LMFDB - Embedded newform 5445.2.a.cb.1.3

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			1.3
Root			$-0.226007$ 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)

$f(q)$	$=$	$q-1.22601 q^{2} -0.496906 q^{4} +1.00000 q^{5} +0.451695 q^{7} +3.06123 q^{8} +O(q^{10})$	$q-1.22601 q^{2} -0.496906 q^{4} +1.00000 q^{5} +0.451695 q^{7} +3.06123 q^{8} -1.22601 q^{10} +4.84034 q^{13} -0.553781 q^{14} -2.75927 q^{16} +0.740078 q^{17} -6.80375 q^{19} -0.496906 q^{20} +0.00634166 q^{23} +1.00000 q^{25} -5.93429 q^{26} -0.224450 q^{28} -0.323900 q^{29} -5.60583 q^{31} -2.73956 q^{32} -0.907341 q^{34} +0.451695 q^{35} +7.36894 q^{37} +8.34144 q^{38} +3.06123 q^{40} -10.9705 q^{41} +1.80668 q^{43} -0.00777492 q^{46} +1.67948 q^{47} -6.79597 q^{49} -1.22601 q^{50} -2.40520 q^{52} -8.63948 q^{53} +1.38274 q^{56} +0.397104 q^{58} +1.50054 q^{59} -13.0454 q^{61} +6.87279 q^{62} +8.87727 q^{64} +4.84034 q^{65} +9.60773 q^{67} -0.367750 q^{68} -0.553781 q^{70} -11.4126 q^{71} -10.2399 q^{73} -9.03438 q^{74} +3.38083 q^{76} +2.09965 q^{79} -2.75927 q^{80} +13.4499 q^{82} -15.8945 q^{83} +0.740078 q^{85} -2.21500 q^{86} +9.36925 q^{89} +2.18636 q^{91} -0.00315121 q^{92} -2.05906 q^{94} -6.80375 q^{95} +5.87983 q^{97} +8.33191 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.22601	−0.866918	−0.433459	−	0.901173i	$-0.642707\pi$
$2$	−1.22601	−0.866918	−0.433459	+	0.901173i	$0.642707\pi$
$3$	0	0
$3$	0	0
$4$	−0.496906	−0.248453
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.451695	0.170725	0.0853623	−	0.996350i	$-0.472795\pi$
$7$	0.451695	0.170725	0.0853623	+	0.996350i	$0.472795\pi$
$8$	3.06123	1.08231
$9$	0	0
$10$	−1.22601	−0.387698
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.84034	1.34247	0.671235	−	0.741245i	$-0.265764\pi$
$13$	4.84034	1.34247	0.671235	+	0.741245i	$0.265764\pi$
$14$	−0.553781	−0.148004
$15$	0	0
$16$	−2.75927	−0.689818
$17$	0.740078	0.179495	0.0897477	−	0.995965i	$-0.471394\pi$
$17$	0.740078	0.179495	0.0897477	+	0.995965i	$0.471394\pi$
$18$	0	0
$19$	−6.80375	−1.56089	−0.780443	−	0.625227i	$-0.785007\pi$
$19$	−6.80375	−1.56089	−0.780443	+	0.625227i	$0.785007\pi$
$20$	−0.496906	−0.111112
$21$	0	0
$22$	0	0
$23$	0.00634166	0.00132233	0.000661164	−	1.00000i	$-0.499790\pi$
$23$	0.00634166	0.00132233	0.000661164		1.00000i	$0.499790\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.93429	−1.16381
$27$	0	0
$28$	−0.224450	−0.0424171
$29$	−0.323900	−0.0601468	−0.0300734	−	0.999548i	$-0.509574\pi$
$29$	−0.323900	−0.0601468	−0.0300734	+	0.999548i	$0.509574\pi$
$30$	0	0
$31$	−5.60583	−1.00684	−0.503419	−	0.864043i	$-0.667925\pi$
$31$	−5.60583	−1.00684	−0.503419	+	0.864043i	$0.667925\pi$
$32$	−2.73956	−0.484291
$33$	0	0
$34$	−0.907341	−0.155608
$35$	0.451695	0.0763504
$36$	0	0
$37$	7.36894	1.21145	0.605723	−	0.795675i	$-0.292884\pi$
$37$	7.36894	1.21145	0.605723	+	0.795675i	$0.292884\pi$
$38$	8.34144	1.35316
$39$	0	0
$40$	3.06123	0.484022
$41$	−10.9705	−1.71330	−0.856649	−	0.515900i	$-0.827457\pi$
$41$	−10.9705	−1.71330	−0.856649	+	0.515900i	$0.827457\pi$
$42$	0	0
$43$	1.80668	0.275516	0.137758	−	0.990466i	$-0.456010\pi$
$43$	1.80668	0.275516	0.137758	+	0.990466i	$0.456010\pi$
$44$	0	0
$45$	0	0
$46$	−0.00777492	−0.00114635
$47$	1.67948	0.244978	0.122489	−	0.992470i	$-0.460912\pi$
$47$	1.67948	0.244978	0.122489	+	0.992470i	$0.460912\pi$
$48$	0	0
$49$	−6.79597	−0.970853
$50$	−1.22601	−0.173384
$51$	0	0
$52$	−2.40520	−0.333541
$53$	−8.63948	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$53$	−8.63948	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$54$	0	0
$55$	0	0
$56$	1.38274	0.184776
$57$	0	0
$58$	0.397104	0.0521423
$59$	1.50054	0.195353	0.0976766	−	0.995218i	$-0.468859\pi$
$59$	1.50054	0.195353	0.0976766	+	0.995218i	$0.468859\pi$
$60$	0	0
$61$	−13.0454	−1.67029	−0.835147	−	0.550027i	$-0.814617\pi$
$61$	−13.0454	−1.67029	−0.835147	+	0.550027i	$0.814617\pi$
$62$	6.87279	0.872845
$63$	0	0
$64$	8.87727	1.10966
$65$	4.84034	0.600371
$66$	0	0
$67$	9.60773	1.17377	0.586885	−	0.809670i	$-0.300354\pi$
$67$	9.60773	1.17377	0.586885	+	0.809670i	$0.300354\pi$
$68$	−0.367750	−0.0445962
$69$	0	0
$70$	−0.553781	−0.0661895
$71$	−11.4126	−1.35442	−0.677211	−	0.735789i	$-0.736812\pi$
$71$	−11.4126	−1.35442	−0.677211	+	0.735789i	$0.736812\pi$
$72$	0	0
$73$	−10.2399	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$73$	−10.2399	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$74$	−9.03438	−1.05023
$75$	0	0
$76$	3.38083	0.387807
$77$	0	0
$78$	0	0
$79$	2.09965	0.236229	0.118115	−	0.993000i	$-0.462315\pi$
$79$	2.09965	0.236229	0.118115	+	0.993000i	$0.462315\pi$
$80$	−2.75927	−0.308496
$81$	0	0
$82$	13.4499	1.48529
$83$	−15.8945	−1.74465	−0.872323	−	0.488929i	$-0.837388\pi$
