21.5447	−1.61033	−0.805163	−	0.593054i	$-0.797922\pi$
$179$	−21.5447	−1.61033	−0.805163	+	0.593054i	$0.797922\pi$
$180$	0	0
$181$	−20.1704	−1.49925	−0.749627	−	0.661861i	$-0.769767\pi$
$181$	−20.1704	−1.49925	−0.749627	+	0.661861i	$0.769767\pi$
$182$	−1.81915	−0.134844
$183$	0	0
$184$	−0.0334855	−0.00246858
$185$	−3.57997	−0.263204
$186$	0	0
$187$	0	0
$188$	12.7046	0.926575
$189$	0	0
$190$	−13.9757	−1.01390
$191$	16.2891	1.17864	0.589321	−	0.807899i	$-0.299395\pi$
$191$	16.2891	1.17864	0.589321	+	0.807899i	$0.299395\pi$
$192$	0	0
$193$	−3.34948	−0.241101	−0.120551	−	0.992707i	$-0.538466\pi$
$193$	−3.34948	−0.241101	−0.120551	+	0.992707i	$0.538466\pi$
$194$	−13.7404	−0.986506
$195$	0	0
$196$	−14.6249	−1.04464
$197$	9.78158	0.696909	0.348454	−	0.937326i	$-0.386707\pi$
$197$	9.78158	0.696909	0.348454	+	0.937326i	$0.386707\pi$
$198$	0	0
$199$	9.89054	0.701121	0.350561	−	0.936540i	$-0.385991\pi$
$199$	9.89054	0.701121	0.350561	+	0.936540i	$0.385991\pi$
$200$	0.406903	0.0287724
$201$	0	0
$202$	12.1828	0.857181
$203$	−4.96987	−0.348817
$204$	0	0
$205$	−4.07321	−0.284485
$206$	3.96609	0.276331
$207$	0	0
$208$	5.36261	0.371830
$209$	0	0
$210$	0	0
$211$	22.1686	1.52615	0.763076	−	0.646309i	$-0.223688\pi$
$211$	22.1686	1.52615	0.763076	+	0.646309i	$0.223688\pi$
$212$	14.8924	1.02282
$213$	0	0
$214$	−16.9809	−1.16079
$215$	−4.86148	−0.331550
$216$	0	0
$217$	−1.16756	−0.0792592
$218$	−6.53199	−0.442402
$219$	0	0
$220$	0	0
$221$	7.40417	0.498058
$222$	0	0
$223$	12.3398	0.826336	0.413168	−	0.910655i	$-0.364422\pi$
$223$	12.3398	0.826336	0.413168	+	0.910655i	$0.364422\pi$
$224$	−4.78755	−0.319882
$225$	0	0
$226$	23.5447	1.56617
$227$	8.22158	0.545685	0.272843	−	0.962059i	$-0.412036\pi$
$227$	8.22158	0.545685	0.272843	+	0.962059i	$0.412036\pi$
$228$	0	0
$229$	27.0445	1.78715	0.893576	−	0.448911i	$-0.148188\pi$
$229$	27.0445	1.78715	0.893576	+	0.448911i	$0.148188\pi$
$230$	−0.168623	−0.0111187
$231$	0	0
$232$	−3.42793	−0.225055
$233$	−25.0042	−1.63808	−0.819040	−	0.573737i	$-0.805493\pi$
$233$	−25.0042	−1.63808	−0.819040	+	0.573737i	$0.805493\pi$
$234$	0	0
$235$	5.77852	0.376949
$236$	−12.2770	−0.799162
$237$	0	0
$238$	−5.94731	−0.385507
$239$	20.7212	1.34035	0.670173	−	0.742205i	$-0.266220\pi$
$239$	20.7212	1.34035	0.670173	+	0.742205i	$0.266220\pi$
$240$	0	0
$241$	23.4295	1.50923	0.754615	−	0.656168i	$-0.227824\pi$
$241$	23.4295	1.50923	0.754615	+	0.656168i	$0.227824\pi$
$242$	0	0
$243$	0	0
$244$	2.11118	0.135154
$245$	−6.65198	−0.424979
$246$	0	0
$247$	10.2644	0.653109
$248$	−0.805316	−0.0511376
$249$	0	0
$250$	2.04904	0.129593
$251$	−10.2697	−0.648220	−0.324110	−	0.946019i	$-0.605065\pi$
$251$	−10.2697	−0.648220	−0.324110	+	0.946019i	$0.605065\pi$
$252$	0	0
$253$	0	0
$254$	−37.2751	−2.33885
$255$	0	0
$256$	12.3667	0.772918
$257$	−1.94941	−0.121601	−0.0608006	−	0.998150i	$-0.519365\pi$
$257$	−1.94941	−0.121601	−0.0608006	+	0.998150i	$0.519365\pi$
$258$	0	0
$259$	−2.11195	−0.131230
$260$	−3.30868	−0.205195
$261$	0	0
$262$	−38.4725	−2.37684
$263$	−8.80234	−0.542775	−0.271388	−	0.962470i	$-0.587483\pi$
$263$	−8.80234	−0.542775	−0.271388	+	0.962470i	$0.587483\pi$
$264$	0	0
$265$	6.77365	0.416102
$266$	−8.24477	−0.505519
$267$	0	0
$268$	30.0237	1.83399
$269$	−12.1842	−0.742883	−0.371441	−	0.928456i	$-0.621136\pi$
$269$	−12.1842	−0.742883	−0.371441	+	0.928456i	$0.621136\pi$
$270$	0	0
$271$	−12.3655	−0.751152	−0.375576	−	0.926792i	$-0.622555\pi$
$271$	−12.3655	−0.751152	−0.375576	+	0.926792i	$0.622555\pi$
$272$	17.5319	1.06303
$273$	0	0
$274$	−9.68590	−0.585146
$275$	0	0
$276$	0	0
$277$	−20.5289	−1.23346	−0.616731	−	0.787174i	$-0.711543\pi$
$277$	−20.5289	−1.23346	−0.616731	+	0.787174i	$0.711543\pi$
$278$	37.8535	2.27030