$83$	−15.8945	−1.74465	−0.872323	+	0.488929i	$0.837388\pi$
$84$	0	0
$85$	0.740078	0.0802727
$86$	−2.21500	−0.238850
$87$	0	0
$88$	0	0
$89$	9.36925	0.993138	0.496569	−	0.867997i	$-0.334593\pi$
$89$	9.36925	0.993138	0.496569	+	0.867997i	$0.334593\pi$
$90$	0	0
$91$	2.18636	0.229193
$92$	−0.00315121	−0.000328536 0
$93$	0	0
$94$	−2.05906	−0.212375
$95$	−6.80375	−0.698050
$96$	0	0
$97$	5.87983	0.597007	0.298503	−	0.954409i	$-0.403513\pi$
$97$	5.87983	0.597007	0.298503	+	0.954409i	$0.403513\pi$
$98$	8.33191	0.841650
$99$	0	0
$100$	−0.496906	−0.0496906
$101$	−2.20797	−0.219701	−0.109851	−	0.993948i	$-0.535037\pi$
$101$	−2.20797	−0.219701	−0.109851	+	0.993948i	$0.535037\pi$
$102$	0	0
$103$	9.67943	0.953743	0.476871	−	0.878973i	$-0.341771\pi$
$103$	9.67943	0.953743	0.476871	+	0.878973i	$0.341771\pi$
$104$	14.8174	1.45296
$105$	0	0
$106$	10.5921	1.02879
$107$	−11.7564	−1.13654	−0.568269	−	0.822843i	$-0.692387\pi$
$107$	−11.7564	−1.13654	−0.568269	+	0.822843i	$0.692387\pi$
$108$	0	0
$109$	12.5091	1.19816	0.599078	−	0.800691i	$-0.295534\pi$
$109$	12.5091	1.19816	0.599078	+	0.800691i	$0.295534\pi$
$110$	0	0
$111$	0	0
$112$	−1.24635	−0.117769
$113$	−9.90608	−0.931885	−0.465943	−	0.884815i	$-0.654285\pi$
$113$	−9.90608	−0.931885	−0.465943	+	0.884815i	$0.654285\pi$
$114$	0	0
$115$	0.00634166	0.000591363 0
$116$	0.160948	0.0149437
$117$	0	0
$118$	−1.83967	−0.169355
$119$	0.334290	0.0306443
$120$	0	0
$121$	0	0
$122$	15.9938	1.44801
$123$	0	0
$124$	2.78557	0.250152
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.561561	0.0498304	0.0249152	−	0.999690i	$-0.492068\pi$
$127$	0.561561	0.0498304	0.0249152	+	0.999690i	$0.492068\pi$
$128$	−5.40447	−0.477692
$129$	0	0
$130$	−5.93429	−0.520472
$131$	3.03500	0.265170	0.132585	−	0.991172i	$-0.457672\pi$
$131$	3.03500	0.265170	0.132585	+	0.991172i	$0.457672\pi$
$132$	0	0
$133$	−3.07322	−0.266482
$134$	−11.7791	−1.01756
$135$	0	0
$136$	2.26555	0.194269
$137$	20.7004	1.76855	0.884275	−	0.466966i	$-0.154653\pi$
$137$	20.7004	1.76855	0.884275	+	0.466966i	$0.154653\pi$
$138$	0	0
$139$	−3.54873	−0.300999	−0.150500	−	0.988610i	$-0.548088\pi$
$139$	−3.54873	−0.300999	−0.150500	+	0.988610i	$0.548088\pi$
$140$	−0.224450	−0.0189695
$141$	0	0
$142$	13.9919	1.17417
$143$	0	0
$144$	0	0
$145$	−0.323900	−0.0268985
$146$	12.5542	1.03899
$147$	0	0
$148$	−3.66168	−0.300988
$149$	6.30459	0.516492	0.258246	−	0.966079i	$-0.416855\pi$
$149$	6.30459	0.516492	0.258246	+	0.966079i	$0.416855\pi$
$150$	0	0
$151$	3.68432	0.299826	0.149913	−	0.988699i	$-0.452101\pi$
$151$	3.68432	0.299826	0.149913	+	0.988699i	$0.452101\pi$
$152$	−20.8278	−1.68936
$153$	0	0
$154$	0	0
$155$	−5.60583	−0.450271
$156$	0	0
$157$	2.82869	0.225754	0.112877	−	0.993609i	$-0.463993\pi$
$157$	2.82869	0.225754	0.112877	+	0.993609i	$0.463993\pi$
$158$	−2.57419	−0.204792
$159$	0	0
$160$	−2.73956	−0.216582
$161$	0.00286450	0.000225754 0
$162$	0	0
$163$	8.85141	0.693296	0.346648	−	0.937995i	$-0.387320\pi$
$163$	8.85141	0.693296	0.346648	+	0.937995i	$0.387320\pi$
$164$	5.45129	0.425674
$165$	0	0
$166$	19.4868	1.51247
$167$	−19.6842	−1.52321	−0.761605	−	0.648042i	$-0.775588\pi$
$167$	−19.6842	−1.52321	−0.761605	+	0.648042i	$0.775588\pi$
$168$	0	0
$169$	10.4289	0.802224
$170$	−0.907341	−0.0695899
$171$	0	0
$172$	−0.897750	−0.0684528
$173$	−2.91136	−0.221347	−0.110673	−	0.993857i	$-0.535301\pi$
$173$	−2.91136	−0.221347	−0.110673	+	0.993857i	$0.535301\pi$
$174$	0	0
$175$	0.451695	0.0341449
$176$	0	0
$177$	0	0
$178$	−11.4868	−0.860969
$179$	−6.29309	−0.470368	−0.235184	−	0.971951i	$-0.575569\pi$
$179$	−6.29309	−0.470368	−0.235184	+	0.971951i	$0.575569\pi$
$180$	0	0
$181$	−11.1642	−0.829826	−0.414913	−	0.909861i	$-0.636188\pi$
$181$	−11.1642	−0.829826	−0.414913	+	0.909861i	$0.636188\pi$
$182$	−2.68049	−0.198691
$183$	0	0
$184$	0.0194132	0.00143116
$185$	7.36894	0.541776
$186$	0	0
$187$	0	0
$188$	−0.834545	−0.0608655
$189$	0	0
$190$	8.34144	0.605152
$191$	−20.3946	−1.47570	−0.737851	−	0.674964i	$-0.764159\pi$
$191$	−20.3946	−1.47570	−0.737851	+	0.674964i	$0.764159\pi$
$192$	0	0
$193$	2.97433	0.214097	0.107049	−	0.994254i	$-0.465860\pi$
$193$	2.97433	0.214097	0.107049	+	0.994254i	$0.465860\pi$
$194$	−7.20872	−0.517556
$195$	0	0
$196$	3.37696	0.241212
$197$	−5.20127	−0.370575	−0.185288	−	0.982684i	$-0.559322\pi$
$197$	−5.20127	−0.370575	−0.185288	+	0.982684i	$0.559322\pi$
$198$	0	0
$199$	8.10264	0.574381	0.287191	−	0.957873i	$-0.407279\pi$
$199$	8.10264	0.574381	0.287191	+	0.957873i	$0.407279\pi$
$200$	3.06123	0.216461
$201$	0	0
$202$	2.70699	0.190463
$203$	−0.146304	−0.0102685
$204$	0	0
$205$	−10.9705	−0.766210
$206$	−11.8671	−0.826817
$207$	0	0
$208$	−13.3558	−0.926059
$209$	0	0
$210$	0	0
$211$	1.31335	0.0904146	0.0452073	−	0.998978i	$-0.485605\pi$
$211$	1.31335	0.0904146	0.0452073	+	0.998978i	$0.485605\pi$