$279$	0	0
$280$	0.240047	0.0143455
$281$	11.4270	0.681677	0.340838	−	0.940122i	$-0.389289\pi$
$281$	11.4270	0.681677	0.340838	+	0.940122i	$0.389289\pi$
$282$	0	0
$283$	−29.5320	−1.75550	−0.877749	−	0.479121i	$-0.840956\pi$
$283$	−29.5320	−1.75550	−0.877749	+	0.479121i	$0.840956\pi$
$284$	−2.82252	−0.167486
$285$	0	0
$286$	0	0
$287$	−2.40293	−0.141841
$288$	0	0
$289$	7.20636	0.423904
$290$	−17.2620	−1.01366
$291$	0	0
$292$	34.3575	2.01062
$293$	11.5964	0.677471	0.338735	−	0.940882i	$-0.390001\pi$
$293$	11.5964	0.677471	0.338735	+	0.940882i	$0.390001\pi$
$294$	0	0
$295$	−5.58403	−0.325115
$296$	−1.45670	−0.0846690
$297$	0	0
$298$	−31.7624	−1.83995
$299$	0.123844	0.00716211
$300$	0	0
$301$	−2.86796	−0.165307
$302$	−11.3812	−0.654915
$303$	0	0
$304$	24.3045	1.39396
$305$	0.960246	0.0549835
$306$	0	0
$307$	−1.27309	−0.0726593	−0.0363296	−	0.999340i	$-0.511567\pi$
$307$	−1.27309	−0.0726593	−0.0363296	+	0.999340i	$0.511567\pi$
$308$	0	0
$309$	0	0
$310$	−4.05533	−0.230327
$311$	−4.90545	−0.278163	−0.139081	−	0.990281i	$-0.544415\pi$
$311$	−4.90545	−0.278163	−0.139081	+	0.990281i	$0.544415\pi$
$312$	0	0
$313$	−15.4197	−0.871572	−0.435786	−	0.900050i	$-0.643530\pi$
$313$	−15.4197	−0.871572	−0.435786	+	0.900050i	$0.643530\pi$
$314$	−37.9309	−2.14056
$315$	0	0
$316$	24.5190	1.37930
$317$	−33.5845	−1.88630	−0.943148	−	0.332374i	$-0.892151\pi$
$317$	−33.5845	−1.88630	−0.943148	+	0.332374i	$0.892151\pi$
$318$	0	0
$319$	0	0
$320$	−9.50196	−0.531175
$321$	0	0
$322$	−0.0994766	−0.00554361
$323$	33.5573	1.86718
$324$	0	0
$325$	−1.50491	−0.0834776
$326$	16.0124	0.886845
$327$	0	0
$328$	−1.65740	−0.0915148
$329$	3.40896	0.187942
$330$	0	0
$331$	−9.50564	−0.522477	−0.261239	−	0.965274i	$-0.584131\pi$
$331$	−9.50564	−0.522477	−0.261239	+	0.965274i	$0.584131\pi$
$332$	8.82105	0.484118
$333$	0	0
$334$	−12.2162	−0.668440
$335$	13.6559	0.746102
$336$	0	0
$337$	−25.6144	−1.39530	−0.697651	−	0.716437i	$-0.745772\pi$
$337$	−25.6144	−1.39530	−0.697651	+	0.716437i	$0.745772\pi$
$338$	−21.9970	−1.19648
$339$	0	0
$340$	−10.8170	−0.586635
$341$	0	0
$342$	0	0
$343$	−8.05379	−0.434864
$344$	−1.97815	−0.106655
$345$	0	0
$346$	33.9605	1.82573
$347$	−27.9083	−1.49820	−0.749099	−	0.662458i	$-0.769513\pi$
$347$	−27.9083	−1.49820	−0.749099	+	0.662458i	$0.769513\pi$
$348$	0	0
$349$	10.6089	0.567884	0.283942	−	0.958841i	$-0.408358\pi$
$349$	10.6089	0.567884	0.283942	+	0.958841i	$0.408358\pi$
$350$	1.20880	0.0646133
$351$	0	0
$352$	0	0
$353$	29.0702	1.54725	0.773627	−	0.633642i	$-0.218441\pi$
$353$	29.0702	1.54725	0.773627	+	0.633642i	$0.218441\pi$
$354$	0	0
$355$	−1.28379	−0.0681366
$356$	13.3713	0.708676
$357$	0	0
$358$	−44.1460	−2.33319
$359$	−30.2307	−1.59552	−0.797758	−	0.602978i	$-0.793981\pi$
$359$	−30.2307	−1.59552	−0.797758	+	0.602978i	$0.793981\pi$
$360$	0	0
$361$	27.5206	1.44845
$362$	−41.3300	−2.17226
$363$	0	0
$364$	−1.95191	−0.102308
$365$	15.6271	0.817960
$366$	0	0
$367$	−23.6556	−1.23481	−0.617406	−	0.786644i	$-0.711817\pi$
$367$	−23.6556	−1.23481	−0.617406	+	0.786644i	$0.711817\pi$
$368$	0.293244	0.0152864
$369$	0	0
$370$	−7.33551	−0.381355
$371$	3.99602	0.207463
$372$	0	0
$373$	14.5998	0.755949	0.377974	−	0.925816i	$-0.376621\pi$
$373$	14.5998	0.755949	0.377974	+	0.925816i	$0.376621\pi$
$374$	0	0
$375$	0	0
$376$	2.35130	0.121259
$377$	12.6780	0.652952
$378$	0	0
$379$	6.70370	0.344346	0.172173	−	0.985067i	$-0.444921\pi$
$379$	6.70370	0.344346	0.172173	+	0.985067i	$0.444921\pi$
$380$	−14.9956	−0.769260
$381$	0	0
$382$	33.3772	1.70773
$383$	−31.5283	−1.61102	−0.805510	−	0.592582i	$-0.798108\pi$
$383$	−31.5283	−1.61102	−0.805510	+	0.592582i	$0.798108\pi$
$384$	0	0
$385$	0	0
$386$	−6.86324	−0.349330
$387$	0	0
$388$	−14.7432	−0.748473