$212$	4.29301	0.294845
$213$	0	0
$214$	14.4135	0.985286
$215$	1.80668	0.123214
$216$	0	0
$217$	−2.53213	−0.171892
$218$	−15.3363	−1.03870
$219$	0	0
$220$	0	0
$221$	3.58223	0.240967
$222$	0	0
$223$	28.6347	1.91752	0.958761	−	0.284214i	$-0.0917325\pi$
$223$	28.6347	1.91752	0.958761	+	0.284214i	$0.0917325\pi$
$224$	−1.23745	−0.0826804
$225$	0	0
$226$	12.1449	0.807868
$227$	7.86603	0.522087	0.261043	−	0.965327i	$-0.415933\pi$
$227$	7.86603	0.522087	0.261043	+	0.965327i	$0.415933\pi$
$228$	0	0
$229$	−16.1510	−1.06729	−0.533644	−	0.845710i	$-0.679178\pi$
$229$	−16.1510	−1.06729	−0.533644	+	0.845710i	$0.679178\pi$
$230$	−0.00777492	−0.000512663 0
$231$	0	0
$232$	−0.991532	−0.0650973
$233$	−18.4345	−1.20768	−0.603841	−	0.797105i	$-0.706364\pi$
$233$	−18.4345	−1.20768	−0.603841	+	0.797105i	$0.706364\pi$
$234$	0	0
$235$	1.67948	0.109557
$236$	−0.745626	−0.0485361
$237$	0	0
$238$	−0.409841	−0.0265661
$239$	−22.3638	−1.44660	−0.723298	−	0.690536i	$-0.757375\pi$
$239$	−22.3638	−1.44660	−0.723298	+	0.690536i	$0.757375\pi$
$240$	0	0
$241$	13.2213	0.851662	0.425831	−	0.904803i	$-0.359982\pi$
$241$	13.2213	0.851662	0.425831	+	0.904803i	$0.359982\pi$
$242$	0	0
$243$	0	0
$244$	6.48235	0.414990
$245$	−6.79597	−0.434179
$246$	0	0
$247$	−32.9325	−2.09544
$248$	−17.1607	−1.08971
$249$	0	0
$250$	−1.22601	−0.0775395
$251$	−21.0032	−1.32571	−0.662854	−	0.748749i	$-0.730655\pi$
$251$	−21.0032	−1.32571	−0.662854	+	0.748749i	$0.730655\pi$
$252$	0	0
$253$	0	0
$254$	−0.688477	−0.0431989
$255$	0	0
$256$	−11.1286	−0.695539
$257$	−3.36772	−0.210073	−0.105036	−	0.994468i	$-0.533496\pi$
$257$	−3.36772	−0.210073	−0.105036	+	0.994468i	$0.533496\pi$
$258$	0	0
$259$	3.32852	0.206824
$260$	−2.40520	−0.149164
$261$	0	0
$262$	−3.72094	−0.229880
$263$	−21.0450	−1.29769	−0.648845	−	0.760920i	$-0.724748\pi$
$263$	−21.0450	−1.29769	−0.648845	+	0.760920i	$0.724748\pi$
$264$	0	0
$265$	−8.63948	−0.530719
$266$	3.76779	0.231018
$267$	0	0
$268$	−4.77414	−0.291627
$269$	23.3816	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$269$	23.3816	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$270$	0	0
$271$	20.5474	1.24817	0.624084	−	0.781357i	$-0.285472\pi$
$271$	20.5474	1.24817	0.624084	+	0.781357i	$0.285472\pi$
$272$	−2.04208	−0.123819
$273$	0	0
$274$	−25.3788	−1.53319
$275$	0	0
$276$	0	0
$277$	19.5818	1.17656	0.588280	−	0.808658i	$-0.299805\pi$
$277$	19.5818	1.17656	0.588280	+	0.808658i	$0.299805\pi$
$278$	4.35077	0.260942
$279$	0	0
$280$	1.38274	0.0826345
$281$	−13.7926	−0.822800	−0.411400	−	0.911455i	$-0.634960\pi$
$281$	−13.7926	−0.822800	−0.411400	+	0.911455i	$0.634960\pi$
$282$	0	0
$283$	−31.5177	−1.87353	−0.936765	−	0.349958i	$-0.886196\pi$
$283$	−31.5177	−1.87353	−0.936765	+	0.349958i	$0.886196\pi$
$284$	5.67098	0.336511
$285$	0	0
$286$	0	0
$287$	−4.95530	−0.292502
$288$	0	0
$289$	−16.4523	−0.967781
$290$	0.397104	0.0233188
$291$	0	0
$292$	5.08826	0.297768
$293$	23.3131	1.36196	0.680982	−	0.732300i	$-0.261553\pi$
$293$	23.3131	1.36196	0.680982	+	0.732300i	$0.261553\pi$
$294$	0	0
$295$	1.50054	0.0873646
$296$	22.5580	1.31116
$297$	0	0
$298$	−7.72948	−0.447757
$299$	0.0306958	0.00177518
$300$	0	0
$301$	0.816068	0.0470374
$302$	−4.51700	−0.259924
$303$	0	0
$304$	18.7734	1.07673
$305$	−13.0454	−0.746978
$306$	0	0
$307$	10.0938	0.576083	0.288042	−	0.957618i	$-0.406996\pi$
$307$	10.0938	0.576083	0.288042	+	0.957618i	$0.406996\pi$
$308$	0	0
$309$	0	0
$310$	6.87279	0.390348
$311$	−13.3254	−0.755614	−0.377807	−	0.925884i	$-0.623322\pi$
$311$	−13.3254	−0.755614	−0.377807	+	0.925884i	$0.623322\pi$
$312$	0	0
$313$	23.1682	1.30955	0.654773	−	0.755825i	$-0.272764\pi$
$313$	23.1682	1.30955	0.654773	+	0.755825i	$0.272764\pi$
$314$	−3.46799	−0.195710
$315$	0	0
$316$	−1.04333	−0.0586920
$317$	6.93189	0.389334	0.194667	−	0.980869i	$-0.437637\pi$
$317$	6.93189	0.389334	0.194667	+	0.980869i	$0.437637\pi$
$318$	0	0
$319$	0	0
$320$	8.87727	0.496254
$321$	0	0
$322$	−0.00351189	−0.000195710 0
$323$	−5.03530	−0.280172
$324$	0	0
$325$	4.84034	0.268494
$326$	−10.8519	−0.601031
$327$	0	0
$328$	−33.5830	−1.85431
$329$	0.758613	0.0418237
$330$	0	0
$331$	−10.5717	−0.581075	−0.290538	−	0.956864i	$-0.593834\pi$
$331$	−10.5717	−0.581075	−0.290538	+	0.956864i	$0.593834\pi$
$332$	7.89807	0.433463
$333$	0	0
$334$	24.1330	1.32050
$335$	9.60773	0.524926
$336$	0	0
$337$	6.34668	0.345726	0.172863	−	0.984946i	$-0.444698\pi$
$337$	6.34668	0.345726	0.172863	+	0.984946i	$0.444698\pi$
$338$	−12.7859	−0.695463
$339$	0	0
$340$	−0.367750	−0.0199440
$341$	0	0
$342$	0	0
$343$	−6.23157	−0.336473
$344$	5.53065	0.298193
$345$	0	0
$346$	3.56935	0.191890
$347$	−25.2417	−1.35505	−0.677524	−	0.735501i	$-0.736947\pi$
$347$	−25.2417	−1.35505	−0.677524	+	0.735501i	$0.736947\pi$
$348$	0	0
$349$	11.1035	0.594359	0.297179	−	0.954822i	$-0.403954\pi$
$349$	11.1035	0.594359	0.297179	+	0.954822i	$0.403954\pi$
$350$	−0.553781	−0.0296009
$351$	0	0
$352$	0	0
$353$	18.8552	1.00356	0.501780	−	0.864996i	$-0.332679\pi$
$353$	18.8552	1.00356	0.501780	+	0.864996i	$0.332679\pi$
$354$	0	0
$355$	−11.4126	−0.605716
$356$	−4.65564	−0.246748
$357$	0	0
$358$	7.71537	0.407770
$359$	−21.2928	−1.12379	−0.561895	−	0.827208i	$-0.689928\pi$
$359$	−21.2928	−1.12379	−0.561895	+	0.827208i	$0.689928\pi$
$360$	0	0
$361$	27.2910	1.43637
$362$	13.6873	0.719391
$363$	0	0
$364$	−1.08642	−0.0569437
$365$	−10.2399	−0.535980
$366$	0	0
$367$	−31.2857	−1.63310	−0.816550	−	0.577275i	$-0.804116\pi$
$367$	−31.2857	−1.63310	−0.816550	+	0.577275i	$0.804116\pi$
$368$	−0.0174984	−0.000912165 0
$369$	0	0
$370$	−9.03438	−0.469675
$371$	−3.90241	−0.202603
$372$	0	0
$373$	−9.39368	−0.486386	−0.243193	−	0.969978i	$-0.578195\pi$
$373$	−9.39368	−0.486386	−0.243193	+	0.969978i	$0.578195\pi$
$374$	0	0
$375$	0	0
$376$	5.14127	0.265141
$377$	−1.56779	−0.0807452
$378$	0	0
$379$	−8.19727	−0.421065	−0.210533	−	0.977587i	$-0.567520\pi$
$379$	−8.19727	−0.421065	−0.210533	+	0.977587i	$0.567520\pi$
$380$	3.38083	0.173433
$381$	0	0
$382$	25.0039	1.27931
$383$	−33.6830	−1.72112	−0.860560	−	0.509349i	$-0.829886\pi$
$383$	−33.6830	−1.72112	−0.860560	+	0.509349i	$0.829886\pi$
$384$	0	0
$385$	0	0
$386$	−3.64656	−0.185605
$387$	0	0
$388$	−2.92173	−0.148328
$389$	32.9338	1.66981	0.834905	−	0.550395i	$-0.185523\pi$
$389$	32.9338	1.66981	0.834905	+	0.550395i	$0.185523\pi$
$390$	0	0
$391$	0.00469332	0.000237352 0
$392$	−20.8040	−1.05076
$393$	0	0
$394$	6.37680	0.321259
$395$	2.09965	0.105645
$396$	0	0
$397$	−15.3865	−0.772228	−0.386114	−	0.922451i	$-0.626183\pi$
$397$	−15.3865	−0.772228	−0.386114	+	0.922451i	$0.626183\pi$
$398$	−9.93390	−0.497941
$399$	0	0
$400$	−2.75927	−0.137964
$401$	6.64295	0.331733	0.165866	−	0.986148i	$-0.446958\pi$
$401$	6.64295	0.331733	0.165866	+	0.986148i	$0.446958\pi$
$402$	0	0
$403$	−27.1341	−1.35165
$404$	1.09715	0.0545855
$405$	0	0
$406$	0.179370	0.00890198
$407$	0	0
$408$	0	0
$409$	−24.4850	−1.21071	−0.605353	−	0.795957i	$-0.706968\pi$
$409$	−24.4850	−1.21071	−0.605353	+	0.795957i	$0.706968\pi$
$410$	13.4499	0.664241
$411$	0	0
$412$	−4.80977	−0.236960
$413$	0.677784	0.0333516
$414$	0	0
$415$	−15.8945	−0.780230
$416$	−13.2604	−0.650146
$417$	0	0
$418$	0	0
$419$	20.8656	1.01935	0.509676	−	0.860367i	$-0.329765\pi$
$419$	20.8656	1.01935	0.509676	+	0.860367i	$0.329765\pi$
$420$	0	0
$421$	−16.9311	−0.825172	−0.412586	−	0.910919i	$-0.635374\pi$
$421$	−16.9311	−0.825172	−0.412586	+	0.910919i	$0.635374\pi$
$422$	−1.61017	−0.0783821
$423$	0	0
$424$	−26.4474	−1.28440
$425$	0.740078	0.0358991
$426$	0	0
$427$	−5.89255	−0.285160
$428$	5.84186	0.282377
$429$	0	0
$430$	−2.21500	−0.106817
$431$	−26.1647	−1.26031	−0.630154	−	0.776470i	$-0.717008\pi$
$431$	−26.1647	−1.26031	−0.630154	+	0.776470i	$0.717008\pi$
$432$	0	0
$433$	−25.0593	−1.20427	−0.602137	−	0.798393i	$-0.705684\pi$
$433$	−25.0593	−1.20427	−0.602137	+	0.798393i	$0.705684\pi$
$434$	3.10440	0.149016
$435$	0	0
$436$	−6.21586	−0.297686
$437$	−0.0431470	−0.00206400
$438$	0	0
$439$	−5.11234	−0.243999	−0.121999	−	0.992530i	$-0.538931\pi$
$439$	−5.11234	−0.243999	−0.121999	+	0.992530i	$0.538931\pi$
$440$	0	0
$441$	0	0
$442$	−4.39184	−0.208899
$443$	28.0773	1.33399	0.666995	−	0.745062i	$-0.267580\pi$
$443$	28.0773	1.33399	0.666995	+	0.745062i	$0.267580\pi$
$444$	0	0
$445$	9.36925	0.444145
$446$	−35.1064	−1.66233
$447$	0	0
$448$	4.00982	0.189446
$449$	36.9951	1.74591	0.872953	−	0.487804i	$-0.162202\pi$
$449$	36.9951	1.74591	0.872953	+	0.487804i	$0.162202\pi$
$450$	0	0
$451$	0	0
$452$	4.92239	0.231530
$453$	0	0
$454$	−9.64381	−0.452606
$455$	2.18636	0.102498
$456$	0	0
$457$	−36.8462	−1.72359	−0.861795	−	0.507256i	$-0.830660\pi$
$457$	−36.8462	−1.72359	−0.861795	+	0.507256i	$0.830660\pi$
$458$	19.8012	0.925250
$459$	0	0
$460$	−0.00315121	−0.000146926 0
$461$	−23.0013	−1.07128	−0.535640	−	0.844447i	$-0.679929\pi$
$461$	−23.0013	−1.07128	−0.535640	+	0.844447i	$0.679929\pi$
$462$	0	0
$463$	−36.1571	−1.68036	−0.840181	−	0.542306i	$-0.817551\pi$
$463$	−36.1571	−1.68036	−0.840181	+	0.542306i	$0.817551\pi$
$464$	0.893729	0.0414903
$465$	0	0
$466$	22.6008	1.04696
$467$	−2.26498	−0.104811	−0.0524053	−	0.998626i	$-0.516689\pi$
$467$	−2.26498	−0.104811	−0.0524053	+	0.998626i	$0.516689\pi$
$468$	0	0
$469$	4.33976	0.200392
$470$	−2.05906	−0.0949772
$471$	0	0
$472$	4.59348	0.211432
$473$	0	0
$474$	0	0
$475$	−6.80375	−0.312177
$476$	−0.166111	−0.00761367
$477$	0	0
$478$	27.4182	1.25408
$479$	30.5308	1.39499	0.697495	−	0.716590i	$-0.254298\pi$
$479$	30.5308	1.39499	0.697495	+	0.716590i	$0.254298\pi$
$480$	0	0
$481$	35.6682	1.62633