$389$	−38.9837	−1.97655	−0.988277	−	0.152672i	$-0.951212\pi$
$389$	−38.9837	−1.97655	−0.988277	+	0.152672i	$0.951212\pi$
$390$	0	0
$391$	0.404883	0.0204758
$392$	−2.70671	−0.136710
$393$	0	0
$394$	20.0429	1.00975
$395$	11.1522	0.561128
$396$	0	0
$397$	27.1682	1.36353	0.681766	−	0.731570i	$-0.261212\pi$
$397$	27.1682	1.36353	0.681766	+	0.731570i	$0.261212\pi$
$398$	20.2661	1.01585
$399$	0	0
$400$	−3.56340	−0.178170
$401$	13.1942	0.658889	0.329445	−	0.944175i	$-0.393139\pi$
$401$	13.1942	0.658889	0.329445	+	0.944175i	$0.393139\pi$
$402$	0	0
$403$	2.97842	0.148366
$404$	13.0719	0.650353
$405$	0	0
$406$	−10.1835	−0.505398
$407$	0	0
$408$	0	0
$409$	9.92734	0.490875	0.245438	−	0.969412i	$-0.421068\pi$
$409$	9.92734	0.490875	0.245438	+	0.969412i	$0.421068\pi$
$410$	−8.34619	−0.412189
$411$	0	0
$412$	4.25553	0.209655
$413$	−3.29422	−0.162098
$414$	0	0
$415$	4.01215	0.196949
$416$	12.2129	0.598788
$417$	0	0
$418$	0	0
$419$	−26.4274	−1.29107	−0.645533	−	0.763733i	$-0.723365\pi$
$419$	−26.4274	−1.29107	−0.645533	+	0.763733i	$0.723365\pi$
$420$	0	0
$421$	13.7490	0.670086	0.335043	−	0.942203i	$-0.391249\pi$
$421$	13.7490	0.670086	0.335043	+	0.942203i	$0.391249\pi$
$422$	45.4245	2.21123
$423$	0	0
$424$	2.75622	0.133854
$425$	−4.92000	−0.238655
$426$	0	0
$427$	0.566483	0.0274141
$428$	−18.2201	−0.880702
$429$	0	0
$430$	−9.96139	−0.480381
$431$	−19.9719	−0.962011	−0.481005	−	0.876718i	$-0.659728\pi$
$431$	−19.9719	−0.962011	−0.481005	+	0.876718i	$0.659728\pi$
$432$	0	0
$433$	16.6024	0.797858	0.398929	−	0.916982i	$-0.369382\pi$
$433$	16.6024	0.797858	0.398929	+	0.916982i	$0.369382\pi$
$434$	−2.39238	−0.114838
$435$	0	0
$436$	−7.00869	−0.335655
$437$	0.561290	0.0268501
$438$	0	0
$439$	28.3645	1.35376	0.676882	−	0.736091i	$-0.263331\pi$
$439$	28.3645	1.35376	0.676882	+	0.736091i	$0.263331\pi$
$440$	0	0
$441$	0	0
$442$	15.1715	0.721633
$443$	22.6826	1.07768	0.538842	−	0.842407i	$-0.318862\pi$
$443$	22.6826	1.07768	0.538842	+	0.842407i	$0.318862\pi$
$444$	0	0
$445$	6.08177	0.288304
$446$	25.2848	1.19727
$447$	0	0
$448$	−5.60554	−0.264837
$449$	4.58187	0.216232	0.108116	−	0.994138i	$-0.465518\pi$
$449$	4.58187	0.216232	0.108116	+	0.994138i	$0.465518\pi$
$450$	0	0
$451$	0	0
$452$	25.2629	1.18827
$453$	0	0
$454$	16.8464	0.790639
$455$	−0.887802	−0.0416208
$456$	0	0
$457$	−6.72587	−0.314623	−0.157311	−	0.987549i	$-0.550283\pi$
$457$	−6.72587	−0.314623	−0.157311	+	0.987549i	$0.550283\pi$
$458$	55.4154	2.58939
$459$	0	0
$460$	−0.180929	−0.00843585
$461$	2.83529	0.132053	0.0660263	−	0.997818i	$-0.478968\pi$
$461$	2.83529	0.132053	0.0660263	+	0.997818i	$0.478968\pi$
$462$	0	0
$463$	20.5498	0.955029	0.477515	−	0.878624i	$-0.341538\pi$
$463$	20.5498	0.955029	0.477515	+	0.878624i	$0.341538\pi$
$464$	30.0196	1.39363
$465$	0	0
$466$	−51.2347	−2.37340
$467$	−11.7594	−0.544158	−0.272079	−	0.962275i	$-0.587711\pi$
$467$	−11.7594	−0.544158	−0.272079	+	0.962275i	$0.587711\pi$
$468$	0	0
$469$	8.05611	0.371997
$470$	11.8404	0.546159
$471$	0	0
$472$	−2.27216	−0.104585
$473$	0	0
$474$	0	0
$475$	−6.82060	−0.312951
$476$	−6.38134	−0.292488
$477$	0	0
$478$	42.4587	1.94202
$479$	−1.52231	−0.0695563	−0.0347781	−	0.999395i	$-0.511072\pi$
$479$	−1.52231	−0.0695563	−0.0347781	+	0.999395i	$0.511072\pi$
$480$	0	0
$481$	5.38754	0.245651
$482$	48.0082	2.18671
$483$	0	0
$484$	0	0
$485$	−6.70578	−0.304494
$486$	0	0
$487$	35.4979	1.60857	0.804283	−	0.594247i	$-0.202550\pi$
$487$	35.4979	1.60857	0.804283	+	0.594247i	$0.202550\pi$
$488$	0.390727	0.0176874
$489$	0	0
$490$	−13.6302	−0.615749
$491$	−12.6774	−0.572123	−0.286062	−	0.958211i	$-0.592346\pi$
$491$	−12.6774	−0.572123	−0.286062	+	0.958211i	$0.592346\pi$
$492$	0	0
$493$	41.4482	1.86673
$494$	21.0322	0.946285
$495$	0	0
$496$	7.05244	0.316664
$497$	−0.757355	−0.0339720
$498$	0	0
$499$	21.0578	0.942675	0.471337	−	0.881953i	$-0.343771\pi$
$499$	21.0578	0.942675	0.471337	+	0.881953i	$0.343771\pi$
$500$	2.19858	0.0983236
$501$	0	0
$502$	−21.0431	−0.939201
$503$	−25.0768	−1.11812	−0.559059	−	0.829128i	$-0.688837\pi$
$503$	−25.0768	−1.11812	−0.559059	+	0.829128i	$0.688837\pi$
$504$	0	0
$505$	5.94562	0.264577
$506$	0	0
$507$	0	0
$508$	−39.9954	−1.77451
$509$	−36.6853	−1.62605	−0.813024	−	0.582230i	$-0.802180\pi$
$509$	−36.6853	−1.62605	−0.813024	+	0.582230i	$0.802180\pi$
$510$	0	0
$511$	9.21899	0.407824
$512$	31.8182	1.40618
$513$	0	0
$514$	−3.99444	−0.176187
$515$	1.93558	0.0852919
$516$	0	0
$517$	0	0
$518$	−4.32748	−0.190139
$519$	0	0
$520$	−0.612355	−0.0268535
$521$	28.3519	1.24212	0.621059	−	0.783764i	$-0.286703\pi$
$521$	28.3519	1.24212	0.621059	+	0.783764i	$0.286703\pi$
$522$	0	0
$523$	−26.7845	−1.17120	−0.585601	−	0.810599i	$-0.699142\pi$
$523$	−26.7845	−1.17120	−0.585601	+	0.810599i	$0.699142\pi$
$524$	−41.2802	−1.80333
$525$	0	0
$526$	−18.0364	−0.786423
$527$	9.73732	0.424164
$528$	0	0
$529$	−22.9932	−0.999706
$530$	13.8795	0.602888
$531$	0	0
$532$	−8.84647	−0.383543
$533$	6.12983	0.265513
$534$	0	0
$535$	−8.28721	−0.358287
$536$	5.55664	0.240010
$537$	0	0
$538$	−24.9659	−1.07636
$539$	0	0
$540$	0	0
$541$	−16.7331	−0.719412	−0.359706	−	0.933066i	$-0.617123\pi$
$541$	−16.7331	−0.719412	−0.359706	+	0.933066i	$0.617123\pi$
$542$	−25.3375	−1.08834
$543$	0	0
$544$	39.9276	1.71188
$545$	−3.18782	−0.136551
$546$	0	0
$547$	21.4016	0.915067	0.457534	−	0.889192i	$-0.348733\pi$
$547$	21.4016	0.915067	0.457534	+	0.889192i	$0.348733\pi$
$548$	−10.3928	−0.443957
$549$	0	0
$550$	0	0
$551$	57.4596	2.44786
$552$	0	0
$553$	6.57908	0.279771
$554$	−42.0646	−1.78715
$555$	0	0
$556$	40.6160	1.72250
$557$	−42.8947	−1.81751	−0.908754	−	0.417331i	$-0.862965\pi$
$557$	−42.8947	−1.81751	−0.908754	+	0.417331i	$0.862965\pi$
$558$	0	0
$559$	7.31611	0.309439
$560$	−2.10218	−0.0888332
$561$	0	0
$562$	23.4144	0.987677
$563$	−32.9570	−1.38897	−0.694487	−	0.719506i	$-0.744368\pi$
$563$	−32.9570	−1.38897	−0.694487	+	0.719506i	$0.744368\pi$
$564$	0	0
$565$	11.4906	0.483411
$566$	−60.5125	−2.54353
$567$	0	0
$568$	−0.522380	−0.0219186
$569$	−3.36920	−0.141244	−0.0706220	−	0.997503i	$-0.522498\pi$
$569$	−3.36920	−0.141244	−0.0706220	+	0.997503i	$0.522498\pi$
$570$	0	0
$571$	0.267599	0.0111987	0.00559934	−	0.999984i	$-0.498218\pi$
$571$	0.267599	0.0111987	0.00559934	+	0.999984i	$0.498218\pi$
$572$	0	0
$573$	0	0
$574$	−4.92372	−0.205512
$575$	−0.0822934	−0.00343187
$576$	0	0
$577$	37.8943	1.57756	0.788781	−	0.614675i	$-0.210713\pi$
$577$	37.8943	1.57756	0.788781	+	0.614675i	$0.210713\pi$
$578$	14.7662	0.614191
$579$	0	0
$580$	−18.5218	−0.769076
$581$	2.36691	0.0981961
$582$	0	0
$583$	0	0
$584$	6.35872	0.263126
$585$	0	0
$586$	23.7616	0.981583
$587$	−16.8569	−0.695760	−0.347880	−	0.937539i	$-0.613098\pi$
$587$	−16.8569	−0.695760	−0.347880	+	0.937539i	$0.613098\pi$
$588$	0	0
$589$	13.4989	0.556211
$590$	−11.4419	−0.471057
$591$	0	0
$592$	12.7569	0.524304
$593$	−11.0211	−0.452580	−0.226290	−	0.974060i	$-0.572660\pi$
$593$	−11.0211	−0.452580	−0.226290	+	0.974060i	$0.572660\pi$
$594$	0	0
$595$	−2.90248	−0.118990
$596$	−34.0804	−1.39599
$597$	0	0
$598$	0.253763	0.0103771
$599$	16.6951	0.682142	0.341071	−	0.940038i	$-0.389210\pi$
$599$	16.6951	0.682142	0.341071	+	0.940038i	$0.389210\pi$
$600$	0	0
$601$	7.75665	0.316400	0.158200	−	0.987407i	$-0.449431\pi$
$601$	7.75665	0.316400	0.158200	+	0.987407i	$0.449431\pi$
$602$	−5.87658	−0.239512
$603$	0	0
$604$	−12.2118	−0.496891
$605$	0	0
$606$	0	0
$607$	−31.8958	−1.29461	−0.647305	−	0.762231i	$-0.724104\pi$