$482$	−16.2095	−0.738321
$483$	0	0
$484$	0	0
$485$	5.87983	0.266989
$486$	0	0
$487$	−16.1543	−0.732021	−0.366010	−	0.930611i	$-0.619277\pi$
$487$	−16.1543	−0.732021	−0.366010	+	0.930611i	$0.619277\pi$
$488$	−39.9349	−1.80777
$489$	0	0
$490$	8.33191	0.376397
$491$	11.6766	0.526960	0.263480	−	0.964665i	$-0.415130\pi$
$491$	11.6766	0.526960	0.263480	+	0.964665i	$0.415130\pi$
$492$	0	0
$493$	−0.239712	−0.0107961
$494$	40.3754	1.81658
$495$	0	0
$496$	15.4680	0.694534
$497$	−5.15500	−0.231233
$498$	0	0
$499$	−40.9366	−1.83257	−0.916287	−	0.400522i	$-0.868829\pi$
$499$	−40.9366	−1.83257	−0.916287	+	0.400522i	$0.868829\pi$
$500$	−0.496906	−0.0222223
$501$	0	0
$502$	25.7500	1.14928
$503$	0.424478	0.0189266	0.00946328	−	0.999955i	$-0.496988\pi$
$503$	0.424478	0.0189266	0.00946328	+	0.999955i	$0.496988\pi$
$504$	0	0
$505$	−2.20797	−0.0982533
$506$	0	0
$507$	0	0
$508$	−0.279043	−0.0123805
$509$	−19.8053	−0.877854	−0.438927	−	0.898523i	$-0.644641\pi$
$509$	−19.8053	−0.877854	−0.438927	+	0.898523i	$0.644641\pi$
$510$	0	0
$511$	−4.62530	−0.204611
$512$	24.4527	1.08067
$513$	0	0
$514$	4.12885	0.182116
$515$	9.67943	0.426527
$516$	0	0
$517$	0	0
$518$	−4.08078	−0.179299
$519$	0	0
$520$	14.8174	0.649785
$521$	36.8095	1.61265	0.806327	−	0.591470i	$-0.201452\pi$
$521$	36.8095	1.61265	0.806327	+	0.591470i	$0.201452\pi$
$522$	0	0
$523$	10.5015	0.459198	0.229599	−	0.973285i	$-0.426258\pi$
$523$	10.5015	0.459198	0.229599	+	0.973285i	$0.426258\pi$
$524$	−1.50811	−0.0658822
$525$	0	0
$526$	25.8013	1.12499
$527$	−4.14875	−0.180723
$528$	0	0
$529$	−23.0000	−0.999998
$530$	10.5921	0.460090
$531$	0	0
$532$	1.52710	0.0662083
$533$	−53.1008	−2.30005
$534$	0	0
$535$	−11.7564	−0.508276
$536$	29.4114	1.27038
$537$	0	0
$538$	−28.6660	−1.23588
$539$	0	0
$540$	0	0
$541$	−0.654613	−0.0281440	−0.0140720	−	0.999901i	$-0.504479\pi$
$541$	−0.654613	−0.0281440	−0.0140720	+	0.999901i	$0.504479\pi$
$542$	−25.1913	−1.08206
$543$	0	0
$544$	−2.02749	−0.0869280
$545$	12.5091	0.535832
$546$	0	0
$547$	13.4159	0.573621	0.286810	−	0.957987i	$-0.407405\pi$
$547$	13.4159	0.573621	0.286810	+	0.957987i	$0.407405\pi$
$548$	−10.2861	−0.439402
$549$	0	0
$550$	0	0
$551$	2.20374	0.0938823
$552$	0	0
$553$	0.948403	0.0403302
$554$	−24.0075	−1.01998
$555$	0	0
$556$	1.76339	0.0747843
$557$	13.2421	0.561087	0.280544	−	0.959841i	$-0.409485\pi$
$557$	13.2421	0.561087	0.280544	+	0.959841i	$0.409485\pi$
$558$	0	0
$559$	8.74494	0.369872
$560$	−1.24635	−0.0526679
$561$	0	0
$562$	16.9099	0.713300
$563$	21.3619	0.900297	0.450149	−	0.892954i	$-0.351371\pi$
$563$	21.3619	0.900297	0.450149	+	0.892954i	$0.351371\pi$
$564$	0	0
$565$	−9.90608	−0.416752
$566$	38.6409	1.62420
$567$	0	0
$568$	−34.9364	−1.46590
$569$	−45.3375	−1.90065	−0.950323	−	0.311267i	$-0.899247\pi$
$569$	−45.3375	−1.90065	−0.950323	+	0.311267i	$0.899247\pi$
$570$	0	0
$571$	25.1544	1.05268	0.526339	−	0.850275i	$-0.323564\pi$
$571$	25.1544	1.05268	0.526339	+	0.850275i	$0.323564\pi$
$572$	0	0
$573$	0	0
$574$	6.07523	0.253575
$575$	0.00634166	0.000264465 0
$576$	0	0
$577$	7.62653	0.317496	0.158748	−	0.987319i	$-0.449254\pi$
$577$	7.62653	0.317496	0.158748	+	0.987319i	$0.449254\pi$
$578$	20.1706	0.838987
$579$	0	0
$580$	0.160948	0.00668301
$581$	−7.17946	−0.297854
$582$	0	0
$583$	0	0
$584$	−31.3466	−1.29713
$585$	0	0
$586$	−28.5820	−1.18071
$587$	−7.88825	−0.325583	−0.162791	−	0.986660i	$-0.552050\pi$
$587$	−7.88825	−0.325583	−0.162791	+	0.986660i	$0.552050\pi$
$588$	0	0
$589$	38.1407	1.57156
$590$	−1.83967	−0.0757379
$591$	0	0
$592$	−20.3329	−0.835678
$593$	−8.68507	−0.356653	−0.178327	−	0.983971i	$-0.557068\pi$
$593$	−8.68507	−0.356653	−0.178327	+	0.983971i	$0.557068\pi$
$594$	0	0
$595$	0.334290	0.0137045
$596$	−3.13279	−0.128324
$597$	0	0
$598$	−0.0376333	−0.00153894
$599$	14.0016	0.572088	0.286044	−	0.958216i	$-0.407660\pi$
$599$	14.0016	0.572088	0.286044	+	0.958216i	$0.407660\pi$
$600$	0	0
$601$	10.2936	0.419886	0.209943	−	0.977714i	$-0.432672\pi$
$601$	10.2936	0.419886	0.209943	+	0.977714i	$0.432672\pi$
$602$	−1.00050	−0.0407775
$603$	0	0
$604$	−1.83076	−0.0744927
$605$	0	0
$606$	0	0
$607$	−27.4756	−1.11520	−0.557599	−	0.830110i	$-0.688277\pi$
$607$	−27.4756	−1.11520	−0.557599	+	0.830110i	$0.688277\pi$
$608$	18.6393	0.755923
$609$	0	0
$610$	15.9938	0.647569
$611$	8.12926	0.328875
$612$	0	0
$613$	10.8580	0.438550	0.219275	−	0.975663i	$-0.429631\pi$
$613$	10.8580	0.438550	0.219275	+	0.975663i	$0.429631\pi$
$614$	−12.3751	−0.499417
$615$	0	0
$616$	0	0
$617$	−18.2404	−0.734330	−0.367165	−	0.930156i	$-0.619671\pi$
$617$	−18.2404	−0.734330	−0.367165	+	0.930156i	$0.619671\pi$
$618$	0	0
$619$	19.6799	0.791004	0.395502	−	0.918465i	$-0.370571\pi$
$619$	19.6799	0.791004	0.395502	+	0.918465i	$0.370571\pi$
$620$	2.78557	0.111871