$607$	−31.8958	−1.29461	−0.647305	+	0.762231i	$0.724104\pi$
$608$	55.3517	2.24481
$609$	0	0
$610$	1.96759	0.0796652
$611$	−8.69618	−0.351810
$612$	0	0
$613$	42.8029	1.72879	0.864396	−	0.502811i	$-0.167701\pi$
$613$	42.8029	1.72879	0.864396	+	0.502811i	$0.167701\pi$
$614$	−2.60862	−0.105276
$615$	0	0
$616$	0	0
$617$	−21.7979	−0.877550	−0.438775	−	0.898597i	$-0.644588\pi$
$617$	−21.7979	−0.877550	−0.438775	+	0.898597i	$0.644588\pi$
$618$	0	0
$619$	33.6332	1.35183	0.675916	−	0.736979i	$-0.263748\pi$
$619$	33.6332	1.35183	0.675916	+	0.736979i	$0.263748\pi$
$620$	−4.35128	−0.174752
$621$	0	0
$622$	−10.0515	−0.403028
$623$	3.58786	0.143744
$624$	0	0
$625$	1.00000	0.0400000
$626$	−31.5956	−1.26281
$627$	0	0
$628$	−40.6990	−1.62407
$629$	17.6134	0.702293
$630$	0	0
$631$	−21.4552	−0.854120	−0.427060	−	0.904223i	$-0.640451\pi$
$631$	−21.4552	−0.854120	−0.427060	+	0.904223i	$0.640451\pi$
$632$	4.53787	0.180507
$633$	0	0
$634$	−68.8162	−2.73304
$635$	−18.1914	−0.721905
$636$	0	0
$637$	10.0107	0.396636
$638$	0	0
$639$	0	0
$640$	−3.23918	−0.128040
$641$	44.6847	1.76494	0.882469	−	0.470370i	$-0.155880\pi$
$641$	44.6847	1.76494	0.882469	+	0.470370i	$0.155880\pi$
$642$	0	0
$643$	8.59101	0.338796	0.169398	−	0.985548i	$-0.445818\pi$
$643$	8.59101	0.338796	0.169398	+	0.985548i	$0.445818\pi$
$644$	−0.106736	−0.00420600
$645$	0	0
$646$	68.7604	2.70534
$647$	8.94435	0.351639	0.175819	−	0.984422i	$-0.443743\pi$
$647$	8.94435	0.351639	0.175819	+	0.984422i	$0.443743\pi$
$648$	0	0
$649$	0	0
$650$	−3.08363	−0.120950
$651$	0	0
$652$	17.1810	0.672859
$653$	−13.9470	−0.545788	−0.272894	−	0.962044i	$-0.587981\pi$
$653$	−13.9470	−0.545788	−0.272894	+	0.962044i	$0.587981\pi$
$654$	0	0
$655$	−18.7758	−0.733632
$656$	14.5145	0.566696
$657$	0	0
$658$	6.98510	0.272308
$659$	6.61492	0.257681	0.128840	−	0.991665i	$-0.458875\pi$
$659$	6.61492	0.257681	0.128840	+	0.991665i	$0.458875\pi$
$660$	0	0
$661$	23.5635	0.916515	0.458257	−	0.888820i	$-0.348474\pi$
$661$	23.5635	0.916515	0.458257	+	0.888820i	$0.348474\pi$
$662$	−19.4775	−0.757013
$663$	0	0
$664$	1.63256	0.0633556
$665$	−4.02371	−0.156033
$666$	0	0
$667$	0.693275	0.0268437
$668$	−13.1077	−0.507153
$669$	0	0
$670$	27.9816	1.08102
$671$	0	0
$672$	0	0
$673$	−4.21365	−0.162424	−0.0812122	−	0.996697i	$-0.525879\pi$
$673$	−4.21365	−0.162424	−0.0812122	+	0.996697i	$0.525879\pi$
$674$	−52.4849	−2.02164
$675$	0	0
$676$	−23.6023	−0.907781
$677$	17.0035	0.653497	0.326748	−	0.945111i	$-0.394047\pi$
$677$	17.0035	0.653497	0.326748	+	0.945111i	$0.394047\pi$
$678$	0	0
$679$	−3.95598	−0.151816
$680$	−2.00196	−0.0767718
$681$	0	0
$682$	0	0
$683$	19.6114	0.750409	0.375204	−	0.926942i	$-0.377573\pi$
$683$	19.6114	0.750409	0.375204	+	0.926942i	$0.377573\pi$
$684$	0	0
$685$	−4.72703	−0.180611
$686$	−16.5026	−0.630071
$687$	0	0
$688$	17.3234	0.660449
$689$	−10.1938	−0.388352
$690$	0	0
$691$	7.66999	0.291780	0.145890	−	0.989301i	$-0.453395\pi$
$691$	7.66999	0.291780	0.145890	+	0.989301i	$0.453395\pi$
$692$	36.4390	1.38520
$693$	0	0
$694$	−57.1854	−2.17073
$695$	18.4737	0.700748
$696$	0	0
$697$	20.0402	0.759076
$698$	21.7382	0.822803
$699$	0	0
$700$	1.29702	0.0490228
$701$	−21.9331	−0.828403	−0.414202	−	0.910185i	$-0.635939\pi$
$701$	−21.9331	−0.828403	−0.414202	+	0.910185i	$0.635939\pi$
$702$	0	0
$703$	24.4175	0.920924
$704$	0	0
$705$	0	0
$706$	59.5662	2.24180
$707$	3.50753	0.131914
$708$	0	0
$709$	−7.22476	−0.271332	−0.135666	−	0.990755i	$-0.543317\pi$
$709$	−7.22476	−0.271332	−0.135666	+	0.990755i	$0.543317\pi$
$710$	−2.63055	−0.0987227
$711$	0	0
$712$	2.47469	0.0927431
$713$	0.162869	0.00609951
$714$	0	0
$715$	0	0
$716$	−47.3678	−1.77022
$717$	0	0
$718$	−61.9440	−2.31173
$719$	−19.2225	−0.716879	−0.358439	−	0.933553i	$-0.616691\pi$
$719$	−19.2225	−0.716879	−0.358439	+	0.933553i	$0.616691\pi$
$720$	0	0
$721$	1.14187	0.0425254
$722$	56.3909	2.09865
$723$	0	0
$724$	−44.3463	−1.64812
$725$	−8.42443	−0.312875
$726$	0	0
$727$	−33.9775	−1.26015	−0.630077	−	0.776532i	$-0.716977\pi$
$727$	−33.9775	−1.26015	−0.630077	+	0.776532i	$0.716977\pi$
$728$	−0.361250	−0.0133888
$729$	0	0
$730$	32.0206	1.18514
$731$	23.9185	0.884657
$732$	0	0
$733$	−6.49826	−0.240019	−0.120009	−	0.992773i	$-0.538292\pi$
$733$	−6.49826	−0.240019	−0.120009	+	0.992773i	$0.538292\pi$
$734$	−48.4714	−1.78911
$735$	0	0
$736$	0.667841	0.0246170
$737$	0	0
$738$	0	0
$739$	18.7140	0.688405	0.344203	−	0.938895i	$-0.388149\pi$
$739$	18.7140	0.688405	0.344203	+	0.938895i	$0.388149\pi$
$740$	−7.87085	−0.289338
$741$	0	0
$742$	8.18802	0.300592
$743$	−8.79889	−0.322800	−0.161400	−	0.986889i	$-0.551601\pi$
$743$	−8.79889	−0.322800	−0.161400	+	0.986889i	$0.551601\pi$
$744$	0	0
$745$	−15.5011	−0.567915
$746$	29.9156	1.09529
$747$	0	0
$748$	0	0
$749$	−4.88892	−0.178637
$750$	0	0
$751$	25.7867	0.940971	0.470485	−	0.882408i	$-0.344079\pi$
$751$	25.7867	0.940971	0.470485	+	0.882408i	$0.344079\pi$
$752$	−20.5912	−0.750884
$753$	0	0
$754$	25.9779	0.946058
$755$	−5.55440	−0.202145
$756$	0	0
$757$	40.0699	1.45636	0.728182	−	0.685384i	$-0.240366\pi$
$757$	40.0699	1.45636	0.728182	+	0.685384i	$0.240366\pi$
$758$	13.7362	0.498920
$759$	0	0
$760$	−2.77532	−0.100672
$761$	−17.9355	−0.650162	−0.325081	−	0.945686i	$-0.605392\pi$
$761$	−17.9355	−0.650162	−0.325081	+	0.945686i	$0.605392\pi$
$762$	0	0
$763$	−1.88061	−0.0680827
$764$	35.8130	1.29567
$765$	0	0
$766$	−64.6028	−2.33420
$767$	8.40349	0.303432
$768$	0	0
$769$	10.8431	0.391011	0.195506	−	0.980703i	$-0.437365\pi$
$769$	10.8431	0.391011	0.195506	+	0.980703i	$0.437365\pi$
$770$	0	0
$771$	0	0
$772$	−7.36412	−0.265040
$773$	−6.86871	−0.247050	−0.123525	−	0.992341i	$-0.539420\pi$
$773$	−6.86871	−0.247050	−0.123525	+	0.992341i	$0.539420\pi$
$774$	0	0
$775$	−1.97913	−0.0710925
$776$	−2.72860	−0.0979512
$777$	0	0
$778$	−79.8794	−2.86381
$779$	27.7817	0.995384
$780$	0	0
$781$	0	0
$782$	0.829623	0.0296673
$783$	0	0
$784$	23.7037	0.846559
$785$	−18.5115	−0.660704
$786$	0	0
$787$	−2.26967	−0.0809050	−0.0404525	−	0.999181i	$-0.512880\pi$
$787$	−2.26967	−0.0809050	−0.0404525	+	0.999181i	$0.512880\pi$
$788$	21.5056	0.766106
$789$	0	0
$790$	22.8513	0.813014
$791$	6.77869	0.241022
$792$	0	0
$793$	−1.44509	−0.0513166
$794$	55.6688	1.97561
$795$	0	0
$796$	21.7452	0.770737
$797$	11.0564	0.391636	0.195818	−	0.980640i	$-0.437264\pi$
$797$	11.0564	0.391636	0.195818	+	0.980640i	$0.437264\pi$
$798$	0	0
$799$	−28.4303	−1.00579
$800$	−8.11537	−0.286922
$801$	0	0
$802$	27.0356	0.954660
$803$	0	0
$804$	0	0
$805$	−0.0485478	−0.00171109
$806$	6.10292	0.214966
$807$	0	0
$808$	2.41929	0.0851104
$809$	−4.30271	−0.151275	−0.0756377	−	0.997135i	$-0.524099\pi$
$809$	−4.30271	−0.151275	−0.0756377	+	0.997135i	$0.524099\pi$
$810$	0	0
$811$	18.7204	0.657361	0.328681	−	0.944441i	$-0.393396\pi$
$811$	18.7204	0.657361	0.328681	+	0.944441i	$0.393396\pi$
$812$	−10.9267	−0.383451
$813$	0	0
$814$	0	0
$815$	7.81457	0.273733
$816$	0	0
$817$	33.1582	1.16006
$818$	20.3416	0.711226
$819$	0	0
$820$	−8.95529	−0.312732
$821$	−36.4237	−1.27119	−0.635597	−	0.772021i	$-0.719246\pi$
$821$	−36.4237	−1.27119	−0.635597	+	0.772021i	$0.719246\pi$
$822$	0	0
$823$	−1.52202	−0.0530542	−0.0265271	−	0.999648i	$-0.508445\pi$
$823$	−1.52202	−0.0530542	−0.0265271	+	0.999648i	$0.508445\pi$
$824$	0.787594	0.0274371
$825$	0	0
$826$	−6.75000	−0.234863
$827$	−25.4190	−0.883904	−0.441952	−	0.897039i	$-0.645714\pi$
$827$	−25.4190	−0.883904	−0.441952	+	0.897039i	$0.645714\pi$
$828$	0	0