$621$	0	0
$622$	16.3370	0.655056
$623$	4.23204	0.169553
$624$	0	0
$625$	1.00000	0.0400000
$626$	−28.4044	−1.13527
$627$	0	0
$628$	−1.40559	−0.0560893
$629$	5.45359	0.217449
$630$	0	0
$631$	−30.4249	−1.21120	−0.605598	−	0.795771i	$-0.707066\pi$
$631$	−30.4249	−1.21120	−0.605598	+	0.795771i	$0.707066\pi$
$632$	6.42751	0.255673
$633$	0	0
$634$	−8.49854	−0.337520
$635$	0.561561	0.0222849
$636$	0	0
$637$	−32.8948	−1.30334
$638$	0	0
$639$	0	0
$640$	−5.40447	−0.213630
$641$	−37.3528	−1.47535	−0.737674	−	0.675157i	$-0.764076\pi$
$641$	−37.3528	−1.47535	−0.737674	+	0.675157i	$0.764076\pi$
$642$	0	0
$643$	1.06494	0.0419971	0.0209986	−	0.999780i	$-0.493315\pi$
$643$	1.06494	0.0419971	0.0209986	+	0.999780i	$0.493315\pi$
$644$	−0.00142339	−5.60893e−5 0
$645$	0	0
$646$	6.17332	0.242886
$647$	15.8049	0.621354	0.310677	−	0.950516i	$-0.399444\pi$
$647$	15.8049	0.621354	0.310677	+	0.950516i	$0.399444\pi$
$648$	0	0
$649$	0	0
$650$	−5.93429	−0.232762
$651$	0	0
$652$	−4.39832	−0.172252
$653$	−35.3513	−1.38340	−0.691701	−	0.722184i	$-0.743138\pi$
$653$	−35.3513	−1.38340	−0.691701	+	0.722184i	$0.743138\pi$
$654$	0	0
$655$	3.03500	0.118587
$656$	30.2705	1.18186
$657$	0	0
$658$	−0.930065	−0.0362577
$659$	−28.4474	−1.10815	−0.554077	−	0.832465i	$-0.686929\pi$
$659$	−28.4474	−1.10815	−0.554077	+	0.832465i	$0.686929\pi$
$660$	0	0
$661$	39.5989	1.54022	0.770110	−	0.637911i	$-0.220201\pi$
$661$	39.5989	1.54022	0.770110	+	0.637911i	$0.220201\pi$
$662$	12.9610	0.503745
$663$	0	0
$664$	−48.6566	−1.88824
$665$	−3.07322	−0.119174
$666$	0	0
$667$	−0.00205407	−7.95337e−5 0
$668$	9.78121	0.378446
$669$	0	0
$670$	−11.7791	−0.455068
$671$	0	0
$672$	0	0
$673$	17.5621	0.676969	0.338484	−	0.940972i	$-0.390086\pi$
$673$	17.5621	0.676969	0.338484	+	0.940972i	$0.390086\pi$
$674$	−7.78108	−0.299716
$675$	0	0
$676$	−5.18220	−0.199315
$677$	−45.0690	−1.73214	−0.866071	−	0.499922i	$-0.833362\pi$
$677$	−45.0690	−1.73214	−0.866071	+	0.499922i	$0.833362\pi$
$678$	0	0
$679$	2.65589	0.101924
$680$	2.26555	0.0868797
$681$	0	0
$682$	0	0
$683$	−15.7677	−0.603334	−0.301667	−	0.953413i	$-0.597543\pi$
$683$	−15.7677	−0.603334	−0.301667	+	0.953413i	$0.597543\pi$
$684$	0	0
$685$	20.7004	0.790920
$686$	7.63995	0.291695
$687$	0	0
$688$	−4.98512	−0.190056
$689$	−41.8180	−1.59314
$690$	0	0
$691$	2.96619	0.112839	0.0564197	−	0.998407i	$-0.482032\pi$
$691$	2.96619	0.112839	0.0564197	+	0.998407i	$0.482032\pi$
$692$	1.44668	0.0549944
$693$	0	0
$694$	30.9465	1.17471
$695$	−3.54873	−0.134611
$696$	0	0
$697$	−8.11899	−0.307529
$698$	−13.6130	−0.515260
$699$	0	0
$700$	−0.224450	−0.00848342
$701$	37.1301	1.40238	0.701192	−	0.712973i	$-0.252652\pi$
$701$	37.1301	1.40238	0.701192	+	0.712973i	$0.252652\pi$
$702$	0	0
$703$	−50.1364	−1.89093
$704$	0	0
$705$	0	0
$706$	−23.1166	−0.870003
$707$	−0.997329	−0.0375084
$708$	0	0
$709$	−31.7505	−1.19241	−0.596207	−	0.802831i	$-0.703326\pi$
$709$	−31.7505	−1.19241	−0.596207	+	0.802831i	$0.703326\pi$
$710$	13.9919	0.525106
$711$	0	0
$712$	28.6814	1.07488
$713$	−0.0355503	−0.00133137
$714$	0	0
$715$	0	0
$716$	3.12708	0.116864
$717$	0	0
$718$	26.1051	0.974234
$719$	31.8772	1.18882	0.594409	−	0.804163i	$-0.297386\pi$
$719$	31.8772	1.18882	0.594409	+	0.804163i	$0.297386\pi$
$720$	0	0
$721$	4.37215	0.162827
$722$	−33.4589	−1.24521
$723$	0	0
$724$	5.54755	0.206173
$725$	−0.323900	−0.0120294
$726$	0	0
$727$	41.4129	1.53592	0.767959	−	0.640499i	$-0.221272\pi$
$727$	41.4129	1.53592	0.767959	+	0.640499i	$0.221272\pi$
$728$	6.69294	0.248057
$729$	0	0
$730$	12.5542	0.464650
$731$	1.33708	0.0494538
$732$	0	0
$733$	−47.6920	−1.76155	−0.880773	−	0.473540i	$-0.842976\pi$
$733$	−47.6920	−1.76155	−0.880773	+	0.473540i	$0.842976\pi$
$734$	38.3565	1.41576
$735$	0	0
$736$	−0.0173734	−0.000640391 0
$737$	0	0
$738$	0	0
$739$	12.4567	0.458228	0.229114	−	0.973400i	$-0.426417\pi$
$739$	12.4567	0.458228	0.229114	+	0.973400i	$0.426417\pi$
$740$	−3.66168	−0.134606
$741$	0	0
$742$	4.78438	0.175640
$743$	−27.3723	−1.00419	−0.502095	−	0.864812i	$-0.667437\pi$
$743$	−27.3723	−1.00419	−0.502095	+	0.864812i	$0.667437\pi$
$744$	0	0
$745$	6.30459	0.230982
$746$	11.5167	0.421657
$747$	0	0
$748$	0	0
$749$	−5.31033	−0.194035
$750$	0	0
$751$	−41.8725	−1.52795	−0.763975	−	0.645246i	$-0.776755\pi$
$751$	−41.8725	−1.52795	−0.763975	+	0.645246i	$0.776755\pi$
$752$	−4.63414	−0.168990
$753$	0	0
$754$	1.92212	0.0699995
$755$	3.68432	0.134086
$756$	0	0
$757$	16.1192	0.585863	0.292932	−	0.956133i	$-0.405369\pi$
$757$	16.1192	0.585863	0.292932	+	0.956133i	$0.405369\pi$
$758$	10.0499	0.365029
$759$	0	0
$760$	−20.8278	−0.755504
$761$	10.9239	0.395990	0.197995	−	0.980203i	$-0.436557\pi$
$761$	10.9239	0.395990	0.197995	+	0.980203i	$0.436557\pi$
$762$	0	0
$763$	5.65030	0.204555
$764$	10.1342	0.366643
$765$	0	0
$766$	41.2956	1.49207
$767$	7.26311	0.262256
$768$	0	0
$769$	34.2074	1.23355	0.616775	−	0.787139i	$-0.288439\pi$
$769$	34.2074	1.23355	0.616775	+	0.787139i	$0.288439\pi$
$770$	0	0
$771$	0	0
$772$	−1.47797	−0.0531932
$773$	8.63688	0.310647	0.155323	−	0.987864i	$-0.450358\pi$
$773$	8.63688	0.310647	0.155323	+	0.987864i	$0.450358\pi$
$774$	0	0
$775$	−5.60583	−0.201367
$776$	17.9995	0.646144
$777$	0	0
$778$	−40.3771	−1.44759
$779$	74.6402	2.67426
$780$	0	0
$781$	0	0
$782$	−0.00575405	−0.000205764 0
$783$	0	0
$784$	18.7519	0.669712
$785$	2.82869	0.100960
$786$	0	0
$787$	23.1043	0.823581	0.411790	−	0.911279i	$-0.364904\pi$
$787$	23.1043	0.823581	0.411790	+	0.911279i	$0.364904\pi$
$788$	2.58455	0.0920707
$789$	0	0
$790$	−2.57419	−0.0915856
$791$	−4.47453	−0.159096
$792$	0	0
$793$	−63.1443	−2.24232
$794$	18.8640	0.669458
$795$	0	0
$796$	−4.02626	−0.142707
$797$	−39.7333	−1.40743	−0.703714	−	0.710484i	$-0.748476\pi$
$797$	−39.7333	−1.40743	−0.703714	+	0.710484i	$0.748476\pi$
$798$	0	0
$799$	1.24295	0.0439723
$800$	−2.73956	−0.0968582
$801$	0	0
$802$	−8.14430	−0.287585
$803$	0	0
$804$	0	0
$805$	0.00286450	0.000100960 0
$806$	33.2667	1.17177
$807$	0	0
$808$	−6.75909	−0.237784
$809$	31.0140	1.09039	0.545197	−	0.838308i	$-0.316455\pi$
$809$	31.0140	1.09039	0.545197	+	0.838308i	$0.316455\pi$
$810$	0	0
$811$	−7.45922	−0.261929	−0.130964	−	0.991387i	$-0.541807\pi$
$811$	−7.45922	−0.261929	−0.130964	+	0.991387i	$0.541807\pi$
$812$	0.0726995	0.00255125
$813$	0	0
$814$	0	0
$815$	8.85141	0.310051
$816$	0	0
$817$	−12.2922	−0.430049
$818$	30.0188	1.04958
$819$	0	0
$820$	5.45129	0.190367
$821$	12.6556	0.441683	0.220841	−	0.975310i	$-0.429120\pi$
$821$	12.6556	0.441683	0.220841	+	0.975310i	$0.429120\pi$
$822$	0	0
$823$	−27.2890	−0.951234	−0.475617	−	0.879652i	$-0.657775\pi$
$823$	−27.2890	−0.951234	−0.475617	+	0.879652i	$0.657775\pi$
$824$	29.6309	1.03224
$825$	0	0
$826$	−0.830969	−0.0289131
$827$	16.9544	0.589563	0.294781	−	0.955565i	$-0.404753\pi$
$827$	16.9544	0.589563	0.294781	+	0.955565i	$0.404753\pi$
$828$	0	0
$829$	19.7794	0.686967	0.343483	−	0.939159i	$-0.388393\pi$
$829$	19.7794	0.686967	0.343483	+	0.939159i	$0.388393\pi$
$830$	19.4868	0.676395
$831$	0	0
$832$	42.9690	1.48968
$833$	−5.02955	−0.174264
$834$	0	0
$835$	−19.6842	−0.681200
$836$	0	0
$837$	0	0
$838$	−25.5814	−0.883694
$839$	−4.97974	−0.171920	−0.0859598	−	0.996299i	$-0.527396\pi$
$839$	−4.97974	−0.171920	−0.0859598	+	0.996299i	$0.527396\pi$
$840$	0	0
$841$	−28.8951	−0.996382
$842$	20.7577	0.715357
$843$	0	0
$844$	−0.652611	−0.0224638
$845$	10.4289	0.358766
$846$	0	0
$847$	0	0
$848$	23.8387	0.818623
$849$	0	0
$850$	−0.907341	−0.0311215
$851$	0.0467313	0.00160193
$852$	0	0
$853$	23.2464	0.795943	0.397971	−	0.917398i	$-0.369714\pi$
$853$	23.2464	0.795943	0.397971	+	0.917398i	$0.369714\pi$
$854$	7.22430	0.247211
$855$	0	0
$856$	−35.9891	−1.23008
$857$	30.6976	1.04861	0.524306	−	0.851530i	$-0.324325\pi$
$857$	30.6976	1.04861	0.524306	+	0.851530i	$0.324325\pi$
$858$	0	0
$859$	−11.1830	−0.381560	−0.190780	−	0.981633i	$-0.561102\pi$
$859$	−11.1830	−0.381560	−0.190780	+	0.981633i	$0.561102\pi$
$860$	−0.897750	−0.0306130
$861$	0	0
$862$	32.0781	1.09258
$863$	−3.66335	−0.124702	−0.0623510	−	0.998054i	$-0.519860\pi$
$863$	−3.66335	−0.124702	−0.0623510	+	0.998054i	$0.519860\pi$
$864$	0	0
$865$	−2.91136	−0.0989894
$866$	30.7229	1.04401
$867$	0	0
$868$	1.25823	0.0427071
$869$	0	0
$870$	0	0
$871$	46.5047	1.57575
$872$	38.2932	1.29677
$873$	0	0
$874$	0.0528986	0.00178932
$875$	0.451695	0.0152701
$876$	0	0
$877$	−7.36271	−0.248621	−0.124310	−	0.992243i	$-0.539672\pi$
$877$	−7.36271	−0.248621	−0.124310	+	0.992243i	$0.539672\pi$
$878$	6.26776	0.211527
$879$	0	0
$880$	0	0
$881$	6.07165	0.204559	0.102280	−	0.994756i	$-0.467386\pi$
$881$	6.07165	0.204559	0.102280	+	0.994756i	$0.467386\pi$
$882$	0	0
$883$	31.1767	1.04918	0.524589	−	0.851356i	$-0.324219\pi$
$883$	31.1767	1.04918	0.524589	+	0.851356i	$0.324219\pi$
$884$	−1.78003	−0.0598690
$885$	0	0
$886$	−34.4229	−1.15646
$887$	5.68999	0.191051	0.0955255	−	0.995427i	$-0.469547\pi$
$887$	5.68999	0.191051	0.0955255	+	0.995427i	$0.469547\pi$
$888$	0	0
$889$	0.253654	0.00850729
$890$	−11.4868	−0.385037
$891$	0	0
$892$	−14.2288	−0.476414
$893$	−11.4268	−0.382382
$894$	0	0
$895$	−6.29309	−0.210355
$896$	−2.44117	−0.0815538
$897$	0	0
$898$	−45.3563	−1.51356
$899$	1.81573	0.0605580
$900$	0	0
$901$	−6.39389	−0.213011
$902$	0	0
$903$	0	0
$904$	−30.3247	−1.00859
$905$	−11.1642	−0.371109
$906$	0	0
$907$	42.4917	1.41091	0.705456	−	0.708754i	$-0.250742\pi$
$907$	42.4917	1.41091	0.705456	+	0.708754i	$0.250742\pi$
$908$	−3.90868	−0.129714
$909$	0	0
$910$	−2.68049	−0.0888574
$911$	−2.14405	−0.0710356	−0.0355178	−	0.999369i	$-0.511308\pi$