$829$	−8.50118	−0.295258	−0.147629	−	0.989043i	$-0.547164\pi$
$829$	−8.50118	−0.295258	−0.147629	+	0.989043i	$0.547164\pi$
$830$	8.22108	0.285358
$831$	0	0
$832$	14.2996	0.495750
$833$	32.7277	1.13395
$834$	0	0
$835$	−5.96190	−0.206320
$836$	0	0
$837$	0	0
$838$	−54.1510	−1.87062
$839$	−44.4716	−1.53533	−0.767666	−	0.640850i	$-0.778582\pi$
$839$	−44.4716	−1.53533	−0.767666	+	0.640850i	$0.778582\pi$
$840$	0	0
$841$	41.9710	1.44728
$842$	28.1723	0.970883
$843$	0	0
$844$	48.7396	1.67769
$845$	−10.7352	−0.369303
$846$	0	0
$847$	0	0
$848$	−24.1372	−0.828877
$849$	0	0
$850$	−10.0813	−0.345785
$851$	0.294608	0.0100990
$852$	0	0
$853$	−27.8719	−0.954315	−0.477158	−	0.878818i	$-0.658333\pi$
$853$	−27.8719	−0.954315	−0.477158	+	0.878818i	$0.658333\pi$
$854$	1.16075	0.0397200
$855$	0	0
$856$	−3.37209	−0.115256
$857$	−23.1342	−0.790249	−0.395125	−	0.918628i	$-0.629299\pi$
$857$	−23.1342	−0.790249	−0.395125	+	0.918628i	$0.629299\pi$
$858$	0	0
$859$	51.9912	1.77392	0.886959	−	0.461849i	$-0.152814\pi$
$859$	51.9912	1.77392	0.886959	+	0.461849i	$0.152814\pi$
$860$	−10.6884	−0.364470
$861$	0	0
$862$	−40.9232	−1.39385
$863$	−29.2084	−0.994266	−0.497133	−	0.867674i	$-0.665614\pi$
$863$	−29.2084	−0.994266	−0.497133	+	0.867674i	$0.665614\pi$
$864$	0	0
$865$	16.5738	0.563528
$866$	34.0190	1.15601
$867$	0	0
$868$	−2.56698	−0.0871289
$869$	0	0
$870$	0	0
$871$	−20.5510	−0.696343
$872$	−1.29714	−0.0439266
$873$	0	0
$874$	1.15011	0.0389030
$875$	0.589936	0.0199435
$876$	0	0
$877$	−34.5757	−1.16754	−0.583769	−	0.811920i	$-0.698423\pi$
$877$	−34.5757	−1.16754	−0.583769	+	0.811920i	$0.698423\pi$
$878$	58.1202	1.96146
$879$	0	0
$880$	0	0
$881$	44.3728	1.49496	0.747479	−	0.664285i	$-0.231264\pi$
$881$	44.3728	1.49496	0.747479	+	0.664285i	$0.231264\pi$
$882$	0	0
$883$	−28.7786	−0.968477	−0.484238	−	0.874936i	$-0.660903\pi$
$883$	−28.7786	−0.968477	−0.484238	+	0.874936i	$0.660903\pi$
$884$	16.2787	0.547511
$885$	0	0
$886$	46.4777	1.56145
$887$	9.30878	0.312558	0.156279	−	0.987713i	$-0.450050\pi$
$887$	9.30878	0.312558	0.156279	+	0.987713i	$0.450050\pi$
$888$	0	0
$889$	−10.7318	−0.359932
$890$	12.4618	0.417721
$891$	0	0
$892$	27.1301	0.908383
$893$	−39.4130	−1.31891
$894$	0	0
$895$	−21.5447	−0.720160
$896$	−1.91091	−0.0638390
$897$	0	0
$898$	9.38845	0.313296
$899$	16.6731	0.556078
$900$	0	0
$901$	−33.3264	−1.11026
$902$	0	0
$903$	0	0
$904$	4.67555	0.155506
$905$	−20.1704	−0.670487
$906$	0	0
$907$	−44.4044	−1.47442	−0.737212	−	0.675662i	$-0.763858\pi$
$907$	−44.4044	−1.47442	−0.737212	+	0.675662i	$0.763858\pi$
$908$	18.0758	0.599867
$909$	0	0
$910$	−1.81915	−0.0603041
$911$	23.6358	0.783088	0.391544	−	0.920159i	$-0.371941\pi$
$911$	23.6358	0.783088	0.391544	+	0.920159i	$0.371941\pi$
$912$	0	0
$913$	0	0
$914$	−13.7816	−0.455855
$915$	0	0
$916$	59.4596	1.96460
$917$	−11.0765	−0.365779
$918$	0	0
$919$	−6.85429	−0.226102	−0.113051	−	0.993589i	$-0.536062\pi$
$919$	−6.85429	−0.226102	−0.113051	+	0.993589i	$0.536062\pi$
$920$	−0.0334855	−0.00110398
$921$	0	0
$922$	5.80963	0.191330
$923$	1.93200	0.0635925
$924$	0	0
$925$	−3.57997	−0.117709
$926$	42.1074	1.38374
$927$	0	0
$928$	68.3674	2.24427
$929$	−28.4759	−0.934265	−0.467132	−	0.884187i	$-0.654713\pi$
$929$	−28.4759	−0.934265	−0.467132	+	0.884187i	$0.654713\pi$
$930$	0	0
$931$	45.3705	1.48696
$932$	−54.9738	−1.80073
$933$	0	0
$934$	−24.0954	−0.788427
$935$	0	0
$936$	0	0
$937$	39.1324	1.27840	0.639199	−	0.769041i	$-0.279266\pi$
$937$	39.1324	1.27840	0.639199	+	0.769041i	$0.279266\pi$
$938$	16.5073	0.538983
$939$	0	0
$940$	12.7046	0.414377
$941$	38.0674	1.24096	0.620480	−	0.784222i	$-0.286938\pi$
$941$	38.0674	1.24096	0.620480	+	0.784222i	$0.286938\pi$
$942$	0	0
$943$	0.335198	0.0109156