$911$	−2.14405	−0.0710356	−0.0355178	+	0.999369i	$0.511308\pi$
$912$	0	0
$913$	0	0
$914$	45.1737	1.49421
$915$	0	0
$916$	8.02553	0.265171
$917$	1.37090	0.0452710
$918$	0	0
$919$	−16.0683	−0.530044	−0.265022	−	0.964242i	$-0.585379\pi$
$919$	−16.0683	−0.530044	−0.265022	+	0.964242i	$0.585379\pi$
$920$	0.0194132	0.000640036 0
$921$	0	0
$922$	28.1998	0.928711
$923$	−55.2407	−1.81827
$924$	0	0
$925$	7.36894	0.242289
$926$	44.3289	1.45674
$927$	0	0
$928$	0.887346	0.0291286
$929$	−37.4770	−1.22958	−0.614791	−	0.788690i	$-0.710760\pi$
$929$	−37.4770	−1.22958	−0.614791	+	0.788690i	$0.710760\pi$
$930$	0	0
$931$	46.2381	1.51539
$932$	9.16021	0.300053
$933$	0	0
$934$	2.77688	0.0908621
$935$	0	0
$936$	0	0
$937$	26.0054	0.849560	0.424780	−	0.905297i	$-0.360351\pi$
$937$	26.0054	0.849560	0.424780	+	0.905297i	$0.360351\pi$
$938$	−5.32058	−0.173723
$939$	0	0
$940$	−0.834545	−0.0272199
$941$	−8.58482	−0.279857	−0.139929	−	0.990162i	$-0.544687\pi$
$941$	−8.58482	−0.279857	−0.139929	+	0.990162i	$0.544687\pi$
$942$	0	0
$943$	−0.0695709	−0.00226554
$944$	−4.14038	−0.134758
$945$	0	0
$946$	0	0
$947$	−36.0304	−1.17083	−0.585416	−	0.810733i	$-0.699069\pi$
$947$	−36.0304	−1.17083	−0.585416	+	0.810733i	$0.699069\pi$
$948$	0	0
$949$	−49.5645	−1.60893
$950$	8.34144	0.270632
$951$	0	0
$952$	1.02334	0.0331665
$953$	43.8525	1.42052	0.710261	−	0.703938i	$-0.248577\pi$
$953$	43.8525	1.42052	0.710261	+	0.703938i	$0.248577\pi$
$954$	0	0
$955$	−20.3946	−0.659954
$956$	11.1127	0.359412
$957$	0	0
$958$	−37.4310	−1.20934
$959$	9.35025	0.301935
$960$	0	0
$961$	0.425347	0.0137209
$962$	−43.7295	−1.40990
$963$	0	0
$964$	−6.56977	−0.211598
$965$	2.97433	0.0957472
$966$	0	0
$967$	48.1271	1.54766	0.773831	−	0.633392i	$-0.218338\pi$
$967$	48.1271	1.54766	0.773831	+	0.633392i	$0.218338\pi$
$968$	0	0
$969$	0	0
$970$	−7.20872	−0.231458
$971$	31.7334	1.01837	0.509187	−	0.860656i	$-0.329946\pi$
$971$	31.7334	1.01837	0.509187	+	0.860656i	$0.329946\pi$
$972$	0	0
$973$	−1.60294	−0.0513880
$974$	19.8053	0.634602
$975$	0	0
$976$	35.9958	1.15220
$977$	43.1280	1.37979	0.689893	−	0.723911i	$-0.257658\pi$
$977$	43.1280	1.37979	0.689893	+	0.723911i	$0.257658\pi$
$978$	0	0
$979$	0	0
$980$	3.37696	0.107873
$981$	0	0
$982$	−14.3156	−0.456831
$983$	44.6922	1.42546	0.712729	−	0.701439i	$-0.247459\pi$
$983$	44.6922	1.42546	0.712729	+	0.701439i	$0.247459\pi$
$984$	0	0
$985$	−5.20127	−0.165726
$986$	0.293888	0.00935931
$987$	0	0
$988$	16.3644	0.520619
$989$	0.0114573	0.000364322 0
$990$	0	0
$991$	13.7657	0.437282	0.218641	−	0.975805i	$-0.429838\pi$
$991$	13.7657	0.437282	0.218641	+	0.975805i	$0.429838\pi$
$992$	15.3575	0.487602
$993$	0	0
$994$	6.32007	0.200460
$995$	8.10264	0.256871
$996$	0	0
$997$	−4.89609	−0.155061	−0.0775303	−	0.996990i	$-0.524703\pi$
$997$	−4.89609	−0.155061	−0.0775303	+	0.996990i	$0.524703\pi$
$998$	50.1886	1.58869
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.cb.1.3		8
3.2	odd	2		5445.2.a.cc.1.6		8
11.2	odd	10		495.2.n.g.136.2	yes	16
11.6	odd	10		495.2.n.g.91.2	✓	16
11.10	odd	2		5445.2.a.cd.1.6		8
33.2	even	10		495.2.n.h.136.3	yes	16
33.17	even	10		495.2.n.h.91.3	yes	16
33.32	even	2		5445.2.a.ca.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.n.g.91.2	✓	16	11.6	odd	10
495.2.n.g.136.2	yes	16	11.2	odd	10
495.2.n.h.91.3	yes	16	33.17	even	10
495.2.n.h.136.3	yes	16	33.2	even	10
5445.2.a.ca.1.3		8	33.32	even	2
5445.2.a.cb.1.3		8	1.1	even	1	trivial
5445.2.a.cc.1.6		8	3.2	odd	2
5445.2.a.cd.1.6		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)

\(f(q)\)	\(=\)	\(q-1.22601 q^{2} -0.496906 q^{4} +1.00000 q^{5} +0.451695 q^{7} +3.06123 q^{8} +O(q^{10})\)	\(q-1.22601 q^{2} -0.496906 q^{4} +1.00000 q^{5} +0.451695 q^{7} +3.06123 q^{8} -1.22601 q^{10} +4.84034 q^{13} -0.553781 q^{14} -2.75927 q^{16} +0.740078 q^{17} -6.80375 q^{19} -0.496906 q^{20} +0.00634166 q^{23} +1.00000 q^{25} -5.93429 q^{26} -0.224450 q^{28} -0.323900 q^{29} -5.60583 q^{31} -2.73956 q^{32} -0.907341 q^{34} +0.451695 q^{35} +7.36894 q^{37} +8.34144 q^{38} +3.06123 q^{40} -10.9705 q^{41} +1.80668 q^{43} -0.00777492 q^{46} +1.67948 q^{47} -6.79597 q^{49} -1.22601 q^{50} -2.40520 q^{52} -8.63948 q^{53} +1.38274 q^{56} +0.397104 q^{58} +1.50054 q^{59} -13.0454 q^{61} +6.87279 q^{62} +8.87727 q^{64} +4.84034 q^{65} +9.60773 q^{67} -0.367750 q^{68} -0.553781 q^{70} -11.4126 q^{71} -10.2399 q^{73} -9.03438 q^{74} +3.38083 q^{76} +2.09965 q^{79} -2.75927 q^{80} +13.4499 q^{82} -15.8945 q^{83} +0.740078 q^{85} -2.21500 q^{86} +9.36925 q^{89} +2.18636 q^{91} -0.00315121 q^{92} -2.05906 q^{94} -6.80375 q^{95} +5.87983 q^{97} +8.33191 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8}+O(q^{10}) \) 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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