$944$	19.8981	0.647629
$945$	0	0
$946$	0	0
$947$	6.37140	0.207043	0.103521	−	0.994627i	$-0.466989\pi$
$947$	6.37140	0.207043	0.103521	+	0.994627i	$0.466989\pi$
$948$	0	0
$949$	−23.5175	−0.763409
$950$	−13.9757	−0.453432
$951$	0	0
$952$	−1.18103	−0.0382774
$953$	2.90339	0.0940500	0.0470250	−	0.998894i	$-0.485026\pi$
$953$	2.90339	0.0940500	0.0470250	+	0.998894i	$0.485026\pi$
$954$	0	0
$955$	16.2891	0.527104
$956$	45.5573	1.47343
$957$	0	0
$958$	−3.11929	−0.100780
$959$	−2.78864	−0.0900500
$960$	0	0
$961$	−27.0830	−0.873646
$962$	11.0393	0.355922
$963$	0	0
$964$	51.5118	1.65908
$965$	−3.34948	−0.107824
$966$	0	0
$967$	9.58404	0.308202	0.154101	−	0.988055i	$-0.450752\pi$
$967$	9.58404	0.308202	0.154101	+	0.988055i	$0.450752\pi$
$968$	0	0
$969$	0	0
$970$	−13.7404	−0.441179
$971$	−53.8394	−1.72779	−0.863895	−	0.503672i	$-0.831982\pi$
$971$	−53.8394	−1.72779	−0.863895	+	0.503672i	$0.831982\pi$
$972$	0	0
$973$	10.8983	0.349384
$974$	72.7369	2.33064
$975$	0	0
$976$	−3.42174	−0.109527
$977$	−29.4618	−0.942568	−0.471284	−	0.881982i	$-0.656209\pi$
$977$	−29.4618	−0.942568	−0.471284	+	0.881982i	$0.656209\pi$
$978$	0	0
$979$	0	0
$980$	−14.6249	−0.467176
$981$	0	0
$982$	−25.9766	−0.828946
$983$	44.9732	1.43442	0.717211	−	0.696856i	$-0.245419\pi$
$983$	44.9732	1.43442	0.717211	+	0.696856i	$0.245419\pi$
$984$	0	0
$985$	9.78158	0.311667
$986$	84.9291	2.70469
$987$	0	0
$988$	22.5672	0.717957
$989$	0.400068	0.0127214
$990$	0	0
$991$	30.3468	0.963997	0.481998	−	0.876172i	$-0.339911\pi$
$991$	30.3468	0.963997	0.481998	+	0.876172i	$0.339911\pi$
$992$	16.0614	0.509950
$993$	0	0
$994$	−1.55185	−0.0492218
$995$	9.89054	0.313551
$996$	0	0
$997$	24.2808	0.768981	0.384490	−	0.923129i	$-0.374377\pi$
$997$	24.2808	0.768981	0.384490	+	0.923129i	$0.374377\pi$
$998$	43.1483	1.36583
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.cb.1.8		8
3.2	odd	2		5445.2.a.cc.1.1		8
11.7	odd	10		495.2.n.g.181.1	✓	16
11.8	odd	10		495.2.n.g.361.1	yes	16
11.10	odd	2		5445.2.a.cd.1.1		8
33.8	even	10		495.2.n.h.361.4	yes	16
33.29	even	10		495.2.n.h.181.4	yes	16
33.32	even	2		5445.2.a.ca.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.181.1	✓	16	11.7	odd	10
495.2.n.g.361.1	yes	16	11.8	odd	10
495.2.n.h.181.4	yes	16	33.29	even	10
495.2.n.h.361.4	yes	16	33.8	even	10
5445.2.a.ca.1.8		8	33.32	even	2
5445.2.a.cb.1.8		8	1.1	even	1	trivial
5445.2.a.cc.1.1		8	3.2	odd	2
5445.2.a.cd.1.1		8	11.10	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)

\(f(q)\)	\(=\)	\(q+2.04904 q^{2} +2.19858 q^{4} +1.00000 q^{5} +0.589936 q^{7} +0.406903 q^{8} +O(q^{10})\)	\(q+2.04904 q^{2} +2.19858 q^{4} +1.00000 q^{5} +0.589936 q^{7} +0.406903 q^{8} +2.04904 q^{10} -1.50491 q^{13} +1.20880 q^{14} -3.56340 q^{16} -4.92000 q^{17} -6.82060 q^{19} +2.19858 q^{20} -0.0822934 q^{23} +1.00000 q^{25} -3.08363 q^{26} +1.29702 q^{28} -8.42443 q^{29} -1.97913 q^{31} -8.11537 q^{32} -10.0813 q^{34} +0.589936 q^{35} -3.57997 q^{37} -13.9757 q^{38} +0.406903 q^{40} -4.07321 q^{41} -4.86148 q^{43} -0.168623 q^{46} +5.77852 q^{47} -6.65198 q^{49} +2.04904 q^{50} -3.30868 q^{52} +6.77365 q^{53} +0.240047 q^{56} -17.2620 q^{58} -5.58403 q^{59} +0.960246 q^{61} -4.05533 q^{62} -9.50196 q^{64} -1.50491 q^{65} +13.6559 q^{67} -10.8170 q^{68} +1.20880 q^{70} -1.28379 q^{71} +15.6271 q^{73} -7.33551 q^{74} -14.9956 q^{76} +11.1522 q^{79} -3.56340 q^{80} -8.34619 q^{82} +4.01215 q^{83} -4.92000 q^{85} -9.96139 q^{86} +6.08177 q^{89} -0.887802 q^{91} -0.180929 q^{92} +11.8404 q^{94} -6.82060 q^{95} -6.70578 q^{97} -13.6302 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8}+O(q^{10}) \) 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

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